Integrand size = 29, antiderivative size = 1098 \[ \int \frac {(f+g x)^3 \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx=-\frac {\left (3 \left (7 b^5 e^5 g^3-512 c^5 d^2 (e f-d g)^3+128 c^4 e (5 b d-4 a e) (e f-d g)^3-4 b^3 c e^4 g^2 (9 b e f-3 b d g+8 a e g)+8 b c^2 e^3 g \left (2 a^2 e^2 g^2+6 a b e g (3 e f-d g)+3 b^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )-32 b c^3 e^2 \left (2 b (e f-d g)^3+3 a e g \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right )+2 c e \left (8 c e (2 c d-b e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-2 \left (8 c^2 d^2-4 b c d e-\frac {3 b^2 e^2}{2}+6 a c e^2\right ) g \left (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{1536 c^4 e^6}+\frac {\left (7 b^3 e^3 g^3+64 c^3 (e f-d g)^3-4 b c e^2 g^2 (9 b e f-3 b d g+a e g)+24 b c^2 e g \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )+2 c e g \left (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{192 c^3 e^4}+\frac {g^2 (36 c e f-22 c d g-7 b e g) \left (a+b x+c x^2\right )^{5/2}}{60 c^2 e^2}+\frac {g^3 (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c e^2}+\frac {\left (4 c e (2 c d-b e) \left (8 c e (b d-2 a e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-d \left (8 b c d-3 b^2 e-4 a c e\right ) g \left (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right )-2 \left (4 c^2 d^2-\frac {b^2 e^2}{2}-2 c e (b d-a e)\right ) \left (8 c e (2 c d-b e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-2 \left (8 c^2 d^2-4 b c d e-\frac {3 b^2 e^2}{2}+6 a c e^2\right ) g \left (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{3072 c^{9/2} e^7}+\frac {\left (c d^2-b d e+a e^2\right )^{3/2} (e f-d g)^3 \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^7} \]
1/192*(7*b^3*e^3*g^3+64*c^3*(-d*g+e*f)^3-4*b*c*e^2*g^2*(a*e*g-3*b*d*g+9*b* e*f)+24*b*c^2*e*g*(d^2*g^2-3*d*e*f*g+3*e^2*f^2)+2*c*e*g*(7*b^2*e^2*g^2-4*c *e*g*(a*e*g-3*b*d*g+9*b*e*f)+24*c^2*(d^2*g^2-3*d*e*f*g+3*e^2*f^2))*x)*(c*x ^2+b*x+a)^(3/2)/c^3/e^4+1/60*g^2*(-7*b*e*g-22*c*d*g+36*c*e*f)*(c*x^2+b*x+a )^(5/2)/c^2/e^2+1/6*g^3*(e*x+d)*(c*x^2+b*x+a)^(5/2)/c/e^2+1/3072*(4*c*e*(- b*e+2*c*d)*(8*c*e*(-2*a*e+b*d)*(24*c^2*e^2*f^3+7*b^2*d*e*g^3-4*c*d*g^2*(a* e*g-3*b*d*g+9*b*e*f))-d*(-4*a*c*e-3*b^2*e+8*b*c*d)*g*(7*b^2*e^2*g^2-4*c*e* g*(a*e*g-3*b*d*g+9*b*e*f)+24*c^2*(d^2*g^2-3*d*e*f*g+3*e^2*f^2)))-2*(4*c^2* d^2-1/2*b^2*e^2-2*c*e*(-a*e+b*d))*(8*c*e*(-b*e+2*c*d)*(24*c^2*e^2*f^3+7*b^ 2*d*e*g^3-4*c*d*g^2*(a*e*g-3*b*d*g+9*b*e*f))-2*(8*c^2*d^2-4*b*c*d*e-3/2*b^ 2*e^2+6*a*c*e^2)*g*(7*b^2*e^2*g^2-4*c*e*g*(a*e*g-3*b*d*g+9*b*e*f)+24*c^2*( d^2*g^2-3*d*e*f*g+3*e^2*f^2))))*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a )^(1/2))/c^(9/2)/e^7+(a*e^2-b*d*e+c*d^2)^(3/2)*(-d*g+e*f)^3*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^ 7-1/1536*(21*b^5*e^5*g^3-1536*c^5*d^2*(-d*g+e*f)^3+384*c^4*e*(-4*a*e+5*b*d )*(-d*g+e*f)^3-12*b^3*c*e^4*g^2*(8*a*e*g-3*b*d*g+9*b*e*f)+24*b*c^2*e^3*g*( 2*a^2*e^2*g^2+6*a*b*e*g*(-d*g+3*e*f)+3*b^2*(d^2*g^2-3*d*e*f*g+3*e^2*f^2))- 96*b*c^3*e^2*(2*b*(-d*g+e*f)^3+3*a*e*g*(d^2*g^2-3*d*e*f*g+3*e^2*f^2))+2*c* e*(8*c*e*(-b*e+2*c*d)*(24*c^2*e^2*f^3+7*b^2*d*e*g^3-4*c*d*g^2*(a*e*g-3*b*d *g+9*b*e*f))-2*(8*c^2*d^2-4*b*c*d*e-3/2*b^2*e^2+6*a*c*e^2)*g*(7*b^2*e^2...
Time = 11.51 (sec) , antiderivative size = 743, normalized size of antiderivative = 0.68 \[ \int \frac {(f+g x)^3 \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx=\frac {5120 (e f-d g)^3 (a+x (b+c x))^{3/2}+\frac {1920 e g (e f-d g)^2 (b+2 c x) (a+x (b+c x))^{3/2}}{c}+\frac {3072 e^2 g^2 (e f-d g) (a+x (b+c x))^{5/2}}{c}+\frac {2560 e^3 g^2 (f+g x) (a+x (b+c x))^{5/2}}{c}+\frac {360 \left (b^2-4 a c\right ) e g (e f-d g)^2 \left (-2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)}+\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )}{c^{5/2}}-\frac {60 e^2 g (-2 c f+b g) (e f-d g) \left (2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)} \left (-3 b^2+8 b c x+4 c \left (5 a+2 c x^2\right )\right )+3 \left (b^2-4 a c\right )^2 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )}{c^{7/2}}+\frac {e^3 g \left (1792 g (2 c f-b g) (a+x (b+c x))^{5/2}+5 \left (24 c^2 f^2+7 b^2 g^2-4 c g (6 b f+a g)\right ) \left (\frac {16 (b+2 c x) (a+x (b+c x))^{3/2}}{c}+\frac {3 \left (b^2-4 a c\right ) \left (-2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)}+\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )}{c^{5/2}}\right )\right )}{c^2}+\frac {960 (e f-d g)^3 \left (-\left ((2 c d-b e) \left (8 c^2 d^2-b^2 e^2+4 c e (-2 b d+3 a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )-2 \sqrt {c} \left (e \sqrt {a+x (b+c x)} \left (-b^2 e^2+4 c^2 d (-2 d+e x)-2 c e (-5 b d+4 a e+b e x)\right )+8 c \left (c d^2+e (-b d+a e)\right )^{3/2} \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )\right )\right )}{c^{3/2} e^3}}{15360 e^4} \]
(5120*(e*f - d*g)^3*(a + x*(b + c*x))^(3/2) + (1920*e*g*(e*f - d*g)^2*(b + 2*c*x)*(a + x*(b + c*x))^(3/2))/c + (3072*e^2*g^2*(e*f - d*g)*(a + x*(b + c*x))^(5/2))/c + (2560*e^3*g^2*(f + g*x)*(a + x*(b + c*x))^(5/2))/c + (36 0*(b^2 - 4*a*c)*e*g*(e*f - d*g)^2*(-2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)] + (b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)] )]))/c^(5/2) - (60*e^2*g*(-2*c*f + b*g)*(e*f - d*g)*(2*Sqrt[c]*(b + 2*c*x) *Sqrt[a + x*(b + c*x)]*(-3*b^2 + 8*b*c*x + 4*c*(5*a + 2*c*x^2)) + 3*(b^2 - 4*a*c)^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/c^(7/2) + (e^3*g*(1792*g*(2*c*f - b*g)*(a + x*(b + c*x))^(5/2) + 5*(24*c^2*f^2 + 7*b^2*g^2 - 4*c*g*(6*b*f + a*g))*((16*(b + 2*c*x)*(a + x*(b + c*x))^(3/2)) /c + (3*(b^2 - 4*a*c)*(-2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)] + (b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/c^(5/2) )))/c^2 + (960*(e*f - d*g)^3*(-((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 + x*(b + c*x)])]) - 2*Sqrt[c]*(e*Sqrt[a + x*(b + c*x)]*(-(b^2*e^2) + 4*c^2*d*(-2*d + e*x) - 2*c*e*(-5*b*d + 4*a*e + b*e*x)) + 8*c*(c*d^2 + e*(-(b*d) + a*e))^(3/2)*Ar cTanh[(-(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)])])))/(c^(3/2)*e^3))/(15360*e^4)
Time = 3.18 (sec) , antiderivative size = 1139, normalized size of antiderivative = 1.04, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.448, Rules used = {1267, 27, 2184, 27, 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.
