Optimal. Leaf size=213 \[ \frac {x \left (a e^2-b d e+c d^2\right )}{3 d e^2 \left (d+e x^3\right )}+\frac {\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (4 c d^2-e (2 a e+b d)\right )}{18 d^{5/3} e^{7/3}}-\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (4 c d^2-e (2 a e+b d)\right )}{9 d^{5/3} e^{7/3}}+\frac {\tan ^{-1}\left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right ) \left (4 c d^2-e (2 a e+b d)\right )}{3 \sqrt {3} d^{5/3} e^{7/3}}+\frac {c x}{e^2} \]
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Rubi [A] time = 0.23, antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {1409, 388, 200, 31, 634, 617, 204, 628} \[ \frac {x \left (a e^2-b d e+c d^2\right )}{3 d e^2 \left (d+e x^3\right )}+\frac {\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (4 c d^2-e (2 a e+b d)\right )}{18 d^{5/3} e^{7/3}}-\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (4 c d^2-e (2 a e+b d)\right )}{9 d^{5/3} e^{7/3}}+\frac {\tan ^{-1}\left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right ) \left (4 c d^2-e (2 a e+b d)\right )}{3 \sqrt {3} d^{5/3} e^{7/3}}+\frac {c x}{e^2} \]
Antiderivative was successfully verified.
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Rule 31
Rule 200
Rule 204
Rule 388
Rule 617
Rule 628
Rule 634
Rule 1409
Rubi steps
\begin {align*} \int \frac {a+b x^3+c x^6}{\left (d+e x^3\right )^2} \, dx &=\frac {\left (c d^2-b d e+a e^2\right ) x}{3 d e^2 \left (d+e x^3\right )}-\frac {\int \frac {c d^2-e (b d+2 a e)-3 c d e x^3}{d+e x^3} \, dx}{3 d e^2}\\ &=\frac {c x}{e^2}+\frac {\left (c d^2-b d e+a e^2\right ) x}{3 d e^2 \left (d+e x^3\right )}-\frac {\left (4 c d^2-e (b d+2 a e)\right ) \int \frac {1}{d+e x^3} \, dx}{3 d e^2}\\ &=\frac {c x}{e^2}+\frac {\left (c d^2-b d e+a e^2\right ) x}{3 d e^2 \left (d+e x^3\right )}-\frac {\left (4 c d^2-e (b d+2 a e)\right ) \int \frac {1}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{9 d^{5/3} e^2}-\frac {\left (4 c d^2-e (b d+2 a e)\right ) \int \frac {2 \sqrt [3]{d}-\sqrt [3]{e} x}{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx}{9 d^{5/3} e^2}\\ &=\frac {c x}{e^2}+\frac {\left (c d^2-b d e+a e^2\right ) x}{3 d e^2 \left (d+e x^3\right )}-\frac {\left (4 c d^2-e (b d+2 a e)\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{9 d^{5/3} e^{7/3}}+\frac {\left (4 c d^2-e (b d+2 a e)\right ) \int \frac {-\sqrt [3]{d} \sqrt [3]{e}+2 e^{2/3} x}{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx}{18 d^{5/3} e^{7/3}}-\frac {\left (4 c d^2-e (b d+2 a e)\right ) \int \frac {1}{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx}{6 d^{4/3} e^2}\\ &=\frac {c x}{e^2}+\frac {\left (c d^2-b d e+a e^2\right ) x}{3 d e^2 \left (d+e x^3\right )}-\frac {\left (4 c d^2-e (b d+2 a e)\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{9 d^{5/3} e^{7/3}}+\frac {\left (4 c d^2-e (b d+2 a e)\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{18 d^{5/3} e^{7/3}}-\frac {\left (4 c d^2-e (b d+2 a e)\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} e^{7/3}}\\ &=\frac {c x}{e^2}+\frac {\left (c d^2-b d e+a e^2\right ) x}{3 d e^2 \left (d+e x^3\right )}+\frac {\left (4 c d^2-e (b d+2 a e)\right ) \tan ^{-1}\left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{3 \sqrt {3} d^{5/3} e^{7/3}}-\frac {\left (4 c d^2-e (b d+2 a e)\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{9 d^{5/3} e^{7/3}}+\frac {\left (4 c d^2-e (b d+2 a e)\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{18 d^{5/3} e^{7/3}}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 199, normalized size = 0.93 \[ \frac {\frac {6 \sqrt [3]{e} x \left (e (a e-b d)+c d^2\right )}{d \left (d+e x^3\right )}+\frac {\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (4 c d^2-e (2 a e+b d)\right )}{d^{5/3}}-\frac {2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (4 c d^2-e (2 a e+b d)\right )}{d^{5/3}}+\frac {2 \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}}{\sqrt {3}}\right ) \left (4 c d^2-e (2 a e+b d)\right )}{d^{5/3}}+18 c \sqrt [3]{e} x}{18 e^{7/3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 697, normalized size = 3.27 \[ \left [\frac {18 \, c d^{3} e^{2} x^{4} - 3 \, \sqrt {\frac {1}{3}} {\left (4 \, c d^{4} e - b d^{3} e^{2} - 2 \, a d^{2} e^{3} + {\left (4 \, c d^{3} e^{2} - b d^{2} e^{3} - 2 \, a d e^{4}\right )} x^{3}\right )} \sqrt {-\frac {\left (d^{2} e\right )^{\frac {1}{3}}}{e}} \log \left (\frac {2 \, d e x^{3} - 3 \, \left (d^{2} e\right )^{\frac {1}{3}} d x - d^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, d e x^{2} + \left (d^{2} e\right )^{\frac {2}{3}} x - \left (d^{2} e\right )^{\frac {1}{3}} d\right )} \sqrt {-\frac {\left (d^{2} e\right )^{\frac {1}{3}}}{e}}}{e x^{3} + d}\right ) + {\left (4 \, c d^{3} - b d^{2} e - 2 \, a d e^{2} + {\left (4 \, c d^{2} e - b d e^{2} - 2 \, a e^{3}\right )} x^{3}\right )} \left (d^{2} e\right )^{\frac {2}{3}} \log \left (d e x^{2} - \left (d^{2} e\right )^{\frac {2}{3}} x + \left (d^{2} e\right )^{\frac {1}{3}} d\right ) - 2 \, {\left (4 \, c d^{3} - b d^{2} e - 2 \, a d e^{2} + {\left (4 \, c d^{2} e - b d e^{2} - 2 \, a e^{3}\right )} x^{3}\right )} \left (d^{2} e\right )^{\frac {2}{3}} \log \left (d e x + \left (d^{2} e\right )^{\frac {2}{3}}\right ) + 6 \, {\left (4 \, c d^{4} e - b d^{3} e^{2} + a d^{2} e^{3}\right )} x}{18 \, {\left (d^{3} e^{4} x^{3} + d^{4} e^{3}\right )}}, \frac {18 \, c d^{3} e^{2} x^{4} - 6 \, \sqrt {\frac {1}{3}} {\left (4 \, c d^{4} e - b d^{3} e^{2} - 2 \, a d^{2} e^{3} + {\left (4 \, c d^{3} e^{2} - b d^{2} e^{3} - 2 \, a d e^{4}\right )} x^{3}\right )} \sqrt {\frac {\left (d^{2} e\right )^{\frac {1}{3}}}{e}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (d^{2} e\right )^{\frac {2}{3}} x - \left (d^{2} e\right )^{\frac {1}{3}} d\right )} \sqrt {\frac {\left (d^{2} e\right )^{\frac {1}{3}}}{e}}}{d^{2}}\right ) + {\left (4 \, c d^{3} - b d^{2} e - 2 \, a d e^{2} + {\left (4 \, c d^{2} e - b d e^{2} - 2 \, a e^{3}\right )} x^{3}\right )} \left (d^{2} e\right )^{\frac {2}{3}} \log \left (d e x^{2} - \left (d^{2} e\right )^{\frac {2}{3}} x + \left (d^{2} e\right )^{\frac {1}{3}} d\right ) - 2 \, {\left (4 \, c d^{3} - b d^{2} e - 2 \, a d e^{2} + {\left (4 \, c d^{2} e - b d e^{2} - 2 \, a e^{3}\right )} x^{3}\right )} \left (d^{2} e\right )^{\frac {2}{3}} \log \left (d e x + \left (d^{2} e\right )^{\frac {2}{3}}\right ) + 6 \, {\left (4 \, c d^{4} e - b d^{3} e^{2} + a d^{2} e^{3}\right )} x}{18 \, {\left (d^{3} e^{4} x^{3} + d^{4} e^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 199, normalized size = 0.93 \[ c x e^{\left (-2\right )} + \frac {\sqrt {3} {\left (4 \, c d^{2} - b d e - 2 \, a e^{2}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-d e^{\left (-1\right )}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-d e^{\left (-1\right )}\right )^{\frac {1}{3}}}\right ) e^{\left (-1\right )}}{9 \, \left (-d e^{2}\right )^{\frac {2}{3}} d} + \frac {{\left (4 \, c d^{2} - b d e - 2 \, a e^{2}\right )} e^{\left (-1\right )} \log \left (x^{2} + \left (-d e^{\left (-1\right )}\right )^{\frac {1}{3}} x + \left (-d e^{\left (-1\right )}\right )^{\frac {2}{3}}\right )}{18 \, \left (-d e^{2}\right )^{\frac {2}{3}} d} + \frac {{\left (4 \, c d^{2} - b d e - 2 \, a e^{2}\right )} \left (-d e^{\left (-1\right )}\right )^{\frac {1}{3}} e^{\left (-2\right )} \log \left ({\left | x - \left (-d e^{\left (-1\right )}\right )^{\frac {1}{3}} \right |}\right )}{9 \, d^{2}} + \frac {{\left (c d^{2} x - b d x e + a x e^{2}\right )} e^{\left (-2\right )}}{3 \, {\left (x^{3} e + d\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 