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

Optimal. Leaf size=66 \[ -\frac{\left (b+2 c x^2\right ) (3 b B-4 A c)}{3 b^3 \sqrt{b x^2+c x^4}}-\frac{A}{3 b x^2 \sqrt{b x^2+c x^4}} \]

[Out]

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

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Rubi [A]  time = 0.310948, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ -\frac{\left (b+2 c x^2\right ) (3 b B-4 A c)}{3 b^3 \sqrt{b x^2+c x^4}}-\frac{A}{3 b x^2 \sqrt{b x^2+c x^4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 19.0681, size = 60, normalized size = 0.91 \[ - \frac{A}{3 b x^{2} \sqrt{b x^{2} + c x^{4}}} + \frac{\left (2 b + 4 c x^{2}\right ) \left (4 A c - 3 B b\right )}{6 b^{3} \sqrt{b x^{2} + c x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A/(3*b*x**2*sqrt(b*x**2 + c*x**4)) + (2*b + 4*c*x**2)*(4*A*c - 3*B*b)/(6*b**3*s
qrt(b*x**2 + c*x**4))

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Mathematica [A]  time = 0.0817957, size = 64, normalized size = 0.97 \[ \frac{A \left (-b^2+4 b c x^2+8 c^2 x^4\right )-3 b B x^2 \left (b+2 c x^2\right )}{3 b^3 x^2 \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

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

[Out]

(-3*b*B*x^2*(b + 2*c*x^2) + A*(-b^2 + 4*b*c*x^2 + 8*c^2*x^4))/(3*b^3*x^2*Sqrt[x^
2*(b + c*x^2)])

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Maple [A]  time = 0.009, size = 66, normalized size = 1. \[ -{\frac{ \left ( c{x}^{2}+b \right ) \left ( -8\,A{c}^{2}{x}^{4}+6\,B{x}^{4}bc-4\,Abc{x}^{2}+3\,B{b}^{2}{x}^{2}+{b}^{2}A \right ) }{3\,{b}^{3}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(c*x^2+b)*(-8*A*c^2*x^4+6*B*b*c*x^4-4*A*b*c*x^2+3*B*b^2*x^2+A*b^2)/b^3/(c*x
^4+b*x^2)^(3/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.234245, size = 97, normalized size = 1.47 \[ -\frac{{\left (2 \,{\left (3 \, B b c - 4 \, A c^{2}\right )} x^{4} + A b^{2} +{\left (3 \, B b^{2} - 4 \, A b c\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{3 \,{\left (b^{3} c x^{6} + b^{4} x^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(2*(3*B*b*c - 4*A*c^2)*x^4 + A*b^2 + (3*B*b^2 - 4*A*b*c)*x^2)*sqrt(c*x^4 +
b*x^2)/(b^3*c*x^6 + b^4*x^4)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{B x^{2} + A}{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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