Integrand size = 22, antiderivative size = 94 \[ \int \left (a-b x^3\right )^2 \sqrt [3]{a+b x^3} \, dx=-\frac {3}{8} a x \left (a+b x^3\right )^{4/3}-\frac {1}{8} x \left (a-b x^3\right ) \left (a+b x^3\right )^{4/3}+\frac {3 a^2 x \sqrt [3]{a+b x^3} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},\frac {1}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{2 \sqrt [3]{1+\frac {b x^3}{a}}} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {427, 396, 252, 251} \[ \int \left (a-b x^3\right )^2 \sqrt [3]{a+b x^3} \, dx=\frac {3 a^2 x \sqrt [3]{a+b x^3} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},\frac {1}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{2 \sqrt [3]{\frac {b x^3}{a}+1}}-\frac {3}{8} a x \left (a+b x^3\right )^{4/3}-\frac {1}{8} x \left (a-b x^3\right ) \left (a+b x^3\right )^{4/3} \]
[In]
[Out]
Rule 251
Rule 252
Rule 396
Rule 427
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{8} x \left (a-b x^3\right ) \left (a+b x^3\right )^{4/3}+\frac {\int \sqrt [3]{a+b x^3} \left (9 a^2 b-15 a b^2 x^3\right ) \, dx}{8 b} \\ & = -\frac {3}{8} a x \left (a+b x^3\right )^{4/3}-\frac {1}{8} x \left (a-b x^3\right ) \left (a+b x^3\right )^{4/3}+\frac {1}{2} \left (3 a^2\right ) \int \sqrt [3]{a+b x^3} \, dx \\ & = -\frac {3}{8} a x \left (a+b x^3\right )^{4/3}-\frac {1}{8} x \left (a-b x^3\right ) \left (a+b x^3\right )^{4/3}+\frac {\left (3 a^2 \sqrt [3]{a+b x^3}\right ) \int \sqrt [3]{1+\frac {b x^3}{a}} \, dx}{2 \sqrt [3]{1+\frac {b x^3}{a}}} \\ & = -\frac {3}{8} a x \left (a+b x^3\right )^{4/3}-\frac {1}{8} x \left (a-b x^3\right ) \left (a+b x^3\right )^{4/3}+\frac {3 a^2 x \sqrt [3]{a+b x^3} \, _2F_1\left (-\frac {1}{3},\frac {1}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{2 \sqrt [3]{1+\frac {b x^3}{a}}} \\ \end{align*}
Time = 6.71 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.90 \[ \int \left (a-b x^3\right )^2 \sqrt [3]{a+b x^3} \, dx=\frac {x \left (2 a^3-a^2 b x^3-2 a b^2 x^6+b^3 x^9+6 a^3 \left (1+\frac {b x^3}{a}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )\right )}{8 \left (a+b x^3\right )^{2/3}} \]
[In]
[Out]
\[\int \left (-b \,x^{3}+a \right )^{2} \left (b \,x^{3}+a \right )^{\frac {1}{3}}d x\]
[In]
[Out]
\[ \int \left (a-b x^3\right )^2 \sqrt [3]{a+b x^3} \, dx=\int { {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (b x^{3} - a\right )}^{2} \,d x } \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 1.46 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.34 \[ \int \left (a-b x^3\right )^2 \sqrt [3]{a+b x^3} \, dx=\frac {a^{\frac {7}{3}} x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} - \frac {2 a^{\frac {4}{3}} b x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {\sqrt [3]{a} b^{2} x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {10}{3}\right )} \]
[In]
[Out]
\[ \int \left (a-b x^3\right )^2 \sqrt [3]{a+b x^3} \, dx=\int { {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (b x^{3} - a\right )}^{2} \,d x } \]
[In]
[Out]
\[ \int \left (a-b x^3\right )^2 \sqrt [3]{a+b x^3} \, dx=\int { {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (b x^{3} - a\right )}^{2} \,d x } \]
[In]
[Out]
Timed out. \[ \int \left (a-b x^3\right )^2 \sqrt [3]{a+b x^3} \, dx=\int {\left (b\,x^3+a\right )}^{1/3}\,{\left (a-b\,x^3\right )}^2 \,d x \]
[In]
[Out]