| \(\displaystyle \int \frac {(f+g x)^3 \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx\) |
| \(\Big \downarrow \) 1267 |
| \(\displaystyle \frac {\int \frac {\left (c x^2+b x+a\right )^{3/2} \left (e^2 g^2 (36 c e f-22 c d g-7 b e g) x^2-2 e g \left (e (6 b d+a e) g^2-c \left (18 e^2 f^2-5 d^2 g^2\right )\right ) x+e \left (12 c e^2 f^3-d (5 b d+2 a e) g^3\right )\right )}{2 (d+e x)}dx}{6 c e^3}+\frac {g^3 (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\left (c x^2+b x+a\right )^{3/2} \left (e^2 g^2 (36 c e f-22 c d g-7 b e g) x^2-2 e g \left (e (6 b d+a e) g^2-c \left (18 e^2 f^2-5 d^2 g^2\right )\right ) x+e \left (12 c e^2 f^3-d (5 b d+2 a e) g^3\right )\right )}{d+e x}dx}{12 c e^3}+\frac {g^3 (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c e^2}\) |
| \(\Big \downarrow \) 2184 |
| \(\displaystyle \frac {\frac {\int \frac {5 e^3 \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)+g \left (24 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{2 (d+e x)}dx}{5 c e^2}+\frac {e g^2 \left (a+b x+c x^2\right )^{5/2} (-7 b e g-22 c d g+36 c e f)}{5 c}}{12 c e^3}+\frac {g^3 (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {e \int \frac {\left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)+g \left (24 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{d+e x}dx}{2 c}+\frac {e g^2 \left (a+b x+c x^2\right )^{5/2} (-7 b e g-22 c d g+36 c e f)}{5 c}}{12 c e^3}+\frac {g^3 (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c e^2}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {\frac {e \left (\frac {\left (a+b x+c x^2\right )^{3/2} \left (2 c e g x \left (-4 c e g (a e g-3 b d g+9 b e f)+7 b^2 e^2 g^2+24 c^2 \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )\right )-4 b c e^2 g^2 (a e g-3 b d g+9 b e f)+7 b^3 e^3 g^3+24 b c^2 e g \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )+64 c^3 (e f-d g)^3\right )}{8 c e^2}-\frac {\int \frac {\left (8 c e (b d-2 a e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-d \left (-3 e b^2+8 c d b-4 a c e\right ) g \left (24 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right )+\left (8 c e (2 c d-b e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-2 \left (8 c^2 d^2-4 b c e d-\frac {3 b^2 e^2}{2}+6 a c e^2\right ) g \left (24 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{2 (d+e x)}dx}{8 c e^2}\right )}{2 c}+\frac {e g^2 \left (a+b x+c x^2\right )^{5/2} (-7 b e g-22 c d g+36 c e f)}{5 c}}{12 c e^3}+\frac {g^3 (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {e \left (\frac {\left (a+b x+c x^2\right )^{3/2} \left (2 c e g x \left (-4 c e g (a e g-3 b d g+9 b e f)+7 b^2 e^2 g^2+24 c^2 \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )\right )-4 b c e^2 g^2 (a e g-3 b d g+9 b e f)+7 b^3 e^3 g^3+24 b c^2 e g \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )+64 c^3 (e f-d g)^3\right )}{8 c e^2}-\frac {\int \frac {\left (8 c e (b d-2 a e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-d \left (-3 e b^2+8 c d b-4 a c e\right ) g \left (24 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right )+\left (8 c e (2 c d-b e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-2 \left (8 c^2 d^2-4 b c e d-\frac {3 b^2 e^2}{2}+6 a c e^2\right ) g \left (24 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{d+e x}dx}{16 c e^2}\right )}{2 c}+\frac {e g^2 \left (a+b x+c x^2\right )^{5/2} (-7 b e g-22 c d g+36 c e f)}{5 c}}{12 c e^3}+\frac {g^3 (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c e^2}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {(d+e x) \left (c x^2+b x+a\right )^{5/2} g^3}{6 c e^2}+\frac {\frac {e g^2 (36 c e f-22 c d g-7 b e g) \left (c x^2+b x+a\right )^{5/2}}{5 c}+\frac {e \left (\frac {\left (7 b^3 e^3 g^3-4 b c e^2 (9 