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

Optimal. Leaf size=58 \[ \frac{\sqrt{a x^2+b x^4}}{2 b}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a x^2+b x^4}}\right )}{2 b^{3/2}} \]

[Out]

Sqrt[a*x^2 + b*x^4]/(2*b) - (a*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a*x^2 + b*x^4]])/(2*b^
(3/2))

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Rubi [A]  time = 0.153705, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{\sqrt{a x^2+b x^4}}{2 b}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a x^2+b x^4}}\right )}{2 b^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^3/Sqrt[a*x^2 + b*x^4],x]

[Out]

Sqrt[a*x^2 + b*x^4]/(2*b) - (a*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a*x^2 + b*x^4]])/(2*b^
(3/2))

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Rubi in Sympy [A]  time = 13.4634, size = 48, normalized size = 0.83 \[ - \frac{a \operatorname{atanh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a x^{2} + b x^{4}}} \right )}}{2 b^{\frac{3}{2}}} + \frac{\sqrt{a x^{2} + b x^{4}}}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(b*x**4+a*x**2)**(1/2),x)

[Out]

-a*atanh(sqrt(b)*x**2/sqrt(a*x**2 + b*x**4))/(2*b**(3/2)) + sqrt(a*x**2 + b*x**4
)/(2*b)

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Mathematica [A]  time = 0.0618191, size = 76, normalized size = 1.31 \[ \frac{x \left (\sqrt{b} x \left (a+b x^2\right )-a \sqrt{a+b x^2} \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )\right )}{2 b^{3/2} \sqrt{x^2 \left (a+b x^2\right )}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 64, normalized size = 1.1 \[ -{\frac{x}{2}\sqrt{b{x}^{2}+a} \left ( -x\sqrt{b{x}^{2}+a}{b}^{{\frac{3}{2}}}+a\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) b \right ){\frac{1}{\sqrt{b{x}^{4}+a{x}^{2}}}}{b}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(b*x^4+a*x^2)^(1/2),x)

[Out]

-1/2*x*(b*x^2+a)^(1/2)*(-x*(b*x^2+a)^(1/2)*b^(3/2)+a*ln(x*b^(1/2)+(b*x^2+a)^(1/2
))*b)/(b*x^4+a*x^2)^(1/2)/b^(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^3/sqrt(b*x^4 + a*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.276668, size = 1, normalized size = 0.02 \[ \left [\frac{a \sqrt{b} \log \left (-{\left (2 \, b x^{2} + a\right )} \sqrt{b} + 2 \, \sqrt{b x^{4} + a x^{2}} b\right ) + 2 \, \sqrt{b x^{4} + a x^{2}} b}{4 \, b^{2}}, \frac{a \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x^{2}}{\sqrt{b x^{4} + a x^{2}}}\right ) + \sqrt{b x^{4} + a x^{2}} b}{2 \, b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/sqrt(b*x^4 + a*x^2),x, algorithm="fricas")

[Out]

[1/4*(a*sqrt(b)*log(-(2*b*x^2 + a)*sqrt(b) + 2*sqrt(b*x^4 + a*x^2)*b) + 2*sqrt(b
*x^4 + a*x^2)*b)/b^2, 1/2*(a*sqrt(-b)*arctan(sqrt(-b)*x^2/sqrt(b*x^4 + a*x^2)) +
sqrt(b*x^4 + a*x^2)*b)/b^2]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(b*x**4+a*x**2)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.296976, size = 80, normalized size = 1.38 \[ \frac{a{\rm ln}\left ({\left | -2 \,{\left (\sqrt{b} x^{2} - \sqrt{b x^{4} + a x^{2}}\right )} \sqrt{b} - a \right |}\right )}{4 \, b^{\frac{3}{2}}} + \frac{\sqrt{b x^{4} + a x^{2}}}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/sqrt(b*x^4 + a*x^2),x, algorithm="giac")

[Out]

1/4*a*ln(abs(-2*(sqrt(b)*x^2 - sqrt(b*x^4 + a*x^2))*sqrt(b) - a))/b^(3/2) + 1/2*
sqrt(b*x^4 + a*x^2)/b