Integrand size = 32, antiderivative size = 1092 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{d+e x+f x^2} \, dx=-\frac {\left (2 A c f (4 c e-5 b f)-B \left (b^2 f^2-2 c f (5 b e-4 a f)+8 c^2 \left (e^2-d f\right )\right )+2 c f (2 B c e-b B f-2 A c f) x\right ) \sqrt {a+b x+c x^2}}{8 c f^3}+\frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}+\frac {\left (2 A c f \left (3 b^2 f^2-12 c f (b e-a f)+8 c^2 \left (e^2-d f\right )\right )-B \left (b^3 f^3+6 b c f^2 (b e-2 a f)-24 c^2 f \left (b e^2-b d f-a e f\right )+16 c^3 \left (e^3-2 d e f\right )\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{3/2} f^4}-\frac {\left (2 c f \left (B d (c e-b f) \left (c e^2-2 c d f-b e f+2 a f^2\right )+A f \left (2 c d f (b e-a f)-f^2 \left (b^2 d-a^2 f\right )-c^2 d \left (e^2-d f\right )\right )\right )-c \left (e-\sqrt {e^2-4 d f}\right ) \left (A f (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+B \left (c^2 \left (e^4-3 d e^2 f+d^2 f^2\right )-f^2 \left (2 a b e f-a^2 f^2-b^2 \left (e^2-d f\right )\right )+2 c f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right )\right )\right )\right ) \text {arctanh}\left (\frac {4 a f-b \left (e-\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-2 c d f-b e f+2 a f^2-(c e-b f) \sqrt {e^2-4 d f}} \sqrt {a+b x+c x^2}}\right )}{\sqrt {2} c f^4 \sqrt {e^2-4 d f} \sqrt {c e^2-2 c d f-b e f+2 a f^2-(c e-b f) \sqrt {e^2-4 d f}}}+\frac {\left (2 f \left (B d (c e-b f) \left (c e^2-2 c d f-b e f+2 a f^2\right )+A f \left (2 c d f (b e-a f)-f^2 \left (b^2 d-a^2 f\right )-c^2 d \left (e^2-d f\right )\right )\right )-\left (e+\sqrt {e^2-4 d f}\right ) \left (A f (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+B \left (c^2 \left (e^4-3 d e^2 f+d^2 f^2\right )-f^2 \left (2 a b e f-a^2 f^2-b^2 \left (e^2-d f\right )\right )+2 c f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right )\right )\right )\right ) \text {arctanh}\left (\frac {4 a f-b \left (e+\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-2 c d f-b e f+2 a f^2+(c e-b f) \sqrt {e^2-4 d f}} \sqrt {a+b x+c x^2}}\right )}{\sqrt {2} f^4 \sqrt {e^2-4 d f} \sqrt {c e^2-2 c d f-b e f+2 a f^2+(c e-b f) \sqrt {e^2-4 d f}}} \]
1/3*B*(c*x^2+b*x+a)^(3/2)/f+1/16*(2*A*c*f*(3*b^2*f^2-12*c*f*(-a*f+b*e)+8*c ^2*(-d*f+e^2))-B*(b^3*f^3+6*b*c*f^2*(-2*a*f+b*e)-24*c^2*f*(-a*e*f-b*d*f+b* e^2)+16*c^3*(-2*d*e*f+e^3)))*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^( 1/2))/c^(3/2)/f^4-1/8*(2*A*c*f*(-5*b*f+4*c*e)-B*(b^2*f^2-2*c*f*(-4*a*f+5*b *e)+8*c^2*(-d*f+e^2))+2*c*f*(-2*A*c*f-B*b*f+2*B*c*e)*x)*(c*x^2+b*x+a)^(1/2 )/c/f^3-1/2*arctanh(1/4*(4*a*f+2*x*(b*f-c*(e-(-4*d*f+e^2)^(1/2)))-b*(e-(-4 *d*f+e^2)^(1/2)))*2^(1/2)/(c*x^2+b*x+a)^(1/2)/(c*e^2-2*c*d*f-b*e*f+2*a*f^2 -(-b*f+c*e)*(-4*d*f+e^2)^(1/2))^(1/2))*(2*c*f*(B*d*(-b*f+c*e)*(2*a*f^2-b*e *f-2*c*d*f+c*e^2)+A*f*(2*c*d*f*(-a*f+b*e)-f^2*(-a^2*f+b^2*d)-c^2*d*(-d*f+e ^2)))-c*(A*f*(-b*f+c*e)*(f*(-2*a*f+b*e)-c*(-2*d*f+e^2))+B*(c^2*(d^2*f^2-3* d*e^2*f+e^4)-f^2*(2*a*b*e*f-a^2*f^2-b^2*(-d*f+e^2))+2*c*f*(a*f*(-d*f+e^2)- b*(-2*d*e*f+e^3))))*(e-(-4*d*f+e^2)^(1/2)))/c/f^4*2^(1/2)/(-4*d*f+e^2)^(1/ 2)/(c*e^2-2*c*d*f-b*e*f+2*a*f^2-(-b*f+c*e)*(-4*d*f+e^2)^(1/2))^(1/2)+1/2*a rctanh(1/4*(4*a*f-b*(e+(-4*d*f+e^2)^(1/2))+2*x*(b*f-c*(e+(-4*d*f+e^2)^(1/2 ))))*2^(1/2)/(c*x^2+b*x+a)^(1/2)/(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(-b*f+c*e)*( -4*d*f+e^2)^(1/2))^(1/2))*(2*f*(B*d*(-b*f+c*e)*(2*a*f^2-b*e*f-2*c*d*f+c*e^ 2)+A*f*(2*c*d*f*(-a*f+b*e)-f^2*(-a^2*f+b^2*d)-c^2*d*(-d*f+e^2)))-(A*f*(-b* f+c*e)*(f*(-2*a*f+b*e)-c*(-2*d*f+e^2))+B*(c^2*(d^2*f^2-3*d*e^2*f+e^4)-f^2* (2*a*b*e*f-a^2*f^2-b^2*(-d*f+e^2))+2*c*f*(a*f*(-d*f+e^2)-b*(-2*d*e*f+e^3)) ))*(e+(-4*d*f+e^2)^(1/2)))/f^4*2^(1/2)/(-4*d*f+e^2)^(1/2)/(c*e^2-2*c*d*...
