Integrand size = 27, antiderivative size = 608 \[ \int \frac {(A+B x) (d+e x)^4}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 (d+e x)^3 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {2 (d+e x) \left (b^3 B e \left (c d^2-5 a e^2\right )-8 a c^2 e \left (A c d^2+8 a B d e+3 a A e^2\right )-2 b^2 c \left (2 B c d^3+5 A c d^2 e-2 a B d e^2-a A e^3\right )+4 b c \left (2 A c d \left (c d^2+3 a e^2\right )+a B e \left (5 c d^2+7 a e^2\right )\right )-\left (5 b^4 B e^3-2 b^3 c e^2 (3 B d+A e)-4 b^2 c e \left (B c d^2+A c d e+8 a B e^2\right )+8 b c^2 \left (B c d^3+3 A c d^2 e+6 a B d e^2+2 a A e^3\right )-16 c^2 \left (2 a B e \left (c d^2-a e^2\right )+A c d \left (c d^2+2 a e^2\right )\right )\right ) x\right )}{3 c^2 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}+\frac {e \left (15 b^4 B e^3-2 b^3 c e^2 (7 B d+3 A e)-4 b^2 c e \left (2 B c d^2+A c d e+25 a B e^2\right )+8 b c^2 \left (2 B c d^3+6 A c d^2 e+13 a B d e^2+5 a A e^3\right )-16 c^2 \left (4 a B e \left (c d^2-2 a e^2\right )+A c d \left (2 c d^2+5 a e^2\right )\right )\right ) \sqrt {a+b x+c x^2}}{3 c^3 \left (b^2-4 a c\right )^2}+\frac {e^3 (8 B c d-5 b B e+2 A c e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 c^{7/2}} \]
2/3*(e*x+d)^3*(2*a*c*(A*e+B*d)-b*(A*c*d+B*a*e)-(b^2*B*e-b*c*(A*e+B*d)+2*c* (A*c*d-B*a*e))*x)/c/(-4*a*c+b^2)/(c*x^2+b*x+a)^(3/2)+1/2*e^3*(2*A*c*e-5*B* b*e+8*B*c*d)*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(7/2)+2/ 3*(e*x+d)*(b^3*B*e*(-5*a*e^2+c*d^2)-8*a*c^2*e*(3*A*a*e^2+A*c*d^2+8*B*a*d*e )-2*b^2*c*(-A*a*e^3+5*A*c*d^2*e-2*B*a*d*e^2+2*B*c*d^3)+4*b*c*(2*A*c*d*(3*a *e^2+c*d^2)+a*B*e*(7*a*e^2+5*c*d^2))-(5*b^4*B*e^3-2*b^3*c*e^2*(A*e+3*B*d)- 4*b^2*c*e*(A*c*d*e+8*B*a*e^2+B*c*d^2)+8*b*c^2*(2*A*a*e^3+3*A*c*d^2*e+6*B*a *d*e^2+B*c*d^3)-16*c^2*(2*a*B*e*(-a*e^2+c*d^2)+A*c*d*(2*a*e^2+c*d^2)))*x)/ c^2/(-4*a*c+b^2)^2/(c*x^2+b*x+a)^(1/2)+1/3*e*(15*b^4*B*e^3-2*b^3*c*e^2*(3* A*e+7*B*d)-4*b^2*c*e*(A*c*d*e+25*B*a*e^2+2*B*c*d^2)+8*b*c^2*(5*A*a*e^3+6*A *c*d^2*e+13*B*a*d*e^2+2*B*c*d^3)-16*c^2*(4*a*B*e*(-2*a*e^2+c*d^2)+A*c*d*(5 *a*e^2+2*c*d^2)))*(c*x^2+b*x+a)^(1/2)/c^3/(-4*a*c+b^2)^2
Time = 6.29 (sec) , antiderivative size = 851, normalized size of antiderivative = 1.40 \[ \int \frac {(A+B x) (d+e x)^4}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {-2 A c \left (3 b^5 e^4 x^2+2 b^4 e^4 x \left (3 a+2 c x^2\right )+b^3 \left (3 a^2 e^4-18 a c e^4 x^2+c^2 d \left (d^3+12 d^2 e x-18 d e^2 x^2-4 e^3 x^3\right )\right )-4 b c \left (5 a^3 e^4+2 c^3 d^3 x^2 (3 d-4 e x)+12 a^2 c d e^2 (d-2 e x)+3 a c^2 d \left (d^3-4 d^2 e x+6 d e^2 x^2-4 e^3 x^3\right )\right )+8 c^2 \left (-2 c^3 d^4 x^3+a^3 e^3 (8 d+3 e x)-3 a c^2 d^2 x \left (d^2+2 e^2 x^2\right )+4 a^2 c e \left (d^3+3 d e^2 x^2+e^3 