3.625 \(\int \frac{x^3}{\sqrt{a^2+2 a b x^2+b^2 x^4}} \, dx\)

Optimal. Leaf size=75 \[ \frac{\sqrt{a^2+2 a b x^2+b^2 x^4}}{2 b^2}-\frac{a \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

[Out]

Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]/(2*b^2) - (a*(a + b*x^2)*Log[a + b*x^2])/(2*b^2*
Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])

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Rubi [A]  time = 0.146885, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{\sqrt{a^2+2 a b x^2+b^2 x^4}}{2 b^2}-\frac{a \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

[In]  Int[x^3/Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]

[Out]

Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]/(2*b^2) - (a*(a + b*x^2)*Log[a + b*x^2])/(2*b^2*
Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/((b*x**2+a)**2)**(1/2),x)

[Out]

Integral(x**3/sqrt((a + b*x**2)**2), x)

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Mathematica [A]  time = 0.020955, size = 44, normalized size = 0.59 \[ \frac{\left (a+b x^2\right ) \left (b x^2-a \log \left (a+b x^2\right )\right )}{2 b^2 \sqrt{\left (a+b x^2\right )^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]

[Out]

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

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Maple [A]  time = 0.011, size = 41, normalized size = 0.6 \[ -{\frac{ \left ( b{x}^{2}+a \right ) \left ( -b{x}^{2}+a\ln \left ( b{x}^{2}+a \right ) \right ) }{2\,{b}^{2}}{\frac{1}{\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/((b*x^2+a)^2)^(1/2),x)

[Out]

-1/2*(b*x^2+a)*(-b*x^2+a*ln(b*x^2+a))/((b*x^2+a)^2)^(1/2)/b^2

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Maxima [A]  time = 0.704752, size = 63, normalized size = 0.84 \[ -\frac{a \sqrt{\frac{1}{b^{2}}} \log \left (x^{2} + \frac{a}{b}\right )}{2 \, b} + \frac{\sqrt{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*a*sqrt(b^(-2))*log(x^2 + a/b)/b + 1/2*sqrt(b^2*x^4 + 2*a*b*x^2 + a^2)/b^2

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Fricas [A]  time = 0.257117, size = 30, normalized size = 0.4 \[ \frac{b x^{2} - a \log \left (b x^{2} + a\right )}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(b*x^2 - a*log(b*x^2 + a))/b^2

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Sympy [A]  time = 1.16651, size = 20, normalized size = 0.27 \[ - \frac{a \log{\left (a + b x^{2} \right )}}{2 b^{2}} + \frac{x^{2}}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/((b*x**2+a)**2)**(1/2),x)

[Out]

-a*log(a + b*x**2)/(2*b**2) + x**2/(2*b)

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GIAC/XCAS [A]  time = 0.270255, size = 45, normalized size = 0.6 \[ \frac{1}{2} \,{\left (\frac{x^{2}}{b} - \frac{a{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{b^{2}}\right )}{\rm sign}\left (b x^{2} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(x^2/b - a*ln(abs(b*x^2 + a))/b^2)*sign(b*x^2 + a)