Optimal. Leaf size=311 \[ \frac {d x}{c}-\frac {\sqrt [6]{a} d \tan ^{-1}\left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 c^{7/6}}+\frac {\left (\sqrt {a} d-\sqrt {3} \sqrt {c} e\right ) \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 \sqrt [3]{a} c^{7/6}}-\frac {\left (\sqrt {a} d+\sqrt {3} \sqrt {c} e\right ) \tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 \sqrt [3]{a} c^{7/6}}-\frac {e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{6 \sqrt [3]{a} c^{2/3}}+\frac {\left (\sqrt {3} \sqrt {a} d+\sqrt {c} e\right ) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 \sqrt [3]{a} c^{7/6}}-\frac {\left (\sqrt {3} \sqrt {a} d-\sqrt {c} e\right ) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 \sqrt [3]{a} c^{7/6}} \]
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Rubi [A]
time = 0.20, antiderivative size = 311, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 10, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.588, Rules used = {1408, 1517,
1430, 649, 209, 266, 648, 631, 210, 642} \begin {gather*} \frac {\text {ArcTan}\left (\sqrt {3}-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right ) \left (\sqrt {a} d-\sqrt {3} \sqrt {c} e\right )}{6 \sqrt [3]{a} c^{7/6}}-\frac {\text {ArcTan}\left (\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}+\sqrt {3}\right ) \left (\sqrt {a} d+\sqrt {3} \sqrt {c} e\right )}{6 \sqrt [3]{a} c^{7/6}}-\frac {\sqrt [6]{a} d \text {ArcTan}\left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 c^{7/6}}+\frac {\left (\sqrt {3} \sqrt {a} d+\sqrt {c} e\right ) \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{12 \sqrt [3]{a} c^{7/6}}-\frac {\left (\sqrt {3} \sqrt {a} d-\sqrt {c} e\right ) \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{12 \sqrt [3]{a} c^{7/6}}-\frac {e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{6 \sqrt [3]{a} c^{2/3}}+\frac {d x}{c} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 210
Rule 266
Rule 631
Rule 642
Rule 648
Rule 649
Rule 1408
Rule 1430
Rule 1517
Rubi steps
\begin {align*} \int \frac {d+\frac {e}{x^3}}{c+\frac {a}{x^6}} \, dx &=\int \frac {x^3 \left (e+d x^3\right )}{a+c x^6} \, dx\\ &=\frac {d x}{c}-\frac {\int \frac {a d-c e x^3}{a+c x^6} \, dx}{c}\\ &=\frac {d x}{c}-\frac {\int \frac {2 a^{2/3} \sqrt [3]{c} d-\left (\sqrt {3} \sqrt {a} \sqrt {c} d+c e\right ) x}{1-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}} \, dx}{6 a^{2/3} c^{4/3}}-\frac {\int \frac {2 a^{2/3} \sqrt [3]{c} d+\left (\sqrt {3} \sqrt {a} \sqrt {c} d-c e\right ) x}{1+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}} \, dx}{6 a^{2/3} c^{4/3}}-\frac {\int \frac {a^{2/3} \sqrt [3]{c} d+c e x}{1+\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}} \, dx}{3 a^{2/3} c^{4/3}}\\ &=\frac {d x}{c}-\frac {d \int \frac {1}{1+\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}} \, dx}{3 c}-\frac {e \int \frac {x}{1+\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}} \, dx}{3 a^{2/3} \sqrt [3]{c}}-\frac {\left (\sqrt {3} \sqrt {a} d-\sqrt {c} e\right ) \int \frac {\frac {\sqrt {3} \sqrt [6]{c}}{\sqrt [6]{a}}+\frac {2 \sqrt [3]{c} x}{\sqrt [3]{a}}}{1+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}} \, dx}{12 \sqrt [3]{a} c^{7/6}}+\frac {\left (\sqrt {3} \sqrt {a} d+\sqrt {c} e\right ) \int \frac {-\frac {\sqrt {3} \sqrt [6]{c}}{\sqrt [6]{a}}+\frac {2 \sqrt [3]{c} x}{\sqrt [3]{a}}}{1-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}} \, dx}{12 \sqrt [3]{a} c^{7/6}}-\frac {\left (d-\frac {\sqrt {3} \sqrt {c} e}{\sqrt {a}}\right ) \int \frac {1}{1-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}} \, dx}{12 c}-\frac {\left (d+\frac {\sqrt {3} \sqrt {c} e}{\sqrt {a}}\right ) \int \frac {1}{1+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}} \, dx}{12 c}\\ &=\frac {d x}{c}-\frac {\sqrt [6]{a} d \tan ^{-1}\left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 c^{7/6}}-\frac {e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{6 \sqrt [3]{a} c^{2/3}}+\frac {\left (\sqrt {3} \sqrt {a} d+\sqrt {c} e\right ) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 \sqrt [3]{a} c^{7/6}}-\frac {\left (\sqrt {3} \sqrt {a} d-\sqrt {c} e\right ) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 \sqrt [3]{a} c^{7/6}}-\frac {\left (\sqrt {3} \sqrt {a} d-3 \sqrt {c} e\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1-\frac {2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}\right )}{18 \sqrt [3]{a} c^{7/6}}+\frac {\left (\sqrt {3} \sqrt {a} d+3 \sqrt {c} e\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1+\frac {2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}\right )}{18 \sqrt [3]{a} c^{7/6}}\\ &=\frac {d x}{c}-\frac {\sqrt [6]{a} d \tan ^{-1}\left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 c^{7/6}}+\frac {\left (\sqrt {a} d-\sqrt {3} \sqrt {c} e\right ) \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 \sqrt [3]{a} c^{7/6}}-\frac {\left (\sqrt {a} d+\sqrt {3} \sqrt {c} e\right ) \tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 \sqrt [3]{a} c^{7/6}}-\frac {e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{6 \sqrt [3]{a} c^{2/3}}+\frac {\left (\sqrt {3} \sqrt {a} d+\sqrt {c} e\right ) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 \sqrt [3]{a} c^{7/6}}-\frac {\left (\sqrt {3} \sqrt {a} d-\sqrt {c} e\right ) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 \sqrt [3]{a} c^{7/6}}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 346, normalized size = 1.11 \begin {gather*} \frac {d x}{c}-\frac {\sqrt [6]{a} d \tan ^{-1}\left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 c^{7/6}}+\frac {\left (-a^{7/6} \sqrt {c} d+\sqrt {3} a^{2/3} c e\right ) \tan ^{-1}\left (\frac {-\sqrt {3} \sqrt [6]{a}+2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 a c^{5/3}}+\frac {\left (-a^{7/6} \sqrt {c} d-\sqrt {3} a^{2/3} c e\right ) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a}+2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 a c^{5/3}}-\frac {e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{6 \sqrt [3]{a} c^{2/3}}-\frac {\left (-\sqrt {3} a^{7/6} \sqrt {c} d-a^{2/3} c e\right ) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 a c^{5/3}}-\frac {\left (\sqrt {3} a^{7/6} \sqrt {c} d-a^{2/3} c e\right ) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 a c^{5/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 339, normalized size = 1.09
method | result | size |
risch | \(\frac {d x}{c}+\frac {\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{6} c +a \right )}{\sum }\frac {\left (\textit {\_R}^{3} c e -a d \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}}}{6 c^{2}}\) | \(45\) |
default | \(\frac {d x}{c}+\frac {-\frac {c \left (\frac {a}{c}\right )^{\frac {7}{6}} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, d}{12 a}+\frac {c \left (\frac {a}{c}\right )^{\frac {2}{3}} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) e}{12 a}-\frac {\left (\frac {a}{c}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) d}{6}-\frac {c \left (\frac {a}{c}\right )^{\frac {2}{3}} \arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) \sqrt {3}\, e}{6 a}-\frac {c \left (\frac {a}{c}\right )^{\frac {2}{3}} e \ln \left (x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{6 a}-\frac {\left (\frac {a}{c}\right )^{\frac {1}{6}} d \arctan \left (\frac {x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{3}+\frac {\ln \left (-x^{2}+\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} x -\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) \left (\frac {a}{c}\right )^{\frac {1}{6}} \sqrt {3}\, d}{12}+\frac {c \ln \left (-x^{2}+\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} x -\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) \left (\frac {a}{c}\right )^{\frac {2}{3}} e}{12 a}-\frac {\left (\frac {a}{c}\right )^{\frac {1}{6}} d \arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6}+\frac {c \left (\frac {a}{c}\right )^{\frac {2}{3}} \sqrt {3}\, e \arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 a}}{c}\) | \(339\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 300, normalized size = 0.96 \begin {gather*} \frac {d x}{c} - \frac {\frac {2 \, c^{\frac {1}{3}} e \log \left (c^{\frac {1}{3}} x^{2} + a^{\frac {1}{3}}\right )}{a^{\frac {1}{3}}} + \frac {4 \, a^{\frac {1}{3}} d \arctan \left (\frac {c^{\frac {1}{3}} x}{\sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}}\right )}{\sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}} + \frac {{\left (\sqrt {3} a^{\frac {7}{6}} \sqrt {c} d - a^{\frac {2}{3}} c e\right )} \log \left (c^{\frac {1}{3}} x^{2} + \sqrt {3} a^{\frac {1}{6}} c^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{a c^{\frac {2}{3}}} - \frac {{\left (\sqrt {3} a^{\frac {7}{6}} \sqrt {c} d + a^{\frac {2}{3}} c e\right )} \log \left (c^{\frac {1}{3}} x^{2} - \sqrt {3} a^{\frac {1}{6}} c^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{a c^{\frac {2}{3}}} + \frac {2 \, {\left (\sqrt {3} a^{\frac {5}{6}} c^{\frac {7}{6}} e + a^{\frac {4}{3}} c^{\frac {2}{3}} d\right )} \arctan \left (\frac {2 \, c^{\frac {1}{3}} x + \sqrt {3} a^{\frac {1}{6}} c^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}}\right )}{a c^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}} - \frac {2 \, {\left (\sqrt {3} a^{\frac {5}{6}} c^{\frac {7}{6}} e - a^{\frac {4}{3}} c^{\frac {2}{3}} d\right )} \arctan \left (\frac {2 \, c^{\frac {1}{3}} x - \sqrt {3} a^{\frac {1}{6}} c^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}}\right )}{a c^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}}}{12 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3543 vs.
