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

Optimal. Leaf size=352 \[ -\frac{2 e \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (e f-d g) \left (a e^2-b d e+c d^2\right )}+\frac{2 g \left (2 a c g+b^2 (-g)+c x (2 c f-b g)+b c f\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (e f-d g) \left (a g^2-b f g+c f^2\right )}+\frac{e^3 \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{(e f-d g) \left (a e^2-b d e+c d^2\right )^{3/2}}-\frac{g^3 \tanh ^{-1}\left (\frac{-2 a g+x (2 c f-b g)+b f}{2 \sqrt{a+b x+c x^2} \sqrt{a g^2-b f g+c f^2}}\right )}{(e f-d g) \left (a g^2-b f g+c f^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 1.01051, antiderivative size = 352, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172 \[ -\frac{2 e \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (e f-d g) \left (a e^2-b d e+c d^2\right )}+\frac{2 g \left (2 a c g+b^2 (-g)+c x (2 c f-b g)+b c f\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (e f-d g) \left (a g^2-b f g+c f^2\right )}+\frac{e^3 \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{(e f-d g) \left (a e^2-b d e+c d^2\right )^{3/2}}-\frac{g^3 \tanh ^{-1}\left (\frac{-2 a g+x (2 c f-b g)+b f}{2 \sqrt{a+b x+c x^2} \sqrt{a g^2-b f g+c f^2}}\right )}{(e f-d g) \left (a g^2-b f g+c f^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 152.818, size = 318, normalized size = 0.9 \[ \frac{e^{3} \operatorname{atanh}{\left (\frac{2 a e - b d + x \left (b e - 2 c d\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{\left (d g - e f\right ) \left (a e^{2} - b d e + c d^{2}\right )^{\frac{3}{2}}} - \frac{2 e \left (- 2 a c e + b^{2} e - b c d + c x \left (b e - 2 c d\right )\right )}{\left (- 4 a c + b^{2}\right ) \left (d g - e f\right ) \sqrt{a + b x + c x^{2}} \left (a e^{2} - b d e + c d^{2}\right )} - \frac{g^{3} \operatorname{atanh}{\left (\frac{2 a g - b f + x \left (b g - 2 c f\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a g^{2} - b f g + c f^{2}}} \right )}}{\left (d g - e f\right ) \left (a g^{2} - b f g + c f^{2}\right )^{\frac{3}{2}}} + \frac{2 g \left (- 2 a c g + b^{2} g - b c f + c x \left (b g - 2 c f\right )\right )}{\left (- 4 a c + b^{2}\right ) \left (d g - e f\right ) \sqrt{a + b x + c x^{2}} \left (a g^{2} - b f g + c f^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e**3*atanh((2*a*e - b*d + x*(b*e - 2*c*d))/(2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2
- b*d*e + c*d**2)))/((d*g - e*f)*(a*e**2 - b*d*e + c*d**2)**(3/2)) - 2*e*(-2*a*
c*e + b**2*e - b*c*d + c*x*(b*e - 2*c*d))/((-4*a*c + b**2)*(d*g - e*f)*sqrt(a +
b*x + c*x**2)*(a*e**2 - b*d*e + c*d**2)) - g**3*atanh((2*a*g - b*f + x*(b*g - 2*
c*f))/(2*sqrt(a + b*x + c*x**2)*sqrt(a*g**2 - b*f*g + c*f**2)))/((d*g - e*f)*(a*
g**2 - b*f*g + c*f**2)**(3/2)) + 2*g*(-2*a*c*g + b**2*g - b*c*f + c*x*(b*g - 2*c
*f))/((-4*a*c + b**2)*(d*g - e*f)*sqrt(a + b*x + c*x**2)*(a*g**2 - b*f*g + c*f**
2))

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Mathematica [A]  time = 2.75508, size = 393, normalized size = 1.12 \[ \frac{2 \left (b c (3 a e g+c (-d f+d g x+e f x))-2 c^2 (a d g+a e (f-g x)+c d f x)+b^3 (-e) g+b^2 c (d g+e (f-g x))\right )}{\left (b^2-4 a c\right ) \sqrt{a+x (b+c x)} \left (e (b d-a e)-c d^2\right ) \left (g (b f-a g)-c f^2\right )}+\frac{e^3 \log (d+e x)}{(e f-d g) \left (e (a e-b d)+c d^2\right )^{3/2}}+\frac{e^3 \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{(d g-e f) \left (e (a e-b d)+c d^2\right )^{3/2}}+\frac{g^3 \log (f+g x)}{(d g-e f) \left (g (a g-b f)+c f^2\right )^{3/2}}+\frac{g^3 \log \left (2 \sqrt{a+x (b+c x)} \sqrt{g (a g-b f)+c f^2}+2 a g-b f+b g x-2 c f x\right )}{(e f-d g) \left (g (a g-b f)+c f^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.026, size = 1343, normalized size = 3.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(d*g-e*f)/(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)-2/(d*g-e*f)*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)*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-1/(d*g-e*f)*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)*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-1/(d*g-e*f)/(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))-1/(d*g-e*f)/(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)+2/(d*g-e*f)*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*b*c-4/(d*g-e*f
)*e/(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+1/(d*g-e*f)*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^2-2/(d*g-e*f)*
e/(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+1/(d*g-e*f)/(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))

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

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)),x, algorithm="maxima")

[Out]

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

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)),x, algorithm="giac")

[Out]

Exception raised: TypeError