Integrand size = 29, antiderivative size = 886 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x) (f+g x)} \, dx=\frac {\left (c d^2-b d e+a e^2\right ) \left (8 c^2 d^2+b^2 e^2-2 c e (5 b d-4 a e)-2 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{8 c e^4 (e f-d g)}-\frac {\left (64 c^3 e f^4-16 c^2 e f^2 g (9 b f-8 a g)-b^2 g^3 (5 b e f+3 b d g-8 a e g)+4 c g^2 \left (22 b^2 e f^2+16 a^2 e g^2-3 a b g (13 e f-d g)\right )-2 c g \left (16 c^2 e f^3+b g^2 (5 b e f+3 b d g-8 a e g)-4 c g \left (6 b e f^2-a g (7 e f-3 d g)\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{64 c e g^4 (e f-d g)}+\frac {\left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{3/2}}{3 e^2 (e f-d g)}-\frac {\left (8 c e f^2-g (11 b e f-3 b d g-8 a e g)-6 c g (e f-d g) x\right ) \left (a+b x+c x^2\right )^{3/2}}{24 e g^2 (e f-d g)}-\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (8 c^2 d^2-b^2 e^2-4 c e (2 b d-3 a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{3/2} e^5 (e f-d g)}+\frac {\left (128 c^4 e f^5-320 c^3 e f^3 g (b f-a g)-b^3 g^4 (5 b e f+3 b d g-8 a e g)+48 c^2 g^2 \left (5 b^2 e f^3-10 a b e f^2 g+a^2 g^2 (5 e f-d g)\right )-8 b c g^3 \left (5 b^2 e f^2+12 a^2 e g^2-3 a b g (5 e f+d g)\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 c^{3/2} e g^5 (e f-d g)}+\frac {\left (c d^2-b d e+a e^2\right )^{5/2} \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{e^5 (e f-d g)}-\frac {\left (c f^2-b f g+a g^2\right )^{5/2} \text {arctanh}\left (\frac {b f-2 a g+(2 c f-b g) x}{2 \sqrt {c f^2-b f g+a g^2} \sqrt {a+b x+c x^2}}\right )}{g^5 (e f-d g)} \]
1/3*(a*e^2-b*d*e+c*d^2)*(c*x^2+b*x+a)^(3/2)/e^2/(-d*g+e*f)-1/24*(8*c*e*f^2 -g*(-8*a*e*g-3*b*d*g+11*b*e*f)-6*c*g*(-d*g+e*f)*x)*(c*x^2+b*x+a)^(3/2)/e/g ^2/(-d*g+e*f)-1/16*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)*(8*c^2*d^2-b^2*e^2-4*c *e*(-3*a*e+2*b*d))*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(3 /2)/e^5/(-d*g+e*f)+1/128*(128*c^4*e*f^5-320*c^3*e*f^3*g*(-a*g+b*f)-b^3*g^4 *(-8*a*e*g+3*b*d*g+5*b*e*f)+48*c^2*g^2*(5*b^2*e*f^3-10*a*b*e*f^2*g+a^2*g^2 *(-d*g+5*e*f))-8*b*c*g^3*(5*b^2*e*f^2+12*a^2*e*g^2-3*a*b*g*(d*g+5*e*f)))*a rctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(3/2)/e/g^5/(-d*g+e*f) +(a*e^2-b*d*e+c*d^2)^(5/2)*arctanh(1/2*(b*d-2*a*e+(-b*e+2*c*d)*x)/(a*e^2-b *d*e+c*d^2)^(1/2)/(c*x^2+b*x+a)^(1/2))/e^5/(-d*g+e*f)-(a*g^2-b*f*g+c*f^2)^ (5/2)*arctanh(1/2*(b*f-2*a*g+(-b*g+2*c*f)*x)/(a*g^2-b*f*g+c*f^2)^(1/2)/(c* x^2+b*x+a)^(1/2))/g^5/(-d*g+e*f)+1/8*(a*e^2-b*d*e+c*d^2)*(8*c^2*d^2+b^2*e^ 2-2*c*e*(-4*a*e+5*b*d)-2*c*e*(-b*e+2*c*d)*x)*(c*x^2+b*x+a)^(1/2)/c/e^4/(-d *g+e*f)-1/64*(64*c^3*e*f^4-16*c^2*e*f^2*g*(-8*a*g+9*b*f)-b^2*g^3*(-8*a*e*g +3*b*d*g+5*b*e*f)+4*c*g^2*(22*b^2*e*f^2+16*a^2*e*g^2-3*a*b*g*(-d*g+13*e*f) )-2*c*g*(16*c^2*e*f^3+b*g^2*(-8*a*e*g+3*b*d*g+5*b*e*f)-4*c*g*(6*b*e*f^2-a* g*(-3*d*g+7*e*f)))*x)*(c*x^2+b*x+a)^(1/2)/c/e/g^4/(-d*g+e*f)
