3.778 \(\int \frac{1}{(c x)^{23/3} \left (a+b x^2\right )^{2/3}} \, dx\)

Optimal. Leaf size=113 \[ \frac{243 \left (a+b x^2\right )^{10/3}}{280 a^4 c (c x)^{20/3}}-\frac{81 \left (a+b x^2\right )^{7/3}}{28 a^3 c (c x)^{20/3}}+\frac{27 \left (a+b x^2\right )^{4/3}}{8 a^2 c (c x)^{20/3}}-\frac{3 \sqrt [3]{a+b x^2}}{2 a c (c x)^{20/3}} \]

[Out]

(-3*(a + b*x^2)^(1/3))/(2*a*c*(c*x)^(20/3)) + (27*(a + b*x^2)^(4/3))/(8*a^2*c*(c
*x)^(20/3)) - (81*(a + b*x^2)^(7/3))/(28*a^3*c*(c*x)^(20/3)) + (243*(a + b*x^2)^
(10/3))/(280*a^4*c*(c*x)^(20/3))

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Rubi [A]  time = 0.123032, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{243 \left (a+b x^2\right )^{10/3}}{280 a^4 c (c x)^{20/3}}-\frac{81 \left (a+b x^2\right )^{7/3}}{28 a^3 c (c x)^{20/3}}+\frac{27 \left (a+b x^2\right )^{4/3}}{8 a^2 c (c x)^{20/3}}-\frac{3 \sqrt [3]{a+b x^2}}{2 a c (c x)^{20/3}} \]

Antiderivative was successfully verified.

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

[Out]

(-3*(a + b*x^2)^(1/3))/(2*a*c*(c*x)^(20/3)) + (27*(a + b*x^2)^(4/3))/(8*a^2*c*(c
*x)^(20/3)) - (81*(a + b*x^2)^(7/3))/(28*a^3*c*(c*x)^(20/3)) + (243*(a + b*x^2)^
(10/3))/(280*a^4*c*(c*x)^(20/3))

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Rubi in Sympy [A]  time = 15.3909, size = 99, normalized size = 0.88 \[ - \frac{3 \sqrt [3]{a + b x^{2}}}{2 a c \left (c x\right )^{\frac{20}{3}}} + \frac{27 \left (a + b x^{2}\right )^{\frac{4}{3}}}{8 a^{2} c \left (c x\right )^{\frac{20}{3}}} - \frac{81 \left (a + b x^{2}\right )^{\frac{7}{3}}}{28 a^{3} c \left (c x\right )^{\frac{20}{3}}} + \frac{243 \left (a + b x^{2}\right )^{\frac{10}{3}}}{280 a^{4} c \left (c x\right )^{\frac{20}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*(a + b*x**2)**(1/3)/(2*a*c*(c*x)**(20/3)) + 27*(a + b*x**2)**(4/3)/(8*a**2*c*
(c*x)**(20/3)) - 81*(a + b*x**2)**(7/3)/(28*a**3*c*(c*x)**(20/3)) + 243*(a + b*x
**2)**(10/3)/(280*a**4*c*(c*x)**(20/3))

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Mathematica [A]  time = 0.0557839, size = 63, normalized size = 0.56 \[ \frac{3 \sqrt [3]{c x} \sqrt [3]{a+b x^2} \left (-14 a^3+18 a^2 b x^2-27 a b^2 x^4+81 b^3 x^6\right )}{280 a^4 c^8 x^7} \]

Antiderivative was successfully verified.

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

[Out]

(3*(c*x)^(1/3)*(a + b*x^2)^(1/3)*(-14*a^3 + 18*a^2*b*x^2 - 27*a*b^2*x^4 + 81*b^3
*x^6))/(280*a^4*c^8*x^7)

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Maple [A]  time = 0.01, size = 53, normalized size = 0.5 \[ -{\frac{3\,x \left ( -81\,{b}^{3}{x}^{6}+27\,a{b}^{2}{x}^{4}-18\,{a}^{2}b{x}^{2}+14\,{a}^{3} \right ) }{280\,{a}^{4}}\sqrt [3]{b{x}^{2}+a} \left ( cx \right ) ^{-{\frac{23}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/280*x*(b*x^2+a)^(1/3)*(-81*b^3*x^6+27*a*b^2*x^4-18*a^2*b*x^2+14*a^3)/a^4/(c*x
)^(23/3)

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Maxima [A]  time = 1.36288, size = 86, normalized size = 0.76 \[ \frac{3 \,{\left (81 \, b^{4} x^{9} + 54 \, a b^{3} x^{7} - 9 \, a^{2} b^{2} x^{5} + 4 \, a^{3} b x^{3} - 14 \, a^{4} x\right )}}{280 \,{\left (b x^{2} + a\right )}^{\frac{2}{3}} a^{4} c^{\frac{23}{3}} x^{\frac{23}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/280*(81*b^4*x^9 + 54*a*b^3*x^7 - 9*a^2*b^2*x^5 + 4*a^3*b*x^3 - 14*a^4*x)/((b*x
^2 + a)^(2/3)*a^4*c^(23/3)*x^(23/3))

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Fricas [A]  time = 0.233513, size = 77, normalized size = 0.68 \[ \frac{3 \,{\left (81 \, b^{3} x^{6} - 27 \, a b^{2} x^{4} + 18 \, a^{2} b x^{2} - 14 \, a^{3}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{3}} \left (c x\right )^{\frac{1}{3}}}{280 \, a^{4} c^{8} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/280*(81*b^3*x^6 - 27*a*b^2*x^4 + 18*a^2*b*x^2 - 14*a^3)*(b*x^2 + a)^(1/3)*(c*x
)^(1/3)/(a^4*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(1/(c*x)**(23/3)/(b*x**2+a)**(2/3),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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