Integrand size = 19, antiderivative size = 79 \[ \int \left (a+b x^3\right )^m \left (c+d x^3\right )^p \, dx=x \left (a+b x^3\right )^m \left (1+\frac {b x^3}{a}\right )^{-m} \left (c+d x^3\right )^p \left (1+\frac {d x^3}{c}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{3},-m,-p,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right ) \]
[Out]
Time = 0.03 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {441, 440} \[ \int \left (a+b x^3\right )^m \left (c+d x^3\right )^p \, dx=x \left (a+b x^3\right )^m \left (\frac {b x^3}{a}+1\right )^{-m} \left (c+d x^3\right )^p \left (\frac {d x^3}{c}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{3},-m,-p,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right ) \]
[In]
[Out]
Rule 440
Rule 441
Rubi steps \begin{align*} \text {integral}& = \left (\left (a+b x^3\right )^m \left (1+\frac {b x^3}{a}\right )^{-m}\right ) \int \left (1+\frac {b x^3}{a}\right )^m \left (c+d x^3\right )^p \, dx \\ & = \left (\left (a+b x^3\right )^m \left (1+\frac {b x^3}{a}\right )^{-m} \left (c+d x^3\right )^p \left (1+\frac {d x^3}{c}\right )^{-p}\right ) \int \left (1+\frac {b x^3}{a}\right )^m \left (1+\frac {d x^3}{c}\right )^p \, dx \\ & = x \left (a+b x^3\right )^m \left (1+\frac {b x^3}{a}\right )^{-m} \left (c+d x^3\right )^p \left (1+\frac {d x^3}{c}\right )^{-p} F_1\left (\frac {1}{3};-m,-p;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right ) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(172\) vs. \(2(79)=158\).
Time = 0.30 (sec) , antiderivative size = 172, normalized size of antiderivative = 2.18 \[ \int \left (a+b x^3\right )^m \left (c+d x^3\right )^p \, dx=\frac {4 a c x \left (a+b x^3\right )^m \left (c+d x^3\right )^p \operatorname {AppellF1}\left (\frac {1}{3},-m,-p,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{4 a c \operatorname {AppellF1}\left (\frac {1}{3},-m,-p,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+3 x^3 \left (b c m \operatorname {AppellF1}\left (\frac {4}{3},1-m,-p,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+a d p \operatorname {AppellF1}\left (\frac {4}{3},-m,1-p,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )} \]
[In]
[Out]
\[\int \left (b \,x^{3}+a \right )^{m} \left (d \,x^{3}+c \right )^{p}d x\]
[In]
[Out]
\[ \int \left (a+b x^3\right )^m \left (c+d x^3\right )^p \, dx=\int { {\left (b x^{3} + a\right )}^{m} {\left (d x^{3} + c\right )}^{p} \,d x } \]
[In]
[Out]
Timed out. \[ \int \left (a+b x^3\right )^m \left (c+d x^3\right )^p \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \left (a+b x^3\right )^m \left (c+d x^3\right )^p \, dx=\int { {\left (b x^{3} + a\right )}^{m} {\left (d x^{3} + c\right )}^{p} \,d x } \]
[In]
[Out]
\[ \int \left (a+b x^3\right )^m \left (c+d x^3\right )^p \, dx=\int { {\left (b x^{3} + a\right )}^{m} {\left (d x^{3} + c\right )}^{p} \,d x } \]
[In]
[Out]
Timed out. \[ \int \left (a+b x^3\right )^m \left (c+d x^3\right )^p \, dx=\int {\left (b\,x^3+a\right )}^m\,{\left (d\,x^3+c\right )}^p \,d x \]
[In]
[Out]