3.9.85 \(\int \frac {1}{(d+e x) (f+g x)^3 (a+b x+c x^2)^{3/2}} \, dx\) [885]

3.9.85.1 Optimal result

Integrand size = 29, antiderivative size = 1064 \[ \int \frac {1}{(d+e x) (f+g x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 e^3 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (e f-d g)^3 \sqrt {a+b x+c x^2}}+\frac {2 e^2 g \left (b c f-b^2 g+2 a c g+c (2 c f-b g) x\right )}{\left (b^2-4 a c\right ) (e f-d g)^3 \left (c f^2-b f g+a g^2\right ) \sqrt {a+b x+c x^2}}+\frac {2 g \left (b c f-b^2 g+2 a c g+c (2 c f-b g) x\right )}{\left (b^2-4 a c\right ) (e f-d g) \left (c f^2-b f g+a g^2\right ) (f+g x)^2 \sqrt {a+b x+c x^2}}+\frac {2 e g \left (b c f-b^2 g+2 a c g+c (2 c f-b g) x\right )}{\left (b^2-4 a c\right ) (e f-d g)^2 \left (c f^2-b f g+a g^2\right ) (f+g x) \sqrt {a+b x+c x^2}}+\frac {g^2 \left (8 c^2 f^2+5 b^2 g^2-4 c g (2 b f+3 a g)\right ) \sqrt {a+b x+c x^2}}{2 \left (b^2-4 a c\right ) (e f-d g) \left (c f^2-b f g+a g^2\right )^2 (f+g x)^2}+\frac {e g^2 \left (4 c^2 f^2+3 b^2 g^2-4 c g (b f+2 a g)\right ) \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right ) (e f-d g)^2 \left (c f^2-b f g+a g^2\right )^2 (f+g x)}+\frac {g^2 (2 c f-b g) \left (8 c^2 f^2+15 b^2 g^2-4 c g (2 b f+13 a g)\right ) \sqrt {a+b x+c x^2}}{4 \left (b^2-4 a c\right ) (e f-d g) \left (c f^2-b f g+a g^2\right )^3 (f+g x)}+\frac {e^5 \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 )}{\left (c d^2-b d e+a e^2\right )^{3/2} (e f-d g)^3}-\frac {3 e g^3 (2 c f-b g) \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 (e f-d g)^2 \left (c f^2-b f g+a g^2\right )^{5/2}}-\frac {e^2 g^3 \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 )}{(e f-d g)^3 \left (c f^2-b f g+a g^2\right )^{3/2}}-\frac {3 g^3 \left (16 c^2 f^2+5 b^2 g^2-4 c g (4 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 (e f-d g) \left (c f^2-b f g+a g^2\right )^{7/2}} \]

output

e^5*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))/(a*e^2-b*d*e+c*d^2)^(3/2)/(-d*g+e*f)^3-3/2*e*g^3*(-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))/(-d*g+e*f)^2/(a*g^2-b*f*g+c*f^2)^(5/2)-e^2*g^3*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)) 
/(-d*g+e*f)^3/(a*g^2-b*f*g+c*f^2)^(3/2)-3/8*g^3*(16*c^2*f^2+5*b^2*g^2-4*c* 
g*(a*g+4*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))/(-d*g+e*f)/(a*g^2-b*f*g+c*f^2)^(7/2)-2*e^3*(b*c 
*d-b^2*e+2*a*c*e+c*(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)/(-d*g+ 
e*f)^3/(c*x^2+b*x+a)^(1/2)+2*e^2*g*(b*c*f-b^2*g+2*a*c*g+c*(-b*g+2*c*f)*x)/ 
(-4*a*c+b^2)/(-d*g+e*f)^3/(a*g^2-b*f*g+c*f^2)/(c*x^2+b*x+a)^(1/2)+2*g*(b*c 
*f-b^2*g+2*a*c*g+c*(-b*g+2*c*f)*x)/(-4*a*c+b^2)/(-d*g+e*f)/(a*g^2-b*f*g+c* 
f^2)/(g*x+f)^2/(c*x^2+b*x+a)^(1/2)+2*e*g*(b*c*f-b^2*g+2*a*c*g+c*(-b*g+2*c* 
f)*x)/(-4*a*c+b^2)/(-d*g+e*f)^2/(a*g^2-b*f*g+c*f^2)/(g*x+f)/(c*x^2+b*x+a)^ 
(1/2)+1/2*g^2*(8*c^2*f^2+5*b^2*g^2-4*c*g*(3*a*g+2*b*f))*(c*x^2+b*x+a)^(1/2 
)/(-4*a*c+b^2)/(-d*g+e*f)/(a*g^2-b*f*g+c*f^2)^2/(g*x+f)^2+e*g^2*(4*c^2*f^2 
+3*b^2*g^2-4*c*g*(2*a*g+b*f))*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)/(-d*g+e*f)^ 
2/(a*g^2-b*f*g+c*f^2)^2/(g*x+f)+1/4*g^2*(-b*g+2*c*f)*(8*c^2*f^2+15*b^2*g^2 
-4*c*g*(13*a*g+2*b*f))*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)/(-d*g+e*f)/(a*g^2- 
b*f*g+c*f^2)^3/(g*x+f)
 

