Integrand size = 29, antiderivative size = 1066 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x) (f+g x)^3} \, dx=\frac {\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 f-d g)^3}+\frac {3 e (4 c f-3 b g-2 c g x) \sqrt {a+b x+c x^2}}{4 g^2 (e f-d g)^2}-\frac {3 (4 c f-b g+2 c g x) \sqrt {a+b x+c x^2}}{4 g^2 (e f-d g) (f+g x)}-\frac {e^2 \left (8 c^2 f^2+b^2 g^2-2 c g (5 b f-4 a g)-2 c g (2 c f-b g) x\right ) \sqrt {a+b x+c x^2}}{8 c g^2 (e f-d g)^3}+\frac {\left (a+b x+c x^2\right )^{3/2}}{2 (e f-d g) (f+g x)^2}+\frac {e \left (a+b x+c x^2\right )^{3/2}}{(e f-d g)^2 (f+g x)}-\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 {a+b x+c x^2}}\right )}{16 c^{3/2} e (e f-d g)^3}+\frac {3 \sqrt {c} (2 c f-b g) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 g^3 (e f-d g)}+\frac {e^2 (2 c f-b g) \left (8 c^2 f^2-b^2 g^2-4 c g (2 b f-3 a g)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{3/2} g^3 (e f-d g)^3}-\frac {3 e \left (8 c^2 f^2+b^2 g^2-4 c g (2 b f-a g)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {c} g^3 (e f-d g)^2}+\frac {\left (c d^2-b d e+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 d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{e (e f-d g)^3}+\frac {3 e (2 c f-b g) \sqrt {c f^2-b f g+a g^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 )}{2 g^3 (e f-d g)^2}-\frac {e^2 \left (c f^2-b f g+a g^2\right )^{3/2} \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^3 (e f-d g)^3}-\frac {3 \left (8 c^2 f^2+b^2 g^2-4 c g (2 b f-a g)\right ) \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 )}{8 g^3 (e f-d g) \sqrt {c f^2-b f g+a g^2}} \]
1/2*(c*x^2+b*x+a)^(3/2)/(-d*g+e*f)/(g*x+f)^2+e*(c*x^2+b*x+a)^(3/2)/(-d*g+e *f)^2/(g*x+f)-1/16*(-b*e+2*c*d)*(8*c^2*d^2-b^2*e^2-4*c*e*(-3*a*e+2*b*d))*a rctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(3/2)/e/(-d*g+e*f)^3+1 /16*e^2*(-b*g+2*c*f)*(8*c^2*f^2-b^2*g^2-4*c*g*(-3*a*g+2*b*f))*arctanh(1/2* (2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(3/2)/g^3/(-d*g+e*f)^3+(a*e^2-b*d *e+c*d^2)^(3/2)*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/(-d*g+e*f)^3-e^2*(a*g^2-b*f*g+c*f^2)^(3/2)*a rctanh(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^3/(-d*g+e*f)^3-3/8*e*(8*c^2*f^2+b^2*g^2-4*c*g*(-a*g+2*b*f))*a rctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/g^3/(-d*g+e*f)^2/c^(1/2) +3/2*(-b*g+2*c*f)*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))*c^(1/ 2)/g^3/(-d*g+e*f)-3/8*(8*c^2*f^2+b^2*g^2-4*c*g*(-a*g+2*b*f))*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 ^3/(-d*g+e*f)/(a*g^2-b*f*g+c*f^2)^(1/2)+3/2*e*(-b*g+2*c*f)*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))*(a* g^2-b*f*g+c*f^2)^(1/2)/g^3/(-d*g+e*f)^2+1/8*(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/(-d*g+e*f)^3+3/4*e*( -2*c*g*x-3*b*g+4*c*f)*(c*x^2+b*x+a)^(1/2)/g^2/(-d*g+e*f)^2-3/4*(2*c*g*x-b* g+4*c*f)*(c*x^2+b*x+a)^(1/2)/g^2/(-d*g+e*f)/(g*x+f)-1/8*e^2*(8*c^2*f^2+b^2 *g^2-2*c*g*(-4*a*g+5*b*f)-2*c*g*(-b*g+2*c*f)*x)*(c*x^2+b*x+a)^(1/2)/c/g...
