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

Optimal. Leaf size=37 \[ -\frac{A b-x^2 (b B-2 A c)}{b^2 \sqrt{b x^2+c x^4}} \]

[Out]

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

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Rubi [A]  time = 0.197311, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{A b-x^2 (b B-2 A c)}{b^2 \sqrt{b x^2+c x^4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.7607, size = 37, normalized size = 1. \[ - \frac{2 A b + x^{2} \left (4 A c - 2 B b\right )}{2 b^{2} \sqrt{b x^{2} + c x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0582011, size = 37, normalized size = 1. \[ \frac{b B x^2-A \left (b+2 c x^2\right )}{b^2 \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 47, normalized size = 1.3 \[ -{\frac{ \left ( c{x}^{2}+b \right ){x}^{2} \left ( 2\,A{x}^{2}c-Bb{x}^{2}+Ab \right ) }{{b}^{2}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.39912, size = 88, normalized size = 2.38 \[ -A{\left (\frac{2 \, c x^{2}}{\sqrt{c x^{4} + b x^{2}} b^{2}} + \frac{1}{\sqrt{c x^{4} + b x^{2}} b}\right )} + \frac{B x^{2}}{\sqrt{c x^{4} + b x^{2}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.227884, size = 66, normalized size = 1.78 \[ \frac{\sqrt{c x^{4} + b x^{2}}{\left ({\left (B b - 2 \, A c\right )} x^{2} - A b\right )}}{b^{2} c x^{4} + b^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.232652, size = 49, normalized size = 1.32 \[ \frac{\frac{{\left (B b - 2 \, A c\right )} x^{2}}{b^{2}} - \frac{A}{b}}{\sqrt{c x^{4} + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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