3.283 \(\int \frac{x^2}{\left (b x^2+c x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=51 \[ \frac{x}{b \sqrt{b x^2+c x^4}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{b^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.1007, antiderivative size = 51, 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 \[ \frac{x}{b \sqrt{b x^2+c x^4}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{b^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(b*x^2 + c*x^4)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 13.8143, size = 42, normalized size = 0.82 \[ \frac{x}{b \sqrt{b x^{2} + c x^{4}}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{b x^{2} + c x^{4}}} \right )}}{b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0590084, size = 75, normalized size = 1.47 \[ \frac{x \left (\log (x) \sqrt{b+c x^2}-\sqrt{b+c x^2} \log \left (\sqrt{b} \sqrt{b+c x^2}+b\right )+\sqrt{b}\right )}{b^{3/2} \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(b*x^2 + c*x^4)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.008, size = 67, normalized size = 1.3 \[ -{{x}^{3} \left ( c{x}^{2}+b \right ) \left ( \ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ) b\sqrt{c{x}^{2}+b}-{b}^{{\frac{3}{2}}} \right ) \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{2}}}{b}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(c*x^4+b*x^2)^(3/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.275976, size = 1, normalized size = 0.02 \[ \left [\frac{{\left (c x^{3} + b x\right )} \sqrt{b} \log \left (-\frac{{\left (c x^{3} + 2 \, b x\right )} \sqrt{b} - 2 \, \sqrt{c x^{4} + b x^{2}} b}{x^{3}}\right ) + 2 \, \sqrt{c x^{4} + b x^{2}} b}{2 \,{\left (b^{2} c x^{3} + b^{3} x\right )}}, \frac{{\left (c x^{3} + b x\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{c x^{4} + b x^{2}}}\right ) + \sqrt{c x^{4} + b x^{2}} b}{b^{2} c x^{3} + b^{3} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*((c*x^3 + b*x)*sqrt(b)*log(-((c*x^3 + 2*b*x)*sqrt(b) - 2*sqrt(c*x^4 + b*x^2
)*b)/x^3) + 2*sqrt(c*x^4 + b*x^2)*b)/(b^2*c*x^3 + b^3*x), ((c*x^3 + b*x)*sqrt(-b
)*arctan(sqrt(-b)*x/sqrt(c*x^4 + b*x^2)) + sqrt(c*x^4 + b*x^2)*b)/(b^2*c*x^3 + b
^3*x)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**2/(x**2*(b + c*x**2))**(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError