3.776 \(\int \frac {1}{(c x)^{11/3} (a+b x^2)^{2/3}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {273, 264} \[ \frac {9 \left (a+b x^2\right )^{4/3}}{8 a^2 c (c x)^{8/3}}-\frac {3 \sqrt [3]{a+b x^2}}{2 a c (c x)^{8/3}} \]

Antiderivative was successfully verified.

[In]

Int[1/((c*x)^(11/3)*(a + b*x^2)^(2/3)),x]

[Out]

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

Rule 264

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]

Rule 273

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

Rubi steps

\begin {align*} \int \frac {1}{(c x)^{11/3} \left (a+b x^2\right )^{2/3}} \, dx &=-\frac {3 \sqrt [3]{a+b x^2}}{2 a c (c x)^{8/3}}-\frac {3 \int \frac {\sqrt [3]{a+b x^2}}{(c x)^{11/3}} \, dx}{a}\\ &=-\frac {3 \sqrt [3]{a+b x^2}}{2 a c (c x)^{8/3}}+\frac {9 \left (a+b x^2\right )^{4/3}}{8 a^2 c (c x)^{8/3}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 34, normalized size = 0.60 \[ -\frac {3 x \left (a-3 b x^2\right ) \sqrt [3]{a+b x^2}}{8 a^2 (c x)^{11/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((c*x)^(11/3)*(a + b*x^2)^(2/3)),x]

[Out]

(-3*x*(a - 3*b*x^2)*(a + b*x^2)^(1/3))/(8*a^2*(c*x)^(11/3))

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fricas [A]  time = 1.14, size = 35, normalized size = 0.61 \[ \frac {3 \, {\left (3 \, b x^{2} - a\right )} {\left (b x^{2} + a\right )}^{\frac {1}{3}} \left (c x\right )^{\frac {1}{3}}}{8 \, a^{2} c^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)^(11/3)/(b*x^2+a)^(2/3),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {2}{3}} \left (c x\right )^{\frac {11}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)^(11/3)/(b*x^2+a)^(2/3),x, algorithm="giac")

[Out]

integrate(1/((b*x^2 + a)^(2/3)*(c*x)^(11/3)), x)

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maple [A]  time = 0.00, size = 29, normalized size = 0.51 \[ -\frac {3 \left (b \,x^{2}+a \right )^{\frac {1}{3}} \left (-3 b \,x^{2}+a \right ) x}{8 \left (c x \right )^{\frac {11}{3}} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*x)^(11/3)/(b*x^2+a)^(2/3),x)

[Out]

-3/8*x*(b*x^2+a)^(1/3)*(-3*b*x^2+a)/a^2/(c*x)^(11/3)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {2}{3}} \left (c x\right )^{\frac {11}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)^(11/3)/(b*x^2+a)^(2/3),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + a)^(2/3)*(c*x)^(11/3)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\left (c\,x\right )}^{11/3}\,{\left (b\,x^2+a\right )}^{2/3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((c*x)^(11/3)*(a + b*x^2)^(2/3)),x)

[Out]

int(1/((c*x)^(11/3)*(a + b*x^2)^(2/3)), x)

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sympy [A]  time = 109.45, size = 78, normalized size = 1.37 \[ - \frac {\sqrt [3]{b} \sqrt [3]{\frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {4}{3}\right )}{6 a c^{\frac {11}{3}} x^{2} \Gamma \left (\frac {2}{3}\right )} + \frac {b^{\frac {4}{3}} \sqrt [3]{\frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {4}{3}\right )}{2 a^{2} c^{\frac {11}{3}} \Gamma \left (\frac {2}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)**(11/3)/(b*x**2+a)**(2/3),x)

[Out]

-b**(1/3)*(a/(b*x**2) + 1)**(1/3)*gamma(-4/3)/(6*a*c**(11/3)*x**2*gamma(2/3)) + b**(4/3)*(a/(b*x**2) + 1)**(1/
3)*gamma(-4/3)/(2*a**2*c**(11/3)*gamma(2/3))

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