b e f-3 b d g+a e g) g^2+24 b c^2 e \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) g+2 c e \left (24 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right ) x g+64 c^3 (e f-d g)^3\right ) \left (c x^2+b x+a\right )^{3/2}}{8 c e^2}-\frac {\frac {\left (3 \left (-512 d^2 (e f-d g)^3 c^5+128 e (5 b d-4 a e) (e f-d g)^3 c^4-32 b e^2 \left (2 b (e f-d g)^3+3 a e g \left (3 e^2 f^2-3 d e g f+d^2 g^2\right )\right ) c^3+8 b e^3 g \left (3 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) b^2+6 a e g (3 e f-d g) b+2 a^2 e^2 g^2\right ) c^2-4 b^3 e^4 g^2 (9 b e f-3 b d g+8 a e g) c+7 b^5 e^5 g^3\right )+2 c e \left (8 c e (2 c d-b e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-2 \left (8 c^2 d^2-4 b c e d-\frac {3 b^2 e^2}{2}+6 a c e^2\right ) g \left (24 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{4 c e^2}-\frac {\int \frac {4 c e (b d-2 a e) \left (8 c e (b d-2 a e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-d \left (-3 e b^2+8 c d b-4 a c e\right ) g \left (24 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right )\right )-d \left (-e b^2+4 c d b-4 a c e\right ) \left (8 c e (2 c d-b e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-2 \left (8 c^2 d^2-4 b c e d-\frac {3 b^2 e^2}{2}+6 a c e^2\right ) g \left (24 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right )\right )+\left (4 c e (2 c d-b e) \left (8 c e (b d-2 a e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-d \left (-3 e b^2+8 c d b-4 a c e\right ) g \left (24 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right )\right )-2 \left (4 c^2 d^2-\frac {b^2 e^2}{2}-2 c e (b d-a e)\right ) \left (8 c e (2 c d-b e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-2 \left (8 c^2 d^2-4 b c e d-\frac {3 b^2 e^2}{2}+6 a c e^2\right ) g \left (24 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right )\right )\right ) x}{2 (d+e x) \sqrt {c x^2+b x+a}}dx}{4 c e^2}}{16 c e^2}\right )}{2 c}}{12 c e^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(d+e x) \left (c x^2+b x+a\right )^{5/2} g^3}{6 c e^2}+\frac {\frac {e g^2 (36 c e f-22 c d g-7 b e g) \left (c x^2+b x+a\right )^{5/2}}{5 c}+\frac {e \left (\frac {\left (7 b^3 e^3 g^3-4 b c e^2 (9 b e f-3 b d g+a e g) g^2+24 b c^2 e \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) g+2 c e \left (24 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right ) x g+64 c^3 (e f-d g)^3\right ) \left (c x^2+b x+a\right )^{3/2}}{8 c e^2}-\frac {\frac {\left (3 \left (-512 d^2 (e f-d g)^3 c^5+128 e (5 b d-4 a e) (e f-d g)^3 c^4-32 b e^2 \left (2 b (e f-d g)^3+3 a e g \left (3 e^2 f^2-3 d e g f+d^2 g^2\right )\right ) c^3+8 b e^3 g \left (3 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) b^2+6 a e g (3 e f-d g) b+2 a^2 e^2 g^2\right ) c^2-4 b^3 e^4 g^2 (9 b e f-3 b d g+8 a e g) c+7 b^5 e^5 g^3\right )+2 c e \left (8 c e (2 c d-b e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-2 \left (8 c^2 d^2-4 b c e d-\frac {3 b^2 e^2}{2}+6 a c e^2\right ) g \left (24 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{4 c e^2}-\frac {\int \frac {4 c e (b d-2 a e) \left (8 c e (b d-2 a e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-d \left (-3 e b^2+8 c d b-4 a c e\right ) g \left (24 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right )\right )-d \left (-e b^2+4 c d b-4 a c e\right ) \left (8 c e (2 c d-b e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-2 \left (8 c^2 d^2-4 b c e d-\frac {3 b^2 e^2}{2}+6 a c e^2\right ) g \left (24 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right )\right )+\left (4 c e (2 c d-b e) \left (8 c e (b d-2 a e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-d \left (-3 e b^2+8 c d b-4 a c e\right ) g \left (24 