3.743 \(\int \frac{\sqrt [3]{a+b x^2}}{(c x)^{17/3}} \, dx\)

Optimal. Leaf size=57 \[ \frac{9 \left (a+b x^2\right )^{7/3}}{56 a^2 c (c x)^{14/3}}-\frac{3 \left (a+b x^2\right )^{4/3}}{8 a c (c x)^{14/3}} \]

[Out]

(-3*(a + b*x^2)^(4/3))/(8*a*c*(c*x)^(14/3)) + (9*(a + b*x^2)^(7/3))/(56*a^2*c*(c
*x)^(14/3))

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Rubi [A]  time = 0.0581911, 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 \[ \frac{9 \left (a+b x^2\right )^{7/3}}{56 a^2 c (c x)^{14/3}}-\frac{3 \left (a+b x^2\right )^{4/3}}{8 a c (c x)^{14/3}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(1/3)/(c*x)^(17/3),x]

[Out]

(-3*(a + b*x^2)^(4/3))/(8*a*c*(c*x)^(14/3)) + (9*(a + b*x^2)^(7/3))/(56*a^2*c*(c
*x)^(14/3))

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Rubi in Sympy [A]  time = 6.72123, size = 48, normalized size = 0.84 \[ - \frac{3 \left (a + b x^{2}\right )^{\frac{4}{3}}}{8 a c \left (c x\right )^{\frac{14}{3}}} + \frac{9 \left (a + b x^{2}\right )^{\frac{7}{3}}}{56 a^{2} c \left (c x\right )^{\frac{14}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(1/3)/(c*x)**(17/3),x)

[Out]

-3*(a + b*x**2)**(4/3)/(8*a*c*(c*x)**(14/3)) + 9*(a + b*x**2)**(7/3)/(56*a**2*c*
(c*x)**(14/3))

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Mathematica [A]  time = 0.0422864, size = 51, normalized size = 0.89 \[ -\frac{3 \sqrt [3]{c x} \sqrt [3]{a+b x^2} \left (4 a^2+a b x^2-3 b^2 x^4\right )}{56 a^2 c^6 x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^(1/3)/(c*x)^(17/3),x]

[Out]

(-3*(c*x)^(1/3)*(a + b*x^2)^(1/3)*(4*a^2 + a*b*x^2 - 3*b^2*x^4))/(56*a^2*c^6*x^5
)

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Maple [A]  time = 0.006, size = 31, normalized size = 0.5 \[ -{\frac{3\,x \left ( -3\,b{x}^{2}+4\,a \right ) }{56\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{4}{3}}} \left ( cx \right ) ^{-{\frac{17}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(1/3)/(c*x)^(17/3),x)

[Out]

-3/56*x*(b*x^2+a)^(4/3)*(-3*b*x^2+4*a)/a^2/(c*x)^(17/3)

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Maxima [A]  time = 1.39592, size = 51, normalized size = 0.89 \[ \frac{3 \,{\left (\frac{7 \,{\left (b x^{2} + a\right )}^{\frac{4}{3}} b}{x^{\frac{8}{3}}} - \frac{4 \,{\left (b x^{2} + a\right )}^{\frac{7}{3}}}{x^{\frac{14}{3}}}\right )}}{56 \, a^{2} c^{\frac{17}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(1/3)/(c*x)^(17/3),x, algorithm="maxima")

[Out]

3/56*(7*(b*x^2 + a)^(4/3)*b/x^(8/3) - 4*(b*x^2 + a)^(7/3)/x^(14/3))/(a^2*c^(17/3
))

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Fricas [A]  time = 0.229433, size = 62, normalized size = 1.09 \[ \frac{3 \,{\left (3 \, b^{2} x^{4} - a b x^{2} - 4 \, a^{2}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{3}} \left (c x\right )^{\frac{1}{3}}}{56 \, a^{2} c^{6} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(1/3)/(c*x)^(17/3),x, algorithm="fricas")

[Out]

3/56*(3*b^2*x^4 - a*b*x^2 - 4*a^2)*(b*x^2 + a)^(1/3)*(c*x)^(1/3)/(a^2*c^6*x^5)

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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((b*x**2+a)**(1/3)/(c*x)**(17/3),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{\frac{1}{3}}}{\left (c x\right )^{\frac{17}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(1/3)/(c*x)^(17/3),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^(1/3)/(c*x)^(17/3), x)