3.760 \(\int \frac{(a+b x^2)^{4/3}}{(c x)^{23/3}} \, dx\)

Optimal. Leaf size=57 \[ \frac{9 \left (a+b x^2\right )^{10/3}}{140 a^2 c (c x)^{20/3}}-\frac{3 \left (a+b x^2\right )^{7/3}}{14 a c (c x)^{20/3}} \]

[Out]

(-3*(a + b*x^2)^(7/3))/(14*a*c*(c*x)^(20/3)) + (9*(a + b*x^2)^(10/3))/(140*a^2*c*(c*x)^(20/3))

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Rubi [A]  time = 0.0148183, antiderivative size = 57, normalized size of antiderivative = 1., 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} \[ \frac{9 \left (a+b x^2\right )^{10/3}}{140 a^2 c (c x)^{20/3}}-\frac{3 \left (a+b x^2\right )^{7/3}}{14 a c (c x)^{20/3}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(4/3)/(c*x)^(23/3),x]

[Out]

(-3*(a + b*x^2)^(7/3))/(14*a*c*(c*x)^(20/3)) + (9*(a + b*x^2)^(10/3))/(140*a^2*c*(c*x)^(20/3))

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]

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]

Rubi steps

\begin{align*} \int \frac{\left (a+b x^2\right )^{4/3}}{(c x)^{23/3}} \, dx &=-\frac{3 \left (a+b x^2\right )^{7/3}}{14 a c (c x)^{20/3}}-\frac{3 \int \frac{\left (a+b x^2\right )^{7/3}}{(c x)^{23/3}} \, dx}{7 a}\\ &=-\frac{3 \left (a+b x^2\right )^{7/3}}{14 a c (c x)^{20/3}}+\frac{9 \left (a+b x^2\right )^{10/3}}{140 a^2 c (c x)^{20/3}}\\ \end{align*}

Mathematica [A]  time = 0.0179027, size = 41, normalized size = 0.72 \[ \frac{3 \sqrt [3]{c x} \left (a+b x^2\right )^{7/3} \left (3 b x^2-7 a\right )}{140 a^2 c^8 x^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(4/3)/(c*x)^(23/3),x]

[Out]

(3*(c*x)^(1/3)*(a + b*x^2)^(7/3)*(-7*a + 3*b*x^2))/(140*a^2*c^8*x^7)

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Maple [A]  time = 0.004, size = 31, normalized size = 0.5 \begin{align*} -{\frac{3\,x \left ( -3\,b{x}^{2}+7\,a \right ) }{140\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{3}}} \left ( cx \right ) ^{-{\frac{23}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(4/3)/(c*x)^(23/3),x)

[Out]

-3/140*x*(b*x^2+a)^(7/3)*(-3*b*x^2+7*a)/a^2/(c*x)^(23/3)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{2} + a\right )}^{\frac{4}{3}}}{\left (c x\right )^{\frac{23}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(4/3)/(c*x)^(23/3),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^(4/3)/(c*x)^(23/3), x)

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Fricas [A]  time = 2.11848, size = 132, normalized size = 2.32 \begin{align*} \frac{3 \,{\left (3 \, b^{3} x^{6} - a b^{2} x^{4} - 11 \, a^{2} b x^{2} - 7 \, a^{3}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{3}} \left (c x\right )^{\frac{1}{3}}}{140 \, a^{2} c^{8} x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(4/3)/(c*x)^(23/3),x, algorithm="fricas")

[Out]

3/140*(3*b^3*x^6 - a*b^2*x^4 - 11*a^2*b*x^2 - 7*a^3)*(b*x^2 + a)^(1/3)*(c*x)^(1/3)/(a^2*c^8*x^7)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(4/3)/(c*x)**(23/3),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{2} + a\right )}^{\frac{4}{3}}}{\left (c x\right )^{\frac{23}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(4/3)/(c*x)^(23/3),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^(4/3)/(c*x)^(23/3), x)