3.8.75 \(\int \frac {1}{(c x)^{5/3} (a+b x^2)^{2/3}} \, dx\) [775]

Optimal. Leaf size=28 \[ -\frac {3 \sqrt [3]{a+b x^2}}{2 a c (c x)^{2/3}} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {270} \begin {gather*} -\frac {3 \sqrt [3]{a+b x^2}}{2 a c (c x)^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((c*x)^(5/3)*(a + b*x^2)^(2/3)),x]

[Out]

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

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {1}{(c x)^{5/3} \left (a+b x^2\right )^{2/3}} \, dx &=-\frac {3 \sqrt [3]{a+b x^2}}{2 a c (c x)^{2/3}}\\ \end {align*}

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Mathematica [A]
time = 0.81, size = 26, normalized size = 0.93 \begin {gather*} -\frac {3 x \sqrt [3]{a+b x^2}}{2 a (c x)^{5/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((c*x)^(5/3)*(a + b*x^2)^(2/3)),x]

[Out]

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

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Maple [A]
time = 0.04, size = 21, normalized size = 0.75

method	result	size
gosper	\(-\frac {3 x \left (b \,x^{2}+a \right )^{\frac {1}{3}}}{2 a \left (c x \right )^{\frac {5}{3}}}\)	\(21\)
risch	\(-\frac {3 \left (b \,x^{2}+a \right )^{\frac {1}{3}}}{2 a c \left (c x \right )^{\frac {2}{3}}}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*x)^(5/3)/(b*x^2+a)^(2/3),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.30, size = 35, normalized size = 1.25 \begin {gather*} -\frac {3 \, {\left (b c^{\frac {1}{3}} x^{3} + a c^{\frac {1}{3}} x\right )}}{2 \, {\left (b x^{2} + a\right )}^{\frac {2}{3}} a c^{2} x^{\frac {5}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)^(5/3)/(b*x^2+a)^(2/3),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 2.22, size = 25, normalized size = 0.89 \begin {gather*} -\frac {3 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} \left (c x\right )^{\frac {1}{3}}}{2 \, a c^{2} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)^(5/3)/(b*x^2+a)^(2/3),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 2.13, size = 36, normalized size = 1.29 \begin {gather*} \frac {\sqrt [3]{b} \sqrt [3]{\frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {1}{3}\right )}{2 a c^{\frac {5}{3}} \Gamma \left (\frac {2}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)**(5/3)/(b*x**2+a)**(2/3),x)

[Out]

b**(1/3)*(a/(b*x**2) + 1)**(1/3)*gamma(-1/3)/(2*a*c**(5/3)*gamma(2/3))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)^(5/3)/(b*x^2+a)^(2/3),x, algorithm="giac")

[Out]

integrate(1/((b*x^2 + a)^(2/3)*(c*x)^(5/3)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {1}{{\left (c\,x\right )}^{5/3}\,{\left (b\,x^2+a\right )}^{2/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((c*x)^(5/3)*(a + b*x^2)^(2/3)),x)

[Out]

int(1/((c*x)^(5/3)*(a + b*x^2)^(2/3)), x)

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