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right )\right )-2 \left (4 c^2 d^2-\frac {b^2 e^2}{2}-2 c e (b d-a e)\right ) \left (8 c e (2 c d-b e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-2 \left (8 c^2 d^2-4 b c e d-\frac {3 b^2 e^2}{2}+6 a c e^2\right ) g \left (24 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right )\right )\right ) x}{(d+e x) \sqrt {c x^2+b x+a}}dx}{8 c e^2}}{16 c e^2}\right )}{2 c}}{12 c e^3}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {(d+e x) \left (c x^2+b x+a\right )^{5/2} g^3}{6 c e^2}+\frac {\frac {e g^2 (36 c e f-22 c d g-7 b e g) \left (c x^2+b x+a\right )^{5/2}}{5 c}+\frac {e \left (\frac {\left (7 b^3 e^3 g^3-4 b c e^2 (9 b e f-3 b d g+a e g) g^2+24 b c^2 e \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) g+2 c e \left (24 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right ) x g+64 c^3 (e f-d g)^3\right ) \left (c x^2+b x+a\right )^{3/2}}{8 c e^2}-\frac {\frac {\left (3 \left (-512 d^2 (e f-d g)^3 c^5+128 e (5 b d-4 a e) (e f-d g)^3 c^4-32 b e^2 \left (2 b (e f-d g)^3+3 a e g \left (3 e^2 f^2-3 d e g f+d^2 g^2\right )\right ) c^3+8 b e^3 g \left (3 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) b^2+6 a e g (3 e f-d g) b+2 a^2 e^2 g^2\right ) c^2-4 b^3 e^4 g^2 (9 b e f-3 b d g+8 a e g) c+7 b^5 e^5 g^3\right )+2 c e \left (8 c e (2 c d-b e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-2 \left (8 c^2 d^2-4 b c e d-\frac {3 b^2 e^2}{2}+6 a c e^2\right ) g \left (24 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{4 c e^2}-\frac {\frac {3072 \left (c d^2-b e d+a e^2\right )^2 (e f-d g)^3 \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx c^4}{e}+\frac {\left (4 c e (2 c d-b e) \left (8 c e (b d-2 a e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-d \left (-3 e b^2+8 c d b-4 a c e\right ) g \left (24 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right )\right )-2 \left (4 c^2 d^2-\frac {b^2 e^2}{2}-2 c e (b d-a e)\right ) \left (8 c e (2 c d-b e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-2 \left (8 c^2 d^2-4 b c e d-\frac {3 b^2 e^2}{2}+6 a c e^2\right ) g \left (24 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right )\right )\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{e}}{8 c e^2}}{16 c e^2}\right )}{2 c}}{12 c e^3}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {(d+e x) \left (c x^2+b x+a\right )^{5/2} g^3}{6 c e^2}+\frac {\frac {e g^2 (36 c e f-22 c d g-7 b e g) \left (c x^2+b x+a\right )^{5/2}}{5 c}+\frac {e \left (\frac {\left (7 b^3 e^3 g^3-4 b c e^2 (9 b e f-3 b d g+a e g) g^2+24 b c^2 e \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) g+2 c e \left (24 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right ) x g+64 c^3 (e f-d g)^3\right ) \left (c x^2+b x+a\right )^{3/2}}{8 c e^2}-\frac {\frac {\left (3 \left (-512 d^2 (e f-d g)^3 c^5+128 e (5 b d-4 a e) (e f-d g)^3 c^4-32 b e^2 \left (2 b (e f-d g)^3+3 a e g \left (3 e^2 f^2-3 d e g f+d^2 g^2\right )\right ) c^3+8 b e^3 g \left (3 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) b^2+6 a e g (3 e f-d g) b+2 a^2 e^2 g^2\right ) c^2-4 b^3 e^4 g^2 (9 b e f-3 b d g+8 a e g) c+7 b^5 e^5 g^3\right )+2 c e \left (8 c e (2 c d-b e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-2 \left (8 c^2 d^2-4 b c e d-\frac {3 b^2 e^2}{2}+6 a c e^2\right ) g \left (24 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{4 c e^2}-\frac {\frac {3072 \left (c d^2-b e d+a e^2\right )^2 (e f-d g)^3 \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx c^4}{e}+\frac {2 \left (4 c e (2 c d-b e) \left (8 c e (b d-2 a e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-d \left (-3 e b^2+8 c d b-4 a c e\right ) g \left (24 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right )\right )-2 \left (4 c^2 d^2-\frac {b^2 e^2}{2}-2 c e (b d-a e)\right ) \left (8 c e (2 c d-b e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-2 \left (8 c^2 d^2-4 b c e d-\frac {3 b^2 e^2}{2}+6 a c e^2\right ) g \left (24 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right )\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}}}{e}}{8 c e^2}}{16 c e^2}\right )}{2 c}}{12 c e^3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {(d+e x) \left (c x^2+b x+a\right )^{5/2} g^3}{6 c e^2}+\frac {\frac {e g^2 (36 c e f-22 c d g-7 b e g) \left (c x^2+b x+a\right )^{5/2}}{5 c}+\frac {e \left (\frac {\left (7 b^3 e^3 g^3-4 b c e^2 (9 b e f-3 b d g+a e g) g^2+24 b c^2 e \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) g+2 c e \left (24 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right ) x g+64 c^3 (e f-d g)^3\right ) \left (c x^2+b x+a\right )^{3/2}}{8 c e^2}-\frac {\frac {\left (3 \left (-512 d^2 (e f-d g)^3 c^5+128 e (5 b d-4 a e) (e f-d g)^3 c^4-32 b e^2 \left (2 b (e f-d g)^3+3 a e g \left (3 e^2 f^2-3 d e g f+d^2 g^2\right )\right ) c^3+8 b e^3 g \left (3 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) b^2+6 a e g (3 e f-d g) b+2 a^2 e^2 g^2\right ) c^2-4 b^3 e^4 g^2 (9 b e f-3 b d g+8 a e g) c+7 b^5 e^5 g^3\right )+2 c e \left (8 c e (2 c d-b e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-2 \left (8 c^2 d^2-4 b c e d-\frac {3 b^2 e^2}{2}+6 a c e^2\right ) g \left (24 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{4 c e^2}-\frac {\frac {3072 \left (c d^2-b e d+a e^2\right )^2 (e f-d g)^3 \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx c^4}{e}+\frac {\left (4 c e (2 c d-b e) \left (8 c e (b d-2 a e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-d \left (-3 e b^2+8 c d b-4 a c e\right ) g \left (24 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right )\right )-2 \left (4 c^2 d^2-\frac {b^2 e^2}{2}-2 c e (b d-a e)\right ) \left (8 c e (2 c d-b e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-2 \left (8 c^2 d^2-4 b c e d-\frac {3 b^2 e^2}{2}+6 a c e^2\right ) g \left (24 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right )\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right )}{e \sqrt {c}}}{8 c e^2}}{16 c e^2}\right )}{2 c}}{12 c e^3}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {(d+e x) \left (c x^2+b x+a\right )^{5/2} g^3}{6 c e^2}+\frac {\frac {e g^2 (36 c e f-22 c d g-7 b e g) \left (c x^2+b x+a\right )^{5/2}}{5 c}+\frac {e \left (\frac {\left (7 b^3 e^3 g^3-4 b c e^2 (9 b e f-3 b d g+a e g) g^2+24 b c^2 e \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) g+2 c e \left (24 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right ) x g+64 c^3 (e f-d g)^3\right ) \left (c x^2+b x+a\right )^{3/2}}{8 c e^2}-\frac {\frac {\left (3 \left (-512 d^2 (e f-d g)^3 c^5+128 e (5 b d-4 a e) (e f-d g)^3 c^4-32 b e^2 \left (2 b (e f-d g)^3+3 a e g \left (3 e^2 f^2-3 d e g f+d^2 g^2\right )\right ) c^3+8 b e^3 g \left (3 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) b^2+6 a e g (3 e f-d g) b+2 a^2 e^2 g^2\right ) c^2-4 b^3 e^4 g^2 (9 b e f-3 b d g+8 a e g) c+7 b^5 e^5 g^3\right )+2 c e \left (8 c e (2 c d-b e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-2 \left (8 c^2 d^2-4 b c e d-\frac {3 b^2 e^2}{2}+6 a c e^2\right ) g \left (24 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{4 c e^2}-\frac {\frac {\left (4 c e (2 c d-b e) \left (8 c e (b d-2 a e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-d \left (-3 e b^2+8 c d b-4 a c e\right ) g \left (24 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right )\right )-2 \left (4 c^2 d^2-\frac {b^2 e^2}{2}-2 c e (b d-a e)\right ) \left (8 c e (2 c d-b e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-2 \left (8 c^2 d^2-4 