Time = 11.70 (sec) , antiderivative size = 647, normalized size of antiderivative = 0.73 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x) (f+g x)} \, dx=\frac {3 \left (5 b^4 e^4 g^4 (-e f+d g)-40 b^2 c e^3 g^3 (e f-d g) (b e f+b d g-3 a e g)+320 c^3 e g \left (-b e^4 f^4+a e^4 f^3 g+b d^4 g^4-a d^3 e g^4\right )+128 c^4 \left (e^5 f^5-d^5 g^5\right )+240 c^2 e^2 g^2 (e f-d g) \left (a^2 e^2 g^2-2 a b e g (e f+d g)+b^2 \left (e^2 f^2+d e f g+d^2 g^2\right )\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )+2 \sqrt {c} \left (-e g (-e f+d g) \sqrt {a+x (b+c x)} \left (15 b^3 e^3 g^3+2 b c e^2 g^2 (278 a e g+b (-132 e f-132 d g+59 e g x))-16 c^3 \left (12 d^3 g^3-6 d^2 e g^2 (-2 f+g x)+2 d e^2 g \left (6 f^2-3 f g x+2 g^2 x^2\right )+e^3 \left (12 f^3-6 f^2 g x+4 f g^2 x^2-3 g^3 x^3\right )\right )+8 c^2 e g \left (a e g (-56 e f-56 d g+27 e g x)+b \left (54 d^2 g^2+2 d e g (27 f-13 g x)+e^2 \left (54 f^2-26 f g x+17 g^2 x^2\right )\right )\right )\right )-192 c \left (c d^2+e (-b d+a e)\right )^{5/2} g^5 \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )+192 c e^5 \left (c f^2+g (-b f+a g)\right )^{5/2} \text {arctanh}\left (\frac {-b f+2 a g-2 c f x+b g x}{2 \sqrt {c f^2+g (-b f+a g)} \sqrt {a+x (b+c x)}}\right )\right )}{384 c^{3/2} e^5 g^5 (e f-d g)} \]
(3*(5*b^4*e^4*g^4*(-(e*f) + d*g) - 40*b^2*c*e^3*g^3*(e*f - d*g)*(b*e*f + b *d*g - 3*a*e*g) + 320*c^3*e*g*(-(b*e^4*f^4) + a*e^4*f^3*g + b*d^4*g^4 - a* d^3*e*g^4) + 128*c^4*(e^5*f^5 - d^5*g^5) + 240*c^2*e^2*g^2*(e*f - d*g)*(a^ 2*e^2*g^2 - 2*a*b*e*g*(e*f + d*g) + b^2*(e^2*f^2 + d*e*f*g + d^2*g^2)))*Ar cTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] + 2*Sqrt[c]*(-(e*g*(- (e*f) + d*g)*Sqrt[a + x*(b + c*x)]*(15*b^3*e^3*g^3 + 2*b*c*e^2*g^2*(278*a* e*g + b*(-132*e*f - 132*d*g + 59*e*g*x)) - 16*c^3*(12*d^3*g^3 - 6*d^2*e*g^ 2*(-2*f + g*x) + 2*d*e^2*g*(6*f^2 - 3*f*g*x + 2*g^2*x^2) + e^3*(12*f^3 - 6 *f^2*g*x + 4*f*g^2*x^2 - 3*g^3*x^3)) + 8*c^2*e*g*(a*e*g*(-56*e*f - 56*d*g + 27*e*g*x) + b*(54*d^2*g^2 + 2*d*e*g*(27*f - 13*g*x) + e^2*(54*f^2 - 26*f *g*x + 17*g^2*x^2))))) - 192*c*(c*d^2 + e*(-(b*d) + a*e))^(5/2)*g^5*ArcTan h[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqr t[a + x*(b + c*x)])] + 192*c*e^5*(c*f^2 + g*(-(b*f) + a*g))^(5/2)*ArcTanh[ (-(b*f) + 2*a*g - 2*c*f*x + b*g*x)/(2*Sqrt[c*f^2 + g*(-(b*f) + a*g)]*Sqrt[ a + x*(b + c*x)])]))/(384*c^(3/2)*e^5*g^5*(e*f - d*g))
Time = 1.95 (sec) , antiderivative size = 834, normalized size of antiderivative = 0.94, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {1270, 1162, 1231, 27, 1231, 27, 1269, 1092, 219, 1154, 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.
