Optimal. Leaf size=242 \[ -\frac {x \left (7 c d^2-e (5 a e+b d)\right )}{18 d^2 e^2 \left (d+e x^3\right )}+\frac {x \left (a e^2-b d e+c d^2\right )}{6 d e^2 \left (d+e x^3\right )^2}-\frac {\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (e (5 a e+b d)+2 c d^2\right )}{54 d^{8/3} e^{7/3}}+\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (e (5 a e+b d)+2 c d^2\right )}{27 d^{8/3} e^{7/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right ) \left (e (5 a e+b d)+2 c d^2\right )}{9 \sqrt {3} d^{8/3} e^{7/3}} \]
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Rubi [A] time = 0.26, antiderivative size = 242, 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, 385, 200, 31, 634, 617, 204, 628} \[ -\frac {x \left (7 c d^2-e (5 a e+b d)\right )}{18 d^2 e^2 \left (d+e x^3\right )}+\frac {x \left (a e^2-b d e+c d^2\right )}{6 d e^2 \left (d+e x^3\right )^2}-\frac {\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (e (5 a e+b d)+2 c d^2\right )}{54 d^{8/3} e^{7/3}}+\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (e (5 a e+b d)+2 c d^2\right )}{27 d^{8/3} e^{7/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right ) \left (e (5 a e+b d)+2 c d^2\right )}{9 \sqrt {3} d^{8/3} e^{7/3}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 200
Rule 204
Rule 385
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 )^3} \, dx &=\frac {\left (c d^2-b d e+a e^2\right ) x}{6 d e^2 \left (d+e x^3\right )^2}-\frac {\int \frac {c d^2-e (b d+5 a e)-6 c d e x^3}{\left (d+e x^3\right )^2} \, dx}{6 d e^2}\\ &=\frac {\left (c d^2-b d e+a e^2\right ) x}{6 d e^2 \left (d+e x^3\right )^2}-\frac {\left (7 c d^2-e (b d+5 a e)\right ) x}{18 d^2 e^2 \left (d+e x^3\right )}+\frac {\left (2 c d^2+e (b d+5 a e)\right ) \int \frac {1}{d+e x^3} \, dx}{9 d^2 e^2}\\ &=\frac {\left (c d^2-b d e+a e^2\right ) x}{6 d e^2 \left (d+e x^3\right )^2}-\frac {\left (7 c d^2-e (b d+5 a e)\right ) x}{18 d^2 e^2 \left (d+e x^3\right )}+\frac {\left (2 c d^2+e (b d+5 a e)\right ) \int \frac {1}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{27 d^{8/3} e^2}+\frac {\left (2 c d^2+e (b d+5 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}{27 d^{8/3} e^2}\\ &=\frac {\left (c d^2-b d e+a e^2\right ) x}{6 d e^2 \left (d+e x^3\right )^2}-\frac {\left (7 c d^2-e (b d+5 a e)\right ) x}{18 d^2 e^2 \left (d+e x^3\right )}+\frac {\left (2 c d^2+e (b d+5 a e)\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{27 d^{8/3} e^{7/3}}-\frac {\left (2 c d^2+e (b d+5 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}{54 d^{8/3} e^{7/3}}+\frac {\left (2 c d^2+e (b d+5 a e)\right ) \int \frac {1}{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx}{18 d^{7/3} e^2}\\ &=\frac {\left (c d^2-b d e+a e^2\right ) x}{6 d e^2 \left (d+e x^3\right )^2}-\frac {\left (7 c d^2-e (b d+5 a e)\right ) x}{18 d^2 e^2 \left (d+e x^3\right )}+\frac {\left (2 c d^2+e (b d+5 a e)\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{27 d^{8/3} e^{7/3}}-\frac {\left (2 c d^2+e (b d+5 a e)\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{54 d^{8/3} e^{7/3}}+\frac {\left (2 c d^2+e (b d+5 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 )}{9 d^{8/3} e^{7/3}}\\ &=\frac {\left (c d^2-b d e+a e^2\right ) x}{6 d e^2 \left (d+e x^3\right )^2}-\frac {\left (7 c d^2-e (b d+5 a e)\right ) x}{18 d^2 e^2 \left (d+e x^3\right )}-\frac {\left (2 c d^2+e (b d+5 a e)\right ) \tan ^{-1}\left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{9 \sqrt {3} d^{8/3} e^{7/3}}+\frac {\left (2 c d^2+e (b d+5 a e)\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{27 d^{8/3} e^{7/3}}-\frac {\left (2 c d^2+e (b d+5 a e)\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{54 d^{8/3} e^{7/3}}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 209, normalized size = 0.86 \[ \frac {2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (e (5 a e+b