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

Optimal. Leaf size=642 \[ \frac{g^2 \sqrt{a+b x+c x^2} \left (-4 c g (2 a g+b f)+3 b^2 g^2+4 c^2 f^2\right )}{\left (b^2-4 a c\right ) (f+g x) (e f-d g) \left (a g^2-b f g+c f^2\right )^2}-\frac{2 e^2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (e f-d g)^2 \left (a e^2-b d e+c d^2\right )}+\frac{2 e g \left (2 a c g+b^2 (-g)+c x (2 c f-b g)+b c f\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (e f-d g)^2 \left (a g^2-b f g+c f^2\right )}+\frac{2 g \left (2 a c g+b^2 (-g)+c x (2 c f-b g)+b c f\right )}{\left (b^2-4 a c\right ) (f+g x) \sqrt{a+b x+c x^2} (e f-d g) \left (a g^2-b f g+c f^2\right )}+\frac{e^4 \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{(e f-d g)^2 \left (a e^2-b d e+c d^2\right )^{3/2}}-\frac{e g^3 \tanh ^{-1}\left (\frac{-2 a g+x (2 c f-b g)+b f}{2 \sqrt{a+b x+c x^2} \sqrt{a g^2-b f g+c f^2}}\right )}{(e f-d g)^2 \left (a g^2-b f g+c f^2\right )^{3/2}}-\frac{3 g^3 (2 c f-b g) \tanh ^{-1}\left (\frac{-2 a g+x (2 c f-b g)+b f}{2 \sqrt{a+b x+c x^2} \sqrt{a g^2-b f g+c f^2}}\right )}{2 (e f-d g) \left (a g^2-b f g+c f^2\right )^{5/2}} \]

[Out]

(-2*e^2*(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)^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)*
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)*Sqrt[a + b*x + c*x^2])
+ (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)*(c*f^2 - b*f*g + a*g^2)^2*(f + g*x)) + (e^4*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)^2) - (3*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)*(c*f^2 - b*f*g + a*g^2)^(5/2)) - (e*g^3*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])])/
((e*f - d*g)^2*(c*f^2 - b*f*g + a*g^2)^(3/2))

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Rubi [A]  time = 2.02871, antiderivative size = 642, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ \frac{g^2 \sqrt{a+b x+c x^2} \left (-4 c g (2 a g+b f)+3 b^2 g^2+4 c^2 f^2\right )}{\left (b^2-4 a c\right ) (f+g x) (e f-d g) \left (a g^2-b f g+c f^2\right )^2}-\frac{2 e^2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (e f-d g)^2 \left (a e^2-b d e+c d^2\right )}+\frac{2 e g \left (2 a c g+b^2 (-g)+c x (2 c f-b g)+b c f\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (e f-d g)^2 \left (a g^2-b f g+c f^2\right )}+\frac{2 g \left (2 a c g+b^2 (-g)+c x (2 c f-b g)+b c f\right )}{\left (b^2-4 a c\right ) (f+g x) \sqrt{a+b x+c x^2} (e f-d g) \left (a g^2-b f g+c f^2\right )}+\frac{e^4 \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{(e f-d g)^2 \left (a e^2-b d e+c d^2\right )^{3/2}}-\frac{e g^3 \tanh ^{-1}\left (\frac{-2 a g+x (2 c f-b g)+b f}{2 \sqrt{a+b x+c x^2} \sqrt{a g^2-b f g+c f^2}}\right )}{(e f-d g)^2 \left (a g^2-b f g+c f^2\right )^{3/2}}-\frac{3 g^3 (2 c f-b g) \tanh ^{-1}\left (\frac{-2 a g+x (2 c f-b g)+b f}{2 \sqrt{a+b x+c x^2} \sqrt{a g^2-b f g+c f^2}}\right )}{2 (e f-d g) \left (a g^2-b f g+c f^2\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*e^2*(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)^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)*
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)*Sqrt[a + b*x + c*x^2])
+ (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)*(c*f^2 - b*f*g + a*g^2)^2*(f + g*x)) + (e^4*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)^2) - (3*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)*(c*f^2 - b*f*g + a*g^2)^(5/2)) - (e*g^3*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])])/
((e*f - d*g)^2*(c*f^2 - b*f*g + a*g^2)^(3/2))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 9.43007, size = 575, normalized size = 0.9 \[ \frac{\sqrt{a+x (b+c x)} \left (\frac{g^4}{(f+g x) (e f-d g)}-\frac{2 \left (2 c^2 \left (a^2 e g^2+a c (d g (g x-2 f)-e f (f-2 g x))-c^2 d f^2 x\right )+b^2 c \left (-4 a e g^2+c d g (2 f-g x)+c e f (f-2 g x)\right )+b c^2 (3 a g (d g+2 e f-e g x)+c f (-d f+2 d g x+e f x))+b^4 e g^2+b^3 c g (-d g-2 e f+e g x)\right )}{\left (b^2-4 a c\right ) (a+x (b+c x)) \left (e (b d-a e)-c d^2\right )}\right )}{\left (g (a g-b f)+c f^2\right )^2}+\frac{e^4 \log (d+e x)}{(e f-d g)^2 \left (e (a e-b d)+c d^2\right )^{3/2}}-\frac{e^4 \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{(e f-d g)^2 \left (e (a e-b d)+c d^2\right )^{3/2}}-\frac{g^3 \log (f+g x) (g (2 a e g+3 b d g-5 b e f)+2 c f (4 e f-3 d g))}{2 (e f-d g)^2 \left (g (a g-b f)+c f^2\right )^{5/2}}+\frac{g^3 (g (2 a e g+3 b d g-5 b e f)+2 c f (4 e f-3 d g)) \log \left (2 \sqrt{a+x (b+c x)} \sqrt{g (a g-b f)+c f^2}+2 a g-b f+b g x-2 c f x\right )}{2 (e f-d g)^2 \left (g (a g-b f)+c f^2\right )^{5/2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(Sqrt[a + x*(b + c*x)]*(g^4/((e*f - d*g)*(f + g*x)) - (2*(b^4*e*g^2 + b^3*c*g*(-
2*e*f - d*g + e*g*x) + b^2*c*(-4*a*e*g^2 + c*e*f*(f - 2*g*x) + c*d*g*(2*f - g*x)
) + b*c^2*(c*f*(-(d*f) + e*f*x + 2*d*g*x) + 3*a*g*(2*e*f + d*g - e*g*x)) + 2*c^2
*(a^2*e*g^2 - c^2*d*f^2*x + a*c*(-(e*f*(f - 2*g*x)) + d*g*(-2*f + g*x)))))/((b^2
- 4*a*c)*(-(c*d^2) + e*(b*d - a*e))*(a + x*(b + c*x)))))/(c*f^2 + g*(-(b*f) + a
*g))^2 + (e^4*Log[d + e*x])/((c*d^2 + e*(-(b*d) + a*e))^(3/2)*(e*f - d*g)^2) - (
g^3*(2*c*f*(4*e*f - 3*d*g) + g*(-5*b*e*f + 3*b*d*g + 2*a*e*g))*Log[f + g*x])/(2*
(e*f - d*g)^2*(c*f^2 + g*(-(b*f) + a*g))^(5/2)) - (e^4*Log[-(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*d^2 +
e*(-(b*d) + a*e))^(3/2)*(e*f - d*g)^2) + (g^3*(2*c*f*(4*e*f - 3*d*g) + g*(-5*b*
e*f + 3*b*d*g + 2*a*e*g))*Log[-(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*(e*f - d*g)^2*(c*f^2 + g*(-(b*f) +
a*g))^(5/2))