Time = 12.29 (sec) , antiderivative size = 1036, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x) (f+g x)^3} \, dx=\frac {1}{4} \left (\frac {2 (a+x (b+c x))^{3/2}}{(e f-d g) (f+g x)^2}+\frac {4 e (a+x (b+c x))^{3/2}}{(e f-d g)^2 (f+g x)}+\frac {-\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 )}{4 c^{3/2} e (e f-d g)^3}-\frac {3 e \left (\left (8 c^2 f^2+b^2 g^2+4 c g (-2 b f+a g)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )+2 \sqrt {c} \left (g (-4 c f+3 b g+2 c g x) \sqrt {a+x (b+c x)}+2 (2 c f-b g) \sqrt {c f^2+g (-b f+a g)} \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 )\right )}{2 \sqrt {c} g^3 (e f-d g)^2}+\frac {3 \left (\frac {(-2 c f+b g) (a+x (b+c x))^{3/2}}{f+g x}-\frac {\sqrt {a+x (b+c x)} \left (b^2 g^2+2 c^2 f (2 f-g x)+c g (-5 b f+2 a g+b g x)\right )}{g^2}+\frac {4 \sqrt {c} (2 c f-b g) \left (c f^2+g (-b f+a g)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )+\left (8 c^2 f^2+b^2 g^2+4 c g (-2 b f+a g)\right ) \sqrt {c f^2+g (-b f+a g)} \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 )}{2 g^3}\right )}{(e f-d g) \left (c f^2+g (-b f+a g)\right )}-\frac {e^2 \left ((2 c f-b g) \left (8 c^2 f^2-b^2 g^2+4 c g (-2 b f+3 a g)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )+2 \sqrt {c} \left (g \sqrt {a+x (b+c x)} \left (-b^2 g^2+4 c^2 f (-2 f+g x)-2 c g (-5 b f+4 a g+b g x)\right )+8 c \left (c f^2+g (-b f+a g)\right )^{3/2} \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 )\right )}{4 c^{3/2} g^3 (-e f+d g)^3}\right ) \]
((2*(a + x*(b + c*x))^(3/2))/((e*f - d*g)*(f + g*x)^2) + (4*e*(a + x*(b + c*x))^(3/2))/((e*f - d*g)^2*(f + g*x)) + (-((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)*ArcTanh[(-(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)])]))/(4*c^(3/2)*e*(e*f - d*g)^3) - (3*e* ((8*c^2*f^2 + b^2*g^2 + 4*c*g*(-2*b*f + a*g))*ArcTanh[(b + 2*c*x)/(2*Sqrt[ c]*Sqrt[a + x*(b + c*x)])] + 2*Sqrt[c]*(g*(-4*c*f + 3*b*g + 2*c*g*x)*Sqrt[ a + x*(b + c*x)] + 2*(2*c*f - b*g)*Sqrt[c*f^2 + g*(-(b*f) + a*g)]*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)])])))/(2*Sqrt[c]*g^3*(e*f - d*g)^2) + (3*(((-2*c*f + b*g)* (a + x*(b + c*x))^(3/2))/(f + g*x) - (Sqrt[a + x*(b + c*x)]*(b^2*g^2 + 2*c ^2*f*(2*f - g*x) + c*g*(-5*b*f + 2*a*g + b*g*x)))/g^2 + (4*Sqrt[c]*(2*c*f - b*g)*(c*f^2 + g*(-(b*f) + a*g))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] + (8*c^2*f^2 + b^2*g^2 + 4*c*g*(-2*b*f + a*g))*Sqrt[c*f^2 + g*(-(b*f) + a*g)]*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)])])/(2*g^3)))/((e*f - d*g)*(c*f ^2 + g*(-(b*f) + a*g))) - (e^2*((2*c*f - b*g)*(8*c^2*f^2 - b^2*g^2 + 4*c*g *(-2*b*f + 3*a*g))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]...