d)+2 c d^2\right )-2 \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}}{\sqrt {3}}\right ) \left (e (5 a e+b d)+2 c d^2\right )-\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (e (5 a e+b d)+2 c d^2\right )-\frac {3 d^{2/3} \sqrt [3]{e} x \left (c d^2 \left (4 d+7 e x^3\right )-e \left (a e \left (8 d+5 e x^3\right )+b d \left (e x^3-2 d\right )\right )\right )}{\left (d+e x^3\right )^2}}{54 d^{8/3} e^{7/3}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.56, size = 941, normalized size = 3.89 \[ \left [-\frac {3 \, {\left (7 \, c d^{4} e^{2} - b d^{3} e^{3} - 5 \, a d^{2} e^{4}\right )} x^{4} - 3 \, \sqrt {\frac {1}{3}} {\left (2 \, c d^{5} e + b d^{4} e^{2} + 5 \, a d^{3} e^{3} + {\left (2 \, c d^{3} e^{3} + b d^{2} e^{4} + 5 \, a d e^{5}\right )} x^{6} + 2 \, {\left (2 \, c d^{4} e^{2} + b d^{3} e^{3} + 5 \, a d^{2} 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 ({\left (2 \, c d^{2} e^{2} + b d e^{3} + 5 \, a e^{4}\right )} x^{6} + 2 \, c d^{4} + b d^{3} e + 5 \, a d^{2} e^{2} + 2 \, {\left (2 \, c d^{3} e + b d^{2} e^{2} + 5 \, a d 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 ({\left (2 \, c d^{2} e^{2} + b d e^{3} + 5 \, a e^{4}\right )} x^{6} + 2 \, c d^{4} + b d^{3} e + 5 \, a d^{2} e^{2} + 2 \, {\left (2 \, c d^{3} e + b d^{2} e^{2} + 5 \, a d 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 (2 \, c d^{5} e + b d^{4} e^{2} - 4 \, a d^{3} e^{3}\right )} x}{54 \, {\left (d^{4} e^{5} x^{6} + 2 \, d^{5} e^{4} x^{3} + d^{6} e^{3}\right )}}, -\frac {3 \, {\left (7 \, c d^{4} e^{2} - b d^{3} e^{3} - 5 \, a d^{2} e^{4}\right )} x^{4} - 6 \, \sqrt {\frac {1}{3}} {\left (2 \, c d^{5} e + b d^{4} e^{2} + 5 \, a d^{3} e^{3} + {\left (2 \, c d^{3} e^{3} + b d^{2} e^{4} + 5 \, a d e^{5}\right )} x^{6} + 2 \, {\left (2 \, c d^{4} e^{2} + b d^{3} e^{3} + 5 \, a d^{2} 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 ({\left (2 \, c d^{2} e^{2} + b d e^{3} + 5 \, a e^{4}\right )} x^{6} + 2 \, c d^{4} + b d^{3} e + 5 \, a d^{2} e^{2} + 2 \, {\left (2 \, c d^{3} e + b d^{2} e^{2} + 5 \, a d 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 ({\left (2 \, c d^{2} e^{2} + b d e^{3} + 5 \, a e^{4}\right )} x^{6} + 2 \, c d^{4} + b d^{3} e + 5 \, a d^{2} e^{2} + 2 \, {\left (2 \, c d^{3} e + b d^{2} e^{2} + 5 \, a d 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 (2 \, c d^{5} e + b d^{4} e^{2} - 4 \, a d^{3} e^{3}\right )} x}{54 \, {\left (d^{4} e^{5} x^{6} + 2 \, d^{5} e^{4} x^{3} + d^{6} e^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 224, normalized size = 0.93 \[ -\frac {\sqrt {3} {\left (2 \, c d^{2} + b d e + 5 \, a e^{2}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-d e^{\left (-1\right )}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-d e^{\left (-1\right )}\right )^{\frac {1}{3}}}\right ) e^{\left (-1\right )}}{27 \, \left (-d e^{2}\right )^{\frac {2}{3}} d^{2}} - \frac {{\left (2 \, c d^{2} + b d e + 5 \, a e^{2}\right )} e^{\left (-1\right )} \log \left (x^{2} + \left (-d e^{\left (-1\right )}\right )^{\frac {1}{3}} x + \left (-d e^{\left (-1\right )}\right )^{\frac {2}{3}}\right )}{54 \, \left (-d e^{2}\right )^{\frac {2}{3}} d^{2}} - \frac {{\left (2 \, c d^{2} + b d e + 5 \, a e^{2}\right )} \left (-d e^{\left (-1\right )}\right )^{\frac {1}{3}} e^{\left (-2\right )} \log \left ({\left | x - \left (-d e^{\left (-1\right )}\right )^{\frac {1}{3}} \right |}\right )}{27 \, d^{3}} - \frac {{\left (7 \, c d^{2} x^{4} e - b d x^{4} e^{2} - 5 \, a x^{4} e^{3} + 4 \, c d^{3} x + 2 \, b d^{2} x e - 8 \, a d x e^{2}\right )} e^{\left (-2\right )}}{18 \, {\left (x^{3} e + d\right )}^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 362, normalized size = 1.50 \[ \frac {5 \sqrt {3}\, a \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {d}{e}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {d}{e}\right )^{\frac {2}{3}} d^{2} e}+\frac {5 a \ln \left (x +\left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {d}{e}\right )^{\frac {2}{3}} d^{2} e}-\frac {5 a \ln \left (x^{2}-\left (\frac {d}{e}\right )^{\frac {1}{3}} x +\left (\frac {d}{e}\right )^{\frac {2}{3}}\right )}{54 \left (\frac {d}{e}\right )^{\frac {2}{3}} d^{2} e}+\frac {\sqrt {3}\, b \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {d}{e}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {d}{e}\right )^{\frac {2}{3}} d \,e^{2}}+\frac {b \ln \left (x +\left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {d}{e}\right )^{\frac {2}{3}} d \,e^{2}}-\frac {b \ln \left (x^{2}-\left (\frac {d}{e}\right )^{\frac {1}{3}} x +\left (\frac {d}{e}\right )^{\frac {2}{3}}\right )}{54 \left (\frac {d}{e}\right )^{\frac {2}{3}} d \,e^{2}}+\frac {2 \sqrt {3}\, c \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {d}{e}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {d}{e}\right )^{\frac {2}{3}} e^{3}}+\frac {2 c \ln \left (x +\left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {d}{e}\right )^{\frac {2}{3}} e^{3}}-\frac {c \ln \left (x^{2}-\left (\frac {d}{e}\right )^{\frac {1}{3}} x +\left (\frac {d}{e}\right )^{\frac {2}{3}}\right )}{27 \left (\frac {d}{e}\right )^{\frac {2}{3}} e^{3}}+\frac {\frac {\left (5 a \,e^{2}+d e b -7 c \,d^{2}\right ) x^{4}}{18 d^{2} e}+\frac {\left (4 a \,e^{2}-d e b -2 c \,d^{2}\right ) x}{9 d \,e^{2}}}{\left (e \,x^{3}+d \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.69, size = 240, normalized size = 0.99 \[ -\frac {{\left (7 \, c d^{2} e - b d e^{2} - 5 \, a e^{3}\right )} x^{4} + 2 \, {\left (2 \, c d^{3} + b d^{2} e - 4 \, a d e^{2}\right )} x}{18 \, {\left (d^{2} e^{4} x^{6} + 2 \, d^{3} e^{3} x^{3} + d^{4} e^{2}\right )}} + \frac {\sqrt {3} {\left (2 \, c d^{2} + b d e + 5 \, a e^{2}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {d}{e}\right )^{\frac {1}{3}}}\right )}{27 \, d^{2} e^{3} \left (\frac {d}{e}\right )^{\frac {2}{3}}} - \frac {{\left (2 \, c d^{2} + b d e + 5 \, a e^{2}\right )} \log \left (x^{2} - x \left (\frac {d}{e}\right )^{\frac {1}{3}} + \left (\frac {d}{e}\right )^{\frac {2}{3}}\right )}{54 \, d^{2} e^{3} \left (\frac {d}{e}\right )^{\frac {2}{3}}} + \frac {{\left (2 \, c d^{2} + b d e + 5 \, a e^{2}\right )} \log \left (x + \left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}{27 \, d^{2} e^{3} \left (\frac {d}{e}\right )^{\frac {2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.29, size = 221, normalized size = 0.91 \[ \frac {\ln \left (e^{1/3}\,x+d^{1/3}\right )\,\left (2\,c\,d^2+b\,d\,e+5\,a\,e^2\right )}{27\,d^{8/3}\,e^{7/3}}-\frac {\frac {x\,\left (2\,c\,d^2+b\,d\,e-4\,a\,e^2\right )}{9\,d\,e^2}-\frac {x^4\,\left (-7\,c\,d^2+b\,d\,e+5\,a\,e^2\right )}{18\,d^2\,e}}{d^2+2\,d\,e\,x^3+e^2\,x^6}+\frac {\ln \left (2\,e^{1/3}\,x-d^{1/3}+\sqrt {3}\,d^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (2\,c\,d^2+b\,d\,e+5\,a\,e^2\right )}{27\,d^{8/3}\,e^{7/3}}-\frac {\ln \left (d^{1/3}-2\,e^{1/3}\,x+\sqrt {3}\,d^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (2\,c\,d^2+b\,d\,e+5\,a\,e^2\right )}{27\,d^{8/3}\,e^{7/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.23, size = 246, normalized size = 1.02 \[ \frac {x^{4} \left (5 a e^{3} + b d e^{2} - 7 c d^{2} e\right ) + x \left (8 a d e^{2} - 2 b d^{2} e - 4 c d^{3}\right )}{18 d^{4} e^{2} + 36 d^{3} e^{3} x^{3} + 18 d^{2} e^{4} x^{6}} + \operatorname {RootSum} {\left (19683 t^{3} d^{8} e^{7} - 125 a^{3} e^{6} - 75 a^{2} b d e^{5} - 150 a^{2} c d^{2} e^{4} - 15 a b^{2} d^{2} e^{4} - 60 a b c d^{3} e^{3} - 60 a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} - 6 b^{2} c d^{4} e^{2} - 12 b c^{2} d^{5} e - 8 c^{3} d^{6}, \left (t \mapsto t \log {\left (\frac {27 t d^{3} e^{2}}{5 a e^{2} + b d e + 2 c d^{2}} + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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