3.9.85.2 Mathematica [A] (verified)

Time = 15.23 (sec) , antiderivative size = 1013, normalized size of antiderivative = 0.95 \[ \int \frac {1}{(d+e x) (f+g x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 e^3 \left (b^2 e-2 c (a e+c d x)+b c (-d+e x)\right )}{\left (b^2-4 a c\right ) \left (-c d^2+e (b d-a e)\right ) (e f-d g)^3 \sqrt {a+x (b+c x)}}-\frac {2 e^2 g \left (b^2 g-2 c (a g+c f x)+b c (-f+g x)\right )}{\left (b^2-4 a c\right ) (-e f+d g)^3 \left (-c f^2+g (b f-a g)\right ) \sqrt {a+x (b+c x)}}-\frac {2 g \left (b^2 g-2 c (a g+c f x)+b c (-f+g x)\right )}{\left (b^2-4 a c\right ) (-e f+d g) \left (-c f^2+g (b f-a g)\right ) (f+g x)^2 \sqrt {a+x (b+c x)}}+\frac {2 e g \left (b^2 g-2 c (a g+c f x)+b c (-f+g x)\right )}{\left (b^2-4 a c\right ) (e f-d g)^2 \left (-c f^2+g (b f-a g)\right ) (f+g x) \sqrt {a+x (b+c x)}}+\frac {e g^2 \left (\frac {2 \left (4 c^2 f^2+3 b^2 g^2-4 c g (b f+2 a g)\right ) \sqrt {a+x (b+c x)}}{\left (b^2-4 a c\right ) \left (c f^2+g (-b f+a g)\right )^2 (f+g x)}+\frac {3 g (2 c f-b 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 )}{\left (c f^2+g (-b f+a g)\right )^{5/2}}\right )}{2 (e f-d g)^2}-\frac {g^2 \left (\frac {4 \left (8 c^2 f^2+5 b^2 g^2-4 c g (2 b f+3 a g)\right ) \sqrt {a+x (b+c x)}}{(f+g x)^2}+\frac {2 (2 c f-b g) \left (8 c^2 f^2+15 b^2 g^2-4 c g (2 b f+13 a g)\right ) \sqrt {a+x (b+c x)}}{\left (c f^2+g (-b f+a g)\right ) (f+g x)}+\frac {3 \left (b^2-4 a c\right ) g \left (16 c^2 f^2+5 b^2 g^2-4 c g (4 b f+a g)\right ) \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 )}{\left (c f^2+g (-b f+a g)\right )^{3/2}}\right )}{8 \left (b^2-4 a c\right ) (-e f+d g) \left (c f^2+g (-b f+a g)\right )^2}-\frac {e^5 \text {arctanh}\left (\frac {-2 a e+2 c d x+b (d-e x)}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{\left (c d^2+e (-b d+a e)\right )^{3/2} (-e f+d g)^3}-\frac {e^2 g^3 \text {arctanh}\left (\frac {-2 a g+2 c f x+b (f-g x)}{2 \sqrt {c f^2+g (-b f+a g)} \sqrt {a+x (b+c x)}}\right )}{(e f-d g)^3 \left (c f^2+g (-b f+a g)\right )^{3/2}} \]