| \(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x) (f+g x)} \, dx\) |
| \(\Big \downarrow \) 1270 |
| \(\displaystyle \frac {\left (a e^2-b d e+c d^2\right ) \int \frac {\left (c x^2+b x+a\right )^{3/2}}{d+e x}dx}{e (e f-d g)}-\frac {\int \frac {(c d f-b e f+a e g-c (e f-d g) x) \left (c x^2+b x+a\right )^{3/2}}{f+g x}dx}{e (e f-d g)}\) |
| \(\Big \downarrow \) 1162 |
| \(\displaystyle \frac {\left (a e^2-b d e+c d^2\right ) \left (\frac {\left (a+b x+c x^2\right )^{3/2}}{3 e}-\frac {\int \frac {(b d-2 a e+(2 c d-b e) x) \sqrt {c x^2+b x+a}}{d+e x}dx}{2 e}\right )}{e (e f-d g)}-\frac {\int \frac {(c d f-b e f+a e g-c (e f-d g) x) \left (c x^2+b x+a\right )^{3/2}}{f+g x}dx}{e (e f-d g)}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {\left (a e^2-b d e+c d^2\right ) \left (\frac {\left (a+b x+c x^2\right )^{3/2}}{3 e}-\frac {-\frac {\int \frac {4 c e (b d-2 a e)^2-d (2 c d-b e) \left (-e b^2+4 c d b-4 a c e\right )-(2 c d-b e) \left (8 c^2 d^2-b^2 e^2-4 c e (2 b d-3 a e)\right ) x}{2 (d+e x) \sqrt {c x^2+b x+a}}dx}{4 c e^2}-\frac {\sqrt {a+b x+c x^2} \left (-2 c e (5 b d-4 a e)+b^2 e^2-2 c e x (2 c d-b e)+8 c^2 d^2\right )}{4 c e^2}}{2 e}\right )}{e (e f-d g)}-\frac {\frac {\left (a+b x+c x^2\right )^{3/2} \left (-g (-8 a e g-3 b d g+11 b e f)-6 c g x (e f-d g)+8 c e f^2\right )}{24 g^2}-\frac {\int \frac {c \left (f \left (-3 g b^2+8 c f b-4 a c g\right ) (e f-d g)+8 g (b f-2 a g) (c d f-b e f+a e g)+\left (16 c^2 e f^3+b g^2 (5 b e f+3 b d g-8 a e g)-4 c g \left (6 b e f^2-a g (7 e f-3 d g)\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{2 (f+g x)}dx}{8 c g^2}}{e (e f-d g)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a e^2-b d e+c d^2\right ) \left (\frac {\left (a+b x+c x^2\right )^{3/2}}{3 e}-\frac {-\frac {\int \frac {4 c e (b d-2 a e)^2-d (2 c d-b e) \left (-e b^2+4 c d b-4 a c e\right )-(2 c d-b e) \left (8 c^2 d^2-b^2 e^2-4 c e (2 b d-3 a e)\right ) x}{(d+e x) \sqrt {c x^2+b x+a}}dx}{8 c e^2}-\frac {\sqrt {a+b x+c x^2} \left (-2 c e (5 b d-4 a e)+b^2 e^2-2 c e x (2 c d-b e)+8 c^2 d^2\right )}{4 c e^2}}{2 e}\right )}{e (e f-d g)}-\frac {\frac {\left (a+b x+c x^2\right )^{3/2} \left (-g (-8 a e g-3 b d g+11 b e f)-6 c g x (e f-d g)+8 c e f^2\right )}{24 g^2}-\frac {\int \frac {\left (f \left (-3 g b^2+8 c f b-4 a c g\right ) (e f-d g)+8 g (b f-2 a g) (c d f-b e f+a e g)+\left (16 c^2 e f^3+b g^2 (5 b e f+3 b d g-8 a e g)-4 c g \left (6 b e f^2-a g (7 e f-3 d g)\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{f+g x}dx}{16 g^2}}{e (e f-d g)}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {\left (c d^2-b e d+a e^2\right ) \left (\frac {\left (c x^2+b x+a\right )^{3/2}}{3 e}-\frac {-\frac {\sqrt {c x^2+b x+a} \left (8 c^2 d^2+b^2 e^2-2 c e (5 b d-4 a e)-2 c e (2 c d-b e) x\right )}{4 c e^2}-\frac {\int \frac {4 c e (b d-2 a e)^2-d (2 c d-b e) \left (-e b^2+4 c d b-4 a c e\right )-(2 c d-b e) \left (8 c^2 d^2-b^2 e^2-4 c e (2 b d-3 a e)\right ) x}{(d+e x) \sqrt {c x^2+b x+a}}dx}{8 c e^2}}{2 e}\right )}{e (e f-d g)}-\frac {\frac {\left (8 c e f^2-g (11 b e f-3 b d g-8 a e g)-6 c g (e f-d g) x\right ) \left (c x^2+b x+a\right )^{3/2}}{24 g^2}-\frac {-\frac {\sqrt {c x^2+b x+a} \left (64 c^3 e f^4-16 c^2 e g (9 b f-8 a g) f^2-b^2 g^3 (5 b e f+3 b d g-8 a e g)+4 c g^2 \left (22 b^2 e f^2+16 a^2 e g^2-3 a b g (13 e f-d g)\right )-2 c g \left (16 c^2 e f^3+b g^2 (5 b e f+3 b d g-8 a e g)-4 c g \left (6 b e f^2-a g (7 e f-3 d g)\right )\right ) x\right )}{4 c g^2}-\frac {\int \frac {f g^3 (5 e f+3 d g) b^4-8 e f g^2 \left (11 c f^2+a g^2\right ) b^3+24 c f g \left (6 c e f^3+a g^2 (11 e f-d g)\right ) b^2-32 c e f \left (2 c^2 f^4+9 a c g^2 f^2+9 a^2 g^4\right ) b+16 a c g \left (4 c^2 e f^4+3 a c g^2 (3 e f+d g) f+8 a^2 e g^4\right )-\left (128 c^4 e f^5-320 c^3 e g (b f-a g) f^3-b^3 g^4 (5 b e f+3 b d g-8 a e g)+48 c^2 g^2 \left (5 b^2 e f^3-10 a b e g f^2+a^2 g^2 (5 e f-d g)\right )-8 b c g^3 \left (5 b^2 e f^2+12 a^2 e g^2-3 a b g (5 e f+d g)\right )\right ) x}{2 (f+g x) \sqrt {c x^2+b x+a}}dx}{4 c g^2}}{16 g^2}}{e (e f-d g)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (c d^2-b e d+a e^2\right ) \left (\frac {\left (c x^2+b x+a\right )^{3/2}}{3 e}-\frac {-\frac {\sqrt {c x^2+b x+a} \left (8 c^2 d^2+b^2 e^2-2 c e (5 b d-4 a e)-2 c e (2 c d-b e) x\right )}{4 c e^2}-\frac {\int \frac {4 c e (b d-2 a e)^2-d (2 c d-b e) \left (-e b^2+4 c d b-4 a c e\right )-(2 c d-b e) \left (8 c^2 d^2-b^2 e^2-4 c e (2 b d-3 a e)\right ) x}{(d+e x) \sqrt {c x^2+b x+a}}dx}{8 c e^2}}{2 e}\right )}{e (e f-d g)}-\frac {\frac {\left (8 c e f^2-g (11 b e f-3 b d g-8 a e g)-6 c g (e f-d g) x\right ) \left (c x^2+b x+a\right )^{3/2}}{24 g^2}-\frac {-\frac {\sqrt {c x^2+b x+a} \left (64 c^3 e f^4-16 c^2 e g (9 b f-8 a g) f^2-b^2 g^3 (5 b e f+3 b d g-8 a e g)+4 c g^2 \left (22 b^2 e f^2+16 a^2 e g^2-3 a b g (13 e f-d g)\right )-2 c g \left (16 c^2 e f^3+b g^2 (5 b e f+3 b d g-8 a e g)-4 c g \left (6 b e f^2-a g (7 e f-3 d g)\right )\right ) x\right )}{4 c g^2}-\frac {\int \frac {f g^3 (5 e f+3 d g) b^4-8 e f g^2 \left (11 c f^2+a g^2\right ) b^3+24 c f g \left (6 c e f^3+a g^2 (11 e f-d g)\right ) b^2-32 c e f \left (2 c^2 f^4+9 a c g^2 f^2+9 a^2 g^4\right ) b+16 a c g \left (4 c^2 e f^4+3 a c g^2 (3 e f+d g) f+8 a^2 e g^4\right )-\left (128 c^4 e f^5-320 c^3 e g (b f-a g) f^3-b^3 g^4 (5 b e f+3 b d g-8 a e g)+48 c^2 g^2 \left (5 b^2 e f^3-10 a b e g f^2+a^2 g^2 (5 e f-d g)\right )-8 b c g^3 \left (5 b^2 e f^2+12 a^2 e g^2-3 a b g (5 e f+d g)\right )\right ) x}{(f+g x) \sqrt {c x^2+b x+a}}dx}{8 c g^2}}{16 g^2}}{e (e f-d g)}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {\left (a e^2-b d e+c d^2\right ) \left (\frac {\left (a+b x+c x^2\right )^{3/2}}{3 e}-\frac {-\frac {\frac {16 c \left (a e^2-b d e+c d^2\right )^2 \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}-\frac {(2 c d-b e) \left (-4 c e (2 b d-3 a e)-b^2 e^2+8 c^2 