x^3\right )\right )-2 b^2 c \left (21 a^2 e^4 x+3 c^2 d^2 x \left (d^2-8 d e x+2 e^2 x^2\right )+2 a c e \left (-2 d^3+18 d^2 e x-6 d e^2 x^2+7 e^3 x^3\right )\right )\right )+B \left (128 a^4 c^2 e^4+b x \left (15 b^5 e^4 x-16 c^5 d^4 x^2+8 b c^4 d^3 x (-3 d+2 e x)+b^3 c^2 e^3 x^2 (-32 d+3 e x)+4 b^4 c e^3 x (-6 d+5 e x)-6 b^2 c^3 d^2 \left (d^2-4 d e x-2 e^2 x^2\right )\right )+4 a^3 c e^2 \left (-25 b^2 e^2+2 b c e (20 d+39 e x)-48 c^2 \left (d^2+d e x-e^2 x^2\right )\right )+a^2 \left (15 b^4 e^4+48 b^2 c^2 e^3 x (7 d+e x)-6 b^3 c e^3 (4 d+35 e x)+32 b c^3 e \left (2 d^3-9 d^2 e x+8 e^3 x^3\right )-16 c^4 \left (d^4+18 d^2 e^2 x^2+16 d e^3 x^3-3 e^4 x^4\right )\right )+2 a \left (15 b^5 e^4 x+32 c^5 d^3 e x^3+2 b^3 c^2 e^3 x^2 (36 d-37 e x)-3 b^4 c e^3 x (8 d+15 e x)-12 b c^4 d^2 x \left (d^2-4 d e x+6 e^2 x^2\right )-2 b^2 c^3 \left (d^4-24 d^3 e x+18 d^2 e^2 x^2-56 d e^3 x^3+6 e^4 x^4\right )\right )\right )}{3 c^3 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}}-\frac {e^3 (8 B c d-5 b B e+2 A c e) \log \left (c^3 \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )\right )}{2 c^{7/2}} \]
(-2*A*c*(3*b^5*e^4*x^2 + 2*b^4*e^4*x*(3*a + 2*c*x^2) + b^3*(3*a^2*e^4 - 18 *a*c*e^4*x^2 + c^2*d*(d^3 + 12*d^2*e*x - 18*d*e^2*x^2 - 4*e^3*x^3)) - 4*b* c*(5*a^3*e^4 + 2*c^3*d^3*x^2*(3*d - 4*e*x) + 12*a^2*c*d*e^2*(d - 2*e*x) + 3*a*c^2*d*(d^3 - 4*d^2*e*x + 6*d*e^2*x^2 - 4*e^3*x^3)) + 8*c^2*(-2*c^3*d^4 *x^3 + a^3*e^3*(8*d + 3*e*x) - 3*a*c^2*d^2*x*(d^2 + 2*e^2*x^2) + 4*a^2*c*e *(d^3 + 3*d*e^2*x^2 + e^3*x^3)) - 2*b^2*c*(21*a^2*e^4*x + 3*c^2*d^2*x*(d^2 - 8*d*e*x + 2*e^2*x^2) + 2*a*c*e*(-2*d^3 + 18*d^2*e*x - 6*d*e^2*x^2 + 7*e ^3*x^3))) + B*(128*a^4*c^2*e^4 + b*x*(15*b^5*e^4*x - 16*c^5*d^4*x^2 + 8*b* c^4*d^3*x*(-3*d + 2*e*x) + b^3*c^2*e^3*x^2*(-32*d + 3*e*x) + 4*b^4*c*e^3*x *(-6*d + 5*e*x) - 6*b^2*c^3*d^2*(d^2 - 4*d*e*x - 2*e^2*x^2)) + 4*a^3*c*e^2 *(-25*b^2*e^2 + 2*b*c*e*(20*d + 39*e*x) - 48*c^2*(d^2 + d*e*x - e^2*x^2)) + a^2*(15*b^4*e^4 + 48*b^2*c^2*e^3*x*(7*d + e*x) - 6*b^3*c*e^3*(4*d + 35*e *x) + 32*b*c^3*e*(2*d^3 - 9*d^2*e*x + 8*e^3*x^3) - 16*c^4*(d^4 + 18*d^2*e^ 2*x^2 + 16*d*e^3*x^3 - 3*e^4*x^4)) + 2*a*(15*b^5*e^4*x + 32*c^5*d^3*e*x^3 + 2*b^3*c^2*e^3*x^2*(36*d - 37*e*x) - 3*b^4*c*e^3*x*(8*d + 15*e*x) - 12*b* c^4*d^2*x*(d^2 - 4*d*e*x + 6*e^2*x^2) - 2*b^2*c^3*(d^4 - 24*d^3*e*x + 18*d ^2*e^2*x^2 - 56*d*e^3*x^3 + 6*e^4*x^4))))/(3*c^3*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^(3/2)) - (e^3*(8*B*c*d - 5*b*B*e + 2*A*c*e)*Log[c^3*(b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(2*c^(7/2))
Time = 1.23 (sec) , antiderivative size = 637, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {1233, 27, 1233, 27, 1160, 1092, 219}
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.