input

Integrate[1/((d + e*x)*(f + g*x)^3*(a + b*x + c*x^2)^(3/2)),x]
 

output

(-2*e^3*(b^2*e - 2*c*(a*e + c*d*x) + b*c*(-d + e*x)))/((b^2 - 4*a*c)*(-(c* 
d^2) + e*(b*d - a*e))*(e*f - d*g)^3*Sqrt[a + x*(b + c*x)]) - (2*e^2*g*(b^2 
*g - 2*c*(a*g + c*f*x) + b*c*(-f + g*x)))/((b^2 - 4*a*c)*(-(e*f) + d*g)^3* 
(-(c*f^2) + g*(b*f - a*g))*Sqrt[a + x*(b + c*x)]) - (2*g*(b^2*g - 2*c*(a*g 
 + c*f*x) + b*c*(-f + g*x)))/((b^2 - 4*a*c)*(-(e*f) + d*g)*(-(c*f^2) + g*( 
b*f - a*g))*(f + g*x)^2*Sqrt[a + x*(b + c*x)]) + (2*e*g*(b^2*g - 2*c*(a*g 
+ c*f*x) + b*c*(-f + g*x)))/((b^2 - 4*a*c)*(e*f - d*g)^2*(-(c*f^2) + g*(b* 
f - a*g))*(f + g*x)*Sqrt[a + x*(b + c*x)]) + (e*g^2*((2*(4*c^2*f^2 + 3*b^2 
*g^2 - 4*c*g*(b*f + 2*a*g))*Sqrt[a + x*(b + c*x)])/((b^2 - 4*a*c)*(c*f^2 + 
 g*(-(b*f) + a*g))^2*(f + g*x)) + (3*g*(2*c*f - b*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)])])/(c*f^2 + g*(-(b*f) + a*g))^(5/2)))/(2*(e*f - d*g)^2) - (g^2*((4*(8* 
c^2*f^2 + 5*b^2*g^2 - 4*c*g*(2*b*f + 3*a*g))*Sqrt[a + x*(b + c*x)])/(f + g 
*x)^2 + (2*(2*c*f - b*g)*(8*c^2*f^2 + 15*b^2*g^2 - 4*c*g*(2*b*f + 13*a*g)) 
*Sqrt[a + x*(b + c*x)])/((c*f^2 + g*(-(b*f) + a*g))*(f + g*x)) + (3*(b^2 - 
 4*a*c)*g*(16*c^2*f^2 + 5*b^2*g^2 - 4*c*g*(4*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)])])/(c*f^2 + g*(-(b*f) + a*g))^(3/2)))/(8*(b^2 - 4*a*c)*(-(e*f) + d 
*g)*(c*f^2 + g*(-(b*f) + a*g))^2) - (e^5*ArcTanh[(-2*a*e + 2*c*d*x + b*(d 
- e*x))/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/((c*...
 

3.9.85.3 Rubi [A] (verified)

Time = 1.83 (sec) , antiderivative size = 1064, 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.

\(\displaystyle \int \frac {1}{(d+e x) (f+g x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 1289