\(2 (218) = 436\).
time = 1.45, size = 3543, normalized size = 11.39 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 5.93, size = 167, normalized size = 0.54 \begin {gather*} \operatorname {RootSum} {\left (46656 t^{6} a^{2} c^{7} + t^{3} \left (- 1296 a^{2} c^{4} d^{2} e + 432 a c^{5} e^{3}\right ) + a^{3} d^{6} + 3 a^{2} c d^{4} e^{2} + 3 a c^{2} d^{2} e^{4} + c^{3} e^{6}, \left ( t \mapsto t \log {\left (x + \frac {- 1296 t^{4} a c^{5} e - 6 t a^{2} c d^{4} + 36 t a c^{2} d^{2} e^{2} - 6 t c^{3} e^{4}}{a^{2} d^{5} - 2 a c d^{3} e^{2} - 3 c^{2} d e^{4}} \right )} \right )\right )} + \frac {d x}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.54, size = 295, normalized size = 0.95 \begin {gather*} -\frac {{\left | c \right |} e \log \left (x^{2} + \left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{6 \, \left (a c^{5}\right )^{\frac {1}{3}}} + \frac {d x}{c} - \frac {\left (a c^{5}\right )^{\frac {1}{6}} d \arctan \left (\frac {x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{3 \, c^{2}} - \frac {{\left (\left (a c^{5}\right )^{\frac {1}{6}} a c^{2} d + \sqrt {3} \left (a c^{5}\right )^{\frac {2}{3}} e\right )} \arctan \left (\frac {2 \, x + \sqrt {3} \left (\frac {a}{c}\right )^{\frac {1}{6}}}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{6 \, a c^{4}} - \frac {{\left (\left (a c^{5}\right )^{\frac {1}{6}} a c^{2} d - \sqrt {3} \left (a c^{5}\right )^{\frac {2}{3}} e\right )} \arctan \left (\frac {2 \, x - \sqrt {3} \left (\frac {a}{c}\right )^{\frac {1}{6}}}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{6 \, a c^{4}} - \frac {{\left (\sqrt {3} \left (a c^{5}\right )^{\frac {1}{6}} a c^{2} d - \left (a c^{5}\right )^{\frac {2}{3}} e\right )} \log \left (x^{2} + \sqrt {3} x \left (\frac {a}{c}\right )^{\frac {1}{6}} + \left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{12 \, a c^{4}} + \frac {{\left (\sqrt {3} \left (a c^{5}\right )^{\frac {1}{6}} a c^{2} d + \left (a c^{5}\right )^{\frac {2}{3}} e\right )} \log \left (x^{2} - \sqrt {3} x \left (\frac {a}{c}\right )^{\frac {1}{6}} + \left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{12 \, a c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.10, size = 1308, normalized size = 4.21 \begin {gather*} \ln \left (e\,x\,\sqrt {-a^3\,c^7}-a^2\,c^4\,{\left (-\frac {a\,c^5\,e^3+a\,d^3\,\sqrt {-a^3\,c^7}-3\,a^2\,c^4\,d^2\,e-3\,c\,d\,e^2\,\sqrt {-a^3\,c^7}}{a^2\,c^7}\right )}^{1/3}+a^2\,c^3\,d\,x\right )\,{\left (-\frac {a\,c^5\,e^3+a\,d^3\,\sqrt {-a^3\,c^7}-3\,a^2\,c^4\,d^2\,e-3\,c\,d\,e^2\,\sqrt {-a^3\,c^7}}{216\,a^2\,c^7}\right )}^{1/3}+\ln \left (e\,x\,\sqrt {-a^3\,c^7}+a^2\,c^4\,{\left (-\frac {a\,c^5\,e^3-a\,d^3\,\sqrt {-a^3\,c^7}-3\,a^2\,c^4\,d^2\,e+3\,c\,d\,e^2\,\sqrt {-a^3\,c^7}}{a^2\,c^7}\right )}^{1/3}-a^2\,c^3\,d\,x\right )\,{\left (-\frac {a\,c^5\,e^3-a\,d^3\,\sqrt {-a^3\,c^7}-3\,a^2\,c^4\,d^2\,e+3\,c\,d\,e^2\,\sqrt {-a^3\,c^7}}{216\,a^2\,c^7}\right )}^{1/3}+\ln \left (2\,e\,x\,\sqrt {-a^3\,c^7}+a^2\,c^4\,{\left (-\frac {a\,c^5\,e^3+a\,d^3\,\sqrt {-a^3\,c^7}-3\,a^2\,c^4\,d^2\,e-3\,c\,d\,e^2\,\sqrt {-a^3\,c^7}}{a^2\,c^7}\right )}^{1/3}+2\,a^2\,c^3\,d\,x-\sqrt {3}\,a^2\,c^4\,{\left (-\frac {a\,c^5\,e^3+a\,d^3\,\sqrt {-a^3\,c^7}-3\,a^2\,c^4\,d^2\,e-3\,c\,d\,e^2\,\sqrt {-a^3\,c^7}}{a^2\,c^7}\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {a\,c^5\,e^3+a\,d^3\,\sqrt {-a^3\,c^7}-3\,a^2\,c^4\,d^2\,e-3\,c\,d\,e^2\,\sqrt {-a^3\,c^7}}{216\,a^2\,c^7}\right )}^{1/3}-\ln \left (2\,e\,x\,\sqrt {-a^3\,c^7}+a^2\,c^4\,{\left (-\frac {a\,c^5\,e^3+a\,d^3\,\sqrt {-a^3\,c^7}-3\,a^2\,c^4\,d^2\,e-3\,c\,d\,e^2\,\sqrt {-a^3\,c^7}}{a^2\,c^7}\right )}^{1/3}+2\,a^2\,c^3\,d\,x+\sqrt {3}\,a^2\,c^4\,{\left (-\frac {a\,c^5\,e^3+a\,d^3\,\sqrt {-a^3\,c^7}-3\,a^2\,c^4\,d^2\,e-3\,c\,d\,e^2\,\sqrt {-a^3\,c^7}}{a^2\,c^7}\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {a\,c^5\,e^3+a\,d^3\,\sqrt {-a^3\,c^7}-3\,a^2\,c^4\,d^2\,e-3\,c\,d\,e^2\,\sqrt {-a^3\,c^7}}{216\,a^2\,c^7}\right )}^{1/3}-\ln \left (a^2\,c^4\,{\left (-\frac {a\,c^5\,e^3-a\,d^3\,\sqrt {-a^3\,c^7}-3\,a^2\,c^4\,d^2\,e+3\,c\,d\,e^2\,\sqrt {-a^3\,c^7}}{a^2\,c^7}\right )}^{1/3}-2\,e\,x\,\sqrt {-a^3\,c^7}+2\,a^2\,c^3\,d\,x+\sqrt {3}\,a^2\,c^4\,{\left (-\frac {a\,c^5\,e^3-a\,d^3\,\sqrt {-a^3\,c^7}-3\,a^2\,c^4\,d^2\,e+3\,c\,d\,e^2\,\sqrt {-a^3\,c^7}}{a^2\,c^7}\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {a\,c^5\,e^3-a\,d^3\,\sqrt {-a^3\,c^7}-3\,a^2\,c^4\,d^2\,e+3\,c\,d\,e^2\,\sqrt {-a^3\,c^7}}{216\,a^2\,c^7}\right )}^{1/3}+\ln \left (2\,e\,x\,\sqrt {-a^3\,c^7}-a^2\,c^4\,{\left (-\frac {a\,c^5\,e^3-a\,d^3\,\sqrt {-a^3\,c^7}-3\,a^2\,c^4\,d^2\,e+3\,c\,d\,e^2\,\sqrt {-a^3\,c^7}}{a^2\,c^7}\right )}^{1/3}-2\,a^2\,c^3\,d\,x+\sqrt {3}\,a^2\,c^4\,{\left (-\frac {a\,c^5\,e^3-a\,d^3\,\sqrt {-a^3\,c^7}-3\,a^2\,c^4\,d^2\,e+3\,c\,d\,e^2\,\sqrt {-a^3\,c^7}}{a^2\,c^7}\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {a\,c^5\,e^3-a\,d^3\,\sqrt {-a^3\,c^7}-3\,a^2\,c^4\,d^2\,e+3\,c\,d\,e^2\,\sqrt {-a^3\,c^7}}{216\,a^2\,c^7}\right )}^{1/3}+\frac {d\,x}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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