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]
________________________________________________________________________________________
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]
[Out]
Rule 273
Rule 264
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]