b c e d-\frac {3 b^2 e^2}{2}+6 a c e^2\right ) g \left (24 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right )\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right )}{\sqrt {c} e}-\frac {6144 c^4 \left (c d^2-b e d+a e^2\right )^2 (e f-d g)^3 \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 )}{e}}{8 c e^2}}{16 c e^2}\right )}{2 c}}{12 c e^3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {(d+e x) \left (c x^2+b x+a\right )^{5/2} g^3}{6 c e^2}+\frac {\frac {e g^2 (36 c e f-22 c d g-7 b e g) \left (c x^2+b x+a\right )^{5/2}}{5 c}+\frac {e \left (\frac {\left (7 b^3 e^3 g^3-4 b c e^2 (9 b e f-3 b d g+a e g) g^2+24 b c^2 e \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) g+2 c e \left (24 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right ) x g+64 c^3 (e f-d g)^3\right ) \left (c x^2+b x+a\right )^{3/2}}{8 c e^2}-\frac {\frac {\left (3 \left (-512 d^2 (e f-d g)^3 c^5+128 e (5 b d-4 a e) (e f-d g)^3 c^4-32 b e^2 \left (2 b (e f-d g)^3+3 a e g \left (3 e^2 f^2-3 d e g f+d^2 g^2\right )\right ) c^3+8 b e^3 g \left (3 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) b^2+6 a e g (3 e f-d g) b+2 a^2 e^2 g^2\right ) c^2-4 b^3 e^4 g^2 (9 b e f-3 b d g+8 a e g) c+7 b^5 e^5 g^3\right )+2 c e \left (8 c e (2 c d-b e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-2 \left (8 c^2 d^2-4 b c e d-\frac {3 b^2 e^2}{2}+6 a c e^2\right ) g \left (24 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{4 c e^2}-\frac {\frac {3072 \left (c d^2-b e d+a e^2\right )^{3/2} (e f-d g)^3 \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 ) c^4}{e}+\frac {\left (4 c e (2 c d-b e) \left (8 c e (b d-2 a e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-d \left (-3 e b^2+8 c d b-4 a c e\right ) g \left (24 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right )\right )-2 \left (4 c^2 d^2-\frac {b^2 e^2}{2}-2 c e (b d-a e)\right ) \left (8 c e (2 c d-b e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-2 \left (8 c^2 d^2-4 b c e d-\frac {3 b^2 e^2}{2}+6 a c e^2\right ) g \left (24 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right )\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right )}{e \sqrt {c}}}{8 c e^2}}{16 c e^2}\right )}{2 c}}{12 c e^3}\) |
(g^3*(d + e*x)*(a + b*x + c*x^2)^(5/2))/(6*c*e^2) + ((e*g^2*(36*c*e*f - 22 *c*d*g - 7*b*e*g)*(a + b*x + c*x^2)^(5/2))/(5*c) + (e*(((7*b^3*e^3*g^3 + 6 4*c^3*(e*f - d*g)^3 - 4*b*c*e^2*g^2*(9*b*e*f - 3*b*d*g + a*e*g) + 24*b*c^2 *e*g*(3*e^2*f^2 - 3*d*e*f*g + d^2*g^2) + 2*c*e*g*(7*b^2*e^2*g^2 - 4*c*e*g* (9*b*e*f - 3*b*d*g + a*e*g) + 24*c^2*(3*e^2*f^2 - 3*d*e*f*g + d^2*g^2))*x) *(a + b*x + c*x^2)^(3/2))/(8*c*e^2) - (((3*(7*b^5*e^5*g^3 - 512*c^5*d^2*(e *f - d*g)^3 + 128*c^4*e*(5*b*d - 4*a*e)*(e*f - d*g)^3 - 4*b^3*c*e^4*g^2*(9 *b*e*f - 3*b*d*g + 8*a*e*g) + 8*b*c^2*e^3*g*(2*a^2*e^2*g^2 + 6*a*b*e*g*(3* e*f - d*g) + 3*b^2*(3*e^2*f^2 - 3*d*e*f*g + d^2*g^2)) - 32*b*c^3*e^2*(2*b* (e*f - d*g)^3 + 3*a*e*g*(3*e^2*f^2 - 3*d*e*f*g + d^2*g^2))) + 2*c*e*(8*c*e *(2*c*d - b*e)*(24*c^2*e^2*f^3 + 7*b^2*d*e*g^3 - 4*c*d*g^2*(9*b*e*f - 3*b* d*g + a*e*g)) - 2*(8*c^2*d^2 - 4*b*c*d*e - (3*b^2*e^2)/2 + 6*a*c*e^2)*g*(7 *b^2*e^2*g^2 - 4*c*e*g*(9*b*e*f - 3*b*d*g + a*e*g) + 24*c^2*(3*e^2*f^2 - 3 *d*e*f*g + d^2*g^2)))*x)*Sqrt[a + b*x + c*x^2])/(4*c*e^2) - (((4*c*e*(2*c* d - b*e)*(8*c*e*(b*d - 2*a*e)*(24*c^2*e^2*f^3 + 7*b^2*d*e*g^3 - 4*c*d*g^2* (9*b*e*f - 3*b*d*g + a*e*g)) - d*(8*b*c*d - 3*b^2*e - 4*a*c*e)*g*(7*b^2*e^ 2*g^2 - 4*c*e*g*(9*b*e*f - 3*b*d*g + a*e*g) + 24*c^2*(3*e^2*f^2 - 3*d*e*f* g + d^2*g^2))) - 2*(4*c^2*d^2 - (b^2*e^2)/2 - 2*c*e*(b*d - a*e))*(8*c*e*(2 *c*d - b*e)*(24*c^2*e^2*f^3 + 7*b^2*d*e*g^3 - 4*c*d*g^2*(9*b*e*f - 3*b*d*g + a*e*g)) - 2*(8*c^2*d^2 - 4*b*c*d*e - (3*b^2*e^2)/2 + 6*a*c*e^2)*g*(7...