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 6.53 (sec) , antiderivative size = 3516, normalized size of antiderivative = 3.22 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{d+e x+f x^2} \, dx=\text {Result too large to show} \]
(Sqrt[c]*f*Sqrt[a + x*(b + c*x)]*(6*A*c*f*(-4*c*e + 5*b*f + 2*c*f*x) + B*( 3*b^2*f^2 + 2*c*f*(-15*b*e + 16*a*f + 7*b*f*x) + 4*c^2*(6*e^2 - 6*d*f - 3* e*f*x + 2*f^2*x^2))) + 3*(8*A*c^2*(2*c*d + 3*b*e)*f^2 + B*(16*c^3*e^3 + 6* b^2*c*e*f^2 + 24*c^2*(b*d + a*e)*f^2 + b^3*f^3))*ArcTanh[(Sqrt[c]*x)/(Sqrt [a] - Sqrt[a + x*(b + c*x)])] + 6*c*f*(2*B*(8*c^2*d*e + 6*b*c*e^2 + 3*a*b* f^2) + A*(8*c^2*e^2 + 3*b^2*f^2 + 12*a*c*f^2))*ArcTanh[(Sqrt[c]*x)/(-Sqrt[ a] + Sqrt[a + x*(b + c*x)])] + 24*c^(3/2)*RootSum[c^2*d - b*c*e + b^2*f + 2*Sqrt[a]*c*e*#1 - 4*Sqrt[a]*b*f*#1 - 2*c*d*#1^2 + b*e*#1^2 + 4*a*f*#1^2 - 2*Sqrt[a]*e*#1^3 + d*#1^4 & , (B*c^3*d*e^3*Log[x] - b*B*c^2*e^4*Log[x] - 2*B*c^3*d^2*e*f*Log[x] + b*B*c^2*d*e^2*f*Log[x] - A*c^3*d*e^2*f*Log[x] + 2 *b^2*B*c*e^3*f*Log[x] + A*b*c^2*e^3*f*Log[x] + b*B*c^2*d^2*f^2*Log[x] + A* c^3*d^2*f^2*Log[x] - 3*b^2*B*c*d*e*f^2*Log[x] + 2*a*B*c^2*d*e*f^2*Log[x] - b^3*B*e^2*f^2*Log[x] - 2*A*b^2*c*e^2*f^2*Log[x] - 2*a*b*B*c*e^2*f^2*Log[x ] + b^3*B*d*f^3*Log[x] + A*b^2*c*d*f^3*Log[x] - 2*a*A*c^2*d*f^3*Log[x] + A *b^3*e*f^3*Log[x] + 2*a*b^2*B*e*f^3*Log[x] + 2*a*A*b*c*e*f^3*Log[x] - 2*a* A*b^2*f^4*Log[x] - a^2*b*B*f^4*Log[x] + a^2*A*c*f^4*Log[x] - B*c^3*d*e^3*L og[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1] + b*B*c^2*e^4*Log[-Sqrt[a] + S qrt[a + b*x + c*x^2] - x*#1] + 2*B*c^3*d^2*e*f*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1] - b*B*c^2*d*e^2*f*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1] + A*c^3*d*e^2*f*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1] - 2...