| \(\displaystyle \int \frac {(A+B x) (d+e x)^4}{\left (a+b x+c x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1233 |
| \(\displaystyle \frac {2 \int -\frac {(d+e x)^2 \left (8 A c^2 d^2-2 b c (2 B d+5 A e) d+b B e (b d-6 a e)+4 a c e (4 B d+3 A e)-e \left (5 B e b^2+4 A c^2 d-2 c (b B d+A b e+8 a B e)\right ) x\right )}{2 \left (c x^2+b x+a\right )^{3/2}}dx}{3 c \left (b^2-4 a c\right )}+\frac {2 (d+e x)^3 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 (d+e x)^3 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {\int \frac {(d+e x)^2 \left (8 A c^2 d^2-2 b c (2 B d+5 A e) d+b B e (b d-6 a e)+4 a c e (4 B d+3 A e)-e \left (5 B e b^2+4 A c^2 d-2 c (b B d+A b e+8 a B e)\right ) x\right )}{\left (c x^2+b x+a\right )^{3/2}}dx}{3 c \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 1233 |
| \(\displaystyle \frac {2 (d+e x)^3 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {\frac {2 \int -\frac {e \left (5 B d e^2 b^4-2 e \left (2 B c d^2+A c e d-5 a B e^2\right ) b^3+4 c \left (2 B c d^3+6 A c e d^2-11 a B e^2 d-a A e^3\right ) b^2-8 c \left (A c d \left (2 c d^2+5 a e^2\right )+a B e \left (4 c d^2+7 a e^2\right )\right ) b+48 a^2 c^2 e^2 (4 B d+A e)+\left (15 B e^3 b^4-2 c e^2 (7 B d+3 A e) b^3-4 c e \left (2 B c d^2+A c e d+25 a B e^2\right ) b^2+8 c^2 \left (2 B c d^3+6 A c e d^2+13 a B e^2 d+5 a A e^3\right ) b-16 c^2 \left (4 a B e \left (c d^2-2 a e^2\right )+A c d \left (2 c d^2+5 a e^2\right )\right )\right ) x\right )}{2 \sqrt {c x^2+b x+a}}dx}{c \left (b^2-4 a c\right )}+\frac {2 (d+e x) \left (2 b^2 c \left (-a A e^3-2 a B d e^2+5 A c d^2 e+2 B c d^3\right )+x \left (-4 b^2 c e \left (8 a B e^2+A c d e+B c d^2\right )+8 b c^2 \left (2 a A e^3+6 a B d e^2+3 A c d^2 e+B c d^3\right )-16 c^2 \left (A c d \left (2 a e^2+c d^2\right )+2 a B e \left (c d^2-a e^2\right )\right )-2 b^3 c e^2 (A e+3 B d)+5 b^4 B e^3\right )-4 b c \left (2 A c d \left (3 a e^2+c d^2\right )+a B e \left (7 a e^2+5 c d^2\right )\right )+8 a c^2 e \left (3 a A e^2+8 a B d e+A c d^2\right )+b^3 (-B) \left (c d^2 e-5 a e^3\right )\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}}{3 c \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 (d+e x)^3 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {\frac {2 (d+e x) \left (2 b^2 c \left (-a A e^3-2 a B d e^2+5 A c d^2 e+2 B c d^3\right )+x \left (-4 b^2 c e \left (8 a B e^2+A c d e+B c d^2\right )+8 b c^2 \left (2 a A e^3+6 a B d e^2+3 A c d^2 e+B c d^3\right )-16 c^2 \left (A c d \left (2 a e^2+c d^2\right )+2 a B e \left (c d^2-a e^2\right )\right )-2 b^3 c e^2 (A e+3 B d)+5 b^4 B e^3\right )-4 b c \left (2 A c d \left (3 a e^2+c d^2\right )+a B e \left (7 a e^2+5 c d^2\right )\right )+8 a c^2 e \left (3 a A e^2+8 a B d e+A c d^2\right )+b^3 (-B) \left (c d^2 e-5 a e^3\right )\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {e \int \frac {5 B d e^2 b^4-2 e \left (2 B c d^2+A c e d-5 a B e^2\right ) b^3+4 c \left (2 B c d^3+6 A c e d^2-11 a B e^2 d-a A e^3\right ) b^2-8 c \left (A c d \left (2 c d^2+5 a e^2\right )+a B e \left (4 c d^2+7 a e^2\right )\right ) b+48 a^2 c^2 e^2 (4 B d+A e)+\left (15 B e^3 b^4-2 c e^2 (7 B d+3 A e) b^3-4 c e \left (2 B c d^2+A c e d+25 a B e^2\right ) b^2+8 c^2 \left (2 B c d^3+6 A c e d^2+13 a B e^2 d+5 a A e^3\right ) b-16 c^2 \left (4 a B e \left (c d^2-2 a e^2\right )+A c d \left (2 c d^2+5 a e^2\right )\right )\right ) x}{\sqrt {c x^2+b x+a}}dx}{c \left (b^2-4 a c\right )}}{3 c \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle \frac {2 (d+e x)^3 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {\frac {2 (d+e x) \left (2 b^2 c \left (-a A e^3-2 a B d e^2+5 A c d^2 e+2 B c d^3\right )+x \left (-4 b^2 c e \left (8 a B e^2+A c d e+B c d^2\right )+8 b c^2 \left (2 a A e^3+6 a B d e^2+3 A c d^2 e+B c d^3\right )-16 c^2 \left (A c d \left (2 a e^2+c d^2\right )+2 a B e \left (c d^2-a e^2\right )\right )-2 b^3 c e^2 (A e+3 B d)+5 b^4 B e^3\right )-4 b c \left (2 A c d \left (3 a e^2+c d^2\right )+a B e \left (7 a e^2+5 c d^2\right )\right )+8 a c^2 e \left (3 a A e^2+8 a B d e+A c d^2\right )+b^3 (-B) \left (c d^2 e-5 a e^3\right )\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {e \left (\frac {3 e^2 \left (b^2-4 a c\right )^2 (2 A c e-5 b B e+8 B c d) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{2 c}+\frac {\sqrt {a+b x+c x^2} \left (-4 b^2 c e \left (25 a B e^2+A c d e+2 B c d^2\right )+8 b c^2 \left (5 a A e^3+13 a B d e^2+6 A c d^2 e+2 B c d^3\right )-16 c^2 \left (A c d \left (5 a e^2+2 c d^2\right )+4 a B e \left (c d^2-2 a e^2\right )\right )-2 b^3 c e^2 (3 A e+7 B d)+15 b^4 B e^3\right )}{c}\right )}{c \left (b^2-4 a c\right )}}{3 c \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {2 (d+e x)^3 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {\frac {2 (d+e x) \left (2 b^2 c \left (-a A e^3-2 a B d e^2+5 A c d^2 e+2 B c d^3\right )+x \left (-4 b^2 c e \left (8 a B e^2+A c d e+B c d^2\right )+8 b c^2 \left (2 a A e^3+6 a B d e^2+3 A c d^2 e+B c d^3\right )-16 c^2 \left (A c d \left (2 a e^2+c d^2\right )+2 a B e \left (c d^2-a e^2\right )\right )-2 b^3 c e^2 (A e+3 B d)+5 b^4 B e^3\right )-4 b c \left (2 A c d \left (3 a e^2+c d^2\right )+a B e \left (7 a e^2+5 c d^2\right )\right )+8 a c^2 e \left (3 a A e^2+8 a B d e+A c d^2\right )+b^3 (-B) \left (c d^2 e-5 a e^3\right )\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {e \left (\frac {3 e^2 \left (b^2-4 a c\right )^2 (2 A c e-5 b B e+8 B c d) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{c}+\frac {\sqrt {a+b x+c x^2} \left (-4 b^2 c e \left (25 a B e^2+A c d e+2 B c d^2\right )+8 b c^2 \left (5 a A e^3+13 a B d e^2+6 A c d^2 e+2 B c d^3\right )-16 c^2 \left (A c d \left (5 a e^2+2 c d^2\right )+4 a B e \left (c d^2-2 a e^2\right )\right )-2 b^3 c e^2 (3 A e+7 B d)+15 b^4 B e^3\right )}{c}\right )}{c \left (b^2-4 a c\right )}}{3 c \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {2 (d+e x)^3 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {\frac {2 (d+e x) \left (2 b^2 c \left (-a A e^3-2 a B d e^2+5 A c d^2 e+2 B c d^3\right )+x \left (-4 b^2 c e \left (8 a B e^2+A c d e+B c d^2\right )+8 b c^2 \left (2 a A e^3+6 a B d e^2+3 A c d^2 e+B c d^3\right )-16 c^2 \left (A c d \left (2 a e^2+c d^2\right )+2 a B e \left (c d^2-a e^2\right )\right )-2 b^3 c e^2 (A e+3 B d)+5 b^4 B e^3\right )-4 b c \left (2 A c d \left (3 a e^2+c d^2\right )+a B e \left (7 a e^2+5 c d^2\right )\right )+8 a c^2 e \left (3 a A e^2+8 a B d e+A c d^2\right )+b^3 (-B) \left (c d^2 e-5 a e^3\right )\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {e \left (\frac {3 e^2 \left (b^2-4 a c\right )^2 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) (2 A c e-5 b B e+8 B c d)}{2 c^{3/2}}+\frac {\sqrt {a+b x+c x^2} \left (-4 b^2 c e \left (25 a B e^2+A c d e+2 B c d^2\right )+8 b c^2 \left (5 a A e^3+13 a B d e^2+6 A c d^2 e+2 B c d^3\right )-16 c^2 \left (A c d \left (5 a e^2+2 c d^2\right )+4 a B e \left (c d^2-2 a e^2\right )\right )-2 b^3 c e^2 (3 A e+7 B d)+15 b^4 B e^3\right )}{c}\right )}{c \left (b^2-4 a c\right )}}{3 c \left (b^2-4 a c\right )}\) |
(2*(d + e*x)^3*(2*a*c*(B*d + A*e) - b*(A*c*d + a*B*e) - (b^2*B*e - b*c*(B* d + A*e) + 2*c*(A*c*d - a*B*e))*x))/(3*c*(b^2 - 4*a*c)*(a + b*x + c*x^2)^( 3/2)) - ((2*(d + e*x)*(8*a*c^2*e*(A*c*d^2 + 8*a*B*d*e + 3*a*A*e^2) - b^3*B *(c*d^2*e - 5*a*e^3) + 2*b^2*c*(2*B*c*d^3 + 5*A*c*d^2*e - 2*a*B*d*e^2 - a* A*e^3) - 4*b*c*(2*A*c*d*(c*d^2 + 3*a*e^2) + a*B*e*(5*c*d^2 + 7*a*e^2)) + ( 5*b^4*B*e^3 - 2*b^3*c*e^2*(3*B*d + A*e) - 4*b^2*c*e*(B*c*d^2 + A*c*d*e + 8 *a*B*e^2) + 8*b*c^2*(B*c*d^3 + 3*A*c*d^2*e + 6*a*B*d*e^2 + 2*a*A*e^3) - 16 *c^2*(2*a*B*e*(c*d^2 - a*e^2) + A*c*d*(c*d^2 + 2*a*e^2)))*x))/(c*(b^2 - 4* a*c)*Sqrt[a + b*x + c*x^2]) - (e*(((15*b^4*B*e^3 - 2*b^3*c*e^2*(7*B*d + 3* A*e) - 4*b^2*c*e*(2*B*c*d^2 + A*c*d*e + 25*a*B*e^2) + 8*b*c^2*(2*B*c*d^3 + 6*A*c*d^2*e + 13*a*B*d*e^2 + 5*a*A*e^3) - 16*c^2*(4*a*B*e*(c*d^2 - 2*a*e^ 2) + A*c*d*(2*c*d^2 + 5*a*e^2)))*Sqrt[a + b*x + c*x^2])/c + (3*(b^2 - 4*a* c)^2*e^2*(8*B*c*d - 5*b*B*e + 2*A*c*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt [a + b*x + c*x^2])])/(2*c^(3/2))))/(c*(b^2 - 4*a*c)))/(3*c*(b^2 - 4*a*c))
3.25.80.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b *e)/(2*c) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^(m - 1))*(a + b*x + c*x^2) ^(p + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g - c *(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Simp[1/(c*( p + 1)*(b^2 - 4*a*c)) Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Sim p[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a*e*(e*f *(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*( m + p + 1) + 2*c^2*d*f*(m + 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2* p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, b, c, d, e, f, g]) | | !ILtQ[m + 2*p + 3, 0])
Leaf count of result is larger than twice the leaf count of optimal. \(1968\) vs. \(2(586)=1172\).
