Optimal. Leaf size=288 \[ -\frac {b^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} (b c-a d)}+\frac {d^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} c^{2/3} (b c-a d)}+\frac {b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} (b c-a d)}-\frac {d^{2/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} (b c-a d)}-\frac {b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} (b c-a d)}+\frac {d^{2/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{2/3} (b c-a d)} \]
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Rubi [A]
time = 0.10, antiderivative size = 288, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {400, 206, 31,
648, 631, 210, 642} \begin {gather*} -\frac {b^{2/3} \text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} (b c-a d)}-\frac {b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} (b c-a d)}+\frac {b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} (b c-a d)}+\frac {d^{2/3} \text {ArcTan}\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)}+\frac {d^{2/3} \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)}-\frac {d^{2/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 206
Rule 210
Rule 400
Rule 631
Rule 642
Rule 648
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx &=\frac {b \int \frac {1}{a+b x^3} \, dx}{b c-a d}-\frac {d \int \frac {1}{c+d x^3} \, dx}{b c-a d}\\ &=\frac {b \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^{2/3} (b c-a d)}+\frac {b \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}{3 a^{2/3} (b c-a d)}-\frac {d \int \frac {1}{\sqrt [3]{c}+\sqrt [3]{d} x} \, dx}{3 c^{2/3} (b c-a d)}-\frac {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} (b c-a d)}\\ &=\frac {b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} (b c-a d)}-\frac {d^{2/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} (b c-a d)}-\frac {b^{2/3} \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^{2/3} (b c-a d)}+\frac {b \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 \sqrt [3]{a} (b c-a d)}+\frac {d^{2/3} \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)}-\frac {d \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)}\\ &=\frac {b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} (b c-a d)}-\frac {d^{2/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} (b c-a d)}-\frac {b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} (b c-a d)}+\frac {d^{2/3} \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)}+\frac {b^{2/3} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{a^{2/3} (b c-a d)}-\frac {d^{2/3} \text {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)}\\ &=-\frac {b^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} (b c-a d)}+\frac {d^{2/3} \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)}+\frac {b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} (b c-a d)}-\frac {d^{2/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} (b c-a d)}-\frac {b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} (b c-a d)}+\frac {d^{2/3} \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)}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 224, normalized size = 0.78 \begin {gather*} \frac {\frac {2 \sqrt {3} b^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{2/3}}-\frac {2 \sqrt {3} d^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{c^{2/3}}-\frac {2 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{2/3}}+\frac {2 d^{2/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{c^{2/3}}+\frac {b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{2/3}}-\frac {d^{2/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{c^{2/3}}}{-6 b c+6 a d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.29, size = 207, normalized size = 0.72
method | result | size |
default | \(-\frac {\left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right ) b}{a d -b c}+\frac {\left (\frac {\ln \left (x +\left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 d \left (\frac {c}{d}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{3}} x +\left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 d \left (\frac {c}{d}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {c}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 d \left (\frac {c}{d}\right )^{\frac {2}{3}}}\right ) d}{a d -b c}\) | \(207\) |
