Optimal. Leaf size=352 \[ -\frac {b^{10/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{10/3} (b c-a d)}+\frac {b^{10/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{10/3} (b c-a d)}+\frac {b^{10/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{10/3} (b c-a d)}+\frac {a d+b c}{4 a^2 c^2 x^4}-\frac {a^2 d^2+a b c d+b^2 c^2}{a^3 c^3 x}+\frac {d^{10/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{10/3} (b c-a d)}-\frac {d^{10/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{10/3} (b c-a d)}-\frac {d^{10/3} \tan ^{-1}\left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} c^{10/3} (b c-a d)}-\frac {1}{7 a c x^7} \]
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Rubi [A] time = 0.50, antiderivative size = 352, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {480, 583, 584, 292, 31, 634, 617, 204, 628} \[ -\frac {a^2 d^2+a b c d+b^2 c^2}{a^3 c^3 x}-\frac {b^{10/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{10/3} (b c-a d)}+\frac {b^{10/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{10/3} (b c-a d)}+\frac {b^{10/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{10/3} (b c-a d)}+\frac {a d+b c}{4 a^2 c^2 x^4}+\frac {d^{10/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{10/3} (b c-a d)}-\frac {d^{10/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{10/3} (b c-a d)}-\frac {d^{10/3} \tan ^{-1}\left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} c^{10/3} (b c-a d)}-\frac {1}{7 a c x^7} \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 292
Rule 480
Rule 583
Rule 584
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {1}{x^8 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx &=-\frac {1}{7 a c x^7}+\frac {\int \frac {-7 (b c+a d)-7 b d x^3}{x^5 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx}{7 a c}\\ &=-\frac {1}{7 a c x^7}+\frac {b c+a d}{4 a^2 c^2 x^4}-\frac {\int \frac {-28 \left (b^2 c^2+a b c d+a^2 d^2\right )-28 b d (b c+a d) x^3}{x^2 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx}{28 a^2 c^2}\\ &=-\frac {1}{7 a c x^7}+\frac {b c+a d}{4 a^2 c^2 x^4}-\frac {b^2 c^2+a b c d+a^2 d^2}{a^3 c^3 x}+\frac {\int \frac {x \left (-28 (b c+a d) \left (b^2 c^2+a^2 d^2\right )-28 b d \left (b^2 c^2+a b c d+a^2 d^2\right ) x^3\right )}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx}{28 a^3 c^3}\\ &=-\frac {1}{7 a c x^7}+\frac {b c+a d}{4 a^2 c^2 x^4}-\frac {b^2 c^2+a b c d+a^2 d^2}{a^3 c^3 x}+\frac {\int \left (-\frac {28 b^4 c^3 x}{(b c-a d) \left (a+b x^3\right )}-\frac {28 a^3 d^4 x}{(-b c+a d) \left (c+d x^3\right )}\right ) \, dx}{28 a^3 c^3}\\ &=-\frac {1}{7 a c x^7}+\frac {b c+a d}{4 a^2 c^2 x^4}-\frac {b^2 c^2+a b c d+a^2 d^2}{a^3 c^3 x}-\frac {b^4 \int \frac {x}{a+b x^3} \, dx}{a^3 (b c-a d)}+\frac {d^4 \int \frac {x}{c+d x^3} \, dx}{c^3 (b c-a d)}\\ &=-\frac {1}{7 a c x^7}+\frac {b c+a d}{4 a^2 c^2 x^4}-\frac {b^2 c^2+a b c d+a^2 d^2}{a^3 c^3 x}+\frac {b^{11/3} \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^{10/3} (b c-a d)}-\frac {b^{11/3} \int \frac {\sqrt [3]{a}+\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{3 a^{10/3} (b c-a d)}-\frac {d^{11/3} \int \frac {1}{\sqrt [3]{c}+\sqrt [3]{d} x} \, dx}{3 c^{10/3} (b c-a d)}+\frac {d^{11/3} \int \frac {\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^{10/3} (b c-a d)}\\ &=-\frac {1}{7 a c x^7}+\frac {b c+a d}{4 a^2 c^2 x^4}-\frac {b^2 c^2+a b c d+a^2 d^2}{a^3 c^3 x}+\frac {b^{10/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{10/3} (b c-a d)}-\frac {d^{10/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{10/3} (b c-a d)}-\frac {b^{10/3} \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}{6 a^{10/3} (b c-a d)}-\frac {b^{11/3} \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 