3.744 \(\int \frac{\sqrt [3]{a+b x^2}}{(c x)^{23/3}} \, dx\)

Optimal. Leaf size=85 \[ -\frac{27 \left (a+b x^2\right )^{10/3}}{280 a^3 c (c x)^{20/3}}+\frac{9 \left (a+b x^2\right )^{7/3}}{28 a^2 c (c x)^{20/3}}-\frac{3 \left (a+b x^2\right )^{4/3}}{8 a c (c x)^{20/3}} \]

[Out]

(-3*(a + b*x^2)^(4/3))/(8*a*c*(c*x)^(20/3)) + (9*(a + b*x^2)^(7/3))/(28*a^2*c*(c
*x)^(20/3)) - (27*(a + b*x^2)^(10/3))/(280*a^3*c*(c*x)^(20/3))

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Rubi [A]  time = 0.0897418, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{27 \left (a+b x^2\right )^{10/3}}{280 a^3 c (c x)^{20/3}}+\frac{9 \left (a+b x^2\right )^{7/3}}{28 a^2 c (c x)^{20/3}}-\frac{3 \left (a+b x^2\right )^{4/3}}{8 a c (c x)^{20/3}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(1/3)/(c*x)^(23/3),x]

[Out]

(-3*(a + b*x^2)^(4/3))/(8*a*c*(c*x)^(20/3)) + (9*(a + b*x^2)^(7/3))/(28*a^2*c*(c
*x)^(20/3)) - (27*(a + b*x^2)^(10/3))/(280*a^3*c*(c*x)^(20/3))

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Rubi in Sympy [A]  time = 10.6772, size = 73, normalized size = 0.86 \[ - \frac{3 \left (a + b x^{2}\right )^{\frac{4}{3}}}{8 a c \left (c x\right )^{\frac{20}{3}}} + \frac{9 \left (a + b x^{2}\right )^{\frac{7}{3}}}{28 a^{2} c \left (c x\right )^{\frac{20}{3}}} - \frac{27 \left (a + b x^{2}\right )^{\frac{10}{3}}}{280 a^{3} c \left (c x\right )^{\frac{20}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(1/3)/(c*x)**(23/3),x)

[Out]

-3*(a + b*x**2)**(4/3)/(8*a*c*(c*x)**(20/3)) + 9*(a + b*x**2)**(7/3)/(28*a**2*c*
(c*x)**(20/3)) - 27*(a + b*x**2)**(10/3)/(280*a**3*c*(c*x)**(20/3))

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Mathematica [A]  time = 0.0486502, size = 63, normalized size = 0.74 \[ -\frac{3 \sqrt [3]{c x} \sqrt [3]{a+b x^2} \left (14 a^3+2 a^2 b x^2-3 a b^2 x^4+9 b^3 x^6\right )}{280 a^3 c^8 x^7} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^(1/3)/(c*x)^(23/3),x]

[Out]

(-3*(c*x)^(1/3)*(a + b*x^2)^(1/3)*(14*a^3 + 2*a^2*b*x^2 - 3*a*b^2*x^4 + 9*b^3*x^
6))/(280*a^3*c^8*x^7)

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Maple [A]  time = 0.009, size = 42, normalized size = 0.5 \[ -{\frac{3\,x \left ( 9\,{b}^{2}{x}^{4}-12\,ab{x}^{2}+14\,{a}^{2} \right ) }{280\,{a}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{4}{3}}} \left ( cx \right ) ^{-{\frac{23}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(1/3)/(c*x)^(23/3),x)

[Out]

-3/280*x*(b*x^2+a)^(4/3)*(9*b^2*x^4-12*a*b*x^2+14*a^2)/a^3/(c*x)^(23/3)

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Maxima [A]  time = 1.40159, size = 74, normalized size = 0.87 \[ -\frac{3 \,{\left (\frac{35 \,{\left (b x^{2} + a\right )}^{\frac{4}{3}} b^{2}}{x^{\frac{8}{3}}} - \frac{40 \,{\left (b x^{2} + a\right )}^{\frac{7}{3}} b}{x^{\frac{14}{3}}} + \frac{14 \,{\left (b x^{2} + a\right )}^{\frac{10}{3}}}{x^{\frac{20}{3}}}\right )}}{280 \, a^{3} c^{\frac{23}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(1/3)/(c*x)^(23/3),x, algorithm="maxima")

[Out]

-3/280*(35*(b*x^2 + a)^(4/3)*b^2/x^(8/3) - 40*(b*x^2 + a)^(7/3)*b/x^(14/3) + 14*
(b*x^2 + a)^(10/3)/x^(20/3))/(a^3*c^(23/3))

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Fricas [A]  time = 0.230776, size = 77, normalized size = 0.91 \[ -\frac{3 \,{\left (9 \, b^{3} x^{6} - 3 \, a b^{2} x^{4} + 2 \, a^{2} b x^{2} + 14 \, a^{3}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{3}} \left (c x\right )^{\frac{1}{3}}}{280 \, a^{3} c^{8} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(1/3)/(c*x)^(23/3),x, algorithm="fricas")

[Out]

-3/280*(9*b^3*x^6 - 3*a*b^2*x^4 + 2*a^2*b*x^2 + 14*a^3)*(b*x^2 + a)^(1/3)*(c*x)^
(1/3)/(a^3*c^8*x^7)

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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)**(23/3),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{\frac{1}{3}}}{\left (c x\right )^{\frac{23}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(1/3)/(c*x)^(23/3),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^(1/3)/(c*x)^(23/3), x)