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

Optimal. Leaf size=102 \[ -\frac{16 c^2 \sqrt{b x^2+c x^4}}{5 b^4 x^2}+\frac{8 c \sqrt{b x^2+c x^4}}{5 b^3 x^4}-\frac{6 \sqrt{b x^2+c x^4}}{5 b^2 x^6}+\frac{1}{b x^4 \sqrt{b x^2+c x^4}} \]

[Out]

1/(b*x^4*Sqrt[b*x^2 + c*x^4]) - (6*Sqrt[b*x^2 + c*x^4])/(5*b^2*x^6) + (8*c*Sqrt[
b*x^2 + c*x^4])/(5*b^3*x^4) - (16*c^2*Sqrt[b*x^2 + c*x^4])/(5*b^4*x^2)

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Rubi [A]  time = 0.295929, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{16 c^2 \sqrt{b x^2+c x^4}}{5 b^4 x^2}+\frac{8 c \sqrt{b x^2+c x^4}}{5 b^3 x^4}-\frac{6 \sqrt{b x^2+c x^4}}{5 b^2 x^6}+\frac{1}{b x^4 \sqrt{b x^2+c x^4}} \]

Antiderivative was successfully verified.

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

[Out]

1/(b*x^4*Sqrt[b*x^2 + c*x^4]) - (6*Sqrt[b*x^2 + c*x^4])/(5*b^2*x^6) + (8*c*Sqrt[
b*x^2 + c*x^4])/(5*b^3*x^4) - (16*c^2*Sqrt[b*x^2 + c*x^4])/(5*b^4*x^2)

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Rubi in Sympy [A]  time = 29.9222, size = 95, normalized size = 0.93 \[ \frac{1}{b x^{4} \sqrt{b x^{2} + c x^{4}}} - \frac{6 \sqrt{b x^{2} + c x^{4}}}{5 b^{2} x^{6}} + \frac{8 c \sqrt{b x^{2} + c x^{4}}}{5 b^{3} x^{4}} - \frac{16 c^{2} \sqrt{b x^{2} + c x^{4}}}{5 b^{4} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(b*x**4*sqrt(b*x**2 + c*x**4)) - 6*sqrt(b*x**2 + c*x**4)/(5*b**2*x**6) + 8*c*s
qrt(b*x**2 + c*x**4)/(5*b**3*x**4) - 16*c**2*sqrt(b*x**2 + c*x**4)/(5*b**4*x**2)

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Mathematica [A]  time = 0.0408813, size = 57, normalized size = 0.56 \[ \frac{-b^3+2 b^2 c x^2-8 b c^2 x^4-16 c^3 x^6}{5 b^4 x^4 \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

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

[Out]

(-b^3 + 2*b^2*c*x^2 - 8*b*c^2*x^4 - 16*c^3*x^6)/(5*b^4*x^4*Sqrt[x^2*(b + c*x^2)]
)

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Maple [A]  time = 0.007, size = 59, normalized size = 0.6 \[ -{\frac{ \left ( c{x}^{2}+b \right ) \left ( 16\,{c}^{3}{x}^{6}+8\,b{c}^{2}{x}^{4}-2\,{b}^{2}c{x}^{2}+{b}^{3} \right ) }{5\,{b}^{4}{x}^{2}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5*(c*x^2+b)*(16*c^3*x^6+8*b*c^2*x^4-2*b^2*c*x^2+b^3)/x^2/b^4/(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^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.270514, size = 85, normalized size = 0.83 \[ -\frac{{\left (16 \, c^{3} x^{6} + 8 \, b c^{2} x^{4} - 2 \, b^{2} c x^{2} + b^{3}\right )} \sqrt{c x^{4} + b x^{2}}}{5 \,{\left (b^{4} c x^{8} + b^{5} x^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5*(16*c^3*x^6 + 8*b*c^2*x^4 - 2*b^2*c*x^2 + b^3)*sqrt(c*x^4 + b*x^2)/(b^4*c*x
^8 + b^5*x^6)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{3} \left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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