a^3 (b c-a d)}+\frac {d^{10/3} \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^{10/3} (b c-a d)}+\frac {d^{11/3} \int \frac {1}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{2 c^3 (b c-a d)}\\ &=-\frac {1}{7 a c x^7}+\frac {b c+a d}{4 a^2 c^2 x^4}-\frac {b^2 c^2+a b c d+a^2 d^2}{a^3 c^3 x}+\frac {b^{10/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{10/3} (b c-a d)}-\frac {d^{10/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{10/3} (b c-a d)}-\frac {b^{10/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{10/3} (b c-a d)}+\frac {d^{10/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{10/3} (b c-a d)}-\frac {b^{10/3} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{a^{10/3} (b c-a d)}+\frac {d^{10/3} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}\right )}{c^{10/3} (b c-a d)}\\ &=-\frac {1}{7 a c x^7}+\frac {b c+a d}{4 a^2 c^2 x^4}-\frac {b^2 c^2+a b c d+a^2 d^2}{a^3 c^3 x}+\frac {b^{10/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{10/3} (b c-a d)}-\frac {d^{10/3} \tan ^{-1}\left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} c^{10/3} (b c-a d)}+\frac {b^{10/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{10/3} (b c-a d)}-\frac {d^{10/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{10/3} (b c-a d)}-\frac {b^{10/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{10/3} (b c-a d)}+\frac {d^{10/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{10/3} (b c-a d)}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 304, normalized size = 0.86 \[ \frac {-\frac {28 b^{10/3} x^7 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{10/3}}-\frac {28 \sqrt {3} b^{10/3} x^7 \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{10/3}}+\frac {14 b^{10/3} x^7 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{10/3}}+\frac {84 b^3 x^6}{a^3}-\frac {21 b^2 x^3}{a^2}+\frac {12 b}{a}+\frac {28 d^{10/3} x^7 \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{c^{10/3}}+\frac {28 \sqrt {3} d^{10/3} x^7 \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{c^{10/3}}-\frac {14 d^{10/3} x^7 \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{c^{10/3}}-\frac {84 d^3 x^6}{c^3}+\frac {21 d^2 x^3}{c^2}-\frac {12 d}{c}}{84 x^7 (a d-b c)} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.37, size = 332, normalized size = 0.94 \[ -\frac {28 \, \sqrt {3} b^{3} c^{3} x^{7} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} x \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) - 28 \, \sqrt {3} a^{3} d^{3} x^{7} \left (\frac {d}{c}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} x \left (\frac {d}{c}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) - 14 \, b^{3} c^{3} x^{7} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x^{2} - a x \left (-\frac {b}{a}\right )^{\frac {2}{3}} - a \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right ) - 14 \, a^{3} d^{3} x^{7} \left (\frac {d}{c}\right )^{\frac {1}{3}} \log \left (d x^{2} - c x \left (\frac {d}{c}\right )^{\frac {2}{3}} + c \left (\frac {d}{c}\right )^{\frac {1}{3}}\right ) + 28 \, b^{3} c^{3} x^{7} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x + a \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right ) + 28 \, a^{3} d^{3} x^{7} \left (\frac {d}{c}\right )^{\frac {1}{3}} \log \left (d x + c \left (\frac {d}{c}\right )^{\frac {2}{3}}\right ) + 84 \, {\left (b^{3} c^{3} - a^{3} d^{3}\right )} x^{6} + 12 \, a^{2} b c^{3} - 12 \, a^{3} c^{2} d - 21 \, {\left (a b^{2} c^{3} - a^{3} c d^{2}\right )} x^{3}}{84 \, {\left (a^{3} b c^{4} - a^{4} c^{3} d\right )} x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 377, normalized size = 1.07 \[ \frac {b^{4} \left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (a^{4} b