3.264 \(\int \frac{x^2}{\left (a x^2+b x^3\right )^{3/2}} \, dx\)

Optimal. Leaf size=52 \[ \frac{2 x}{a \sqrt{a x^2+b x^3}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right )}{a^{3/2}} \]

[Out]

(2*x)/(a*Sqrt[a*x^2 + b*x^3]) - (2*ArcTanh[(Sqrt[a]*x)/Sqrt[a*x^2 + b*x^3]])/a^(
3/2)

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Rubi [A]  time = 0.098892, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{2 x}{a \sqrt{a x^2+b x^3}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right )}{a^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a*x^2 + b*x^3)^(3/2),x]

[Out]

(2*x)/(a*Sqrt[a*x^2 + b*x^3]) - (2*ArcTanh[(Sqrt[a]*x)/Sqrt[a*x^2 + b*x^3]])/a^(
3/2)

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Rubi in Sympy [A]  time = 9.86185, size = 46, normalized size = 0.88 \[ \frac{2 x}{a \sqrt{a x^{2} + b x^{3}}} - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{a} x}{\sqrt{a x^{2} + b x^{3}}} \right )}}{a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(b*x**3+a*x**2)**(3/2),x)

[Out]

2*x/(a*sqrt(a*x**2 + b*x**3)) - 2*atanh(sqrt(a)*x/sqrt(a*x**2 + b*x**3))/a**(3/2
)

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Mathematica [A]  time = 0.0314246, size = 54, normalized size = 1.04 \[ \frac{2 x \left (\sqrt{a}-\sqrt{a+b x} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )\right )}{a^{3/2} \sqrt{x^2 (a+b x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(a*x^2 + b*x^3)^(3/2),x]

[Out]

(2*x*(Sqrt[a] - Sqrt[a + b*x]*ArcTanh[Sqrt[a + b*x]/Sqrt[a]]))/(a^(3/2)*Sqrt[x^2
*(a + b*x)])

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Maple [A]  time = 0.007, size = 54, normalized size = 1. \[ -2\,{\frac{{x}^{3} \left ( bx+a \right ) }{ \left ( b{x}^{3}+a{x}^{2} \right ) ^{3/2}{a}^{5/2}} \left ({\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) a\sqrt{bx+a}-{a}^{3/2} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(b*x^3+a*x^2)^(3/2),x)

[Out]

-2*x^3*(b*x+a)*(arctanh((b*x+a)^(1/2)/a^(1/2))*a*(b*x+a)^(1/2)-a^(3/2))/(b*x^3+a
*x^2)^(3/2)/a^(5/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(b*x^3 + a*x^2)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.231069, size = 1, normalized size = 0.02 \[ \left [\frac{{\left (b x^{2} + a x\right )} \sqrt{a} \log \left (\frac{{\left (b x^{2} + 2 \, a x\right )} \sqrt{a} - 2 \, \sqrt{b x^{3} + a x^{2}} a}{x^{2}}\right ) + 2 \, \sqrt{b x^{3} + a x^{2}} a}{a^{2} b x^{2} + a^{3} x}, -\frac{2 \,{\left ({\left (b x^{2} + a x\right )} \sqrt{-a} \arctan \left (\frac{a x}{\sqrt{b x^{3} + a x^{2}} \sqrt{-a}}\right ) - \sqrt{b x^{3} + a x^{2}} a\right )}}{a^{2} b x^{2} + a^{3} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(b*x^3 + a*x^2)^(3/2),x, algorithm="fricas")

[Out]

[((b*x^2 + a*x)*sqrt(a)*log(((b*x^2 + 2*a*x)*sqrt(a) - 2*sqrt(b*x^3 + a*x^2)*a)/
x^2) + 2*sqrt(b*x^3 + a*x^2)*a)/(a^2*b*x^2 + a^3*x), -2*((b*x^2 + a*x)*sqrt(-a)*
arctan(a*x/(sqrt(b*x^3 + a*x^2)*sqrt(-a))) - sqrt(b*x^3 + a*x^2)*a)/(a^2*b*x^2 +
a^3*x)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(b*x**3+a*x**2)**(3/2),x)

[Out]

Integral(x**2/(x**2*(a + b*x))**(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(x^2/(b*x^3 + a*x^2)^(3/2),x, algorithm="giac")

[Out]

Exception raised: TypeError