3.9.62.3.1 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_)*((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_))^(n_)*((a_.) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[g^n*(d + e*x)^(m + n - 1)*((a + b *x + c*x^2)^(p + 1)/(c*e^(n - 1)*(m + n + 2*p + 1))), x] + Simp[1/(c*e^n*(m + n + 2*p + 1)) Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^n*(m + n + 2*p + 1)*(f + g*x)^n - c*g^n*(m + n + 2*p + 1)*(d + e*x)^n - g^n*(d + e*x)^(n - 2)*(b*d*e*(p + 1) + a*e^2*(m + n - 1) - c*d^2*(m + n + 2*p + 1) - e*(2*c*d - b*e)*(m + n + p)*x), x], x], x] /; FreeQ[{a, b, c, d, e, f, g , m, p}, x] && IGtQ[n, 1] && IntegerQ[m] && NeQ[m + n + 2*p + 1, 0]
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[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p _), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, S imp[f*(d + e*x)^(m + q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(c*e^q*(m + q + 2*p + 1)) Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q - 1) - c *d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[ q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && Pol yQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && !(IGt Q[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Time = 0.97 (sec) , antiderivative size = 1409, normalized size of antiderivative = 1.28
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1409\) |
| risch | \(\text {Expression too large to display}\) | \(2365\) |
(-d^3*g^3+3*d^2*e*f*g^2-3*d*e^2*f^2*g+e^3*f^3)/e^4*(1/3*((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)^(3/2)+1/2*(b*e-2*c*d)/e*(1/4*(2* c*(x+d/e)+(b*e-2*c*d)/e)/c*((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)+1/8*(4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^2)/c^( 3/2)*ln((1/2*(b*e-2*c*d)/e+c*(x+d/e))/c^(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)))+(a*e^2-b*d*e+c*d^2)/e^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)+1/2*(b*e-2*c*d)/e *ln((1/2*(b*e-2*c*d)/e+c*(x+d/e))/c^(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))/c^(1/2)-(a*e^2-b*d*e+c*d^2)/e^2/((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))))+g/e^3*(d^2*g^2*(1/8*(2*c*x+b)/c*( c*x^2+b*x+a)^(3/2)+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)/c*(c*x^2+b*x+a)^(1/2) +1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))+e^2 *g^2*(1/6*x*(c*x^2+b*x+a)^(5/2)/c-7/12*b/c*(1/5*(c*x^2+b*x+a)^(5/2)/c-1/2* b/c*(1/8*(2*c*x+b)/c*(c*x^2+b*x+a)^(3/2)+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b) /c*(c*x^2+b*x+a)^(1/2)+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x ^2+b*x+a)^(1/2)))))-1/6*a/c*(1/8*(2*c*x+b)/c*(c*x^2+b*x+a)^(3/2)+3/16*(4*a *c-b^2)/c*(1/4*(2*c*x+b)/c*(c*x^2+b*x+a)^(1/2)+1/8*(4*a*c-b^2)/c^(3/2)*ln( (1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))))+3*e^2*f^2*(1/8*(2*c*x+b)/c*...
Timed out. \[ \int \frac {(f+g x)^3 \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx=\text {Timed out} \]
\[ \int \frac {(f+g x)^3 \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx=\int \frac {\left (f + g x\right )^{3} \left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{d + e x}\, dx \]
Exception generated. \[ \int \frac {(f+g x)^3 \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Exception generated. \[ \int \frac {(f+g x)^3 \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {(f+g x)^3 \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx=\int \frac {{\left (f+g\,x\right )}^3\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{d+e\,x} \,d x \]