Time = 0.91 (sec) , antiderivative size = 1969, normalized size of antiderivative = 3.24
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1969\) |
| risch | \(\text {Expression too large to display}\) | \(16731\) |
A*d^4*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2* (2*c*x+b)/(c*x^2+b*x+a)^(1/2))+B*e^4*(x^4/c/(c*x^2+b*x+a)^(3/2)-5/2*b/c*(- 1/3*x^3/c/(c*x^2+b*x+a)^(3/2)-1/2*b/c*(-x^2/c/(c*x^2+b*x+a)^(3/2)+1/2*b/c* (-1/2*x/c/(c*x^2+b*x+a)^(3/2)-1/4*b/c*(-1/3/c/(c*x^2+b*x+a)^(3/2)-1/2*b/c* (2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x +b)/(c*x^2+b*x+a)^(1/2)))+1/2*a/c*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a) ^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2)))+2*a/c*(-1/3/c/ (c*x^2+b*x+a)^(3/2)-1/2*b/c*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2) +16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2))))+1/c*(-x/c/(c*x^2+b* x+a)^(1/2)-1/2*b/c*(-1/c/(c*x^2+b*x+a)^(1/2)-b/c*(2*c*x+b)/(4*a*c-b^2)/(c* x^2+b*x+a)^(1/2))+1/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))- 4*a/c*(-x^2/c/(c*x^2+b*x+a)^(3/2)+1/2*b/c*(-1/2*x/c/(c*x^2+b*x+a)^(3/2)-1/ 4*b/c*(-1/3/c/(c*x^2+b*x+a)^(3/2)-1/2*b/c*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^ 2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2)))+1/2*a/ c*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c *x+b)/(c*x^2+b*x+a)^(1/2)))+2*a/c*(-1/3/c/(c*x^2+b*x+a)^(3/2)-1/2*b/c*(2/3 *(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/ (c*x^2+b*x+a)^(1/2)))))+(A*e^4+4*B*d*e^3)*(-1/3*x^3/c/(c*x^2+b*x+a)^(3/2)- 1/2*b/c*(-x^2/c/(c*x^2+b*x+a)^(3/2)+1/2*b/c*(-1/2*x/c/(c*x^2+b*x+a)^(3/2)- 1/4*b/c*(-1/3/c/(c*x^2+b*x+a)^(3/2)-1/2*b/c*(2/3*(2*c*x+b)/(4*a*c-b^2)/...
Leaf count of result is larger than twice the leaf count of optimal. 1589 vs. \(2 (586) = 1172\).
Time = 11.71 (sec) , antiderivative size = 3181, normalized size of antiderivative = 5.23 \[ \int \frac {(A+B x) (d+e x)^4}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Too large to display} \]
[1/12*(3*(8*(B*a^2*b^4*c - 8*B*a^3*b^2*c^2 + 16*B*a^4*c^3)*d*e^3 - (5*B*a^ 2*b^5 - 32*A*a^4*c^3 + 16*(5*B*a^4*b + A*a^3*b^2)*c^2 - 2*(20*B*a^3*b^3 + A*a^2*b^4)*c)*e^4 + (8*(B*b^4*c^3 - 8*B*a*b^2*c^4 + 16*B*a^2*c^5)*d*e^3 - (5*B*b^5*c^2 - 32*A*a^2*c^5 + 16*(5*B*a^2*b + A*a*b^2)*c^4 - 2*(20*B*a*b^3 + A*b^4)*c^3)*e^4)*x^4 + 2*(8*(B*b^5*c^2 - 8*B*a*b^3*c^3 + 16*B*a^2*b*c^4 )*d*e^3 - (5*B*b^6*c - 32*A*a^2*b*c^4 + 16*(5*B*a^2*b^2 + A*a*b^3)*c^3 - 2 *(20*B*a*b^4 + A*b^5)*c^2)*e^4)*x^3 + (8*(B*b^6*c - 6*B*a*b^4*c^2 + 32*B*a ^3*c^4)*d*e^3 - (5*B*b^7 + 12*A*a*b^4*c^2 + 160*B*a^3*b*c^3 - 64*A*a^3*c^4 - 2*(15*B*a*b^5 + A*b^6)*c)*e^4)*x^2 + 2*(8*(B*a*b^5*c - 8*B*a^2*b^3*c^2 + 16*B*a^3*b*c^3)*d*e^3 - (5*B*a*b^6 - 32*A*a^3*b*c^3 + 16*(5*B*a^3*b^2 + A*a^2*b^3)*c^2 - 2*(20*B*a^2*b^4 + A*a*b^5)*c)*e^4)*x)*sqrt(c)*log(-8*c^2* x^2 - 8*b*c*x - b^2 - 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) - 4*(96*(2*B*a^3 - A*a^2*b)*c^4*d^2*e^2 - 3*(B*b^4*c^3 - 8*B*a*b^2*c^4 + 16*B*a^2*c^5)*e^4*x^4 + 2*(4*(2*B*a^2 - 3*A*a*b)*c^5 + (2*B*a*b^2 + A*b^3) *c^4)*d^4 + 16*(4*A*a^2*c^5 - (4*B*a^2*b - A*a*b^2)*c^4)*d^3*e + 8*(3*B*a^ 2*b^3*c^2 - 20*B*a^3*b*c^3 + 16*A*a^3*c^4)*d*e^3 - (15*B*a^2*b^4*c + 8*(16 *B*a^4 + 5*A*a^3*b)*c^3 - 2*(50*B*a^3*b^2 + 3*A*a^2*b^3)*c^2)*e^4 + 4*(4*( B*b*c^6 - 2*A*c^7)*d^4 - 4*(B*b^2*c^5 + 4*(B*a - A*b)*c^6)*d^3*e - 3*(B*b^ 3*c^4 + 8*A*a*c^6 - 2*(6*B*a*b - A*b^2)*c^5)*d^2*e^2 + 2*(4*B*b^4*c^3 + 4* (8*B*a^2 + 3*A*a*b)*c^5 - (28*B*a*b^2 + A*b^3)*c^4)*d*e^3 - (5*B*b^5*c^...
