3.777 \(\int \frac{1}{(c x)^{17/3} \left (a+b x^2\right )^{2/3}} \, dx\)

Optimal. Leaf size=85 \[ -\frac{27 \left (a+b x^2\right )^{7/3}}{28 a^3 c (c x)^{14/3}}+\frac{9 \left (a+b x^2\right )^{4/3}}{4 a^2 c (c x)^{14/3}}-\frac{3 \sqrt [3]{a+b x^2}}{2 a c (c x)^{14/3}} \]

[Out]

(-3*(a + b*x^2)^(1/3))/(2*a*c*(c*x)^(14/3)) + (9*(a + b*x^2)^(4/3))/(4*a^2*c*(c*
x)^(14/3)) - (27*(a + b*x^2)^(7/3))/(28*a^3*c*(c*x)^(14/3))

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Rubi [A]  time = 0.0894068, 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 \[ -\frac{27 \left (a+b x^2\right )^{7/3}}{28 a^3 c (c x)^{14/3}}+\frac{9 \left (a+b x^2\right )^{4/3}}{4 a^2 c (c x)^{14/3}}-\frac{3 \sqrt [3]{a+b x^2}}{2 a c (c x)^{14/3}} \]

Antiderivative was successfully verified.

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

[Out]

(-3*(a + b*x^2)^(1/3))/(2*a*c*(c*x)^(14/3)) + (9*(a + b*x^2)^(4/3))/(4*a^2*c*(c*
x)^(14/3)) - (27*(a + b*x^2)^(7/3))/(28*a^3*c*(c*x)^(14/3))

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Rubi in Sympy [A]  time = 10.7068, size = 73, normalized size = 0.86 \[ - \frac{3 \sqrt [3]{a + b x^{2}}}{2 a c \left (c x\right )^{\frac{14}{3}}} + \frac{9 \left (a + b x^{2}\right )^{\frac{4}{3}}}{4 a^{2} c \left (c x\right )^{\frac{14}{3}}} - \frac{27 \left (a + b x^{2}\right )^{\frac{7}{3}}}{28 a^{3} c \left (c x\right )^{\frac{14}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*(a + b*x**2)**(1/3)/(2*a*c*(c*x)**(14/3)) + 9*(a + b*x**2)**(4/3)/(4*a**2*c*(
c*x)**(14/3)) - 27*(a + b*x**2)**(7/3)/(28*a**3*c*(c*x)**(14/3))

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Mathematica [A]  time = 0.0490489, size = 52, normalized size = 0.61 \[ -\frac{3 \sqrt [3]{c x} \sqrt [3]{a+b x^2} \left (2 a^2-3 a b x^2+9 b^2 x^4\right )}{28 a^3 c^6 x^5} \]

Antiderivative was successfully verified.

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

[Out]

(-3*(c*x)^(1/3)*(a + b*x^2)^(1/3)*(2*a^2 - 3*a*b*x^2 + 9*b^2*x^4))/(28*a^3*c^6*x
^5)

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Maple [A]  time = 0.007, size = 42, normalized size = 0.5 \[ -{\frac{3\,x \left ( 9\,{b}^{2}{x}^{4}-3\,ab{x}^{2}+2\,{a}^{2} \right ) }{28\,{a}^{3}}\sqrt [3]{b{x}^{2}+a} \left ( cx \right ) ^{-{\frac{17}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/28*x*(b*x^2+a)^(1/3)*(9*b^2*x^4-3*a*b*x^2+2*a^2)/a^3/(c*x)^(17/3)

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Maxima [A]  time = 1.40287, size = 74, normalized size = 0.87 \[ -\frac{3 \,{\left (\frac{14 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} b^{2}}{x^{\frac{2}{3}}} - \frac{7 \,{\left (b x^{2} + a\right )}^{\frac{4}{3}} b}{x^{\frac{8}{3}}} + \frac{2 \,{\left (b x^{2} + a\right )}^{\frac{7}{3}}}{x^{\frac{14}{3}}}\right )}}{28 \, a^{3} c^{\frac{17}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/28*(14*(b*x^2 + a)^(1/3)*b^2/x^(2/3) - 7*(b*x^2 + a)^(4/3)*b/x^(8/3) + 2*(b*x
^2 + a)^(7/3)/x^(14/3))/(a^3*c^(17/3))

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Fricas [A]  time = 0.229017, size = 62, normalized size = 0.73 \[ -\frac{3 \,{\left (9 \, b^{2} x^{4} - 3 \, a b x^{2} + 2 \, a^{2}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{3}} \left (c x\right )^{\frac{1}{3}}}{28 \, a^{3} c^{6} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/28*(9*b^2*x^4 - 3*a*b*x^2 + 2*a^2)*(b*x^2 + a)^(1/3)*(c*x)^(1/3)/(a^3*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(1/(c*x)**(17/3)/(b*x**2+a)**(2/3),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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