Optimal. Leaf size=74 \[ \frac {3 x \left (\frac {b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{5 \left (a+b x^3\right )^{2/3}}+\frac {2 x \left (a-b x^3\right )}{5 \left (a+b x^3\right )^{5/3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 74, normalized size of antiderivative = 1.00, 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, 21, 246, 245} \[ \frac {3 x \left (\frac {b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{5 \left (a+b x^3\right )^{2/3}}+\frac {2 x \left (a-b x^3\right )}{5 \left (a+b x^3\right )^{5/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 21
Rule 245
Rule 246
Rule 413
Rubi steps
\begin {align*} \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{8/3}} \, dx &=\frac {2 x \left (a-b x^3\right )}{5 \left (a+b x^3\right )^{5/3}}+\frac {\int \frac {3 a^2 b+3 a b^2 x^3}{\left (a+b x^3\right )^{5/3}} \, dx}{5 a b}\\ &=\frac {2 x \left (a-b x^3\right )}{5 \left (a+b x^3\right )^{5/3}}+\frac {3}{5} \int \frac {1}{\left (a+b x^3\right )^{2/3}} \, dx\\ &=\frac {2 x \left (a-b x^3\right )}{5 \left (a+b x^3\right )^{5/3}}+\frac {\left (3 \left (1+\frac {b x^3}{a}\right )^{2/3}\right ) \int \frac {1}{\left (1+\frac {b x^3}{a}\right )^{2/3}} \, dx}{5 \left (a+b x^3\right )^{2/3}}\\ &=\frac {2 x \left (a-b x^3\right )}{5 \left (a+b x^3\right )^{5/3}}+\frac {3 x \left (1+\frac {b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{5 \left (a+b x^3\right )^{2/3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 70, normalized size = 0.95 \[ \frac {3 x \left (a+b x^3\right ) \left (\frac {b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^3}{a}\right )+2 x \left (a-b x^3\right )}{5 \left (a+b x^3\right )^{5/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} x^{6} - 2 \, a b x^{3} + a^{2}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{b^{3} x^{9} + 3 \, a b^{2} x^{6} + 3 \, a^{2} b x^{3} + a^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{3} - a\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac {8}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.51, size = 0, normalized size = 0.00 \[ \int \frac {\left (-b \,x^{3}+a \right )^{2}}{\left (b \,x^{3}+a \right )^{\frac {8}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{3} - a\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac {8}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a-b\,x^3\right )}^2}{{\left (b\,x^3+a\right )}^{8/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________