Optimal. Leaf size=234 \[ -\frac {\sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 \sqrt [3]{c} d}+\frac {\sqrt [3]{b c-a d} \log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{c} d}+\frac {\sqrt [3]{b c-a d} \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{c} d}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{2 d}-\frac {\sqrt [3]{b} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} d} \]
________________________________________________________________________________________
Rubi [C] time = 0.04, antiderivative size = 64, normalized size of antiderivative = 0.27, 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} \begin {gather*} \frac {x^2 \sqrt [3]{a+b x^3} F_1\left (\frac {2}{3};-\frac {1}{3},1;\frac {5}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 c \sqrt [3]{\frac {b x^3}{a}+1}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 510
Rule 511
Rubi steps
\begin {align*} \int \frac {x \sqrt [3]{a+b x^3}}{c+d x^3} \, dx &=\frac {\sqrt [3]{a+b x^3} \int \frac {x \sqrt [3]{1+\frac {b x^3}{a}}}{c+d x^3} \, dx}{\sqrt [3]{1+\frac {b x^3}{a}}}\\ &=\frac {x^2 \sqrt [3]{a+b x^3} F_1\left (\frac {2}{3};-\frac {1}{3},1;\frac {5}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 c \sqrt [3]{1+\frac {b x^3}{a}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.04, size = 65, normalized size = 0.28 \begin {gather*} \frac {x^2 \sqrt [3]{a+b x^3} F_1\left (\frac {2}{3};-\frac {1}{3},1;\frac {5}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 c \sqrt [3]{\frac {a+b x^3}{a}}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [C] time = 3.79, size = 490, normalized size = 2.09 \begin {gather*} \frac {\sqrt [3]{b} \log \left (\sqrt [3]{b} x \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}+b^{2/3} x^2\right )}{6 d}+\frac {\left (\sqrt [3]{b c-a d}-i \sqrt {3} \sqrt [3]{b c-a d}\right ) \log \left (\left (\sqrt {3}+i\right ) c^{2/3} \left (a+b x^3\right )^{2/3}+\sqrt [3]{c} \left (-\sqrt {3} x+i x\right ) \sqrt [3]{a+b x^3} \sqrt [3]{b c-a d}-2 i x^2 (b c-a d)^{2/3}\right )}{12 \sqrt [3]{c} d}+\frac {i \left (\sqrt {3} \sqrt [3]{b c-a d}+i \sqrt [3]{b c-a d}\right ) \log \left (2 x \sqrt [3]{b c-a d}+\left (1+i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}\right )}{6 \sqrt [3]{c} d}-\frac {\sqrt {-1-i \sqrt {3}} \sqrt [3]{b c-a d} \tan ^{-1}\left (\frac {3 x \sqrt [3]{b c-a d}}{\sqrt {3} x \sqrt [3]{b c-a d}-\sqrt {3} \sqrt [3]{c} \sqrt [3]{a+b x^3}-3 i \sqrt [3]{c} \sqrt [3]{a+b x^3}}\right )}{\sqrt {6} \sqrt [3]{c} d}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{3 d}-\frac {\sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{b} x}{2 \sqrt [3]{a+b x^3}+\sqrt [3]{b} x}\right )}{\sqrt {3} d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.48, size = 330, normalized size = 1.41 \begin {gather*} \frac {2 \, \sqrt {3} \left (\frac {b c - a d}{c}\right )^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} {\left (b c - a d\right )} x + 2 \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} c \left (\frac {b c - a d}{c}\right )^{\frac {2}{3}}}{3 \, {\left (b c - a d\right )} x}\right ) - 2 \, \sqrt {3} \left (-b\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} b x + 2 \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b\right )^{\frac {2}{3}}}{3 \, b x}\right ) + 2 \, \left (-b\right )^{\frac {1}{3}} \log \left (\frac {\left (-b\right )^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) + 2 \, \left (\frac {b c - a d}{c}\right )^{\frac {1}{3}} \log \left (-\frac {x \left (\frac {b c - a d}{c}\right )^{\frac {1}{3}} - {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) - \left (-b\right )^{\frac {1}{3}} \log \left (\frac {\left (-b\right )^{\frac {2}{3}} x^{2} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b\right )^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right ) - \left (\frac {b c - a d}{c}\right )^{\frac {1}{3}} \log \left (\frac {x^{2} \left (\frac {b c - a d}{c}\right )^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} x \left (\frac {b c - a d}{c}\right )^{\frac {1}{3}} + {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{6 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} x}{d x^{3} + c}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.58, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}} x}{d \,x^{3}+c}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} x}{d x^{3} + c}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x\,{\left (b\,x^3+a\right )}^{1/3}}{d\,x^3+c} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \sqrt [3]{a + b x^{3}}}{c + d x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________