c - a^{5} d\right )}} - \frac {d^{4} \left (-\frac {c}{d}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {c}{d}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (b c^{5} - a c^{4} d\right )}} + \frac {\left (-a b^{2}\right )^{\frac {2}{3}} b^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} a^{4} b c - \sqrt {3} a^{5} d} - \frac {\left (-c d^{2}\right )^{\frac {2}{3}} d^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {c}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {c}{d}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} b c^{5} - \sqrt {3} a c^{4} d} - \frac {\left (-a b^{2}\right )^{\frac {2}{3}} b^{2} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, {\left (a^{4} b c - a^{5} d\right )}} + \frac {\left (-c d^{2}\right )^{\frac {2}{3}} d^{2} \log \left (x^{2} + x \left (-\frac {c}{d}\right )^{\frac {1}{3}} + \left (-\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b c^{5} - a c^{4} d\right )}} - \frac {28 \, b^{2} c^{2} x^{6} + 28 \, a b c d x^{6} + 28 \, a^{2} d^{2} x^{6} - 7 \, a b c^{2} x^{3} - 7 \, a^{2} c d x^{3} + 4 \, a^{2} c^{2}}{28 \, a^{3} c^{3} x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 334, normalized size = 0.95 \[ \frac {\sqrt {3}\, b^{3} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{3}}-\frac {b^{3} \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{3}}+\frac {b^{3} \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{3}}-\frac {\sqrt {3}\, d^{3} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {c}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{3}} c^{3}}+\frac {d^{3} \ln \left (x +\left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{3}} c^{3}}-\frac {d^{3} \ln \left (x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{3}} x +\left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{3}} c^{3}}-\frac {d^{2}}{a \,c^{3} x}-\frac {b d}{a^{2} c^{2} x}-\frac {b^{2}}{a^{3} c x}+\frac {d}{4 a \,c^{2} x^{4}}+\frac {b}{4 a^{2} c \,x^{4}}-\frac {1}{7 a c \,x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.16, size = 376, normalized size = 1.07 \[ -\frac {\sqrt {3} b^{3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, {\left (a^{3} b c - a^{4} d\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {\sqrt {3} d^{3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {c}{d}\right )^{\frac {1}{3}}}\right )}{3 \, {\left (b c^{4} - a c^{3} d\right )} \left (\frac {c}{d}\right )^{\frac {1}{3}}} - \frac {b^{3} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, {\left (a^{3} b c \left (\frac {a}{b}\right )^{\frac {1}{3}} - a^{4} d \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}} + \frac {d^{3} \log \left (x^{2} - x \left (\frac {c}{d}\right )^{\frac {1}{3}} + \left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b c^{4} \left (\frac {c}{d}\right )^{\frac {1}{3}} - a c^{3} d \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}} + \frac {b^{3} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, {\left (a^{3} b c \left (\frac {a}{b}\right )^{\frac {1}{3}} - a^{4} d \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}} - \frac {d^{3} \log \left (x + \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 \, {\left (b c^{4} \left (\frac {c}{d}\right )^{\frac {1}{3}} - a c^{3} d \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}} - \frac {28 \, {\left (b^{2} c^{2} + a b c d + a^{2} d^{2}\right )} x^{6} + 4 \, a^{2} c^{2} - 7 \, {\left (a b c^{2} + a^{2} c d\right )} x^{3}}{28 \, a^{3} c^{3} x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.91, size = 1814, normalized size = 5.15 \[ \text {result too large to display} \]
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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