risch | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (a^{5} d^{3}-3 a^{4} b c \,d^{2}+3 a^{3} b^{2} c^{2} d -b^{3} a^{2} c^{3}\right ) \textit {\_Z}^{3}+b^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-a^{5} d^{5}+3 a^{4} b c \,d^{4}-2 a^{3} b^{2} c^{2} d^{3}-2 a^{2} b^{3} c^{3} d^{2}+3 a \,b^{4} c^{4} d -b^{5} c^{5}\right ) \textit {\_R}^{3}-2 b^{2} d^{2}\right ) x +\left (-a^{6} c \,d^{5}+3 a^{5} b \,c^{2} d^{4}-2 a^{4} b^{2} c^{3} d^{3}-2 a^{3} b^{3} c^{4} d^{2}+3 a^{2} b^{4} c^{5} d -a \,c^{6} b^{5}\right ) \textit {\_R}^{4}+\left (a^{2} b \,d^{3}-2 a \,b^{2} c \,d^{2}+b^{3} c^{2} d \right ) \textit {\_R} \right )\right )}{3}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (a^{3} c^{2} d^{3}-3 a^{2} b \,c^{3} d^{2}+3 a \,b^{2} c^{4} d -b^{3} c^{5}\right ) \textit {\_Z}^{3}-d^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-a^{5} d^{5}+3 a^{4} b c \,d^{4}-2 a^{3} b^{2} c^{2} d^{3}-2 a^{2} b^{3} c^{3} d^{2}+3 a \,b^{4} c^{4} d -b^{5} c^{5}\right ) \textit {\_R}^{3}-2 b^{2} d^{2}\right ) x +\left (-a^{6} c \,d^{5}+3 a^{5} b \,c^{2} d^{4}-2 a^{4} b^{2} c^{3} d^{3}-2 a^{3} b^{3} c^{4} d^{2}+3 a^{2} b^{4} c^{5} d -a \,c^{6} b^{5}\right ) \textit {\_R}^{4}+\left (a^{2} b \,d^{3}-2 a \,b^{2} c \,d^{2}+b^{3} c^{2} d \right ) \textit {\_R} \right )\right )}{3}\) | \(490\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 293, normalized size = 1.02 \begin {gather*} \frac {\sqrt {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 (b c \left (\frac {a}{b}\right )^{\frac {1}{3}} - a d \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {\sqrt {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 \left (\frac {c}{d}\right )^{\frac {1}{3}} - a d \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )} \left (\frac {c}{d}\right )^{\frac {1}{3}}} - \frac {\log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b c \left (\frac {a}{b}\right )^{\frac {2}{3}} - a d \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}} + \frac {\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 \left (\frac {c}{d}\right )^{\frac {2}{3}} - a d \left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}} + \frac {\log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, {\left (b c \left (\frac {a}{b}\right )^{\frac {2}{3}} - a d \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}} - \frac {\log \left (x + \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 \, {\left (b c \left (\frac {c}{d}\right )^{\frac {2}{3}} - a d \left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.06, size = 254, normalized size = 0.88 \begin {gather*} -\frac {2 \, \sqrt {3} \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 ) + 2 \, \sqrt {3} \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 (-\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 ) - \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 (-\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 ) + 2 \, \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 c - a d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.70, size = 278, normalized size = 0.97 \begin {gather*} -\frac {b \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (a b c - a^{2} d\right )}} + \frac {d \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 c^{2} - a c d\right )}} + \frac {\left (-a b^{2}\right )^{\frac {1}{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 b c - \sqrt {3} a^{2} d} - \frac {\left (-c d^{2}\right )^{\frac {1}{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^{2} - \sqrt {3} a c d} + \frac {\left (-a b^{2}\right )^{\frac {1}{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 b c - a^{2} d\right )}} - \frac {\left (-c d^{2}\right )^{\frac {1}{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^{2} - a c d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 9.01, size = 1364, normalized size = 4.74 \begin {gather*} \ln \left (\frac {{\left (-\frac {b^2}{a^2\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}\,\left (9\,a^2\,b^4\,d^6+9\,b^6\,c^2\,d^4-18\,a\,b^5\,c\,d^5-9\,b^3\,d^3\,\left (x+a\,c\,{\left (-\frac {b^2}{a^2\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}\right )\,\left (a\,d+b\,c\right )\,{\left (a\,d-b\,c\right )}^4\,{\left (-\frac {b^2}{a^2\,{\left (a\,d-b\,c\right )}^3}\right )}^{2/3}\right )}{3}-6\,b^5\,d^5\,x\right )\,{\left (-\frac {b^2}{27\,a^5\,d^3-81\,a^4\,b\,c\,d^2+81\,a^3\,b^2\,c^2\,d-27\,a^2\,b^3\,c^3}\right )}^{1/3}+\ln \left (\frac {{\left (\frac {d^2}{c^2\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}\,\left (9\,a^2\,b^4\,d^6+9\,b^6\,c^2\,d^4-18\,a\,b^5\,c\,d^5-9\,b^3\,d^3\,\left (x+a\,c\,{\left (\frac {d^2}{c^2\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}\right )\,\left (a\,d+b\,c\right )\,{\left (a\,d-b\,c\right )}^4\,{\left (\frac {d^2}{c^2\,{\left (a\,d-b\,c\right )}^3}\right )}^{2/3}\right )}{3}-6\,b^5\,d^5\,x\right )\,{\left (-\frac {d^2}{-27\,a^3\,c^2\,d^3+81\,a^2\,b\,c^3\,d^2-81\,a\,b^2\,c^4\,d+27\,b^3\,c^5}\right )}^{1/3}+\frac {\ln \left (6\,b^5\,d^5\,x+\frac {\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-\frac {b^2}{a^2\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}\,\left (\frac {{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (81\,b^3\,d^3\,x\,\left (a\,d+b\,c\right )\,{\left (a\,d-b\,c\right )}^4+\frac {81\,a\,b^3\,c\,d^3\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (a\,d+b\,c\right )\,{\left (a\,d-b\,c\right )}^4\,{\left (-\frac {b^2}{a^2\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}}{2}\right )\,{\left (-\frac {b^2}{a^2\,{\left (a\,d-b\,c\right )}^3}\right )}^{2/3}}{36}-9\,a^2\,b^4\,d^6-9\,b^6\,c^2\,d^4+18\,a\,b^5\,c\,d^5\right )}{6}\right )\,{\left (-\frac {b^2}{27\,a^5\,d^3-81\,a^4\,b\,c\,d^2+81\,a^3\,b^2\,c^2\,d-27\,a^2\,b^3\,c^3}\right )}^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}-\frac {\ln \left (6\,b^5\,d^5\,x-\frac {\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-\frac {b^2}{a^2\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}\,\left (\frac {{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (81\,b^3\,d^3\,x\,\left (a\,d+b\,c\right )\,{\left (a\,d-b\,c\right )}^4-\frac {81\,a\,b^3\,c\,d^3\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (a\,d+b\,c\right )\,{\left (a\,d-b\,c\right )}^4\,{\left (-\frac {b^2}{a^2\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}}{2}\right )\,{\left (-\frac {b^2}{a^2\,{\left (a\,d-b\,c\right )}^3}\right )}^{2/3}}{36}-9\,a^2\,b^4\,d^6-9\,b^6\,c^2\,d^4+18\,a\,b^5\,c\,d^5\right )}{6}\right )\,{\left (-\frac {b^2}{27\,a^5\,d^3-81\,a^4\,b\,c\,d^2+81\,a^3\,b^2\,c^2\,d-27\,a^2\,b^3\,c^3}\right )}^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}+\frac {\ln \left (6\,b^5\,d^5\,x+\frac {\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {d^2}{c^2\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}\,\left (\frac {{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (81\,b^3\,d^3\,x\,\left (a\,d+b\,c\right )\,{\left (a\,d-b\,c\right )}^4+\frac {81\,a\,b^3\,c\,d^3\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (a\,d+b\,c\right )\,{\left (a\,d-b\,c\right )}^4\,{\left (\frac {d^2}{c^2\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}}{2}\right )\,{\left (\frac {d^2}{c^2\,{\left (a\,d-b\,c\right )}^3}\right )}^{2/3}}{36}-9\,a^2\,b^4\,d^6-9\,b^6\,c^2\,d^4+18\,a\,b^5\,c\,d^5\right )}{6}\right )\,{\left (-\frac {d^2}{-27\,a^3\,c^2\,d^3+81\,a^2\,b\,c^3\,d^2-81\,a\,b^2\,c^4\,d+27\,b^3\,c^5}\right )}^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}-\frac {\ln \left (6\,b^5\,d^5\,x-\frac {\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {d^2}{c^2\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}\,\left (\frac {{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (81\,b^3\,d^3\,x\,\left (a\,d+b\,c\right )\,{\left (a\,d-b\,c\right )}^4-\frac {81\,a\,b^3\,c\,d^3\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (a\,d+b\,c\right )\,{\left (a\,d-b\,c\right )}^4\,{\left (\frac {d^2}{c^2\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}}{2}\right )\,{\left (\frac {d^2}{c^2\,{\left (a\,d-b\,c\right )}^3}\right )}^{2/3}}{36}-9\,a^2\,b^4\,d^6-9\,b^6\,c^2\,d^4+18\,a\,b^5\,c\,d^5\right )}{6}\right )\,{\left (-\frac {d^2}{-27\,a^3\,c^2\,d^3+81\,a^2\,b\,c^3\,d^2-81\,a\,b^2\,c^4\,d+27\,b^3\,c^5}\right )}^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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