Optimal. Leaf size=299 \[ -\frac {b^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{4/3} (b c-a d)}+\frac {b^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{4/3} (b c-a d)}+\frac {b^{4/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{4/3} (b c-a d)}+\frac {d^{4/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{4/3} (b c-a d)}-\frac {d^{4/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{4/3} (b c-a d)}-\frac {d^{4/3} \tan ^{-1}\left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} c^{4/3} (b c-a d)}-\frac {1}{a c x} \]
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Rubi [A] time = 0.27, antiderivative size = 299, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {480, 584, 292, 31, 634, 617, 204, 628} \[ -\frac {b^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{4/3} (b c-a d)}+\frac {b^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{4/3} (b c-a d)}+\frac {b^{4/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{4/3} (b c-a d)}+\frac {d^{4/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{4/3} (b c-a d)}-\frac {d^{4/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{4/3} (b c-a d)}-\frac {d^{4/3} \tan ^{-1}\left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} c^{4/3} (b c-a d)}-\frac {1}{a c x} \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 292
Rule 480
Rule 584
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx &=-\frac {1}{a c x}+\frac {\int \frac {x \left (-b c-a d-b d x^3\right )}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx}{a c}\\ &=-\frac {1}{a c x}+\frac {\int \left (-\frac {b^2 c x}{(b c-a d) \left (a+b x^3\right )}-\frac {a d^2 x}{(-b c+a d) \left (c+d x^3\right )}\right ) \, dx}{a c}\\ &=-\frac {1}{a c x}-\frac {b^2 \int \frac {x}{a+b x^3} \, dx}{a (b c-a d)}+\frac {d^2 \int \frac {x}{c+d x^3} \, dx}{c (b c-a d)}\\ &=-\frac {1}{a c x}+\frac {b^{5/3} \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^{4/3} (b c-a d)}-\frac {b^{5/3} \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^{4/3} (b c-a d)}-\frac {d^{5/3} \int \frac {1}{\sqrt [3]{c}+\sqrt [3]{d} x} \, dx}{3 c^{4/3} (b c-a d)}+\frac {d^{5/3} \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^{4/3} (b c-a d)}\\ &=-\frac {1}{a c x}+\frac {b^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{4/3} (b c-a d)}-\frac {d^{4/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{4/3} (b c-a d)}-\frac {b^{4/3} \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^{4/3} (b c-a d)}-\frac {b^{5/3} \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 a (b c-a d)}+\frac {d^{4/3} \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^{4/3} (b c-a d)}+\frac {d^{5/3} \int \frac {1}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{2 c (b c-a d)}\\ &=-\frac {1}{a c x}+\frac {b^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{4/3} (b c-a d)}-\frac {d^{4/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{4/3} (b c-a d)}-\frac {b^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{4/3} (b c-a d)}+\frac {d^{4/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{4/3} (b c-a d)}-\frac {b^{4/3} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{a^{4/3} (b c-a d)}+\frac {d^{4/3} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}\right )}{c^{4/3} (b c-a d)}\\ &=-\frac {1}{a c x}+\frac {b^{4/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{4/3} (b c-a d)}-\frac {d^{4/3} \tan ^{-1}\left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} c^{4/3} (b c-a d)}+\frac {b^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{4/3} (b c-a d)}-\frac {d^{4/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{4/3} (b c-a d)}-\frac {b^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{4/3} (b c-a d)}+\frac {d^{4/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{4/3} (b c-a d)}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 244, normalized size = 0.82 \[ \frac {\frac {b^{4/3} x \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{4/3}}-\frac {2 b^{4/3} x \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{4/3}}-\frac {2 \sqrt {3} b^{4/3} x \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{4/3}}+\frac {6 b}{a}-\frac {d^{4/3} x \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{c^{4/3}}+\frac {2 d^{4/3} x \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{c^{4/3}}+\frac {2 \sqrt {3} d^{4/3} x \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{c^{4/3}}-\frac {6 d}{c}}{6 a d x-6 b c x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 238, normalized size = 0.80 \[ -\frac {2 \, \sqrt {3} b c x \left (-\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} x \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) - 2 \, \sqrt {3} a d x \left (\frac {d}{c}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} x \left (\frac {d}{c}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) - b c x \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x^{2} - a x \left (-\frac {b}{a}\right )^{\frac {2}{3}} - a \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right ) - a d x \left (\frac {d}{c}\right )^{\frac {1}{3}} \log \left (d x^{2} - c x \left (\frac {d}{c}\right )^{\frac {2}{3}} + c \left (\frac {d}{c}\right )^{\frac {1}{3}}\right ) + 2 \, b c x \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x + a \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right ) + 2 \, a d x \left (\frac {d}{c}\right )^{\frac {1}{3}} \log \left (d x + c \left (\frac {d}{c}\right )^{\frac {2}{3}}\right ) + 6 \, b c - 6 \, a d}{6 \, {\left (a b c^{2} - a^{2} c