Optimal. Leaf size=144 \[ \frac {(b c-a d) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{2/3} d^{4/3}}-\frac {(b c-a d) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} d^{4/3}}+\frac {(b c-a d) \tan ^{-1}\left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} c^{2/3} d^{4/3}}+\frac {b x}{d} \]
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Rubi [A] time = 0.09, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {388, 200, 31, 634, 617, 204, 628} \[ \frac {(b c-a d) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{2/3} d^{4/3}}-\frac {(b c-a d) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} d^{4/3}}+\frac {(b c-a d) \tan ^{-1}\left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} c^{2/3} d^{4/3}}+\frac {b x}{d} \]
Antiderivative was successfully verified.
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Rule 31
Rule 200
Rule 204
Rule 388
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {a+b x^3}{c+d x^3} \, dx &=\frac {b x}{d}-\frac {(b c-a d) \int \frac {1}{c+d x^3} \, dx}{d}\\ &=\frac {b x}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt [3]{c}+\sqrt [3]{d} x} \, dx}{3 c^{2/3} d}-\frac {(b c-a d) \int \frac {2 \sqrt [3]{c}-\sqrt [3]{d} x}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{3 c^{2/3} d}\\ &=\frac {b x}{d}-\frac {(b c-a d) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} d^{4/3}}+\frac {(b c-a d) \int \frac {-\sqrt [3]{c} \sqrt [3]{d}+2 d^{2/3} x}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{6 c^{2/3} d^{4/3}}-\frac {(b c-a d) \int \frac {1}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{2 \sqrt [3]{c} d}\\ &=\frac {b x}{d}-\frac {(b c-a d) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} d^{4/3}}+\frac {(b c-a d) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{2/3} d^{4/3}}-\frac {(b c-a d) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}\right )}{c^{2/3} d^{4/3}}\\ &=\frac {b x}{d}+\frac {(b c-a d) \tan ^{-1}\left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} c^{2/3} d^{4/3}}-\frac {(b c-a d) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} d^{4/3}}+\frac {(b c-a d) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{2/3} d^{4/3}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 128, normalized size = 0.89 \[ \frac {(b c-a d) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )-2 (b c-a d) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )+2 \sqrt {3} (b c-a d) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )+6 b c^{2/3} \sqrt [3]{d} x}{6 c^{2/3} d^{4/3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 369, normalized size = 2.56 \[ \left [\frac {6 \, b c^{2} d x - 3 \, \sqrt {\frac {1}{3}} {\left (b c^{2} d - a c d^{2}\right )} \sqrt {-\frac {\left (c^{2} d\right )^{\frac {1}{3}}}{d}} \log \left (\frac {2 \, c d x^{3} - 3 \, \left (c^{2} d\right )^{\frac {1}{3}} c x - c^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, c d x^{2} + \left (c^{2} d\right )^{\frac {2}{3}} x - \left (c^{2} d\right )^{\frac {1}{3}} c\right )} \sqrt {-\frac {\left (c^{2} d\right )^{\frac {1}{3}}}{d}}}{d x^{3} + c}\right ) + \left (c^{2} d\right )^{\frac {2}{3}} {\left (b c - a d\right )} \log \left (c d x^{2} - \left (c^{2} d\right )^{\frac {2}{3}} x + \left (c^{2} d\right )^{\frac {1}{3}} c\right ) - 2 \, \left (c^{2} d\right )^{\frac {2}{3}} {\left (b c - a d\right )} \log \left (c d x + \left (c^{2} d\right )^{\frac {2}{3}}\right )}{6 \, c^{2} d^{2}}, \frac {6 \, b c^{2} d x - 6 \, \sqrt {\frac {1}{3}} {\left (b c^{2} d - a c d^{2}\right )} \sqrt {\frac {\left (c^{2} d\right )^{\frac {1}{3}}}{d}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (c^{2} d\right )^{\frac {2}{3}} x - \left (c^{2} d\right )^{\frac {1}{3}} c\right )} \sqrt {\frac {\left (c^{2} d\right )^{\frac {1}{3}}}{d}}}{c^{2}}\right ) + \left (c^{2} d\right )^{\frac {2}{3}} {\left (b c - a d\right )} \log \left (c d x^{2} - \left (c^{2} d\right )^{\frac {2}{3}} x + \left (c^{2} d\right )^{\frac {1}{3}} c\right ) - 2 \, \left (c^{2} d\right )^{\frac {2}{3}} {\left (b c - a d\right )} \log \left (c d x + \left (c^{2} d\right )^{\frac {2}{3}}\right )}{6 \, c^{2} d^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 133, normalized size = 0.92 \[ \frac {\sqrt {3} {\left (b c - a d\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {c}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {c}{d}\right )^{\frac {1}{3}}}\right )}{3 \, \left (-c d^{2}\right )^{\frac {2}{3}}} + \frac {{\left (b c - a d\right )} \log \left (x^{2} + x \left (-\frac {c}{d}\right )^{\frac {1}{3}} + \left (-\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 \, \left (-c d^{2}\right )^{\frac {2}{3}}} + \frac {b x}{d} + \frac {{\left (b c - a d\right )} \left (-\frac {c}{d}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {c}{d}\right )^{\frac {1}{3}} \right |}\right )}{3 \, c d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 195, normalized size = 1.35 \[ \frac {\sqrt {3}\, a \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {c}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {c}{d}\right )^{\frac {2}{3}} d}+\frac {a \ln \left (x +\left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {c}{d}\right )^{\frac {2}{3}} d}-\frac {a \ln \left (x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{3}} x +\left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {c}{d}\right )^{\frac {2}{3}} d}-\frac {\sqrt {3}\, b c \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {c}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {c}{d}\right )^{\frac {2}{3}} d^{2}}-\frac {b c \ln \left (x +\left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {c}{d}\right )^{\frac {2}{3}} d^{2}}+\frac {b c \ln \left (x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{3}} x +\left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {c}{d}\right )^{\frac {2}{3}} d^{2}}+\frac {b x}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 128, normalized size = 0.89 \[ \frac {b x}{d} - \frac {\sqrt {3} {\left (b c - a d\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {c}{d}\right )^{\frac {1}{3}}}\right )}{3 \, d^{2} \left (\frac {c}{d}\right )^{\frac {2}{3}}} + \frac {{\left (b c - a d\right )} \log \left (x^{2} - x \left (\frac {c}{d}\right )^{\frac {1}{3}} + \left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 \, d^{2} \left (\frac {c}{d}\right )^{\frac {2}{3}}} - \frac {{\left (b c - a d\right )} \log \left (x + \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 \, d^{2} \left (\frac {c}{d}\right )^{\frac {2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.38, size = 123, normalized size = 0.85 \[ \frac {b\,x}{d}+\frac {\ln \left (d^{1/3}\,x+c^{1/3}\right )\,\left (a\,d-b\,c\right )}{3\,c^{2/3}\,d^{4/3}}-\frac {\ln \left (c^{1/3}-2\,d^{1/3}\,x+\sqrt {3}\,c^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (a\,d-b\,c\right )}{3\,c^{2/3}\,d^{4/3}}+\frac {\ln \left (2\,d^{1/3}\,x-c^{1/3}+\sqrt {3}\,c^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (a\,d-b\,c\right )}{3\,c^{2/3}\,d^{4/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.42, size = 71, normalized size = 0.49 \[ \frac {b x}{d} + \operatorname {RootSum} {\left (27 t^{3} c^{2} d^{4} - a^{3} d^{3} + 3 a^{2} b c d^{2} - 3 a b^{2} c^{2} d + b^{3} c^{3}, \left (t \mapsto t \log {\left (\frac {3 t c d}{a d - b c} + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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