\(\displaystyle \int \left (\frac {e^3}{(d+e x) \left (a+b x+c x^2\right )^{3/2} (e f-d g)^3}-\frac {e^2 g}{(f+g x) \left (a+b x+c x^2\right )^{3/2} (e f-d g)^3}-\frac {e g}{(f+g x)^2 \left (a+b x+c x^2\right )^{3/2} (e f-d g)^2}-\frac {g}{(f+g x)^3 \left (a+b x+c x^2\right )^{3/2} (e f-d g)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b e d+a e^2} \sqrt {c x^2+b x+a}}\right ) e^5}{\left (c d^2-b e d+a e^2\right )^{3/2} (e f-d g)^3}-\frac {2 \left (-e b^2+c d b+2 a c e+c (2 c d-b e) x\right ) e^3}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) (e f-d g)^3 \sqrt {c x^2+b x+a}}-\frac {g^3 \text {arctanh}\left (\frac {b f-2 a g+(2 c f-b g) x}{2 \sqrt {c f^2-b g f+a g^2} \sqrt {c x^2+b x+a}}\right ) e^2}{(e f-d g)^3 \left (c f^2-b g f+a g^2\right )^{3/2}}+\frac {2 g \left (-g b^2+c f b+2 a c g+c (2 c f-b g) x\right ) e^2}{\left (b^2-4 a c\right ) (e f-d g)^3 \left (c f^2-b g f+a g^2\right ) \sqrt {c x^2+b x+a}}-\frac {3 g^3 (2 c f-b g) \text {arctanh}\left (\frac {b f-2 a g+(2 c f-b g) x}{2 \sqrt {c f^2-b g f+a g^2} \sqrt {c x^2+b x+a}}\right ) e}{2 (e f-d g)^2 \left (c f^2-b g f+a g^2\right )^{5/2}}+\frac {g^2 \left (4 c^2 f^2+3 b^2 g^2-4 c g (b f+2 a g)\right ) \sqrt {c x^2+b x+a} e}{\left (b^2-4 a c\right ) (e f-d g)^2 \left (c f^2-b g f+a g^2\right )^2 (f+g x)}+\frac {2 g \left (-g b^2+c f b+2 a c g+c (2 c f-b g) x\right ) e}{\left (b^2-4 a c\right ) (e f-d g)^2 \left (c f^2-b g f+a g^2\right ) (f+g x) \sqrt {c x^2+b x+a}}-\frac {3 g^3 \left (16 c^2 f^2+5 b^2 g^2-4 c g (4 b f+a g)\right ) \text {arctanh}\left (\frac {b f-2 a g+(2 c f-b g) x}{2 \sqrt {c f^2-b g f+a g^2} \sqrt {c x^2+b x+a}}\right )}{8 (e f-d g) \left (c f^2-b g f+a g^2\right )^{7/2}}+\frac {g^2 (2 c f-b g) \left (8 c^2 f^2+15 b^2 g^2-4 c g (2 b f+13 a g)\right ) \sqrt {c x^2+b x+a}}{4 \left (b^2-4 a c\right ) (e f-d g) \left (c f^2-b g f+a g^2\right )^3 (f+g x)}+\frac {g^2 \left (8 c^2 f^2+5 b^2 g^2-4 c g (2 b f+3 a g)\right ) \sqrt {c x^2+b x+a}}{2 \left (b^2-4 a c\right ) (e f-d g) \left (c f^2-b g f+a g^2\right )^2 (f+g x)^2}+\frac {2 g \left (-g b^2+c f b+2 a c g+c (2 c f-b g) x\right )}{\left (b^2-4 a c\right ) (e f-d g) \left (c f^2-b g f+a g^2\right ) (f+g x)^2 \sqrt {c x^2+b x+a}}\)

input

Int[1/((d + e*x)*(f + g*x)^3*(a + b*x + c*x^2)^(3/2)),x]
 

output

(-2*e^3*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/((b^2 - 4*a*c)*(c*d 
^2 - b*d*e + a*e^2)*(e*f - d*g)^3*Sqrt[a + b*x + c*x^2]) + (2*e^2*g*(b*c*f 
 - b^2*g + 2*a*c*g + c*(2*c*f - b*g)*x))/((b^2 - 4*a*c)*(e*f - d*g)^3*(c*f 
^2 - b*f*g + a*g^2)*Sqrt[a + b*x + c*x^2]) + (2*g*(b*c*f - b^2*g + 2*a*c*g 
 + c*(2*c*f - b*g)*x))/((b^2 - 4*a*c)*(e*f - d*g)*(c*f^2 - b*f*g + a*g^2)* 
(f + g*x)^2*Sqrt[a + b*x + c*x^2]) + (2*e*g*(b*c*f - b^2*g + 2*a*c*g + c*( 
2*c*f - b*g)*x))/((b^2 - 4*a*c)*(e*f - d*g)^2*(c*f^2 - b*f*g + a*g^2)*(f + 
 g*x)*Sqrt[a + b*x + c*x^2]) + (g^2*(8*c^2*f^2 + 5*b^2*g^2 - 4*c*g*(2*b*f 
+ 3*a*g))*Sqrt[a + b*x + c*x^2])/(2*(b^2 - 4*a*c)*(e*f - d*g)*(c*f^2 - b*f 
*g + a*g^2)^2*(f + g*x)^2) + (e*g^2*(4*c^2*f^2 + 3*b^2*g^2 - 4*c*g*(b*f + 
2*a*g))*Sqrt[a + b*x + c*x^2])/((b^2 - 4*a*c)*(e*f - d*g)^2*(c*f^2 - b*f*g 
 + a*g^2)^2*(f + g*x)) + (g^2*(2*c*f - b*g)*(8*c^2*f^2 + 15*b^2*g^2 - 4*c* 
g*(2*b*f + 13*a*g))*Sqrt[a + b*x + c*x^2])/(4*(b^2 - 4*a*c)*(e*f - d*g)*(c 
*f^2 - b*f*g + a*g^2)^3*(f + g*x)) + (e^5*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])])/((c*d^2 - 
b*d*e + a*e^2)^(3/2)*(e*f - d*g)^3) - (3*e*g^3*(2*c*f - b*g)*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*(e*f - d*g)^2*(c*f^2 - b*f*g + a*g^2)^(5/2)) - (e^2*g^3*ArcTan 
h[(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])])/((e*f - d*g)^3*(c*f^2 - b*f*g + a*g^2)^(3/2)) - (3*g^3*...
 

3.9.85.3.1 Defintions of rubi rules used

rule 1289

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]))
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

3.9.85.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2684\) vs. \(2(1010)=2020\).

Time = 1.14 (sec) , antiderivative size = 2685, normalized size of antiderivative = 2.52

method	result	size
default	\(\text {Expression too large to display}\)	\(2685\)

input

int(1/(e*x+d)/(g*x+f)^3/(c*x^2+b*x+a)^(3/2),x,method=_RETURNVERBOSE)
 

output

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)^(1/2)-5/4*(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)^(1/2)-3/2*(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)^2*c+(b*g-2*c*f)/g*(x+f/g)+(a*g^2-b 
*f*g+c*f^2)/g^2)^(1/2)-(b*g-2*c*f)*g/(a*g^2-b*f*g+c*f^2)*(2*c*(x+f/g)+(b*g 
-2*c*f)/g)/(4*c*(a*g^2-b*f*g+c*f^2)/g^2-(b*g-2*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/(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*(2*c*(x+f/g)+(b*g-2*c*f)/g)/(4*c*(a*g^2-b*f*g+c*f^2)/g^2-(b*g-2*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))-3 
/2*c/(a*g^2-b*f*g+c*f^2)*g^2*(1/(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)-(b*g-2*c*f)*g/(a*g^2-b*f*g 
+c*f^2)*(2*c*(x+f/g)+(b*g-2*c*f)/g)/(4*c*(a*g^2-b*f*g+c*f^2)/g^2-(b*g-2*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/(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...
 

3.9.85.5 Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x) (f+g x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Timed out} \]

input

integrate(1/(e*x+d)/(g*x+f)^3/(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")
 

output

Timed out
 

3.9.85.6 Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x) (f+g x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Timed out} \]

input

integrate(1/(e*x+d)/(g*x+f)**3/(c*x**2+b*x+a)**(3/2),x)
 

output

Timed out
 

3.9.85.7 Maxima [F]

\[ \int \frac {1}{(d+e x) (f+g x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}} {\left (e x + d\right )} {\left (g x + f\right )}^{3}} \,d x } \]

input

integrate(1/(e*x+d)/(g*x+f)^3/(c*x^2+b*x+a)^(3/2),x, algorithm="maxima")
 

output

integrate(1/((c*x^2 + b*x + a)^(3/2)*(e*x + d)*(g*x + f)^3), x)
 

3.9.85.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 14979 vs. \(2 (1010) = 2020\).

