Optimal. Leaf size=32 \[ \frac{3 (a+b x)^{4/3}}{4 b^2}-\frac{3 a \sqrt [3]{a+b x}}{b^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0074868, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {43} \[ \frac{3 (a+b x)^{4/3}}{4 b^2}-\frac{3 a \sqrt [3]{a+b x}}{b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rubi steps
\begin{align*} \int \frac{x}{(a+b x)^{2/3}} \, dx &=\int \left (-\frac{a}{b (a+b x)^{2/3}}+\frac{\sqrt [3]{a+b x}}{b}\right ) \, dx\\ &=-\frac{3 a \sqrt [3]{a+b x}}{b^2}+\frac{3 (a+b x)^{4/3}}{4 b^2}\\ \end{align*}
Mathematica [A] time = 0.0179814, size = 23, normalized size = 0.72 \[ \frac{3 (b x-3 a) \sqrt [3]{a+b x}}{4 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.001, size = 21, normalized size = 0.7 \begin{align*} -{\frac{-3\,bx+9\,a}{4\,{b}^{2}}\sqrt [3]{bx+a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.04654, size = 35, normalized size = 1.09 \begin{align*} \frac{3 \,{\left (b x + a\right )}^{\frac{4}{3}}}{4 \, b^{2}} - \frac{3 \,{\left (b x + a\right )}^{\frac{1}{3}} a}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.48452, size = 50, normalized size = 1.56 \begin{align*} \frac{3 \,{\left (b x + a\right )}^{\frac{1}{3}}{\left (b x - 3 \, a\right )}}{4 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 1.70781, size = 162, normalized size = 5.06 \begin{align*} - \frac{9 a^{\frac{10}{3}} \sqrt [3]{1 + \frac{b x}{a}}}{4 a^{2} b^{2} + 4 a b^{3} x} + \frac{9 a^{\frac{10}{3}}}{4 a^{2} b^{2} + 4 a b^{3} x} - \frac{6 a^{\frac{7}{3}} b x \sqrt [3]{1 + \frac{b x}{a}}}{4 a^{2} b^{2} + 4 a b^{3} x} + \frac{9 a^{\frac{7}{3}} b x}{4 a^{2} b^{2} + 4 a b^{3} x} + \frac{3 a^{\frac{4}{3}} b^{2} x^{2} \sqrt [3]{1 + \frac{b x}{a}}}{4 a^{2} b^{2} + 4 a b^{3} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.10816, size = 31, normalized size = 0.97 \begin{align*} \frac{3 \,{\left ({\left (b x + a\right )}^{\frac{4}{3}} - 4 \,{\left (b x + a\right )}^{\frac{1}{3}} a\right )}}{4 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]