3.141 \(\int \frac{x^2 \left (A+B x^2\right )}{\sqrt{b x^2+c x^4}} \, dx\)

Optimal. Leaf size=59 \[ \frac{B x \sqrt{b x^2+c x^4}}{3 c}-\frac{\sqrt{b x^2+c x^4} (2 b B-3 A c)}{3 c^2 x} \]

[Out]

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

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Rubi [A]  time = 0.204252, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{B x \sqrt{b x^2+c x^4}}{3 c}-\frac{\sqrt{b x^2+c x^4} (2 b B-3 A c)}{3 c^2 x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 21.5912, size = 49, normalized size = 0.83 \[ \frac{B x \sqrt{b x^{2} + c x^{4}}}{3 c} + \frac{\left (3 A c - 2 B b\right ) \sqrt{b x^{2} + c x^{4}}}{3 c^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(B*x**2+A)/(c*x**4+b*x**2)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0372505, size = 40, normalized size = 0.68 \[ \frac{\sqrt{x^2 \left (b+c x^2\right )} \left (3 A c-2 b B+B c x^2\right )}{3 c^2 x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 42, normalized size = 0.7 \[{\frac{ \left ( c{x}^{2}+b \right ) \left ( Bc{x}^{2}+3\,Ac-2\,Bb \right ) x}{3\,{c}^{2}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(B*x^2+A)/(c*x^4+b*x^2)^(1/2),x)

[Out]

1/3*(c*x^2+b)*(B*c*x^2+3*A*c-2*B*b)*x/c^2/(c*x^4+b*x^2)^(1/2)

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Maxima [A]  time = 1.39403, size = 68, normalized size = 1.15 \[ \frac{\sqrt{c x^{2} + b} A}{c} + \frac{{\left (c^{2} x^{4} - b c x^{2} - 2 \, b^{2}\right )} B}{3 \, \sqrt{c x^{2} + b} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^2/sqrt(c*x^4 + b*x^2),x, algorithm="maxima")

[Out]

sqrt(c*x^2 + b)*A/c + 1/3*(c^2*x^4 - b*c*x^2 - 2*b^2)*B/(sqrt(c*x^2 + b)*c^2)

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Fricas [A]  time = 0.226888, size = 49, normalized size = 0.83 \[ \frac{\sqrt{c x^{4} + b x^{2}}{\left (B c x^{2} - 2 \, B b + 3 \, A c\right )}}{3 \, c^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*sqrt(c*x^4 + b*x^2)*(B*c*x^2 - 2*B*b + 3*A*c)/(c^2*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(B*x**2+A)/(c*x**4+b*x**2)**(1/2),x)

[Out]

Integral(x**2*(A + B*x**2)/sqrt(x**2*(b + c*x**2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{2} + A\right )} x^{2}}{\sqrt{c x^{4} + b x^{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((B*x^2 + A)*x^2/sqrt(c*x^4 + b*x^2), x)