Timed out. \[ \int \frac {(A+B x) (d+e x)^4}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(A+B x) (d+e x)^4}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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*a*c-b^2>0)', see `assume?` for more deta
Leaf count of result is larger than twice the leaf count of optimal. 1228 vs. \(2 (586) = 1172\).
Time = 0.31 (sec) , antiderivative size = 1228, normalized size of antiderivative = 2.02 \[ \int \frac {(A+B x) (d+e x)^4}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {{\left ({\left ({\left (\frac {3 \, {\left (B b^{4} c^{2} e^{4} - 8 \, B a b^{2} c^{3} e^{4} + 16 \, B a^{2} c^{4} e^{4}\right )} x}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}} - \frac {4 \, {\left (4 \, B b c^{5} d^{4} - 8 \, A c^{6} d^{4} - 4 \, B b^{2} c^{4} d^{3} e - 16 \, B a c^{5} d^{3} e + 16 \, A b c^{5} d^{3} e - 3 \, B b^{3} c^{3} d^{2} e^{2} + 36 \, B a b c^{4} d^{2} e^{2} - 6 \, A b^{2} c^{4} d^{2} e^{2} - 24 \, A a c^{5} d^{2} e^{2} + 8 \, B b^{4} c^{2} d e^{3} - 56 \, B a b^{2} c^{3} d e^{3} - 2 \, A b^{3} c^{3} d e^{3} + 64 \, B a^{2} c^{4} d e^{3} + 24 \, A a b c^{4} d e^{3} - 5 \, B b^{5} c e^{4} + 37 \, B a b^{3} c^{2} e^{4} + 2 \, A b^{4} c^{2} e^{4} - 64 \, B a^{2} b c^{3} e^{4} - 14 \, A a b^{2} c^{3} e^{4} + 16 \, A a^{2} c^{4} e^{4}\right )}}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}}\right )} x - \frac {3 \, {\left (8 \, B b^{2} c^{4} d^{4} - 16 \, A b c^{5} d^{4} - 8 \, B b^{3} c^{3} d^{3} e - 32 \, B a b c^{4} d^{3} e + 32 \, A b^{2} c^{4} d^{3} e + 24 \, B a b^{2} c^{3} d^{2} e^{2} - 12 \, A b^{3} c^{3} d^{2} e^{2} + 96 \, B a^{2} c^{4} d^{2} e^{2} - 48 \, A a b c^{4} d^{2} e^{2} + 8 \, B b^{5} c d e^{3} - 48 \, B a b^{3} c^{2} d e^{3} + 16 \, A a b^{2} c^{3} d e^{3} + 64 \, A a^{2} c^{4} d e^{3} - 5 \, B b^{6} e^{4} + 30 \, B a b^{4} c e^{4} + 2 \, A b^{5} c e^{4} - 16 \, B a^{2} b^{2} c^{2} e^{4} - 12 \, A a b^{3} c^{2} e^{4} - 64 \, B a^{3} c^{3} e^{4}\right )}}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}}\right )} x - \frac {6 \, {\left (B b^{3} c^{3} d^{4} + 4 \, B a b c^{4} d^{4} - 2 \, A b^{2} c^{4} d^{4} - 8 \, A a c^{5} d^{4} - 16 \, B a b^{2} c^{3} d^{3} e + 4 \, A b^{3} c^{3} d^{3} e + 16 \, A a b c^{4} d^{3} e + 48 \, B a^{2} b c^{3} d^{2} e^{2} - 24 \, A a b^{2} c^{3} d^{2} e^{2} + 8 \, B a b^{4} c d e^{3} - 56 \, B a^{2} b^{2} c^{2} d e^{3} + 32 \, B a^{3} c^{3} d e^{3} + 32 \, A a^{2} b c^{3} d e^{3} - 5 \, B a b^{5} e^{4} + 35 \, B a^{2} b^{3} c e^{4} + 2 \, A a b^{4} c e^{4} - 52 \, B a^{3} b c^{2} e^{4} - 14 \, A a^{2} b^{2} c^{2} e^{4} + 