3.776 \(\int \frac{1}{(c x)^{11/3} \left (a+b x^2\right )^{2/3}} \, dx\)

Optimal. Leaf size=57 \[ \frac{9 \left (a+b x^2\right )^{4/3}}{8 a^2 c (c x)^{8/3}}-\frac{3 \sqrt [3]{a+b x^2}}{2 a c (c x)^{8/3}} \]

[Out]

(-3*(a + b*x^2)^(1/3))/(2*a*c*(c*x)^(8/3)) + (9*(a + b*x^2)^(4/3))/(8*a^2*c*(c*x
)^(8/3))

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Rubi [A]  time = 0.0575739, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{9 \left (a+b x^2\right )^{4/3}}{8 a^2 c (c x)^{8/3}}-\frac{3 \sqrt [3]{a+b x^2}}{2 a c (c x)^{8/3}} \]

Antiderivative was successfully verified.

[In]  Int[1/((c*x)^(11/3)*(a + b*x^2)^(2/3)),x]

[Out]

(-3*(a + b*x^2)^(1/3))/(2*a*c*(c*x)^(8/3)) + (9*(a + b*x^2)^(4/3))/(8*a^2*c*(c*x
)^(8/3))

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Rubi in Sympy [A]  time = 6.6536, size = 48, normalized size = 0.84 \[ - \frac{3 \sqrt [3]{a + b x^{2}}}{2 a c \left (c x\right )^{\frac{8}{3}}} + \frac{9 \left (a + b x^{2}\right )^{\frac{4}{3}}}{8 a^{2} c \left (c x\right )^{\frac{8}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c*x)**(11/3)/(b*x**2+a)**(2/3),x)

[Out]

-3*(a + b*x**2)**(1/3)/(2*a*c*(c*x)**(8/3)) + 9*(a + b*x**2)**(4/3)/(8*a**2*c*(c
*x)**(8/3))

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Mathematica [A]  time = 0.0372911, size = 34, normalized size = 0.6 \[ -\frac{3 x \left (a-3 b x^2\right ) \sqrt [3]{a+b x^2}}{8 a^2 (c x)^{11/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((c*x)^(11/3)*(a + b*x^2)^(2/3)),x]

[Out]

(-3*x*(a - 3*b*x^2)*(a + b*x^2)^(1/3))/(8*a^2*(c*x)^(11/3))

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Maple [A]  time = 0.007, size = 29, normalized size = 0.5 \[ -{\frac{3\,x \left ( -3\,b{x}^{2}+a \right ) }{8\,{a}^{2}}\sqrt [3]{b{x}^{2}+a} \left ( cx \right ) ^{-{\frac{11}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x)^(11/3)/(b*x^2+a)^(2/3),x)

[Out]

-3/8*x*(b*x^2+a)^(1/3)*(-3*b*x^2+a)/a^2/(c*x)^(11/3)

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Maxima [A]  time = 1.39331, size = 51, normalized size = 0.89 \[ \frac{3 \,{\left (\frac{4 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} b}{x^{\frac{2}{3}}} - \frac{{\left (b x^{2} + a\right )}^{\frac{4}{3}}}{x^{\frac{8}{3}}}\right )}}{8 \, a^{2} c^{\frac{11}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(2/3)*(c*x)^(11/3)),x, algorithm="maxima")

[Out]

3/8*(4*(b*x^2 + a)^(1/3)*b/x^(2/3) - (b*x^2 + a)^(4/3)/x^(8/3))/(a^2*c^(11/3))

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Fricas [A]  time = 0.229495, size = 47, normalized size = 0.82 \[ \frac{3 \,{\left (3 \, b x^{2} - a\right )}{\left (b x^{2} + a\right )}^{\frac{1}{3}} \left (c x\right )^{\frac{1}{3}}}{8 \, a^{2} c^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(2/3)*(c*x)^(11/3)),x, algorithm="fricas")

[Out]

3/8*(3*b*x^2 - a)*(b*x^2 + a)^(1/3)*(c*x)^(1/3)/(a^2*c^4*x^3)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c*x)**(11/3)/(b*x**2+a)**(2/3),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{2}{3}} \left (c x\right )^{\frac{11}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(2/3)*(c*x)^(11/3)),x, algorithm="giac")

[Out]

integrate(1/((b*x^2 + a)^(2/3)*(c*x)^(11/3)), x)