Optimal. Leaf size=346 \[ -\frac {b^{2/3} (2 b c-5 a d) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} (b c-a d)^2}+\frac {b^{2/3} (2 b c-5 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} (b c-a d)^2}-\frac {b^{2/3} (2 b c-5 a d) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} (b c-a d)^2}-\frac {d^{5/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{2/3} (b c-a d)^2}+\frac {d^{5/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} (b c-a d)^2}-\frac {d^{5/3} \tan ^{-1}\left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} c^{2/3} (b c-a d)^2}+\frac {b x}{3 a \left (a+b x^3\right ) (b c-a d)} \]
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Rubi [A] time = 0.25, antiderivative size = 346, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {414, 522, 200, 31, 634, 617, 204, 628} \[ -\frac {b^{2/3} (2 b c-5 a d) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} (b c-a d)^2}+\frac {b^{2/3} (2 b c-5 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} (b c-a d)^2}-\frac {b^{2/3} (2 b c-5 a d) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} (b c-a d)^2}-\frac {d^{5/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{2/3} (b c-a d)^2}+\frac {d^{5/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} (b c-a d)^2}-\frac {d^{5/3} \tan ^{-1}\left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} c^{2/3} (b c-a d)^2}+\frac {b x}{3 a \left (a+b x^3\right ) (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 31
Rule 200
Rule 204
Rule 414
Rule 522
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x^3\right )^2 \left (c+d x^3\right )} \, dx &=\frac {b x}{3 a (b c-a d) \left (a+b x^3\right )}-\frac {\int \frac {-2 b c+3 a d-2 b d x^3}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx}{3 a (b c-a d)}\\ &=\frac {b x}{3 a (b c-a d) \left (a+b x^3\right )}+\frac {d^2 \int \frac {1}{c+d x^3} \, dx}{(b c-a d)^2}+\frac {(b (2 b c-5 a d)) \int \frac {1}{a+b x^3} \, dx}{3 a (b c-a d)^2}\\ &=\frac {b x}{3 a (b c-a d) \left (a+b x^3\right )}+\frac {d^2 \int \frac {1}{\sqrt [3]{c}+\sqrt [3]{d} x} \, dx}{3 c^{2/3} (b c-a d)^2}+\frac {d^2 \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} (b c-a d)^2}+\frac {(b (2 b c-5 a d)) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{5/3} (b c-a d)^2}+\frac {(b (2 b c-5 a d)) \int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{9 a^{5/3} (b c-a d)^2}\\ &=\frac {b x}{3 a (b c-a d) \left (a+b x^3\right )}+\frac {b^{2/3} (2 b c-5 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} (b c-a d)^2}+\frac {d^{5/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} (b c-a d)^2}-\frac {d^{5/3} \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} (b c-a d)^2}+\frac {d^2 \int \frac {1}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{2 \sqrt [3]{c} (b c-a d)^2}-\frac {\left (b^{2/3} (2 b c-5 a d)\right ) \int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^{5/3} (b c-a d)^2}+\frac {(b (2 b c-5 a d)) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 a^{4/3} (b c-a d)^2}\\ &=\frac {b x}{3 a (b c-a d) \left (a+b x^3\right )}+\frac {b^{2/3} (2 b c-5 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} (b c-a d)^2}+\frac {d^{5/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} (b c-a d)^2}-\frac {b^{2/3} (2 b c-5 a d) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} (b c-a d)^2}-\frac {d^{5/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{2/3} (b c-a d)^2}+\frac {d^{5/3} \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} (b c-a d)^2}+\frac {\left (b^{2/3} (2 b c-5 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a^{5/3} (b c-a d)^2}\\ &=\frac {b x}{3 a (b c-a d) \left (a+b x^3\right )}-\frac {b^{2/3} (2 b c-5 a d) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} (b c-a d)^2}-\frac {d^{5/3} \tan ^{-1}\left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} c^{2/3} (b c-a d)^2}+\frac {b^{2/3} (2 b c-5 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} (b c-a d)^2}+\frac {d^{5/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} (b c-a d)^2}-\frac {b^{2/3} (2 b c-5 a d) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} (b c-a d)^2}-\frac {d^{5/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{2/3} (b c-a d)^2}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 337, normalized size = 0.97 \[ \frac {-b^{2/3} c^{2/3} \left (a+b x^3\right ) (2 b c-5 a d) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-3 a^{5/3} d^{5/3} \left (a+b x^3\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )+6 a^{2/3} b c^{2/3} x (b c-a d)+6 a^{5/3} d^{5/3} \left (a+b x^3\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )-6 \sqrt {3} a^{5/3} d^{5/3} \left (a+b x^3\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )+2 b^{2/3} c^{2/3} \left (a+b x^3\right ) (2 b c-5 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-2 \sqrt {3} b^{2/3} c^{2/3} \left (a+b x^3\right ) (2 b c-5 a d) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{18 a^{5/3} c^{2/3} \left (a+b x^3\right ) (b c-a d)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 7.38, size = 440, normalized size = 1.27 \[ -\frac {2 \, \sqrt {3} {\left ({\left (2 \, b^{2} c - 5 \, a b d\right )} x^{3} + 2 \, a b c - 5 \, a^{2} d\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) - 6 \, \sqrt {3} {\left (a b d x^{3} + a^{2} d\right )} \left (\frac {d^{2}}{c^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} c x \left (\frac {d^{2}}{c^{2}}\right )^{\frac {2}{3}} - \sqrt {3} d}{3 \, d}\right ) - {\left ({\left (2 \, b^{2} c - 5 \, a b d\right )} x^{3} + 2 \, a b c - 5 \, a^{2} d\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b^{2} x^{2} + a b x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} + a^{2} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}}\right ) + 3 \, {\left (a b d x^{3} + a^{2} d\right )} \left (\frac {d^{2}}{c^{2}}\right )^{\frac {1}{3}} \log \left (d^{2} x^{2} - c d x \left (\frac {d^{2}}{c^{2}}\right )^{\frac {1}{3}} + c^{2} \left (\frac {d^{2}}{c^{2}}\right )^{\frac {2}{3}}\right ) + 2 \, {\left ({\left (2 \, b^{2} c - 5 \, a b d\right )} x^{3} + 2 \, a b c - 5 \, a^{2} d\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b x - a \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right ) - 6 \, {\left (a b d x^{3} + a^{2} d\right )} \left (\frac {d^{2}}{c^{2}}\right )^{\frac {1}{3}} \log \left (d x + c \left (\frac {d^{2}}{c^{2}}\right )^{\frac {1}{3}}\right ) - 6 \, {\left (b^{2} c - a b d\right )} x}{18 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} + {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 443, normalized size = 1.28 \[ -\frac {d^{2} \left (-\frac {c}{d}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {c}{d}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )}} + \frac {\left (-c d^{2}\right )^{\frac {1}{3}} d \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {c}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {c}{d}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} b^{2} c^{3} - 2 \, \sqrt {3} a b c^{2} d + \sqrt {3} a^{2} c d^{2}} + \frac {\left (-c d^{2}\right )^{\frac {1}{3}} d \log \left (x^{2} + x \left (-\frac {c}{d}\right )^{\frac {1}{3}} + \left (-\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )}} - \frac {{\left (2 \, b^{2} c - 5 \, a b d\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )}} + \frac {{\left (2 \, \left (-a b^{2}\right )^{\frac {1}{3}} b c - 5 \, \left (-a b^{2}\right )^{\frac {1}{3}} a d\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, {\left (\sqrt {3} a^{2} b^{2} c^{2} - 2 \, \sqrt {3} a^{3} b c d + \sqrt {3} a^{4} d^{2}\right )}} + \frac {{\left (2 \, \left (-a b^{2}\right )^{\frac {1}{3}} b c - 5 \, \left (-a b^{2}\right )^{\frac {1}{3}} a d\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )}} + \frac {b x}{3 \, {\left (b x^{3} + a\right )} {\left (a b c - a^{2} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 406, normalized size = 1.17 \[ \frac {b^{2} c x}{3 \left (a d -b c \right )^{2} \left (b \,x^{3}+a \right ) a}-\frac {b d x}{3 \left (a d -b c \right )^{2} \left (b \,x^{3}+a \right )}+\frac {2 \sqrt {3}\, b c \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (a d -b c \right )^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} a}+\frac {2 b c \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (a d -b c \right )^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} a}-\frac {b c \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \left (a d -b c \right )^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} a}-\frac {5 \sqrt {3}\, d \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (a d -b c \right )^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, d \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {c}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (a d -b c \right )^{2} \left (\frac {c}{d}\right )^{\frac {2}{3}}}-\frac {5 d \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (a d -b c \right )^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {d \ln \left (x +\left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 \left (a d -b c \right )^{2} \left (\frac {c}{d}\right )^{\frac {2}{3}}}+\frac {5 d \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \left (a d -b c \right )^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {d \ln \left (x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{3}} x +\left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 \left (a d -b c \right )^{2} \left (\frac {c}{d}\right )^{\frac {2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.29, size = 489, normalized size = 1.41 \[ \frac {\sqrt {3} {\left (2 \, b c - 5 \, a d\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, {\left (a b^{2} c^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, a^{2} b c d \left (\frac {a}{b}\right )^{\frac {1}{3}} + a^{3} d^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {\sqrt {3} d \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 (b^{2} c^{2} \left (\frac {c}{d}\right )^{\frac {1}{3}} - 2 \, a b c d \left (\frac {c}{d}\right )^{\frac {1}{3}} + a^{2} d^{2} \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )} \left (\frac {c}{d}\right )^{\frac {1}{3}}} + \frac {b x}{3 \, {\left (a^{2} b c - a^{3} d + {\left (a b^{2} c - a^{2} b d\right )} x^{3}\right )}} - \frac {{\left (2 \, b c - 5 \, a d\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, {\left (a b^{2} c^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a^{2} b c d \left (\frac {a}{b}\right )^{\frac {2}{3}} + a^{3} d^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}} - \frac {d \log \left (x^{2} - x \left (\frac {c}{d}\right )^{\frac {1}{3}} + \left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b^{2} c^{2} \left (\frac {c}{d}\right )^{\frac {2}{3}} - 2 \, a b c d \left (\frac {c}{d}\right )^{\frac {2}{3}} + a^{2} d^{2} \left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}} + \frac {{\left (2 \, b c - 5 \, a d\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, {\left (a b^{2} c^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a^{2} b c d \left (\frac {a}{b}\right )^{\frac {2}{3}} + a^{3} d^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}} + \frac {d \log \left (x + \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 \, {\left (b^{2} c^{2} \left (\frac {c}{d}\right )^{\frac {2}{3}} - 2 \, a b c d \left (\frac {c}{d}\right )^{\frac {2}{3}} + a^{2} d^{2} \left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 15.93, size = 2492, normalized size = 7.20 \[ \text {result too large to display} \]
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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