Optimal. Leaf size=74 \[ \frac{3 b x^4 \left (\frac{b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{4}{3};\frac{7}{3};-\frac{b x^3}{a}\right )}{4 \left (a+b x^3\right )^{2/3}}+\frac{x \left (a-b x^3\right )}{\left (a+b x^3\right )^{2/3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0339424, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {413, 12, 365, 364} \[ \frac{3 b x^4 \left (\frac{b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{4}{3};\frac{7}{3};-\frac{b x^3}{a}\right )}{4 \left (a+b x^3\right )^{2/3}}+\frac{x \left (a-b x^3\right )}{\left (a+b x^3\right )^{2/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 413
Rule 12
Rule 365
Rule 364
Rubi steps
\begin{align*} \int \frac{\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{5/3}} \, dx &=\frac{x \left (a-b x^3\right )}{\left (a+b x^3\right )^{2/3}}+\frac{\int \frac{6 a b^2 x^3}{\left (a+b x^3\right )^{2/3}} \, dx}{2 a b}\\ &=\frac{x \left (a-b x^3\right )}{\left (a+b x^3\right )^{2/3}}+(3 b) \int \frac{x^3}{\left (a+b x^3\right )^{2/3}} \, dx\\ &=\frac{x \left (a-b x^3\right )}{\left (a+b x^3\right )^{2/3}}+\frac{\left (3 b \left (1+\frac{b x^3}{a}\right )^{2/3}\right ) \int \frac{x^3}{\left (1+\frac{b x^3}{a}\right )^{2/3}} \, dx}{\left (a+b x^3\right )^{2/3}}\\ &=\frac{x \left (a-b x^3\right )}{\left (a+b x^3\right )^{2/3}}+\frac{3 b x^4 \left (1+\frac{b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{4}{3};\frac{7}{3};-\frac{b x^3}{a}\right )}{4 \left (a+b x^3\right )^{2/3}}\\ \end{align*}
Mathematica [A] time = 0.0375889, size = 62, normalized size = 0.84 \[ \frac{-3 a x \left (\frac{b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right )+5 a x+b x^4}{2 \left (a+b x^3\right )^{2/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.421, size = 0, normalized size = 0. \begin{align*} \int{ \left ( -b{x}^{3}+a \right ) ^{2} \left ( b{x}^{3}+a \right ) ^{-{\frac{5}{3}}}}\, dx \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^{3} - a\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac{5}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b^{2} x^{6} - 2 \, a b x^{3} + a^{2}\right )}{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{b^{2} x^{6} + 2 \, a b x^{3} + a^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (- a + b x^{3}\right )^{2}}{\left (a + b x^{3}\right )^{\frac{5}{3}}}\, dx \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^{3} - a\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac{5}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]