Optimal. Leaf size=64 \[ -\frac {\sqrt [3]{a+b x^3} F_1\left (-\frac {2}{3};-\frac {1}{3},1;\frac {1}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 c x^2 \sqrt [3]{\frac {b x^3}{a}+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {511, 510} \[ -\frac {\sqrt [3]{a+b x^3} F_1\left (-\frac {2}{3};-\frac {1}{3},1;\frac {1}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 c x^2 \sqrt [3]{\frac {b x^3}{a}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 510
Rule 511
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{a+b x^3}}{x^3 \left (c+d x^3\right )} \, dx &=\frac {\sqrt [3]{a+b x^3} \int \frac {\sqrt [3]{1+\frac {b x^3}{a}}}{x^3 \left (c+d x^3\right )} \, dx}{\sqrt [3]{1+\frac {b x^3}{a}}}\\ &=-\frac {\sqrt [3]{a+b x^3} F_1\left (-\frac {2}{3};-\frac {1}{3},1;\frac {1}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 c x^2 \sqrt [3]{1+\frac {b x^3}{a}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.26, size = 327, normalized size = 5.11 \[ \frac {\frac {c \left (16 a c \left (a \left (c+3 d x^3\right )+b d x^6\right ) F_1\left (\frac {1}{3};\frac {2}{3},1;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )-4 x^3 \left (a+b x^3\right ) \left (c+d x^3\right ) \left (3 a d F_1\left (\frac {4}{3};\frac {2}{3},2;\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+2 b c F_1\left (\frac {4}{3};\frac {5}{3},1;\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )\right )}{\left (c+d x^3\right ) \left (x^3 \left (3 a d F_1\left (\frac {4}{3};\frac {2}{3},2;\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+2 b c F_1\left (\frac {4}{3};\frac {5}{3},1;\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )-4 a c F_1\left (\frac {1}{3};\frac {2}{3},1;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )}-b d x^6 \left (\frac {b x^3}{a}+1\right )^{2/3} F_1\left (\frac {4}{3};\frac {2}{3},1;\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{8 c^2 x^2 \left (a+b x^3\right )^{2/3}} \]
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 \[ \int \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{{\left (d x^{3} + c\right )} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.61, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{\left (d \,x^{3}+c \right ) x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{{\left (d x^{3} + c\right )} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (b\,x^3+a\right )}^{1/3}}{x^3\,\left (d\,x^3+c\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt [3]{a + b x^{3}}}{x^{3} \left (c + d x^{3}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________