Optimal. Leaf size=93 \[ \frac{15 x \left (\frac{b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{8}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{22 a^2 \left (a+b x^3\right )^{2/3}}+\frac{3 x}{22 \left (a+b x^3\right )^{8/3}}+\frac{2 x \left (a-b x^3\right )}{11 \left (a+b x^3\right )^{11/3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.033415, antiderivative size = 93, 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, 385, 246, 245} \[ \frac{15 x \left (\frac{b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{8}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{22 a^2 \left (a+b x^3\right )^{2/3}}+\frac{3 x}{22 \left (a+b x^3\right )^{8/3}}+\frac{2 x \left (a-b x^3\right )}{11 \left (a+b x^3\right )^{11/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 413
Rule 385
Rule 246
Rule 245
Rubi steps
\begin{align*} \int \frac{\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{14/3}} \, dx &=\frac{2 x \left (a-b x^3\right )}{11 \left (a+b x^3\right )^{11/3}}+\frac{\int \frac{9 a^2 b-3 a b^2 x^3}{\left (a+b x^3\right )^{11/3}} \, dx}{11 a b}\\ &=\frac{2 x \left (a-b x^3\right )}{11 \left (a+b x^3\right )^{11/3}}+\frac{3 x}{22 \left (a+b x^3\right )^{8/3}}+\frac{15}{22} \int \frac{1}{\left (a+b x^3\right )^{8/3}} \, dx\\ &=\frac{2 x \left (a-b x^3\right )}{11 \left (a+b x^3\right )^{11/3}}+\frac{3 x}{22 \left (a+b x^3\right )^{8/3}}+\frac{\left (15 \left (1+\frac{b x^3}{a}\right )^{2/3}\right ) \int \frac{1}{\left (1+\frac{b x^3}{a}\right )^{8/3}} \, dx}{22 a^2 \left (a+b x^3\right )^{2/3}}\\ &=\frac{2 x \left (a-b x^3\right )}{11 \left (a+b x^3\right )^{11/3}}+\frac{3 x}{22 \left (a+b x^3\right )^{8/3}}+\frac{15 x \left (1+\frac{b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{8}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{22 a^2 \left (a+b x^3\right )^{2/3}}\\ \end{align*}
Mathematica [A] time = 0.0842305, size = 95, normalized size = 1.02 \[ \frac{x \left (23 a^2 b x^3+16 a^3+21 a b^2 x^6+6 \left (a+b x^3\right )^3 \left (\frac{b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right )+6 b^3 x^9\right )}{22 a^2 \left (a+b x^3\right )^{11/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.37, size = 0, normalized size = 0. \begin{align*} \int{ \left ( -b{x}^{3}+a \right ) ^{2} \left ( b{x}^{3}+a \right ) ^{-{\frac{14}{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{14}{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^{5} x^{15} + 5 \, a b^{4} x^{12} + 10 \, a^{2} b^{3} x^{9} + 10 \, a^{3} b^{2} x^{6} + 5 \, a^{4} b x^{3} + a^{5}}, x\right ) \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^{3} - a\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac{14}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]