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Maple [B]  time = 0.034, size = 2807, normalized size = 4.4 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*g^2/(d*g-e*f)/(a*g^2-b*f*g+c*f^2)^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))*c*f-8*g
/(d*g-e*f)*c^2/(a*g^2-b*f*g+c*f^2)/(4*a*c-b^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-4*g/(d*g-e*f)*c/(a*g^2-b*f*g+c*f^2)/(4*a*c-b^
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+1/(d*g-e*
f)^2*e^3/(a*e^2-b*d*e+c*d^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)+4/(d*g-e*f)^2*e^2/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^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*c^2*d+2/(d*g-e*f)^2*e^2/(a*
e^2-b*d*e+c*d^2)/(4*a*c-b^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)*b*c*d+3*g^3/(d*g-e*f)/(a*g^2-b*f*g+c*f^2)^2/(4*a*c-b^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*b^2*c+12*g/(d*g-e*f)/
(a*g^2-b*f*g+c*f^2)^2/(4*a*c-b^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*c^3*f^2-6*g^2/(d*g-e*f)/(a*g^2-b*f*g+c*f^2)^2/(4*a*c-b^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^2*c*f+6*g/(d
*g-e*f)/(a*g^2-b*f*g+c*f^2)^2/(4*a*c-b^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*c^2*f^2-g/(d*g-e*f)/(a*g^2-b*f*g+c*f^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)-1/(d*g-e*f)^2*e^
3/(a*e^2-b*d*e+c*d^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))+2/(d*g-e*f)^2*e*g^2/(a*g
^2-b*f*g+c*f^2)/(4*a*c-b^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*b*c-4/(d*g-e*f)^2*e*g/(a*g^2-b*f*g+c*f^2)/(4*a*c-b^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*c^2*f-2/(d*g-e*f)^2*e*
g/(a*g^2-b*f*g+c*f^2)/(4*a*c-b^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*c*f-12*g^2/(d*g-e*f)/(a*g^2-b*f*g+c*f^2)^2/(4*a*c-b^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*b*c^2*f+1/(d*g-
e*f)^2*e*g^2/(a*g^2-b*f*g+c*f^2)/(4*a*c-b^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^2-2/(d*g-e*f)^2*e^3/(a*e^2-b*d*e+c*d^2)/(4*a*c-
b^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*b*c-3/2
*g^3/(d*g-e*f)/(a*g^2-b*f*g+c*f^2)^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-1/(d*g-e*f)^2*e/(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/(d*g-e*f)^2*e^3/(a*e^2-b*d*
e+c*d^2)/(4*a*c-b^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)*b^2+3/2*g^3/(d*g-e*f)/(a*g^2-b*f*g+c*f^2)^2/(4*a*c-b^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^3+3/2*g^3/(d*g-e*f)/(a*g^2-b*
f*g+c*f^2)^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))*b+3*g^2/(d*g-e*f)/(a*g^2-b*f*g+c*
f^2)^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*f+1/(
d*g-e*f)^2*e/(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))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}{\left (e x + d\right )}{\left (g x + f\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError