Optimal. Leaf size=134 \[ \frac {a \log \left (d^{2/3}+\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{2/3} \sqrt [3]{e}}-\frac {a \log \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )}{3 d^{2/3} \sqrt [3]{e}}+\frac {a \tan ^{-1}\left (\frac {\sqrt [3]{d}+2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{\sqrt {3} d^{2/3} \sqrt [3]{e}}-\frac {c \log \left (d-e x^3\right )}{3 e} \]
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Rubi [A] time = 0.09, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1871, 12, 200, 31, 634, 617, 204, 628, 260} \[ \frac {a \log \left (d^{2/3}+\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{2/3} \sqrt [3]{e}}-\frac {a \log \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )}{3 d^{2/3} \sqrt [3]{e}}+\frac {a \tan ^{-1}\left (\frac {\sqrt [3]{d}+2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{\sqrt {3} d^{2/3} \sqrt [3]{e}}-\frac {c \log \left (d-e x^3\right )}{3 e} \]
Antiderivative was successfully verified.
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Rule 12
Rule 31
Rule 200
Rule 204
Rule 260
Rule 617
Rule 628
Rule 634
Rule 1871
Rubi steps
\begin {align*} \int \frac {a+c x^2}{d-e x^3} \, dx &=c \int \frac {x^2}{d-e x^3} \, dx+\int \frac {a}{d-e x^3} \, dx\\ &=-\frac {c \log \left (d-e x^3\right )}{3 e}+a \int \frac {1}{d-e x^3} \, dx\\ &=-\frac {c \log \left (d-e x^3\right )}{3 e}+\frac {a \int \frac {1}{\sqrt [3]{d}-\sqrt [3]{e} x} \, dx}{3 d^{2/3}}+\frac {a \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}{3 d^{2/3}}\\ &=-\frac {a \log \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )}{3 d^{2/3} \sqrt [3]{e}}-\frac {c \log \left (d-e x^3\right )}{3 e}+\frac {a \int \frac {1}{d^{2/3}+\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx}{2 \sqrt [3]{d}}+\frac {a \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}{6 d^{2/3} \sqrt [3]{e}}\\ &=-\frac {a \log \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )}{3 d^{2/3} \sqrt [3]{e}}+\frac {a \log \left (d^{2/3}+\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{2/3} \sqrt [3]{e}}-\frac {c \log \left (d-e x^3\right )}{3 e}-\frac {a \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{d^{2/3} \sqrt [3]{e}}\\ &=\frac {a \tan ^{-1}\left (\frac {\sqrt [3]{d}+2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{\sqrt {3} d^{2/3} \sqrt [3]{e}}-\frac {a \log \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )}{3 d^{2/3} \sqrt [3]{e}}+\frac {a \log \left (d^{2/3}+\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{2/3} \sqrt [3]{e}}-\frac {c \log \left (d-e x^3\right )}{3 e}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 123, normalized size = 0.92 \[ \frac {a e^{2/3} \log \left (d^{2/3}+\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )-2 a e^{2/3} \log \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )+2 \sqrt {3} a e^{2/3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}+1}{\sqrt {3}}\right )-2 c d^{2/3} \log \left (d-e x^3\right )}{6 d^{2/3} e} \]
Antiderivative was successfully verified.
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fricas [C] time = 2.37, size = 1267, normalized size = 9.46 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 95, normalized size = 0.71 \[ -\frac {1}{3} \, c e^{\left (-1\right )} \log \left ({\left | x^{3} e - d \right |}\right ) + \frac {\sqrt {3} a \arctan \left (\frac {\sqrt {3} {\left (d^{\frac {1}{3}} e^{\left (-\frac {1}{3}\right )} + 2 \, x\right )} e^{\frac {1}{3}}}{3 \, d^{\frac {1}{3}}}\right ) e^{\left (-\frac {1}{3}\right )}}{3 \, d^{\frac {2}{3}}} + \frac {a e^{\left (-\frac {1}{3}\right )} \log \left (d^{\frac {1}{3}} x e^{\left (-\frac {1}{3}\right )} + x^{2} + d^{\frac {2}{3}} e^{\left (-\frac {2}{3}\right )}\right )}{6 \, d^{\frac {2}{3}}} - \frac {a e^{\left (-\frac {1}{3}\right )} \log \left ({\left | -d^{\frac {1}{3}} e^{\left (-\frac {1}{3}\right )} + x \right |}\right )}{3 \, d^{\frac {2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 111, normalized size = 0.83 \[ \frac {\sqrt {3}\, a \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {d}{e}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{3 \left (\frac {d}{e}\right )^{\frac {2}{3}} e}-\frac {a \ln \left (x -\left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {d}{e}\right )^{\frac {2}{3}} e}+\frac {a \ln \left (x^{2}+\left (\frac {d}{e}\right )^{\frac {1}{3}} x +\left (\frac {d}{e}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {d}{e}\right )^{\frac {2}{3}} e}-\frac {c \ln \left (e \,x^{3}-d \right )}{3 e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.05, size = 144, normalized size = 1.07 \[ -\frac {\sqrt {3} {\left (2 \, c d - {\left (3 \, a \left (\frac {d}{e}\right )^{\frac {1}{3}} + \frac {2 \, c d}{e}\right )} e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {d}{e}\right )^{\frac {1}{3}}}\right )}{9 \, d e} - \frac {{\left (2 \, c \left (\frac {d}{e}\right )^{\frac {2}{3}} - a\right )} \log \left (x^{2} + x \left (\frac {d}{e}\right )^{\frac {1}{3}} + \left (\frac {d}{e}\right )^{\frac {2}{3}}\right )}{6 \, e \left (\frac {d}{e}\right )^{\frac {2}{3}}} - \frac {{\left (c \left (\frac {d}{e}\right )^{\frac {2}{3}} + a\right )} \log \left (x - \left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}{3 \, e \left (\frac {d}{e}\right )^{\frac {2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.01, size = 178, normalized size = 1.33 \[ \sum _{k=1}^3\ln \left (-\left (c+\mathrm {root}\left (27\,d^2\,e^3\,z^3+27\,c\,d^2\,e^2\,z^2+9\,c^2\,d^2\,e\,z+c^3\,d^2+a^3\,e^2,z,k\right )\,e\,3\right )\,\left (c\,d+\mathrm {root}\left (27\,d^2\,e^3\,z^3+27\,c\,d^2\,e^2\,z^2+9\,c^2\,d^2\,e\,z+c^3\,d^2+a^3\,e^2,z,k\right )\,d\,e\,3+a\,e\,x\right )\right )\,\mathrm {root}\left (27\,d^2\,e^3\,z^3+27\,c\,d^2\,e^2\,z^2+9\,c^2\,d^2\,e\,z+c^3\,d^2+a^3\,e^2,z,k\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.59, size = 70, normalized size = 0.52 \[ - \operatorname {RootSum} {\left (27 t^{3} d^{2} e^{3} - 27 t^{2} c d^{2} e^{2} + 9 t c^{2} d^{2} e - a^{3} e^{2} - c^{3} d^{2}, \left (t \mapsto t \log {\left (x + \frac {- 3 t d e + c d}{a e} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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