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{d+e x+f x^2} \, dx\) |
| \(\Big \downarrow \) 1352 |
| \(\displaystyle \frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}-\frac {\int \frac {3 \sqrt {c x^2+b x+a} \left ((2 B c e-b B f-2 A c f) x^2-(2 A b f-B (2 c d+b e-2 a f)) x+b B d-2 a A f\right )}{2 \left (f x^2+e x+d\right )}dx}{3 f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}-\frac {\int \frac {\sqrt {c x^2+b x+a} \left ((2 B c e-b B f-2 A c f) x^2-(2 A b f-B (2 c d+b e-2 a f)) x+b B d-2 a A f\right )}{f x^2+e x+d}dx}{2 f}\) |
| \(\Big \downarrow \) 2138 |
| \(\displaystyle \frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}-\frac {\frac {\sqrt {a+b x+c x^2} \left (-B \left (-2 c f (5 b e-4 a f)+b^2 f^2+8 c^2 \left (e^2-d f\right )\right )+2 c f x (-2 A c f-b B f+2 B c e)+2 A c f (4 c e-5 b f)\right )}{4 c f^2}-\frac {\int -\frac {B d f^2 b^3-10 c d f (B e-A f) b^2-4 c d \left (2 A c e f-B \left (2 c e^2+5 a f^2-2 c d f\right )\right ) b-\left (2 A c f \left (8 \left (e^2-d f\right ) c^2-12 f (b e-a f) c+3 b^2 f^2\right )-B \left (16 \left (e^3-2 d e f\right ) c^3-24 f \left (b e^2-a f e-b d f\right ) c^2+6 b f^2 (b e-2 a f) c+b^3 f^3\right )\right ) x^2-8 a c f (B c d e-A f (c d-2 a f))-\left (2 A c f \left (8 d e c^2-4 a e f c+4 b \left (e^2-4 d f\right ) c-b f (5 b e-16 a f)\right )-B \left (e f^2 b^3+16 c^3 d \left (e^2-d f\right )+2 c f \left (-\left (\left (5 e^2-8 d f\right ) b^2\right )+10 a e f b-8 a^2 f^2\right )-8 c^2 \left (a f \left (e^2-4 d f\right )-b \left (e^3-5 d e f\right )\right )\right )\right ) x}{4 \sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{2 c f^2}}{2 f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}-\frac {\frac {\int \frac {B d f^2 b^3-10 c d f (B e-A f) b^2-4 c d \left (-5 a B f^2+2 A c e f-2 B c \left (e^2-d f\right )\right ) b-\left (2 A c f \left (8 \left (e^2-d f\right ) c^2-12 f (b e-a f) c+3 b^2 f^2\right )-B \left (16 \left (e^3-2 d e f\right ) c^3-24 f \left (b e^2-a f e-b d f\right ) c^2+6 b f^2 (b e-2 a f) c+b^3 f^3\right )\right ) x^2-8 a c f (B c d e-A f (c d-2 a f))-\left (2 A c f \left (8 d e c^2-4 a e f c+4 b \left (e^2-4 d f\right ) c-b f (5 b e-16 a f)\right )-B \left (e f^2 b^3+16 c^3 d \left (e^2-d f\right )+2 c f \left (-\left (\left (5 e^2-8 d f\right ) b^2\right )+10 a e f b-8 a^2 f^2\right )-8 c^2 \left (a f \left (e^2-4 d f\right )-b \left (e^3-5 d e f\right )\right )\right )\right ) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{8 c f^2}+\frac {\sqrt {a+b x+c x^2} \left (-B \left (-2 c f (5 b e-4 a f)+b^2 f^2+8 c^2 \left (e^2-d f\right )\right )+2 c f x (-2 A c f-b B f+2 B c e)+2 A c f (4 c e-5 b f)\right )}{4 c f^2}}{2 f}\) |
| \(\Big \downarrow \) 2143 |
| \(\displaystyle \frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}-\frac {\frac {\frac {\int \frac {16 c \left (B d (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-A f \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )-\left (A f (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+B \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )\right ) x\right )}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{f}-\frac {\left (2 A c f \left (-12 c f (b e-a f)+3 b^2 f^2+8 c^2 \left (e^2-d f\right )\right )-B \left (-24 c^2 f \left (-a e f-b d f+b e^2\right )+6 b c f^2 (b e-2 a f)+b^3 f^3+16 c^3 \left (e^3-2 d e f\right )\right )\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{f}}{8 c f^2}+\frac {\sqrt {a+b x+c x^2} \left (-B \left (-2 c f (5 b e-4 a f)+b^2 f^2+8 c^2 \left (e^2-d f\right )\right )+2 c f x (-2 A c f-b B f+2 B c e)+2 A c f (4 c e-5 b f)\right )}{4 c f^2}}{2 f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}-\frac {\frac {\frac {16 c \int -\frac {B d (c e-b f) \left (c e^2-b f e+2 a f^2-2 c d f\right )+A f \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )+\left (A f (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+B \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )\right ) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{f}-\frac {\left (2 A c f \left (-12 c f (b e-a f)+3 b^2 f^2+8 c^2 \left (e^2-d f\right )\right )-B \left (-24 c^2 f \left (-a e f-b d f+b e^2\right )+6 b c f^2 (b e-2 a f)+b^3 f^3+16 c^3 \left (e^3-2 d e f\right )\right )\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{f}}{8 c f^2}+\frac {\sqrt {a+b x+c x^2} \left (-B \left (-2 c f (5 b e-4 a f)+b^2 f^2+8 c^2 \left (e^2-d f\right )\right )+2 c f x (-2 A c f-b B f+2 B c e)+2 A c f (4 c e-5 b f)\right )}{4 c f^2}}{2 f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}-\frac {\frac {-\frac {16 c \int -\frac {B d (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-A f \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )-\left (A f (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+B \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )\right ) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{f}-\frac {\left (2 A c f \left (-12 c f (b e-a f)+3 b^2 f^2+8 c^2 \left (e^2-d f\right )\right )-B \left (-24 c^2 f \left (-a e f-b d f+b e^2\right )+6 b c f^2 (b e-2 a f)+b^3 f^3+16 c^3 \left (e^3-2 d e f\right )\right )\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{f}}{8 c f^2}+\frac {\sqrt {a+b x+c x^2} \left (-B \left (-2 c f (5 b e-4 a f)+b^2 f^2+8 c^2 \left (e^2-d f\right )\right )+2 c f x (-2 A c f-b B f+2 B c e)+2 A c f (4 c e-5 b f)\right )}{4 c f^2}}{2 f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}-\frac {\frac {\frac {16 c \int -\frac {B d (c e-b f) \left (c e^2-b f e+2 a f^2-2 c d f\right )+A f \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )+\left (A f (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+B \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )\right ) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{f}-\frac {\left (2 A c f \left (-12 c f (b e-a f)+3 b^2 f^2+8 c^2 \left (e^2-d f\right )\right )-B \left (-24 c^2 f \left (-a e f-b d f+b e^2\right )+6 b c f^2 (b e-2 a f)+b^3 f^3+16 c^3 \left (e^3-2 d e f\right )\right )\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{f}}{8 c f^2}+\frac {\sqrt {a+b x+c x^2} \left (-B \left (-2 c f (5 b e-4 a f)+b^2 f^2+8 c^2 \left (e^2-d f\right )\right )+2 c f x (-2 A c f-b B f+2 B c e)+2 A c f (4 c e-5 b f)\right )}{4 c f^2}}{2 f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}-\frac {\frac {-\frac {16 c \int -\frac {B d (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-A f \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )-\left (A f (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+B \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )\right ) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{f}-\frac {\left (2 A c f \left (-12 c f (b e-a f)+3 b^2 f^2+8 c^2 \left (e^2-d f\right )\right )-B \left (-24 c^2 f \left (-a e f-b d f+b e^2\right )+6 b c f^2 (b e-2 a f)+b^3 f^3+16 c^3 \left (e^3-2 d e f\right )\right )\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{f}}{8 c f^2}+\frac {\sqrt {a+b x+c x^2} \left (-B \left (-2 c f (5 b e-4 a f)+b^2 f^2+8 c^2 \left (e^2-d f\right )\right )+2 c f x (-2 A c f-b B f+2 B c e)+2 A c f (4 c e-5 b f)\right )}{4 c f^2}}{2 f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}-\frac {\frac {\frac {16 c \int -\frac {B d (c e-b f) \left (c e^2-b f e+2 a f^2-2 c d f\right )+A f \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )+\left (A f (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+B \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )\right ) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{f}-\frac {\left (2 A c f \left (-12 c f (b e-a f)+3 b^2 f^2+8 c^2 \left (e^2-d f\right )\right )-B \left (-24 c^2 f \left (-a e f-b d f+b e^2\right )+6 b c f^2 (b e-2 a f)+b^3 f^3+16 c^3 \left (e^3-2 d e f\right )\right )\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{f}}{8 c f^2}+\frac {\sqrt {a+b x+c x^2} \left (-B \left (-2 c f (5 b e-4 a f)+b^2 f^2+8 c^2 \left (e^2-d f\right )\right )+2 c f x (-2 A c f-b B f+2 B c e)+2 A c f (4 c e-5 b f)\right )}{4 c f^2}}{2 f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}-\frac {\frac {-\frac {16 c \int -\frac {B d (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-A f \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )-\left (A f (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+B \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )\right ) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{f}-\frac {\left (2 A c f \left (-12 c f (b e-a f)+3 b^2 f^2+8 c^2 \left (e^2-d f\right )\right )-B \left (-24 c^2 f \left (-a e f-b d f+b e^2\right )+6 b c f^2 (b e-2 a f)+b^3 f^3+16 c^3 \left (e^3-2 d e f\right )\right )\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{f}}{8 c f^2}+\frac {\sqrt {a+b x+c x^2} \left (-B \left (-2 c f (5 b e-4 a f)+b^2 f^2+8 c^2 \left (e^2-d f\right )\right )+2 c f x (-2 A c f-b B f+2 B c e)+2 A c f (4 c e-5 b f)\right )}{4 c f^2}}{2 f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}-\frac {\frac {\frac {16 c \int -\frac {B d (c e-b f) \left (c e^2-b f e+2 a f^2-2 c d f\right )+A f \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )+\left (A f (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+B \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )\right ) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{f}-\frac {\left (2 A c f \left (-12 c f (b e-a f)+3 b^2 f^2+8 c^2 \left (e^2-d f\right )\right )-B \left (-24 c^2 f \left (-a e f-b d f+b e^2\right )+6 b c f^2 (b e-2 a f)+b^3 f^3+16 c^3 \left (e^3-2 d e f\right )\right )\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{f}}{8 c f^2}+\frac {\sqrt {a+b x+c x^2} \left (-B \left (-2 c f (5 b e-4 a f)+b^2 f^2+8 c^2 \left (e^2-d f\right )\right )+2 c f x (-2 A c f-b B f+2 B c e)+2 A c f (4 c e-5 b f)\right )}{4 c f^2}}{2 f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}-\frac {\frac {-\frac {16 c \int -\frac {B d (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-A f \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )-\left (A f (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+B \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )\right ) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{f}-\frac {\left (2 A c f \left (-12 c f (b e-a f)+3 b^2 f^2+8 c^2 \left (e^2-d f\right )\right )-B \left (-24 c^2 f \left (-a e f-b d f+b e^2\right )+6 b c f^2 (b e-2 a f)+b^3 f^3+16 c^3 \left (e^3-2 d e f\right )\right )\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{f}}{8 c f^2}+\frac {\sqrt {a+b x+c x^2} \left (-B \left (-2 c f (5 b e-4 a f)+b^2 f^2+8 c^2 \left (e^2-d f\right )\right )+2 c f x (-2 A c f-b B f+2 B c e)+2 A c f (4 c e-5 b f)\right )}{4 c f^2}}{2 f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}-\frac {\frac {\frac {16 c \int -\frac {B d (c e-b f) \left (c e^2-b f e+2 a f^2-2 c d f\right )+A f \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )+\left (A f (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+B \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )\right ) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{f}-\frac {\left (2 A c f \left (-12 c f (b e-a f)+3 b^2 f^2+8 c^2 \left (e^2-d f\right )\right )-B \left (-24 c^2 f \left (-a e f-b d f+b e^2\right )+6 b c f^2 (b e-2 a f)+b^3 f^3+16 c^3 \left (e^3-2 d e f\right )\right )\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{f}}{8 c f^2}+\frac {\sqrt {a+b x+c x^2} \left (-B \left (-2 c f (5 b e-4 a f)+b^2 f^2+8 c^2 \left (e^2-d f\right )\right )+2 c f x (-2 A c f-b B f+2 B c e)+2 A c f (4 c e-5 b f)\right )}{4 c f^2}}{2 f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}-\frac {\frac {-\frac {16 c \int -\frac {B d (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-A f \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )-\left (A f (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+B \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )\right ) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{f}-\frac {\left (2 A c f \left (-12 c f (b e-a f)+3 b^2 f^2+8 c^2 \left (e^2-d f\right )\right )-B \left (-24 c^2 f \left (-a e f-b d f+b e^2\right )+6 b c f^2 (b e-2 a f)+b^3 f^3+16 c^3 \left (e^3-2 d e f\right )\right )\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{f}}{8 c f^2}+\frac {\sqrt {a+b x+c x^2} \left (-B \left (-2 c f (5 b e-4 a f)+b^2 f^2+8 c^2 \left (e^2-d f\right )\right )+2 c f x (-2 A c f-b B f+2 B c e)+2 A c f (4 c e-5 b f)\right )}{4 c f^2}}{2 f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}-\frac {\frac {\frac {16 c \int -\frac {B d (c e-b f) \left (c e^2-b f e+2 a f^2-2 c d f\right )+A f \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )+\left (A f (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+B \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )\right ) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{f}-\frac {\left (2 A c f \left (-12 c f (b e-a f)+3 b^2 f^2+8 c^2 \left (e^2-d f\right )\right )-B \left (-24 c^2 f \left (-a e f-b d f+b e^2\right )+6 b c f^2 (b e-2 a f)+b^3 f^3+16 c^3 \left (e^3-2 d e f\right )\right )\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{f}}{8 c f^2}+\frac {\sqrt {a+b x+c x^2} \left (-B \left (-2 c f (5 b e-4 a f)+b^2 f^2+8 c^2 \left (e^2-d f\right )\right )+2 c f x (-2 A c f-b B f+2 B c e)+2 A c f (4 c e-5 b f)\right )}{4 c f^2}}{2 f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}-\frac {\frac {-\frac {16 c \int -\frac {B d (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-A f \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )-\left (A f (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+B \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )\right ) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{f}-\frac {\left (2 A c f \left (-12 c f (b e-a f)+3 b^2 f^2+8 c^2 \left (e^2-d f\right )\right )-B \left (-24 c^2 f \left (-a e f-b d f+b e^2\right )+6 b c f^2 (b e-2 a f)+b^3 f^3+16 c^3 \left (e^3-2 d e f\right )\right )\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{f}}{8 c f^2}+\frac {\sqrt {a+b x+c x^2} \left (-B \left (-2 c f (5 b e-4 a f)+b^2 f^2+8 c^2 \left (e^2-d f\right )\right )+2 c f x (-2 A c f-b B f+2 B c e)+2 A c f (4 c e-5 b f)\right )}{4 c f^2}}{2 f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}-\frac {\frac {\frac {16 c \int -\frac {B d (c e-b f) \left (c e^2-b f e+2 a f^2-2 c d f\right )+A f \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )+\left (A f (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+B \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )\right ) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{f}-\frac {\left (2 A c f \left (-12 c f (b e-a f)+3 b^2 f^2+8 c^2 \left (e^2-d f\right )\right )-B \left (-24 c^2 f \left (-a e f-b d f+b e^2\right )+6 b c f^2 (b e-2 a f)+b^3 f^3+16 c^3 \left (e^3-2 d e f\right )\right )\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{f}}{8 c f^2}+\frac {\sqrt {a+b x+c x^2} \left (-B \left (-2 c f (5 b e-4 a f)+b^2 f^2+8 c^2 \left (e^2-d f\right )\right )+2 c f x (-2 A c f-b B f+2 B c e)+2 A c f (4 c e-5 b f)\right )}{4 c f^2}}{2 f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}-\frac {\frac {-\frac {16 c \int -\frac {B d (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-A f \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )-\left (A f (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+B \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )\right ) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{f}-\frac {\left (2 A c f \left (-12 c f (b e-a f)+3 b^2 f^2+8 c^2 \left (e^2-d f\right )\right )-B \left (-24 c^2 f \left (-a e f-b d f+b e^2\right )+6 b c f^2 (b e-2 a f)+b^3 f^3+16 c^3 \left (e^3-2 d e f\right )\right )\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{f}}{8 c f^2}+\frac {\sqrt {a+b x+c x^2} \left (-B \left (-2 c f (5 b e-4 a f)+b^2 f^2+8 c^2 \left (e^2-d f\right )\right )+2 c f x (-2 A c f-b B f+2 B c e)+2 A c f (4 c e-5 b f)\right )}{4 c f^2}}{2 f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}-\frac {\frac {\frac {16 c \int -\frac {B d (c e-b f) \left (c e^2-b f e+2 a f^2-2 c d f\right )+A f \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )+\left (A f (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+B \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )\right ) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{f}-\frac {\left (2 A c f \left (-12 c f (b e-a f)+3 b^2 f^2+8 c^2 \left (e^2-d f\right )\right )-B \left (-24 c^2 f \left (-a e f-b d f+b e^2\right )+6 b c f^2 (b e-2 a f)+b^3 f^3+16 c^3 \left (e^3-2 d e f\right )\right )\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{f}}{8 c f^2}+\frac {\sqrt {a+b x+c x^2} \left (-B \left (-2 c f (5 b e-4 a f)+b^2 f^2+8 c^2 \left (e^2-d f\right )\right )+2 c f x (-2 A c f-b B f+2 B c e)+2 A c f (4 c e-5 b f)\right )}{4 c f^2}}{2 f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}-\frac {\frac {-\frac {16 c \int -\frac {B d (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-A f \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )-\left (A f (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+B \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )\right ) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{f}-\frac {\left (2 A c f \left (-12 c f (b e-a f)+3 b^2 f^2+8 c^2 \left (e^2-d f\right )\right )-B \left (-24 c^2 f \left (-a e f-b d f+b e^2\right )+6 b c f^2 (b e-2 a f)+b^3 f^3+16 c^3 \left (e^3-2 d e f\right )\right )\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{f}}{8 c f^2}+\frac {\sqrt {a+b x+c x^2} \left (-B \left (-2 c f (5 b e-4 a f)+b^2 f^2+8 c^2 \left (e^2-d f\right )\right )+2 c f x (-2 A c f-b B f+2 B c e)+2 A c f (4 c e-5 b f)\right )}{4 c f^2}}{2 f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}-\frac {\frac {\frac {16 c \int -\frac {B d (c e-b f) \left (c e^2-b f e+2 a f^2-2 c d f\right )+A f \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )+\left (A f (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+B \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )\right ) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{f}-\frac {\left (2 A c f \left (-12 c f (b e-a f)+3 b^2 f^2+8 c^2 \left (e^2-d f\right )\right )-B \left (-24 c^2 f \left (-a e f-b d f+b e^2\right )+6 b c f^2 (b e-2 a f)+b^3 f^3+16 c^3 \left (e^3-2 d e f\right )\right )\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{f}}{8 c f^2}+\frac {\sqrt {a+b x+c x^2} \left (-B \left (-2 c f (5 b e-4 a f)+b^2 f^2+8 c^2 \left (e^2-d f\right )\right )+2 c f x (-2 A c f-b B f+2 B c e)+2 A c f (4 c e-5 b f)\right )}{4 c f^2}}{2 f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}-\frac {\frac {-\frac {16 c \int -\frac {B d (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-A f \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )-\left (A f (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+B \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )\right ) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{f}-\frac {\left (2 A c f \left (-12 c f (b e-a f)+3 b^2 f^2+8 c^2 \left (e^2-d f\right )\right )-B \left (-24 c^2 f \left (-a e f-b d f+b e^2\right )+6 b c f^2 (b e-2 a f)+b^3 f^3+16 c^3 \left (e^3-2 d e f\right )\right )\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{f}}{8 c f^2}+\frac {\sqrt {a+b x+c x^2} \left (-B \left (-2 c f (5 b e-4 a f)+b^2 f^2+8 c^2 \left (e^2-d f\right )\right )+2 c f x (-2 A c f-b B f+2 B c e)+2 A c f (4 c e-5 b f)\right )}{4 c f^2}}{2 f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}-\frac {\frac {\frac {16 c \int -\frac {B d (c e-b f) \left (c e^2-b f e+2 a f^2-2 c d f\right )+A f \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )+\left (A f (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+B \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )\right ) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{f}-\frac {\left (2 A c f \left (-12 c f (b e-a f)+3 b^2 f^2+8 c^2 \left (e^2-d f\right )\right )-B \left (-24 c^2 f \left (-a e f-b d f+b e^2\right )+6 b c f^2 (b e-2 a f)+b^3 f^3+16 c^3 \left (e^3-2 d e f\right )\right )\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{f}}{8 c f^2}+\frac {\sqrt {a+b x+c x^2} \left (-B \left (-2 c f (5 b e-4 a f)+b^2 f^2+8 c^2 \left (e^2-d f\right )\right )+2 c f x (-2 A c f-b B f+2 B c e)+2 A c f (4 c e-5 b f)\right )}{4 c f^2}}{2 f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}-\frac {\frac {-\frac {16 c \int -\frac {B d (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-A f \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )-\left (A f (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+B \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )\right ) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{f}-\frac {\left (2 A c f \left (-12 c f (b e-a f)+3 b^2 f^2+8 c^2 \left (e^2-d f\right )\right )-B \left (-24 c^2 f \left (-a e f-b d f+b e^2\right )+6 b c f^2 (b e-2 a f)+b^3 f^3+16 c^3 \left (e^3-2 d e f\right )\right )\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{f}}{8 c f^2}+\frac {\sqrt {a+b x+c x^2} \left (-B \left (-2 c f (5 b e-4 a f)+b^2 f^2+8 c^2 \left (e^2-d f\right )\right )+2 c f x (-2 A c f-b B f+2 B c e)+2 A c f (4 c e-5 b f)\right )}{4 c f^2}}{2 f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}-\frac {\frac {\frac {16 c \int -\frac {B d (c e-b f) \left (c e^2-b f e+2 a f^2-2 c d f\right )+A f \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )+\left (A f (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+B \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )\right ) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{f}-\frac {\left (2 A c f \left (-12 c f (b e-a f)+3 b^2 f^2+8 c^2 \left (e^2-d f\right )\right )-B \left (-24 c^2 f \left (-a e f-b d f+b e^2\right )+6 b c f^2 (b e-2 a f)+b^3 f^3+16 c^3 \left (e^3-2 d e f\right )\right )\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{f}}{8 c f^2}+\frac {\sqrt {a+b x+c x^2} \left (-B \left (-2 c f (5 b e-4 a f)+b^2 f^2+8 c^2 \left (e^2-d f\right )\right )+2 c f x (-2 A c f-b B f+2 B c e)+2 A c f (4 c e-5 b f)\right )}{4 c f^2}}{2 f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}-\frac {\frac {-\frac {16 c \int -\frac {B d (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-A f \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )-\left (A f (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+B \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )\right ) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{f}-\frac {\left (2 A c f \left (-12 c f (b e-a f)+3 b^2 f^2+8 c^2 \left (e^2-d f\right )\right )-B \left (-24 c^2 f \left (-a e f-b d f+b e^2\right )+6 b c f^2 (b e-2 a f)+b^3 f^3+16 c^3 \left (e^3-2 d e f\right )\right )\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{f}}{8 c f^2}+\frac {\sqrt {a+b x+c x^2} \left (-B \left (-2 c f (5 b e-4 a f)+b^2 f^2+8 c^2 \left (e^2-d f\right )\right )+2 c f x (-2 A c f-b B f+2 B c e)+2 A c f (4 c e-5 b f)\right )}{4 c f^2}}{2 f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}-\frac {\frac {\frac {16 c \int -\frac {B d (c e-b f) \left (c e^2-b f e+2 a f^2-2 c d f\right )+A f \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )+\left (A f (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+B \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )\right ) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{f}-\frac {\left (2 A c f \left (-12 c f (b e-a f)+3 b^2 f^2+8 c^2 \left (e^2-d f\right )\right )-B \left (-24 c^2 f \left (-a e f-b d f+b e^2\right )+6 b c f^2 (b e-2 a f)+b^3 f^3+16 c^3 \left (e^3-2 d e f\right )\right )\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{f}}{8 c f^2}+\frac {\sqrt {a+b x+c x^2} \left (-B \left (-2 c f (5 b e-4 a f)+b^2 f^2+8 c^2 \left (e^2-d f\right )\right )+2 c f x (-2 A c f-b B f+2 B c e)+2 A c f (4 c e-5 b f)\right )}{4 c f^2}}{2 f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}-\frac {\frac {-\frac {16 c \int -\frac {B d (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-A f \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )-\left (A f (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+B \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )\right ) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{f}-\frac {\left (2 A c f \left (-12 c f (b e-a f)+3 b^2 f^2+8 c^2 \left (e^2-d f\right )\right )-B \left (-24 c^2 f \left (-a e f-b d f+b e^2\right )+6 b c f^2 (b e-2 a f)+b^3 f^3+16 c^3 \left (e^3-2 d e f\right )\right )\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{f}}{8 c f^2}+\frac {\sqrt {a+b x+c x^2} \left (-B \left (-2 c f (5 b e-4 a f)+b^2 f^2+8 c^2 \left (e^2-d f\right )\right )+2 c f x (-2 A c f-b B f+2 B c e)+2 A c f (4 c e-5 b f)\right )}{4 c f^2}}{2 f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}-\frac {\frac {\frac {16 c \int -\frac {B d (c e-b f) \left (c e^2-b f e+2 a f^2-2 c d f\right )+A f \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )+\left (A f (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+B \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )\right ) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{f}-\frac {\left (2 A c f \left (-12 c f (b e-a f)+3 b^2 f^2+8 c^2 \left (e^2-d f\right )\right )-B \left (-24 c^2 f \left (-a e f-b d f+b e^2\right )+6 b c f^2 (b e-2 a f)+b^3 f^3+16 c^3 \left (e^3-2 d e f\right )\right )\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{f}}{8 c f^2}+\frac {\sqrt {a+b x+c x^2} \left (-B \left (-2 c f (5 b e-4 a f)+b^2 f^2+8 c^2 \left (e^2-d f\right )\right )+2 c f x (-2 A c f-b B f+2 B c e)+2 A c f (4 c e-5 b f)\right )}{4 c f^2}}{2 f}\) |
3.1.20.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((g_.) + (h_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e _.)*(x_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp[h*(a + b*x + c*x^2)^p*((d + e*x + f*x^2)^(q + 1)/(2*f*(p + q + 1))), x] - Simp[1/(2*f*(p + q + 1)) Int[(a + b*x + c*x^2)^(p - 1)*(d + e*x + f*x^2)^q*Simp[h*p*(b*d - a*e) + a* (h*e - 2*g*f)*(p + q + 1) + (2*h*p*(c*d - a*f) + b*(h*e - 2*g*f)*(p + q + 1 ))*x + (h*p*(c*e - b*f) + c*(h*e - 2*g*f)*(p + q + 1))*x^2, x], x], x] /; F reeQ[{a, b, c, d, e, f, g, h, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4* d*f, 0] && GtQ[p, 0] && NeQ[p + q + 1, 0]
Int[(Px_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_. )*(x_)^2)^(q_), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1] , C = Coeff[Px, x, 2]}, Simp[(B*c*f*(2*p + 2*q + 3) + C*(b*f*p - c*e*(2*p + q + 2)) + 2*c*C*f*(p + q + 1)*x)*(a + b*x + c*x^2)^p*((d + e*x + f*x^2)^(q + 1)/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3))), x] - Simp[1/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3)) Int[(a + b*x + c*x^2)^(p - 1)*(d + e*x + f*x^2)^q*Si mp[p*(b*d - a*e)*(C*(c*e - b*f)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(b^2*C*d*f*p + a*c*(C*(2*d*f - e^2*(2*p + q + 2)) + f*(B*e - 2* A*f)*(2*p + 2*q + 3))) + (2*p*(c*d - a*f)*(C*(c*e - b*f)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(C*e*f*p*(b^2 - 4*a*c) - b*c*(C*(e^2 - 4*d*f)*(2*p + q + 2) + f*(2*C*d - B*e + 2*A*f)*(2*p + 2*q + 3))))*x + (p*( c*e - b*f)*(C*(c*e - b*f)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(C*f^2*p*(b^2 - 4*a*c) - c^2*(C*(e^2 - 4*d*f)*(2*p + q + 2) + f*(2*C* d - B*e + 2*A*f)*(2*p + 2*q + 3))))*x^2, x], x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && PolyQ[Px, x, 2] && GtQ[p, 0] && NeQ[p + q + 1, 0] && NeQ[2* p + 2*q + 3, 0] && !IGtQ[p, 0] && !IGtQ[q, 0]
Int[(Px_)/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_ .)*(x_)^2]), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[Px, x, 2]}, Simp[C/c Int[1/Sqrt[d + e*x + f*x^2], x], x] + Simp[ 1/c Int[(A*c - a*C + (B*c - b*C)*x)/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x ^2]), x], x]] /; FreeQ[{a, b, c, d, e, f}, x] && PolyQ[Px, x, 2]
Leaf count of result is larger than twice the leaf count of optimal. \(2277\) vs. \(2(1027)=2054\).
