Optimal. Leaf size=67 \[ \frac {x^2 \sqrt [3]{\frac {b x^3}{a}+1} F_1\left (\frac {2}{3};\frac {4}{3},1;\frac {5}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 a c \sqrt [3]{a+b x^3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {511, 510} \[ \frac {x^2 \sqrt [3]{\frac {b x^3}{a}+1} F_1\left (\frac {2}{3};\frac {4}{3},1;\frac {5}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 a c \sqrt [3]{a+b x^3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 510
Rule 511
Rubi steps
\begin {align*} \int \frac {x}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx &=\frac {\sqrt [3]{1+\frac {b x^3}{a}} \int \frac {x}{\left (1+\frac {b x^3}{a}\right )^{4/3} \left (c+d x^3\right )} \, dx}{a \sqrt [3]{a+b x^3}}\\ &=\frac {x^2 \sqrt [3]{1+\frac {b x^3}{a}} F_1\left (\frac {2}{3};\frac {4}{3},1;\frac {5}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 a c \sqrt [3]{a+b x^3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.13, size = 141, normalized size = 2.10 \[ \frac {x^2 \left (2 b d x^3 \sqrt [3]{\frac {b x^3}{a}+1} F_1\left (\frac {5}{3};\frac {1}{3},1;\frac {8}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+5 \sqrt [3]{\frac {b x^3}{a}+1} (a d+b c) F_1\left (\frac {2}{3};\frac {1}{3},1;\frac {5}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )-10 b c\right )}{10 a c \sqrt [3]{a+b x^3} (a d-b c)} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [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]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{{\left (b x^{3} + a\right )}^{\frac {4}{3}} {\left (d x^{3} + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.86, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (b \,x^{3}+a \right )^{\frac {4}{3}} \left (d \,x^{3}+c \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{{\left (b x^{3} + a\right )}^{\frac {4}{3}} {\left (d x^{3} + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x}{{\left (b\,x^3+a\right )}^{4/3}\,\left (d\,x^3+c\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (a + b x^{3}\right )^{\frac {4}{3}} \left (c + d x^{3}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________