Integrand size = 21, antiderivative size = 59 \[ \int \frac {1}{\left (a+b x^3\right )^{2/3} \left (c+d x^3\right )^2} \, dx=\frac {x \left (1+\frac {b x^3}{a}\right )^{2/3} \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},2,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c^2 \left (a+b x^3\right )^{2/3}} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {441, 440} \[ \int \frac {1}{\left (a+b x^3\right )^{2/3} \left (c+d x^3\right )^2} \, dx=\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},2,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c^2 \left (a+b x^3\right )^{2/3}} \]
[In]
[Out]
Rule 440
Rule 441
Rubi steps \begin{align*} \text {integral}& = \frac {\left (1+\frac {b x^3}{a}\right )^{2/3} \int \frac {1}{\left (1+\frac {b x^3}{a}\right )^{2/3} \left (c+d x^3\right )^2} \, dx}{\left (a+b x^3\right )^{2/3}} \\ & = \frac {x \left (1+\frac {b x^3}{a}\right )^{2/3} F_1\left (\frac {1}{3};\frac {2}{3},2;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c^2 \left (a+b x^3\right )^{2/3}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(393\) vs. \(2(59)=118\).
Time = 10.30 (sec) , antiderivative size = 393, normalized size of antiderivative = 6.66 \[ \int \frac {1}{\left (a+b x^3\right )^{2/3} \left (c+d x^3\right )^2} \, dx=\frac {4 a c x \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right ) \left (4 c \left (-3 b c+3 a d+b d x^3\right )+b d x^3 \left (1+\frac {b x^3}{a}\right )^{2/3} \left (c+d x^3\right ) \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )-d x^4 \left (4 c \left (a+b x^3\right )+b x^3 \left (1+\frac {b x^3}{a}\right )^{2/3} \left (c+d x^3\right ) \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right ) \left (3 a d \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},2,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+2 b c \operatorname {AppellF1}\left (\frac {4}{3},\frac {5}{3},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )}{12 c^2 (b c-a d) \left (a+b x^3\right )^{2/3} \left (c+d x^3\right ) \left (-4 a c \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+x^3 \left (3 a d \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},2,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+2 b c \operatorname {AppellF1}\left (\frac {4}{3},\frac {5}{3},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )\right )} \]
[In]
[Out]
\[\int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {2}{3}} \left (d \,x^{3}+c \right )^{2}}d x\]
[In]
[Out]
Timed out. \[ \int \frac {1}{\left (a+b x^3\right )^{2/3} \left (c+d x^3\right )^2} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {1}{\left (a+b x^3\right )^{2/3} \left (c+d x^3\right )^2} \, dx=\int \frac {1}{\left (a + b x^{3}\right )^{\frac {2}{3}} \left (c + d x^{3}\right )^{2}}\, dx \]
[In]
[Out]
\[ \int \frac {1}{\left (a+b x^3\right )^{2/3} \left (c+d x^3\right )^2} \, dx=\int { \frac {1}{{\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (d x^{3} + c\right )}^{2}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{\left (a+b x^3\right )^{2/3} \left (c+d x^3\right )^2} \, dx=\int { \frac {1}{{\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (d x^{3} + c\right )}^{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1}{\left (a+b x^3\right )^{2/3} \left (c+d x^3\right )^2} \, dx=\int \frac {1}{{\left (b\,x^3+a\right )}^{2/3}\,{\left (d\,x^3+c\right )}^2} \,d x \]
[In]
[Out]