d\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 305, normalized size = 1.02 \[ \frac {b^{2} \left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (a^{2} b c - a^{3} d\right )}} - \frac {d^{2} \left (-\frac {c}{d}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {c}{d}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (b c^{3} - a c^{2} d\right )}} + \frac {\left (-a b^{2}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} a^{2} b c - \sqrt {3} a^{3} d} - \frac {\left (-c d^{2}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {c}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {c}{d}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} b c^{3} - \sqrt {3} a c^{2} d} - \frac {\left (-a b^{2}\right )^{\frac {2}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, {\left (a^{2} b c - a^{3} d\right )}} + \frac {\left (-c d^{2}\right )^{\frac {2}{3}} \log \left (x^{2} + x \left (-\frac {c}{d}\right )^{\frac {1}{3}} + \left (-\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b c^{3} - a c^{2} d\right )}} - \frac {1}{a c x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 257, normalized size = 0.86 \[ \frac {\sqrt {3}\, b \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{3}} a}-\frac {b \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{3}} a}+\frac {b \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{3}} a}-\frac {\sqrt {3}\, d \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {c}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{3}} c}+\frac {d \ln \left (x +\left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{3}} c}-\frac {d \ln \left (x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{3}} x +\left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{3}} c}-\frac {1}{a c x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.20, size = 300, normalized size = 1.00 \[ -\frac {\sqrt {3} b \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, {\left (a b c - a^{2} d\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {\sqrt {3} d \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {c}{d}\right )^{\frac {1}{3}}}\right )}{3 \, {\left (b c^{2} - a c d\right )} \left (\frac {c}{d}\right )^{\frac {1}{3}}} - \frac {b \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, {\left (a b c \left (\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} d \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}} + \frac {d \log \left (x^{2} - x \left (\frac {c}{d}\right )^{\frac {1}{3}} + \left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b c^{2} \left (\frac {c}{d}\right )^{\frac {1}{3}} - a c d \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}} + \frac {b \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, {\left (a b c \left (\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} d \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}} - \frac {d \log \left (x + \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 \, {\left (b c^{2} \left (\frac {c}{d}\right )^{\frac {1}{3}} - a c d \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}} - \frac {1}{a c x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.85, size = 716, normalized size = 2.39 \[ \ln \left (b-a^2\,d\,x\,{\left (-\frac {b^4}{a^4\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}+a\,b\,c\,x\,{\left (-\frac {b^4}{a^4\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}\right )\,{\left (-\frac {b^4}{27\,a^7\,d^3-81\,a^6\,b\,c\,d^2+81\,a^5\,b^2\,c^2\,d-27\,a^4\,b^3\,c^3}\right )}^{1/3}+\ln \left (d-b\,c^2\,x\,{\left (\frac {d^4}{c^4\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}+a\,c\,d\,x\,{\left (\frac {d^4}{c^4\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}\right )\,{\left (-\frac {d^4}{-27\,a^3\,c^4\,d^3+81\,a^2\,b\,c^5\,d^2-81\,a\,b^2\,c^6\,d+27\,b^3\,c^7}\right )}^{1/3}-\frac {1}{a\,c\,x}-\frac {\ln \left (b+2\,a^2\,d\,x\,{\left (-\frac {b^4}{a^4\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}-2\,a\,b\,c\,x\,{\left (-\frac {b^4}{a^4\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}-\sqrt {3}\,b\,1{}\mathrm {i}\right )\,{\left (-\frac {b^4}{27\,a^7\,d^3-81\,a^6\,b\,c\,d^2+81\,a^5\,b^2\,c^2\,d-27\,a^4\,b^3\,c^3}\right )}^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}+\frac {\ln \left (b+2\,a^2\,d\,x\,{\left (-\frac {b^4}{a^4\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}-2\,a\,b\,c\,x\,{\left (-\frac {b^4}{a^4\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}+\sqrt {3}\,b\,1{}\mathrm {i}\right )\,{\left (-\frac {b^4}{27\,a^7\,d^3-81\,a^6\,b\,c\,d^2+81\,a^5\,b^2\,c^2\,d-27\,a^4\,b^3\,c^3}\right )}^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}-\frac {\ln \left (d+2\,b\,c^2\,x\,{\left (\frac {d^4}{c^4\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}-2\,a\,c\,d\,x\,{\left (\frac {d^4}{c^4\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}-\sqrt {3}\,d\,1{}\mathrm {i}\right )\,{\left (-\frac {d^4}{-27\,a^3\,c^4\,d^3+81\,a^2\,b\,c^5\,d^2-81\,a\,b^2\,c^6\,d+27\,b^3\,c^7}\right )}^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}+\frac {\ln \left (d+2\,b\,c^2\,x\,{\left (\frac {d^4}{c^4\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}-2\,a\,c\,d\,x\,{\left (\frac {d^4}{c^4\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}+\sqrt {3}\,d\,1{}\mathrm {i}\right )\,{\left (-\frac {d^4}{-27\,a^3\,c^4\,d^3+81\,a^2\,b\,c^5\,d^2-81\,a\,b^2\,c^6\,d+27\,b^3\,c^7}\right )}^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2} \]
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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