d^2\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{e}}{8 c e^2}-\frac {\sqrt {a+b x+c x^2} \left (-2 c e (5 b d-4 a e)+b^2 e^2-2 c e x (2 c d-b e)+8 c^2 d^2\right )}{4 c e^2}}{2 e}\right )}{e (e f-d g)}-\frac {\frac {\left (a+b x+c x^2\right )^{3/2} \left (-g (-8 a e g-3 b d g+11 b e f)-6 c g x (e f-d g)+8 c e f^2\right )}{24 g^2}-\frac {-\frac {\frac {128 c e \left (a g^2-b f g+c f^2\right )^3 \int \frac {1}{(f+g x) \sqrt {c x^2+b x+a}}dx}{g}-\frac {\left (48 c^2 g^2 \left (a^2 g^2 (5 e f-d g)-10 a b e f^2 g+5 b^2 e f^3\right )-8 b c g^3 \left (12 a^2 e g^2-3 a b g (d g+5 e f)+5 b^2 e f^2\right )-b^3 g^4 (-8 a e g+3 b d g+5 b e f)-320 c^3 e f^3 g (b f-a g)+128 c^4 e f^5\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{g}}{8 c g^2}-\frac {\sqrt {a+b x+c x^2} \left (4 c g^2 \left (16 a^2 e g^2-3 a b g (13 e f-d g)+22 b^2 e f^2\right )-b^2 g^3 (-8 a e g+3 b d g+5 b e f)-2 c g x \left (-4 c g \left (6 b e f^2-a g (7 e f-3 d g)\right )+b g^2 (-8 a e g+3 b d g+5 b e f)+16 c^2 e f^3\right )-16 c^2 e f^2 g (9 b f-8 a g)+64 c^3 e f^4\right )}{4 c g^2}}{16 g^2}}{e (e f-d g)}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {\left (c d^2-b e d+a e^2\right ) \left (\frac {\left (c x^2+b x+a\right )^{3/2}}{3 e}-\frac {-\frac {\sqrt {c x^2+b x+a} \left (8 c^2 d^2+b^2 e^2-2 c e (5 b d-4 a e)-2 c e (2 c d-b e) x\right )}{4 c e^2}-\frac {\frac {16 c \left (c d^2-b e d+a e^2\right )^2 \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}-\frac {2 (2 c d-b e) \left (8 c^2 d^2-b^2 e^2-4 c e (2 b d-3 a e)\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{e}}{8 c e^2}}{2 e}\right )}{e (e f-d g)}-\frac {\frac {\left (8 c e f^2-g (11 b e f-3 b d g-8 a e g)-6 c g (e f-d g) x\right ) \left (c x^2+b x+a\right )^{3/2}}{24 g^2}-\frac {-\frac {\sqrt {c x^2+b x+a} \left (64 c^3 e f^4-16 c^2 e g (9 b f-8 a g) f^2-b^2 g^3 (5 b e f+3 b d g-8 a e g)+4 c g^2 \left (22 b^2 e f^2+16 a^2 e g^2-3 a b g (13 e f-d g)\right )-2 c g \left (16 c^2 e f^3+b g^2 (5 b e f+3 b d g-8 a e g)-4 c g \left (6 b e f^2-a g (7 e f-3 d g)\right )\right ) x\right )}{4 c g^2}-\frac {\frac {128 c e \left (c f^2-b g f+a g^2\right )^3 \int \frac {1}{(f+g x) \sqrt {c x^2+b x+a}}dx}{g}-\frac {2 \left (128 c^4 e f^5-320 c^3 e g (b f-a g) f^3-b^3 g^4 (5 b e f+3 b d g-8 a e g)+48 c^2 g^2 \left (5 b^2 e f^3-10 a b e g f^2+a^2 g^2 (5 e f-d g)\right )-8 b c g^3 \left (5 b^2 e f^2+12 a^2 e g^2-3 a b g (5 e f+d g)\right )\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{g}}{8 c g^2}}{16 g^2}}{e (e f-d g)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\left (a e^2-b d e+c d^2\right ) \left (\frac {\left (a+b x+c x^2\right )^{3/2}}{3 e}-\frac {-\frac {\frac {16 c \left (a e^2-b d e+c d^2\right )^2 \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}-\frac {(2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c e (2 b d-3 a e)-b^2 e^2+8 c^2 d^2\right )}{\sqrt {c} e}}{8 c e^2}-\frac {\sqrt {a+b x+c x^2} \left (-2 c e (5 b d-4 a e)+b^2 e^2-2 