Time = 1.08 (sec) , antiderivative size = 2278, normalized size of antiderivative = 2.09
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(2278\) |
| default | \(\text {Expression too large to display}\) | \(2908\) |
1/24/c*(8*B*c^2*f^2*x^2+12*A*c^2*f^2*x+14*B*b*c*f^2*x-12*B*c^2*e*f*x+30*A* b*c*f^2-24*A*c^2*e*f+32*B*a*c*f^2+3*B*b^2*f^2-30*B*b*c*e*f-24*B*c^2*d*f+24 *B*c^2*e^2)*(c*x^2+b*x+a)^(1/2)/f^3+1/16/f^3/c*(1/f*(24*A*a*c^2*f^3+6*A*b^ 2*c*f^3-24*A*b*c^2*e*f^2-16*A*c^3*d*f^2+16*A*c^3*e^2*f+12*B*a*b*c*f^3-24*B *a*c^2*e*f^2-B*b^3*f^3-6*B*b^2*c*e*f^2-24*B*b*c^2*d*f^2+24*B*b*c^2*e^2*f+3 2*B*c^3*d*e*f-16*B*c^3*e^3)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^ (1/2)-8/f^2*c*(-2*A*a*c*e*f^3*(-4*d*f+e^2)^(1/2)-2*A*f^3*b*c*d*(-4*d*f+e^2 )^(1/2)+2*A*b*c*e^2*f^2*(-4*d*f+e^2)^(1/2)+2*A*c^2*d*e*f^2*(-4*d*f+e^2)^(1 /2)-2*B*a*b*e*f^3*(-4*d*f+e^2)^(1/2)-2*B*a*c*d*f^3*(-4*d*f+e^2)^(1/2)+2*B* a*c*e^2*f^2*(-4*d*f+e^2)^(1/2)-2*B*b*c*e^3*f*(-4*d*f+e^2)^(1/2)-3*B*c^2*d* e^2*f*(-4*d*f+e^2)^(1/2)+6*A*b*c*d*e*f^3+6*B*a*c*d*e*f^3-8*B*b*c*d*e^2*f^2 -B*b^2*e^3*f^2+A*c^2*e^4*f-B*a^2*e*f^4-2*A*b^2*d*f^4+A*b^2*e^2*f^3+2*A*c^2 *d^2*f^3+4*B*b*c*d*e*f^2*(-4*d*f+e^2)^(1/2)+2*A*a^2*f^5-B*c^2*e^5-4*A*a*c* d*f^4+2*A*a*c*e^2*f^3-2*A*b*c*e^3*f^2-4*A*c^2*d*e^2*f^2-4*B*a*b*d*f^4+2*B* a*b*e^2*f^3-2*B*a*c*e^3*f^2+3*B*b^2*d*e*f^3+4*B*b*c*d^2*f^3+2*B*b*c*e^4*f- 5*B*c^2*d^2*e*f^2+5*B*c^2*d*e^3*f-2*A*a*b*e*f^4+B*a^2*f^4*(-4*d*f+e^2)^(1/ 2)+B*c^2*e^4*(-4*d*f+e^2)^(1/2)+2*A*a*b*f^4*(-4*d*f+e^2)^(1/2)-A*f^3*b^2*e *(-4*d*f+e^2)^(1/2)-A*f*c^2*e^3*(-4*d*f+e^2)^(1/2)-B*b^2*d*f^3*(-4*d*f+e^2 )^(1/2)+B*f^2*b^2*e^2*(-4*d*f+e^2)^(1/2)+B*c^2*d^2*f^2*(-4*d*f+e^2)^(1/2)) /(-4*d*f+e^2)^(1/2)*2^(1/2)/((b*f*(-4*d*f+e^2)^(1/2)-(-4*d*f+e^2)^(1/2)...
Timed out. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{d+e x+f x^2} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{d+e x+f x^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{d+e x+f x^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*d*f-e^2>0)', see `assume?` for more deta
Exception generated. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{d+e x+f x^2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{d+e x+f x^2} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{f\,x^2+e\,x+d} \,d x \]