Time = 1.80 (sec) , antiderivative size = 1066, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {1289, 2009}
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.
| \(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x) (f+g x)^3} \, dx\) |
| \(\Big \downarrow \) 1289 |
| \(\displaystyle \int \left (\frac {e^3 \left (a+b x+c x^2\right )^{3/2}}{(d+e x) (e f-d g)^3}-\frac {e^2 g \left (a+b x+c x^2\right )^{3/2}}{(f+g x) (e f-d g)^3}-\frac {e g \left (a+b x+c x^2\right )^{3/2}}{(f+g x)^2 (e f-d g)^2}-\frac {g \left (a+b x+c x^2\right )^{3/2}}{(f+g x)^3 (e f-d g)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(2 c f-b g) \left (8 c^2 f^2-b^2 g^2-4 c g (2 b f-3 a g)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right ) e^2}{16 c^{3/2} g^3 (e f-d g)^3}-\frac {\left (c f^2-b g f+a g^2\right )^{3/2} \text {arctanh}\left (\frac {b f-2 a g+(2 c f-b g) x}{2 \sqrt {c f^2-b g f+a g^2} \sqrt {c x^2+b x+a}}\right ) e^2}{g^3 (e f-d g)^3}-\frac {\left (8 c^2 f^2+b^2 g^2-2 c g (5 b f-4 a g)-2 c g (2 c f-b g) x\right ) \sqrt {c x^2+b x+a} e^2}{8 c g^2 (e f-d g)^3}+\frac {\left (c x^2+b x+a\right )^{3/2} e}{(e f-d g)^2 (f+g x)}-\frac {3 \left (8 c^2 f^2+b^2 g^2-4 c g (2 b f-a g)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right ) e}{8 \sqrt {c} g^3 (e f-d g)^2}+\frac {3 (2 c f-b g) \sqrt {c f^2-b g f+a g^2} \text {arctanh}\left (\frac {b f-2 a g+(2 c f-b g) x}{2 \sqrt {c f^2-b g f+a g^2} \sqrt {c x^2+b x+a}}\right ) e}{2 g^3 (e f-d g)^2}+\frac {3 (4 c f-3 b g-2 c g x) \sqrt {c x^2+b x+a} e}{4 g^2 (e f-d g)^2}+\frac {\left (c x^2+b x+a\right )^{3/2}}{2 (e f-d g) (f+g x)^2}+\frac {3 \sqrt {c} (2 c f-b g) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right )}{2 g^3 (e f-d g)}-\frac {3 \left (8 c^2 f^2+b^2 g^2-4 c g (2 b f-a g)\right ) \text {arctanh}\left (\frac {b f-2 a g+(2 c f-b g) x}{2 \sqrt {c f^2-b g f+a g^2} \sqrt {c x^2+b x+a}}\right )}{8 g^3 (e f-d g) \sqrt {c f^2-b g f+a g^2}}+\frac {\left (8 c^2 d^2+b^2 e^2-2 c e (5 b d-4 a e)-2 c e (2 c d-b e) x\right ) \sqrt {c x^2+b x+a}}{8 c (e f-d g)^3}-\frac {3 (4 c f-b g+2 c g x) \sqrt {c x^2+b x+a}}{4 g^2 (e f-d g) (f+g x)}-\frac {(2 c d-b e) \left (8 c^2 d^2-b^2 e^2-4 c e (2 b d-3 a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right )}{16 c^{3/2} (e f-d g)^3 e}+\frac {\left (c d^2-b e d+a e^2\right )^{3/2} \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b e d+a e^2} \sqrt {c x^2+b x+a}}\right )}{(e f-d g)^3 e}\) |
((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)*Sqr t[a + b*x + c*x^2])/(8*c*(e*f - d*g)^3) + (3*e*(4*c*f - 3*b*g - 2*c*g*x)*S qrt[a + b*x + c*x^2])/(4*g^2*(e*f - d*g)^2) - (3*(4*c*f - b*g + 2*c*g*x)*S qrt[a + b*x + c*x^2])/(4*g^2*(e*f - d*g)*(f + g*x)) - (e^2*(8*c^2*f^2 + b^ 2*g^2 - 2*c*g*(5*b*f - 4*a*g) - 2*c*g*(2*c*f - b*g)*x)*Sqrt[a + b*x + c*x^ 2])/(8*c*g^2*(e*f - d*g)^3) + (a + b*x + c*x^2)^(3/2)/(2*(e*f - d*g)*(f + g*x)^2) + (e*(a + b*x + c*x^2)^(3/2))/((e*f - d*g)^2*(f + g*x)) - ((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])])/(16*c^(3/2)*e*(e*f - d*g)^3) + (3*Sqrt[ c]*(2*c*f - b*g)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/( 2*g^3*(e*f - d*g)) + (e^2*(2*c*f - b*g)*(8*c^2*f^2 - b^2*g^2 - 4*c*g*(2*b* f - 3*a*g))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16*c^ (3/2)*g^3*(e*f - d*g)^3) - (3*e*(8*c^2*f^2 + b^2*g^2 - 4*c*g*(2*b*f - a*g) )*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*Sqrt[c]*g^3*( e*f - d*g)^2) + ((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*(e *f - d*g)^3) + (3*e*(2*c*f - b*g)*Sqrt[c*f^2 - b*f*g + a*g^2]*ArcTanh[(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])])/(2*g^3*(e*f - d*g)^2) - (e^2*(c*f^2 - b*f*g + a*g^2)^(3/2)*ArcTa nh[(b*f - 2*a*g + (2*c*f - b*g)*x)/(2*Sqrt[c*f^2 - b*f*g + a*g^2]*Sqrt[...
3.9.68.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && ( IntegerQ[p] || (ILtQ[m, 0] && ILtQ[n, 0]))
Leaf count of result is larger than twice the leaf count of optimal. \(4241\) vs. \(2(968)=1936\).
Time = 0.96 (sec) , antiderivative size = 4242, normalized size of antiderivative = 3.98
1/g^2/(d*g-e*f)*(-1/2/(a*g^2-b*f*g+c*f^2)*g^2/(x+f/g)^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)^(5/2)+1/4*(b*g-2*c*f)*g/(a*g^2-b *f*g+c*f^2)*(-1/(a*g^2-b*f*g+c*f^2)*g^2/(x+f/g)*((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)^(5/2)+3/2*(b*g-2*c*f)*g/(a*g^2-b*f*g+c*f ^2)*(1/3*((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)^(3/2) +1/2*(b*g-2*c*f)/g*(1/4*(2*c*(x+f/g)+(b*g-2*c*f)/g)/c*((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)+1/8*(4*c*(a*g^2-b*f*g+c*f^2) /g^2-(b*g-2*c*f)^2/g^2)/c^(3/2)*ln((1/2*(b*g-2*c*f)/g+c*(x+f/g))/c^(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)))+(a*g^2- b*f*g+c*f^2)/g^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)+1/2*(b*g-2*c*f)/g*ln((1/2*(b*g-2*c*f)/g+c*(x+f/g))/c^(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))/c^(1/2)-(a*g ^2-b*f*g+c*f^2)/g^2/((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))))+4*c/(a *g^2-b*f*g+c*f^2)*g^2*(1/8*(2*c*(x+f/g)+(b*g-2*c*f)/g)/c*((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)^(3/2)+3/16*(4*c*(a*g^2-b*f*g+c* f^2)/g^2-(b*g-2*c*f)^2/g^2)/c*(1/4*(2*c*(x+f/g)+(b*g-2*c*f)/g)/c*((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)+1/8*(4*c*(a*g^2-b *f*g+c*f^2)/g^2-(b*g-2*c*f)^2/g^2)/c^(3/2)*ln((1/2*(b*g-2*c*f)/g+c*(x+f...
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x) (f+g x)^3} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x) (f+g x)^3} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x) (f+g x)^3} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}}{{\left (e x + d\right )} {\left (g x + f\right )}^{3}} \,d x } \]
Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x) (f+g x)^3} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x) (f+g x)^3} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{{\left (f+g\,x\right )}^3\,\left (d+e\,x\right )} \,d x \]