Time = 4.14 (sec) , antiderivative size = 14979, normalized size of antiderivative = 14.08 \[ \int \frac {1}{(d+e x) (f+g x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Too large to display} \]

input

integrate(1/(e*x+d)/(g*x+f)^3/(c*x^2+b*x+a)^(3/2),x, algorithm="giac")
 

output

2*e^5*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x + a))*e + sqrt(c)*d)/sqrt(-c* 
d^2 + b*d*e - a*e^2))/((c*d^2*e^3*f^3 - b*d*e^4*f^3 + a*e^5*f^3 - 3*c*d^3* 
e^2*f^2*g + 3*b*d^2*e^3*f^2*g - 3*a*d*e^4*f^2*g + 3*c*d^4*e*f*g^2 - 3*b*d^ 
3*e^2*f*g^2 + 3*a*d^2*e^3*f*g^2 - c*d^5*g^3 + b*d^4*e*g^3 - a*d^3*e^2*g^3) 
*sqrt(-c*d^2 + b*d*e - a*e^2)) - 2*((2*c^9*d^3*f^9 - 3*b*c^8*d^2*e*f^9 + b 
^2*c^7*d*e^2*f^9 + 2*a*c^8*d*e^2*f^9 - a*b*c^7*e^3*f^9 - 9*b*c^8*d^3*f^8*g 
 + 15*b^2*c^7*d^2*e*f^8*g - 6*a*c^8*d^2*e*f^8*g - 6*b^3*c^6*d*e^2*f^8*g - 
3*a*b*c^7*d*e^2*f^8*g + 6*a*b^2*c^6*e^3*f^8*g - 6*a^2*c^7*e^3*f^8*g + 18*b 
^2*c^7*d^3*f^7*g^2 - 33*b^3*c^6*d^2*e*f^7*g^2 + 24*a*b*c^7*d^2*e*f^7*g^2 + 
 15*b^4*c^5*d*e^2*f^7*g^2 - 6*a*b^2*c^6*d*e^2*f^7*g^2 - 15*a*b^3*c^5*e^3*f 
^7*g^2 + 24*a^2*b*c^6*e^3*f^7*g^2 - 21*b^3*c^6*d^3*f^6*g^3 + 41*b^4*c^5*d^ 
2*e*f^6*g^3 - 34*a*b^2*c^6*d^2*e*f^6*g^3 - 16*a^2*c^7*d^2*e*f^6*g^3 - 20*b 
^5*c^4*d*e^2*f^6*g^3 + 13*a*b^3*c^5*d*e^2*f^6*g^3 + 16*a^2*b*c^6*d*e^2*f^6 
*g^3 + 20*a*b^4*c^4*e^3*f^6*g^3 - 34*a^2*b^2*c^5*e^3*f^6*g^3 - 16*a^3*c^6* 
e^3*f^6*g^3 + 15*b^4*c^5*d^3*f^5*g^4 + 6*a*b^2*c^6*d^3*f^5*g^4 - 12*a^2*c^ 
7*d^3*f^5*g^4 - 30*b^5*c^4*d^2*e*f^5*g^4 + 9*a*b^3*c^5*d^2*e*f^5*g^4 + 66* 
a^2*b*c^6*d^2*e*f^5*g^4 + 15*b^6*c^3*d*e^2*f^5*g^4 - 48*a^2*b^2*c^5*d*e^2* 
f^5*g^4 - 12*a^3*c^6*d*e^2*f^5*g^4 - 15*a*b^5*c^3*e^3*f^5*g^4 + 15*a^2*b^3 
*c^4*e^3*f^5*g^4 + 54*a^3*b*c^5*e^3*f^5*g^4 - 6*b^5*c^4*d^3*f^4*g^5 - 15*a 
*b^3*c^5*d^3*f^4*g^5 + 30*a^2*b*c^6*d^3*f^4*g^5 + 12*b^6*c^3*d^2*e*f^4*...
 

3.9.85.9 Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x) (f+g x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (f+g\,x\right )}^3\,\left (d+e\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \]

input

int(1/((f + g*x)^3*(d + e*x)*(a + b*x + c*x^2)^(3/2)),x)
 

output

int(1/((f + g*x)^3*(d + e*x)*(a + b*x + c*x^2)^(3/2)), x)