3.278 \(\int \frac{1}{x \left (b x^2+c x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=74 \[ \frac{8 c \sqrt{b x^2+c x^4}}{3 b^3 x^2}-\frac{4 \sqrt{b x^2+c x^4}}{3 b^2 x^4}+\frac{1}{b x^2 \sqrt{b x^2+c x^4}} \]

[Out]

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

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Rubi [A]  time = 0.215646, antiderivative size = 74, 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 \[ \frac{8 c \sqrt{b x^2+c x^4}}{3 b^3 x^2}-\frac{4 \sqrt{b x^2+c x^4}}{3 b^2 x^4}+\frac{1}{b x^2 \sqrt{b x^2+c x^4}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(b*x^2 + c*x^4)^(3/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 21.7421, size = 68, normalized size = 0.92 \[ \frac{1}{b x^{2} \sqrt{b x^{2} + c x^{4}}} - \frac{4 \sqrt{b x^{2} + c x^{4}}}{3 b^{2} x^{4}} + \frac{8 c \sqrt{b x^{2} + c x^{4}}}{3 b^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(b*x**2*sqrt(b*x**2 + c*x**4)) - 4*sqrt(b*x**2 + c*x**4)/(3*b**2*x**4) + 8*c*s
qrt(b*x**2 + c*x**4)/(3*b**3*x**2)

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Mathematica [A]  time = 0.0337585, size = 46, normalized size = 0.62 \[ \frac{-b^2+4 b c x^2+8 c^2 x^4}{3 b^3 x^2 \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(b*x^2 + c*x^4)^(3/2)),x]

[Out]

(-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 = 0.007, size = 45, normalized size = 0.6 \[ -{\frac{ \left ( c{x}^{2}+b \right ) \left ( -8\,{c}^{2}{x}^{4}-4\,bc{x}^{2}+{b}^{2} \right ) }{3\,{b}^{3}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(c*x^4+b*x^2)^(3/2),x)

[Out]

-1/3*(c*x^2+b)*(-8*c^2*x^4-4*b*c*x^2+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(1/((c*x^4 + b*x^2)^(3/2)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.268223, size = 73, normalized size = 0.99 \[ \frac{{\left (8 \, c^{2} x^{4} + 4 \, b c x^{2} - b^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{3 \,{\left (b^{3} c x^{6} + b^{4} x^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + b*x^2)^(3/2)*x),x, algorithm="fricas")

[Out]

1/3*(8*c^2*x^4 + 4*b*c*x^2 - b^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. \[ \int \frac{1}{x \left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x*(x**2*(b + c*x**2))**(3/2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + b*x^2)^(3/2)*x),x, algorithm="giac")

[Out]

integrate(1/((c*x^4 + b*x^2)^(3/2)*x), x)