8 \, A a^{3} c^{3} e^{4}\right )}}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}}\right )} x - \frac {4 \, B a b^{2} c^{3} d^{4} + 2 \, A b^{3} c^{3} d^{4} + 16 \, B a^{2} c^{4} d^{4} - 24 \, A a b c^{4} d^{4} - 64 \, B a^{2} b c^{3} d^{3} e + 16 \, A a b^{2} c^{3} d^{3} e + 64 \, A a^{2} c^{4} d^{3} e + 192 \, B a^{3} c^{3} d^{2} e^{2} - 96 \, A a^{2} b c^{3} d^{2} e^{2} + 24 \, B a^{2} b^{3} c d e^{3} - 160 \, B a^{3} b c^{2} d e^{3} + 128 \, A a^{3} c^{3} d e^{3} - 15 \, B a^{2} b^{4} e^{4} + 100 \, B a^{3} b^{2} c e^{4} + 6 \, A a^{2} b^{3} c e^{4} - 128 \, B a^{4} c^{2} e^{4} - 40 \, A a^{3} b c^{2} e^{4}}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}}}{3 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} - \frac {{\left (8 \, B c d e^{3} - 5 \, B b e^{4} + 2 \, A c e^{4}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{2 \, c^{\frac {7}{2}}} \]
1/3*((((3*(B*b^4*c^2*e^4 - 8*B*a*b^2*c^3*e^4 + 16*B*a^2*c^4*e^4)*x/(b^4*c^ 3 - 8*a*b^2*c^4 + 16*a^2*c^5) - 4*(4*B*b*c^5*d^4 - 8*A*c^6*d^4 - 4*B*b^2*c ^4*d^3*e - 16*B*a*c^5*d^3*e + 16*A*b*c^5*d^3*e - 3*B*b^3*c^3*d^2*e^2 + 36* B*a*b*c^4*d^2*e^2 - 6*A*b^2*c^4*d^2*e^2 - 24*A*a*c^5*d^2*e^2 + 8*B*b^4*c^2 *d*e^3 - 56*B*a*b^2*c^3*d*e^3 - 2*A*b^3*c^3*d*e^3 + 64*B*a^2*c^4*d*e^3 + 2 4*A*a*b*c^4*d*e^3 - 5*B*b^5*c*e^4 + 37*B*a*b^3*c^2*e^4 + 2*A*b^4*c^2*e^4 - 64*B*a^2*b*c^3*e^4 - 14*A*a*b^2*c^3*e^4 + 16*A*a^2*c^4*e^4)/(b^4*c^3 - 8* a*b^2*c^4 + 16*a^2*c^5))*x - 3*(8*B*b^2*c^4*d^4 - 16*A*b*c^5*d^4 - 8*B*b^3 *c^3*d^3*e - 32*B*a*b*c^4*d^3*e + 32*A*b^2*c^4*d^3*e + 24*B*a*b^2*c^3*d^2* e^2 - 12*A*b^3*c^3*d^2*e^2 + 96*B*a^2*c^4*d^2*e^2 - 48*A*a*b*c^4*d^2*e^2 + 8*B*b^5*c*d*e^3 - 48*B*a*b^3*c^2*d*e^3 + 16*A*a*b^2*c^3*d*e^3 + 64*A*a^2* c^4*d*e^3 - 5*B*b^6*e^4 + 30*B*a*b^4*c*e^4 + 2*A*b^5*c*e^4 - 16*B*a^2*b^2* c^2*e^4 - 12*A*a*b^3*c^2*e^4 - 64*B*a^3*c^3*e^4)/(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5))*x - 6*(B*b^3*c^3*d^4 + 4*B*a*b*c^4*d^4 - 2*A*b^2*c^4*d^4 - 8* A*a*c^5*d^4 - 16*B*a*b^2*c^3*d^3*e + 4*A*b^3*c^3*d^3*e + 16*A*a*b*c^4*d^3* e + 48*B*a^2*b*c^3*d^2*e^2 - 24*A*a*b^2*c^3*d^2*e^2 + 8*B*a*b^4*c*d*e^3 - 56*B*a^2*b^2*c^2*d*e^3 + 32*B*a^3*c^3*d*e^3 + 32*A*a^2*b*c^3*d*e^3 - 5*B*a *b^5*e^4 + 35*B*a^2*b^3*c*e^4 + 2*A*a*b^4*c*e^4 - 52*B*a^3*b*c^2*e^4 - 14* A*a^2*b^2*c^2*e^4 + 8*A*a^3*c^3*e^4)/(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)) *x - (4*B*a*b^2*c^3*d^4 + 2*A*b^3*c^3*d^4 + 16*B*a^2*c^4*d^4 - 24*A*a*b...
Timed out. \[ \int \frac {(A+B x) (d+e x)^4}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^4}{{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \]