Optimal. Leaf size=304 \[ -\frac {d x^2 \sqrt {\frac {b x^3}{a}+1} F_1\left (\frac {2}{3};\frac {1}{2},1;\frac {5}{3};-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{2 \sqrt {a+b x^3} \left (a c^2-d^2\right )}+\frac {\log \left (-\sqrt [3]{b} c^{2/3} x \sqrt [3]{a c^2-d^2}+\left (a c^2-d^2\right )^{2/3}+b^{2/3} c^{4/3} x^2\right )}{6 b^{2/3} \sqrt [3]{c} \sqrt [3]{a c^2-d^2}}-\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 b^{2/3} \sqrt [3]{c} \sqrt [3]{a c^2-d^2}}-\frac {\tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}}{\sqrt {3}}\right )}{\sqrt {3} b^{2/3} \sqrt [3]{c} \sqrt [3]{a c^2-d^2}} \]
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Rubi [A] time = 0.30, antiderivative size = 304, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2156, 292, 31, 634, 617, 204, 628, 511, 510} \[ -\frac {d x^2 \sqrt {\frac {b x^3}{a}+1} F_1\left (\frac {2}{3};\frac {1}{2},1;\frac {5}{3};-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{2 \sqrt {a+b x^3} \left (a c^2-d^2\right )}+\frac {\log \left (-\sqrt [3]{b} c^{2/3} x \sqrt [3]{a c^2-d^2}+\left (a c^2-d^2\right )^{2/3}+b^{2/3} c^{4/3} x^2\right )}{6 b^{2/3} \sqrt [3]{c} \sqrt [3]{a c^2-d^2}}-\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 b^{2/3} \sqrt [3]{c} \sqrt [3]{a c^2-d^2}}-\frac {\tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}}{\sqrt {3}}\right )}{\sqrt {3} b^{2/3} \sqrt [3]{c} \sqrt [3]{a c^2-d^2}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 292
Rule 510
Rule 511
Rule 617
Rule 628
Rule 634
Rule 2156
Rubi steps
\begin {align*} \int \frac {x}{a c+b c x^3+d \sqrt {a+b x^3}} \, dx &=(a c) \int \frac {x}{a^2 c^2-a d^2+a b c^2 x^3} \, dx-(a d) \int \frac {x}{\sqrt {a+b x^3} \left (a^2 c^2-a d^2+a b c^2 x^3\right )} \, dx\\ &=-\frac {\left (\sqrt [3]{a} \sqrt [3]{c}\right ) \int \frac {1}{\sqrt [3]{a} \sqrt [3]{a c^2-d^2}+\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x} \, dx}{3 \sqrt [3]{b} \sqrt [3]{a c^2-d^2}}+\frac {\left (\sqrt [3]{a} \sqrt [3]{c}\right ) \int \frac {\sqrt [3]{a} \sqrt [3]{a c^2-d^2}+\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x}{a^{2/3} \left (a c^2-d^2\right )^{2/3}-a^{2/3} \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+a^{2/3} b^{2/3} c^{4/3} x^2} \, dx}{3 \sqrt [3]{b} \sqrt [3]{a c^2-d^2}}-\frac {\left (a d \sqrt {1+\frac {b x^3}{a}}\right ) \int \frac {x}{\sqrt {1+\frac {b x^3}{a}} \left (a^2 c^2-a d^2+a b c^2 x^3\right )} \, dx}{\sqrt {a+b x^3}}\\ &=-\frac {d x^2 \sqrt {1+\frac {b x^3}{a}} F_1\left (\frac {2}{3};\frac {1}{2},1;\frac {5}{3};-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{2 \left (a c^2-d^2\right ) \sqrt {a+b x^3}}-\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 b^{2/3} \sqrt [3]{c} \sqrt [3]{a c^2-d^2}}+\frac {\left (a^{2/3} \sqrt [3]{c}\right ) \int \frac {1}{a^{2/3} \left (a c^2-d^2\right )^{2/3}-a^{2/3} \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+a^{2/3} b^{2/3} c^{4/3} x^2} \, dx}{2 \sqrt [3]{b}}+\frac {\int \frac {-a^{2/3} \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2}+2 a^{2/3} b^{2/3} c^{4/3} x}{a^{2/3} \left (a c^2-d^2\right )^{2/3}-a^{2/3} \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+a^{2/3} b^{2/3} c^{4/3} x^2} \, dx}{6 b^{2/3} \sqrt [3]{c} \sqrt [3]{a c^2-d^2}}\\ &=-\frac {d x^2 \sqrt {1+\frac {b x^3}{a}} F_1\left (\frac {2}{3};\frac {1}{2},1;\frac {5}{3};-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{2 \left (a c^2-d^2\right ) \sqrt {a+b x^3}}-\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 b^{2/3} \sqrt [3]{c} \sqrt [3]{a c^2-d^2}}+\frac {\log \left (\left (a c^2-d^2\right )^{2/3}-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+b^{2/3} c^{4/3} x^2\right )}{6 b^{2/3} \sqrt [3]{c} \sqrt [3]{a c^2-d^2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}\right )}{b^{2/3} \sqrt [3]{c} \sqrt [3]{a c^2-d^2}}\\ &=-\frac {d x^2 \sqrt {1+\frac {b x^3}{a}} F_1\left (\frac {2}{3};\frac {1}{2},1;\frac {5}{3};-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{2 \left (a c^2-d^2\right ) \sqrt {a+b x^3}}-\frac {\tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}}{\sqrt {3}}\right )}{\sqrt {3} b^{2/3} \sqrt [3]{c} \sqrt [3]{a c^2-d^2}}-\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 b^{2/3} \sqrt [3]{c} \sqrt [3]{a c^2-d^2}}+\frac {\log \left (\left (a c^2-d^2\right )^{2/3}-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+b^{2/3} c^{4/3} x^2\right )}{6 b^{2/3} \sqrt [3]{c} \sqrt [3]{a c^2-d^2}}\\ \end {align*}
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Mathematica [F] time = 0.16, size = 0, normalized size = 0.00 \[ \int \frac {x}{a c+b c x^3+d \sqrt {a+b x^3}} \, dx \]
Verification is Not applicable to the result.
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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.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{b c x^{3} + a c + \sqrt {b x^{3} + a} d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.08, size = 619, normalized size = 2.04 \[ \frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a \,c^{2}-d^{2}}{b \,c^{2}}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {a \,c^{2}-d^{2}}{b \,c^{2}}\right )^{\frac {1}{3}} b c}-\frac {\ln \left (x +\left (\frac {a \,c^{2}-d^{2}}{b \,c^{2}}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a \,c^{2}-d^{2}}{b \,c^{2}}\right )^{\frac {1}{3}} b c}+\frac {\ln \left (x^{2}-\left (\frac {a \,c^{2}-d^{2}}{b \,c^{2}}\right )^{\frac {1}{3}} x +\left (\frac {a \,c^{2}-d^{2}}{b \,c^{2}}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {a \,c^{2}-d^{2}}{b \,c^{2}}\right )^{\frac {1}{3}} b c}-\frac {i \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (2 x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}-i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right ) b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {\left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right ) b}{-3 \left (-a \,b^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i \left (2 x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right ) b}{2 \left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \left (2 \RootOf \left (b \,c^{2} \textit {\_Z}^{3}+a \,c^{2}-d^{2}\right )^{2} b^{2}+i \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \RootOf \left (b \,c^{2} \textit {\_Z}^{3}+a \,c^{2}-d^{2}\right ) b -\left (-a \,b^{2}\right )^{\frac {1}{3}} \RootOf \left (b \,c^{2} \textit {\_Z}^{3}+a \,c^{2}-d^{2}\right ) b -i \left (-a \,b^{2}\right )^{\frac {2}{3}} \sqrt {3}-\left (-a \,b^{2}\right )^{\frac {2}{3}}\right ) \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, -\frac {\left (2 i \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \RootOf \left (b \,c^{2} \textit {\_Z}^{3}+a \,c^{2}-d^{2}\right )^{2} b +i \sqrt {3}\, a b -3 a b -i \left (-a \,b^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \RootOf \left (b \,c^{2} \textit {\_Z}^{3}+a \,c^{2}-d^{2}\right )-3 \left (-a \,b^{2}\right )^{\frac {2}{3}} \RootOf \left (b \,c^{2} \textit {\_Z}^{3}+a \,c^{2}-d^{2}\right )\right ) c^{2}}{2 b \,d^{2}}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{\left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) b}}\right )}{3 b^{3} d \sqrt {b \,x^{3}+a}\, \RootOf \left (b \,c^{2} \textit {\_Z}^{3}+a \,c^{2}-d^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{b c x^{3} + a c + \sqrt {b x^{3} + a} d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x}{a\,c+d\,\sqrt {b\,x^3+a}+b\,c\,x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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