345, normalized size = 1.62 \[ \frac {a x}{3 \left (e \,x^{3}+d \right ) d}-\frac {b x}{3 \left (e \,x^{3}+d \right ) e}+\frac {c d x}{3 \left (e \,x^{3}+d \right ) e^{2}}+\frac {2 \sqrt {3}\, a \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {d}{e}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {d}{e}\right )^{\frac {2}{3}} d e}+\frac {2 a \ln \left (x +\left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {d}{e}\right )^{\frac {2}{3}} d e}-\frac {a \ln \left (x^{2}-\left (\frac {d}{e}\right )^{\frac {1}{3}} x +\left (\frac {d}{e}\right )^{\frac {2}{3}}\right )}{9 \left (\frac {d}{e}\right )^{\frac {2}{3}} d e}+\frac {\sqrt {3}\, b \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {d}{e}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {d}{e}\right )^{\frac {2}{3}} e^{2}}+\frac {b \ln \left (x +\left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {d}{e}\right )^{\frac {2}{3}} e^{2}}-\frac {b \ln \left (x^{2}-\left (\frac {d}{e}\right )^{\frac {1}{3}} x +\left (\frac {d}{e}\right )^{\frac {2}{3}}\right )}{18 \left (\frac {d}{e}\right )^{\frac {2}{3}} e^{2}}-\frac {4 \sqrt {3}\, c d \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {d}{e}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {d}{e}\right )^{\frac {2}{3}} e^{3}}-\frac {4 c d \ln \left (x +\left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {d}{e}\right )^{\frac {2}{3}} e^{3}}+\frac {2 c d \ln \left (x^{2}-\left (\frac {d}{e}\right )^{\frac {1}{3}} x +\left (\frac {d}{e}\right )^{\frac {2}{3}}\right )}{9 \left (\frac {d}{e}\right )^{\frac {2}{3}} e^{3}}+\frac {c x}{e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.60, size = 204, normalized size = 0.96 \[ \frac {{\left (c d^{2} - b d e + a e^{2}\right )} x}{3 \, {\left (d e^{3} x^{3} + d^{2} e^{2}\right )}} + \frac {c x}{e^{2}} - \frac {\sqrt {3} {\left (4 \, c d^{2} - b d e - 2 \, a e^{2}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {d}{e}\right )^{\frac {1}{3}}}\right )}{9 \, d e^{3} \left (\frac {d}{e}\right )^{\frac {2}{3}}} + \frac {{\left (4 \, c d^{2} - b d e - 2 \, a e^{2}\right )} \log \left (x^{2} - x \left (\frac {d}{e}\right )^{\frac {1}{3}} + \left (\frac {d}{e}\right )^{\frac {2}{3}}\right )}{18 \, d e^{3} \left (\frac {d}{e}\right )^{\frac {2}{3}}} - \frac {{\left (4 \, c d^{2} - b d e - 2 \, a e^{2}\right )} \log \left (x + \left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}{9 \, d e^{3} \left (\frac {d}{e}\right )^{\frac {2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.80, size = 187, normalized size = 0.88 \[ \frac {c\,x}{e^2}+\frac {\ln \left (e^{1/3}\,x+d^{1/3}\right )\,\left (-4\,c\,d^2+b\,d\,e+2\,a\,e^2\right )}{9\,d^{5/3}\,e^{7/3}}+\frac {x\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{3\,d\,\left (e^3\,x^3+d\,e^2\right )}+\frac {\ln \left (2\,e^{1/3}\,x-d^{1/3}+\sqrt {3}\,d^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-4\,c\,d^2+b\,d\,e+2\,a\,e^2\right )}{9\,d^{5/3}\,e^{7/3}}-\frac {\ln \left (d^{1/3}-2\,e^{1/3}\,x+\sqrt {3}\,d^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-4\,c\,d^2+b\,d\,e+2\,a\,e^2\right )}{9\,d^{5/3}\,e^{7/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.68, size = 206, normalized size = 0.97 \[ \frac {c x}{e^{2}} + \frac {x \left (a e^{2} - b d e + c d^{2}\right )}{3 d^{2} e^{2} + 3 d e^{3} x^{3}} + \operatorname {RootSum} {\left (729 t^{3} d^{5} e^{7} - 8 a^{3} e^{6} - 12 a^{2} b d e^{5} + 48 a^{2} c d^{2} e^{4} - 6 a b^{2} d^{2} e^{4} + 48 a b c d^{3} e^{3} - 96 a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} + 12 b^{2} c d^{4} e^{2} - 48 b c^{2} d^{5} e + 64 c^{3} d^{6}, \left (t \mapsto t \log {\left (\frac {9 t d^{2} e^{2}}{2 a e^{2} + b d e - 4 c d^{2}} + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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