Optimal. Leaf size=288 \[ \frac {\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{a} (b c-a d)}-\frac {\sqrt [3]{d} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 \sqrt [3]{c} (b c-a d)}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} (b c-a d)}+\frac {\sqrt [3]{d} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 \sqrt [3]{c} (b c-a d)}-\frac {\sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{a} (b c-a d)}+\frac {\sqrt [3]{d} \tan ^{-1}\left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} \sqrt [3]{c} (b c-a d)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.14, antiderivative size = 288, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {482, 292, 31, 634, 617, 204, 628} \[ \frac {\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{a} (b c-a d)}-\frac {\sqrt [3]{d} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 \sqrt [3]{c} (b c-a d)}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} (b c-a d)}+\frac {\sqrt [3]{d} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 \sqrt [3]{c} (b c-a d)}-\frac {\sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{a} (b c-a d)}+\frac {\sqrt [3]{d} \tan ^{-1}\left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} \sqrt [3]{c} (b c-a d)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 204
Rule 292
Rule 482
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {x}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx &=\frac {b \int \frac {x}{a+b x^3} \, dx}{b c-a d}-\frac {d \int \frac {x}{c+d x^3} \, dx}{b c-a d}\\ &=-\frac {b^{2/3} \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 \sqrt [3]{a} (b c-a d)}+\frac {b^{2/3} \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 \sqrt [3]{a} (b c-a d)}+\frac {d^{2/3} \int \frac {1}{\sqrt [3]{c}+\sqrt [3]{d} x} \, dx}{3 \sqrt [3]{c} (b c-a d)}-\frac {d^{2/3} \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 \sqrt [3]{c} (b c-a d)}\\ &=-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} (b c-a d)}+\frac {\sqrt [3]{d} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 \sqrt [3]{c} (b c-a d)}+\frac {\sqrt [3]{b} \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 \sqrt [3]{a} (b c-a d)}+\frac {b^{2/3} \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 (b c-a d)}-\frac {\sqrt [3]{d} \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 \sqrt [3]{c} (b c-a d)}-\frac {d^{2/3} \int \frac {1}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{2 (b c-a d)}\\ &=-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} (b c-a d)}+\frac {\sqrt [3]{d} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 \sqrt [3]{c} (b c-a d)}+\frac {\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{a} (b c-a d)}-\frac {\sqrt [3]{d} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 \sqrt [3]{c} (b c-a d)}+\frac {\sqrt [3]{b} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{a} (b c-a d)}-\frac {\sqrt [3]{d} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}\right )}{\sqrt [3]{c} (b c-a d)}\\ &=-\frac {\sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{a} (b c-a d)}+\frac {\sqrt [3]{d} \tan ^{-1}\left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} \sqrt [3]{c} (b c-a d)}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} (b c-a d)}+\frac {\sqrt [3]{d} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 \sqrt [3]{c} (b c-a d)}+\frac {\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{a} (b c-a d)}-\frac {\sqrt [3]{d} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 \sqrt [3]{c} (b c-a d)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.14, size = 224, normalized size = 0.78 \[ \frac {-\frac {\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{a}}+\frac {2 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a}}+\frac {2 \sqrt {3} \sqrt [3]{b} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{a}}+\frac {\sqrt [3]{d} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{\sqrt [3]{c}}-\frac {2 \sqrt [3]{d} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt [3]{c}}-\frac {2 \sqrt {3} \sqrt [3]{d} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{\sqrt [3]{c}}}{6 a d-6 b c} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.93, size = 201, normalized size = 0.70 \[ \frac {2 \, \sqrt {3} \left (\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} x \left (\frac {b}{a}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) - 2 \, \sqrt {3} \left (-\frac {d}{c}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} x \left (-\frac {d}{c}\right )^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x^{2} - a x \left (\frac {b}{a}\right )^{\frac {2}{3}} + a \left (\frac {b}{a}\right )^{\frac {1}{3}}\right ) + \left (-\frac {d}{c}\right )^{\frac {1}{3}} \log \left (d x^{2} - c x \left (-\frac {d}{c}\right )^{\frac {2}{3}} - c \left (-\frac {d}{c}\right )^{\frac {1}{3}}\right ) - 2 \, \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x + a \left (\frac {b}{a}\right )^{\frac {2}{3}}\right ) - 2 \, \left (-\frac {d}{c}\right )^{\frac {1}{3}} \log \left (d x + c \left (-\frac {d}{c}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b c - a d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.25, size = 290, normalized size = 1.01 \[ -\frac {b \left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (a b c - a^{2} d\right )}} + \frac {d \left (-\frac {c}{d}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {c}{d}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (b c^{2} - a c d\right )}} - \frac {\left (-a b^{2}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} a b^{2} c - \sqrt {3} a^{2} b d} + \frac {\left (-c d^{2}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {c}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {c}{d}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} b c^{2} d - \sqrt {3} a c d^{2}} + \frac {\left (-a b^{2}\right )^{\frac {2}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, {\left (a b^{2} c - a^{2} b d\right )}} - \frac {\left (-c d^{2}\right )^{\frac {2}{3}} \log \left (x^{2} + x \left (-\frac {c}{d}\right )^{\frac {1}{3}} + \left (-\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b c^{2} d - a c d^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 222, normalized size = 0.77 \[ -\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {c}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{3}}}+\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{3}}}-\frac {\ln \left (x +\left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{3}} x +\left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.29, size = 265, normalized size = 0.92 \[ \frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, {\left (b c - a d\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {c}{d}\right )^{\frac {1}{3}}}\right )}{3 \, {\left (b c - a d\right )} \left (\frac {c}{d}\right )^{\frac {1}{3}}} + \frac {\log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b c \left (\frac {a}{b}\right )^{\frac {1}{3}} - a d \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}} - \frac {\log \left (x^{2} - x \left (\frac {c}{d}\right )^{\frac {1}{3}} + \left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b c \left (\frac {c}{d}\right )^{\frac {1}{3}} - a d \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}} - \frac {\log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, {\left (b c \left (\frac {a}{b}\right )^{\frac {1}{3}} - a d \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}} + \frac {\log \left (x + \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 \, {\left (b c \left (\frac {c}{d}\right )^{\frac {1}{3}} - a d \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 5.42, size = 982, normalized size = 3.41 \[ \ln \left (b\,x+a^3\,d^2\,{\left (\frac {b}{a\,{\left (a\,d-b\,c\right )}^3}\right )}^{2/3}+a\,b^2\,c^2\,{\left (\frac {b}{a\,{\left (a\,d-b\,c\right )}^3}\right )}^{2/3}-2\,a^2\,b\,c\,d\,{\left (\frac {b}{a\,{\left (a\,d-b\,c\right )}^3}\right )}^{2/3}\right )\,{\left (\frac {b}{27\,a^4\,d^3-81\,a^3\,b\,c\,d^2+81\,a^2\,b^2\,c^2\,d-27\,a\,b^3\,c^3}\right )}^{1/3}+\ln \left (d\,x+b^2\,c^3\,{\left (-\frac {d}{c\,{\left (a\,d-b\,c\right )}^3}\right )}^{2/3}+a^2\,c\,d^2\,{\left (-\frac {d}{c\,{\left (a\,d-b\,c\right )}^3}\right )}^{2/3}-2\,a\,b\,c^2\,d\,{\left (-\frac {d}{c\,{\left (a\,d-b\,c\right )}^3}\right )}^{2/3}\right )\,{\left (\frac {d}{-27\,a^3\,c\,d^3+81\,a^2\,b\,c^2\,d^2-81\,a\,b^2\,c^3\,d+27\,b^3\,c^4}\right )}^{1/3}+\frac {\ln \left (b^4\,d^4\,x-\frac {b\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^3\,\left (27\,b^3\,d^3\,x\,\left (a^2\,d^2+b^2\,c^2\right )\,{\left (a\,d-b\,c\right )}^2+\frac {27\,a\,b^3\,c\,d^3\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (a\,d+b\,c\right )\,{\left (a\,d-b\,c\right )}^4\,{\left (\frac {b}{a\,{\left (a\,d-b\,c\right )}^3}\right )}^{2/3}}{4}\right )}{216\,a\,{\left (a\,d-b\,c\right )}^3}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {b}{27\,a^4\,d^3-81\,a^3\,b\,c\,d^2+81\,a^2\,b^2\,c^2\,d-27\,a\,b^3\,c^3}\right )}^{1/3}}{2}-\frac {\ln \left (b^4\,d^4\,x+\frac {b\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^3\,\left (27\,b^3\,d^3\,x\,\left (a^2\,d^2+b^2\,c^2\right )\,{\left (a\,d-b\,c\right )}^2+\frac {27\,a\,b^3\,c\,d^3\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (a\,d+b\,c\right )\,{\left (a\,d-b\,c\right )}^4\,{\left (\frac {b}{a\,{\left (a\,d-b\,c\right )}^3}\right )}^{2/3}}{4}\right )}{216\,a\,{\left (a\,d-b\,c\right )}^3}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {b}{27\,a^4\,d^3-81\,a^3\,b\,c\,d^2+81\,a^2\,b^2\,c^2\,d-27\,a\,b^3\,c^3}\right )}^{1/3}}{2}+\frac {\ln \left (b^4\,d^4\,x+\frac {d\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^3\,\left (27\,b^3\,d^3\,x\,\left (a^2\,d^2+b^2\,c^2\right )\,{\left (a\,d-b\,c\right )}^2+\frac {27\,a\,b^3\,c\,d^3\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (a\,d+b\,c\right )\,{\left (a\,d-b\,c\right )}^4\,{\left (-\frac {d}{c\,{\left (a\,d-b\,c\right )}^3}\right )}^{2/3}}{4}\right )}{216\,c\,{\left (a\,d-b\,c\right )}^3}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {d}{-27\,a^3\,c\,d^3+81\,a^2\,b\,c^2\,d^2-81\,a\,b^2\,c^3\,d+27\,b^3\,c^4}\right )}^{1/3}}{2}-\frac {\ln \left (b^4\,d^4\,x-\frac {d\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^3\,\left (27\,b^3\,d^3\,x\,\left (a^2\,d^2+b^2\,c^2\right )\,{\left (a\,d-b\,c\right )}^2+\frac {27\,a\,b^3\,c\,d^3\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (a\,d+b\,c\right )\,{\left (a\,d-b\,c\right )}^4\,{\left (-\frac {d}{c\,{\left (a\,d-b\,c\right )}^3}\right )}^{2/3}}{4}\right )}{216\,c\,{\left (a\,d-b\,c\right )}^3}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {d}{-27\,a^3\,c\,d^3+81\,a^2\,b\,c^2\,d^2-81\,a\,b^2\,c^3\,d+27\,b^3\,c^4}\right )}^{1/3}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 14.51, size = 515, normalized size = 1.79 \[ \operatorname {RootSum} {\left (t^{3} \left (27 a^{4} d^{3} - 81 a^{3} b c d^{2} + 81 a^{2} b^{2} c^{2} d - 27 a b^{3} c^{3}\right ) - b, \left (t \mapsto t \log {\left (x + \frac {243 t^{5} a^{7} c d^{6} - 1458 t^{5} a^{6} b c^{2} d^{5} + 3645 t^{5} a^{5} b^{2} c^{3} d^{4} - 4860 t^{5} a^{4} b^{3} c^{4} d^{3} + 3645 t^{5} a^{3} b^{4} c^{5} d^{2} - 1458 t^{5} a^{2} b^{5} c^{6} d + 243 t^{5} a b^{6} c^{7} + 9 t^{2} a^{4} d^{4} - 18 t^{2} a^{3} b c d^{3} + 18 t^{2} a^{2} b^{2} c^{2} d^{2} - 18 t^{2} a b^{3} c^{3} d + 9 t^{2} b^{4} c^{4}}{a b d^{2} + b^{2} c d} \right )} \right )\right )} + \operatorname {RootSum} {\left (t^{3} \left (27 a^{3} c d^{3} - 81 a^{2} b c^{2} d^{2} + 81 a b^{2} c^{3} d - 27 b^{3} c^{4}\right ) + d, \left (t \mapsto t \log {\left (x + \frac {243 t^{5} a^{7} c d^{6} - 1458 t^{5} a^{6} b c^{2} d^{5} + 3645 t^{5} a^{5} b^{2} c^{3} d^{4} - 4860 t^{5} a^{4} b^{3} c^{4} d^{3} + 3645 t^{5} a^{3} b^{4} c^{5} d^{2} - 1458 t^{5} a^{2} b^{5} c^{6} d + 243 t^{5} a b^{6} c^{7} + 9 t^{2} a^{4} d^{4} - 18 t^{2} a^{3} b c d^{3} + 18 t^{2} a^{2} b^{2} c^{2} d^{2} - 18 t^{2} a b^{3} c^{3} d + 9 t^{2} b^{4} c^{4}}{a b d^{2} + b^{2} c d} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________