c e x (2 c d-b e)+8 c^2 d^2\right )}{4 c e^2}}{2 e}\right )}{e (e f-d g)}-\frac {\frac {\left (a+b x+c x^2\right )^{3/2} \left (-g (-8 a e g-3 b d g+11 b e f)-6 c g x (e f-d g)+8 c e f^2\right )}{24 g^2}-\frac {-\frac {\frac {128 c e \left (a g^2-b f g+c f^2\right )^3 \int \frac {1}{(f+g x) \sqrt {c x^2+b x+a}}dx}{g}-\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (48 c^2 g^2 \left (a^2 g^2 (5 e f-d g)-10 a b e f^2 g+5 b^2 e f^3\right )-8 b c g^3 \left (12 a^2 e g^2-3 a b g (d g+5 e f)+5 b^2 e f^2\right )-b^3 g^4 (-8 a e g+3 b d g+5 b e f)-320 c^3 e f^3 g (b f-a g)+128 c^4 e f^5\right )}{\sqrt {c} g}}{8 c g^2}-\frac {\sqrt {a+b x+c x^2} \left (4 c g^2 \left (16 a^2 e g^2-3 a b g (13 e f-d g)+22 b^2 e f^2\right )-b^2 g^3 (-8 a e g+3 b d g+5 b e f)-2 c g x \left (-4 c g \left (6 b e f^2-a g (7 e f-3 d g)\right )+b g^2 (-8 a e g+3 b d g+5 b e f)+16 c^2 e f^3\right )-16 c^2 e f^2 g (9 b f-8 a g)+64 c^3 e f^4\right )}{4 c g^2}}{16 g^2}}{e (e f-d g)}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {\left (c d^2-b e d+a e^2\right ) \left (\frac {\left (c x^2+b x+a\right )^{3/2}}{3 e}-\frac {-\frac {\sqrt {c x^2+b x+a} \left (8 c^2 d^2+b^2 e^2-2 c e (5 b d-4 a e)-2 c e (2 c d-b e) x\right )}{4 c e^2}-\frac {-\frac {32 c \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right ) \left (c d^2-b e d+a e^2\right )^2}{e}-\frac {(2 c d-b e) \left (8 c^2 d^2-b^2 e^2-4 c e (2 b d-3 a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right )}{\sqrt {c} e}}{8 c e^2}}{2 e}\right )}{e (e f-d g)}-\frac {\frac {\left (8 c e f^2-g (11 b e f-3 b d g-8 a e g)-6 c g (e f-d g) x\right ) \left (c x^2+b x+a\right )^{3/2}}{24 g^2}-\frac {-\frac {\sqrt {c x^2+b x+a} \left (64 c^3 e f^4-16 c^2 e g (9 b f-8 a g) f^2-b^2 g^3 (5 b e f+3 b d g-8 a e g)+4 c g^2 \left (22 b^2 e f^2+16 a^2 e g^2-3 a b g (13 e f-d g)\right )-2 c g \left (16 c^2 e f^3+b g^2 (5 b e f+3 b d g-8 a e g)-4 c g \left (6 b e f^2-a g (7 e f-3 d g)\right )\right ) x\right )}{4 c g^2}-\frac {-\frac {256 c e \int \frac {1}{4 \left (c f^2-b g f+a g^2\right )-\frac {(b f-2 a g+(2 c f-b g) x)^2}{c x^2+b x+a}}d\left (-\frac {b f-2 a g+(2 c f-b g) x}{\sqrt {c x^2+b x+a}}\right ) \left (c f^2-b g f+a g^2\right )^3}{g}-\frac {\left (128 c^4 e f^5-320 c^3 e g (b f-a g) f^3-b^3 g^4 (5 b e f+3 b d g-8 a e g)+48 c^2 g^2 \left (5 b^2 e f^3-10 a b e g f^2+a^2 g^2 (5 e f-d g)\right )-8 b c g^3 \left (5 b^2 e f^2+12 a^2 e g^2-3 a b g (5 e f+d g)\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right )}{\sqrt {c} g}}{8 c g^2}}{16 g^2}}{e (e f-d g)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\left (c d^2-b e d+a e^2\right ) \left (\frac {\left (c x^2+b x+a\right )^{3/2}}{3 e}-\frac {-\frac {\sqrt {c x^2+b x+a} \left (8 c^2 d^2+b^2 e^2-2 c e (5 b d-4 a e)-2 c e (2 c d-b e) x\right )}{4 c e^2}-\frac {\frac {16 c \left (c d^2-b e d+a e^2\right )^{3/2} \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b e d+a e^2} \sqrt {c x^2+b x+a}}\right )}{e}-\frac {(2 c d-b e) \left (8 c^2 d^2-b^2 e^2-4 c e (2 b d-3 a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right )}{\sqrt {c} e}}{8 c e^2}}{2 e}\right )}{e (e f-d g)}-\frac {\frac {\left (8 c e f^2-g (11 b e f-3 b d g-8 a e g)-6 c g (e f-d g) x\right ) \left (c x^2+b x+a\right )^{3/2}}{24 g^2}-\frac {-\frac {\sqrt {c x^2+b x+a} \left (64 c^3 e f^4-16 c^2 e g (9 b f-8 a g) f^2-b^2 g^3 (5 b e f+3 b d g-8 a e g)+4 c g^2 \left (22 b^2 e f^2+16 a^2 e g^2-3 a b g (13 e f-d g)\right )-2 c g \left (16 c^2 e f^3+b g^2 (5 b e f+3 b d g-8 a e g)-4 c g \left (6 b e f^2-a g (7 e f-3 d g)\right )\right ) x\right )}{4 c g^2}-\frac {\frac {128 c e \left (c f^2-b g f+a g^2\right )^{5/2} \text {arctanh}\left (\frac {b f-2 a g+(2 c f-b g) x}{2 \sqrt {c f^2-b g f+a g^2} \sqrt {c x^2+b x+a}}\right )}{g}-\frac {\left (128 c^4 e f^5-320 c^3 e g (b f-a g) f^3-b^3 g^4 (5 b e f+3 b d g-8 a e g)+48 c^2 g^2 \left (5 b^2 e f^3-10 a b e g f^2+a^2 g^2 (5 e f-d g)\right )-8 b c g^3 \left (5 b^2 e f^2+12 a^2 e g^2-3 a b g (5 e f+d g)\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right )}{\sqrt {c} g}}{8 c g^2}}{16 g^2}}{e (e f-d g)}\) |
((c*d^2 - b*d*e + a*e^2)*((a + b*x + c*x^2)^(3/2)/(3*e) - (-1/4*((8*c^2*d^ 2 + b^2*e^2 - 2*c*e*(5*b*d - 4*a*e) - 2*c*e*(2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(c*e^2) - (-(((2*c*d - b*e)*(8*c^2*d^2 - b^2*e^2 - 4*c*e*(2*b*d - 3*a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(Sqrt[c] *e)) + (16*c*(c*d^2 - b*d*e + a*e^2)^(3/2)*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/e)/(8*c*e ^2))/(2*e)))/(e*(e*f - d*g)) - (((8*c*e*f^2 - g*(11*b*e*f - 3*b*d*g - 8*a* e*g) - 6*c*g*(e*f - d*g)*x)*(a + b*x + c*x^2)^(3/2))/(24*g^2) - (-1/4*((64 *c^3*e*f^4 - 16*c^2*e*f^2*g*(9*b*f - 8*a*g) - b^2*g^3*(5*b*e*f + 3*b*d*g - 8*a*e*g) + 4*c*g^2*(22*b^2*e*f^2 + 16*a^2*e*g^2 - 3*a*b*g*(13*e*f - d*g)) - 2*c*g*(16*c^2*e*f^3 + b*g^2*(5*b*e*f + 3*b*d*g - 8*a*e*g) - 4*c*g*(6*b* e*f^2 - a*g*(7*e*f - 3*d*g)))*x)*Sqrt[a + b*x + c*x^2])/(c*g^2) - (-(((128 *c^4*e*f^5 - 320*c^3*e*f^3*g*(b*f - a*g) - b^3*g^4*(5*b*e*f + 3*b*d*g - 8* a*e*g) + 48*c^2*g^2*(5*b^2*e*f^3 - 10*a*b*e*f^2*g + a^2*g^2*(5*e*f - d*g)) - 8*b*c*g^3*(5*b^2*e*f^2 + 12*a^2*e*g^2 - 3*a*b*g*(5*e*f + d*g)))*ArcTanh [(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(Sqrt[c]*g)) + (128*c*e*( c*f^2 - b*f*g + a*g^2)^(5/2)*ArcTanh[(b*f - 2*a*g + (2*c*f - b*g)*x)/(2*Sq rt[c*f^2 - b*f*g + a*g^2]*Sqrt[a + b*x + c*x^2])])/g)/(8*c*g^2))/(16*g^2)) /(e*(e*f - d*g))
3.9.69.3.1 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[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 2*p + 1))), x ] - Simp[p/(e*(m + 2*p + 1)) Int[(d + e*x)^m*Simp[b*d - 2*a*e + (2*c*d - b*e)*x, x]*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x ] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !RationalQ[m] || LtQ[m, 1]) && !ILtQ[m + 2*p, 0] && IntQuadraticQ[a, b, c, d, e, m, p, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ (c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)) Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c ^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x ] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || !R ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Integer Q[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)/(((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))), x_Symbol] :> Simp[(c*d^2 - b*d*e + a*e^2)/(e*(e*f - d*g)) Int[(a + b*x + c*x^2)^(p - 1)/(d + e*x), x], x] - Simp[1/(e*(e*f - d*g)) Int[Simp[c*d*f - b*e*f + a*e*g - c*(e*f - d*g)*x, x]*((a + b*x + c*x^2)^(p - 1)/(f + g*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && FractionQ[p] && GtQ[p, 0]
Time = 0.89 (sec) , antiderivative size = 1248, normalized size of antiderivative = 1.41
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(1248\) |
| default | \(\text {Expression too large to display}\) | \(2107\) |
1/192/c*(48*c^3*e^3*g^3*x^3+136*b*c^2*e^3*g^3*x^2-64*c^3*d*e^2*g^3*x^2-64* c^3*e^3*f*g^2*x^2+216*a*c^2*e^3*g^3*x+118*b^2*c*e^3*g^3*x-208*b*c^2*d*e^2* g^3*x-208*b*c^2*e^3*f*g^2*x+96*c^3*d^2*e*g^3*x+96*c^3*d*e^2*f*g^2*x+96*c^3 *e^3*f^2*g*x+556*a*b*c*e^3*g^3-448*a*c^2*d*e^2*g^3-448*a*c^2*e^3*f*g^2+15* b^3*e^3*g^3-264*b^2*c*d*e^2*g^3-264*b^2*c*e^3*f*g^2+432*b*c^2*d^2*e*g^3+43 2*b*c^2*d*e^2*f*g^2+432*b*c^2*e^3*f^2*g-192*c^3*d^3*g^3-192*c^3*d^2*e*f*g^ 2-192*c^3*d*e^2*f^2*g-192*c^3*e^3*f^3)*(c*x^2+b*x+a)^(1/2)/e^4/g^4+1/128/g ^4/e^4/c*(128/e^2*g^4*c*(a^3*e^6-3*a^2*b*d*e^5+3*a^2*c*d^2*e^4+3*a*b^2*d^2 *e^4-6*a*b*c*d^3*e^3+3*a*c^2*d^4*e^2-b^3*d^3*e^3+3*b^2*c*d^4*e^2-3*b*c^2*d ^5*e+c^3*d^6)/(d*g-e*f)/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e +c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/ e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))-128* e^4/g^2*c*(a^3*g^6-3*a^2*b*f*g^5+3*a^2*c*f^2*g^4+3*a*b^2*f^2*g^4-6*a*b*c*f ^3*g^3+3*a*c^2*f^4*g^2-b^3*f^3*g^3+3*b^2*c*f^4*g^2-3*b*c^2*f^5*g+c^3*f^6)/ (d*g-e*f)/((a*g^2-b*f*g+c*f^2)/g^2)^(1/2)*ln((2*(a*g^2-b*f*g+c*f^2)/g^2+(b *g-2*c*f)/g*(x+f/g)+2*((a*g^2-b*f*g+c*f^2)/g^2)^(1/2)*((x+f/g)^2*c+(b*g-2* c*f)/g*(x+f/g)+(a*g^2-b*f*g+c*f^2)/g^2)^(1/2))/(x+f/g))+(240*a^2*c^2*e^4*g ^4+120*a*b^2*c*e^4*g^4-480*a*b*c^2*d*e^3*g^4-480*a*b*c^2*e^4*f*g^3+320*a*c ^3*d^2*e^2*g^4+320*a*c^3*d*e^3*f*g^3+320*a*c^3*e^4*f^2*g^2-5*b^4*e^4*g^4-4 0*b^3*c*d*e^3*g^4-40*b^3*c*e^4*f*g^3+240*b^2*c^2*d^2*e^2*g^4+240*b^2*c^...
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x) (f+g x)} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x) (f+g x)} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x) (f+g x)} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(d*g-e*f>0)', see `assume?` for m ore detail
Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x) (f+g x)} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x) (f+g x)} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{5/2}}{\left (f+g\,x\right )\,\left (d+e\,x\right )} \,d x \]