Optimal. Leaf size=188 \[ -\frac {(c d-b e) x}{e^2}+\frac {c x^4}{4 e}-\frac {\left (c d^2-b d e+a e^2\right ) \tan ^{-1}\left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{\sqrt {3} d^{2/3} e^{7/3}}+\frac {\left (c d^2-b d e+a e^2\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{2/3} e^{7/3}}-\frac {\left (c d^2-b d e+a e^2\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{2/3} e^{7/3}} \]
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Rubi [A]
time = 0.14, antiderivative size = 188, 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 = {1425, 396, 206,
31, 648, 631, 210, 642} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right ) \left (a e^2-b d e+c d^2\right )}{\sqrt {3} d^{2/3} e^{7/3}}-\frac {\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (a e^2-b d e+c d^2\right )}{6 d^{2/3} e^{7/3}}+\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (a e^2-b d e+c d^2\right )}{3 d^{2/3} e^{7/3}}-\frac {x (c d-b e)}{e^2}+\frac {c x^4}{4 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 206
Rule 210
Rule 396
Rule 631
Rule 642
Rule 648
Rule 1425
Rubi steps
\begin {align*} \int \frac {a+b x^3+c x^6}{d+e x^3} \, dx &=\frac {c x^4}{4 e}+\frac {\int \frac {4 a e-(4 c d-4 b e) x^3}{d+e x^3} \, dx}{4 e}\\ &=-\frac {(c d-b e) x}{e^2}+\frac {c x^4}{4 e}-\left (-a-\frac {d (c d-b e)}{e^2}\right ) \int \frac {1}{d+e x^3} \, dx\\ &=-\frac {(c d-b e) x}{e^2}+\frac {c x^4}{4 e}+\frac {\left (a+\frac {d (c d-b e)}{e^2}\right ) \int \frac {1}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{3 d^{2/3}}+\frac {\left (a+\frac {d (c d-b e)}{e^2}\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}{3 d^{2/3}}\\ &=-\frac {(c d-b e) x}{e^2}+\frac {c x^4}{4 e}+\frac {\left (c d^2-b d e+a e^2\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{2/3} e^{7/3}}-\frac {\left (c d^2-b d e+a e^2\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}{6 d^{2/3} e^{7/3}}+\frac {\left (a+\frac {d (c d-b e)}{e^2}\right ) \int \frac {1}{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx}{2 \sqrt [3]{d}}\\ &=-\frac {(c d-b e) x}{e^2}+\frac {c x^4}{4 e}+\frac {\left (c d^2-b d e+a e^2\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{2/3} e^{7/3}}-\frac {\left (c d^2-b d e+a e^2\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{2/3} e^{7/3}}+\frac {\left (c d^2-b d e+a e^2\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{d^{2/3} e^{7/3}}\\ &=-\frac {(c d-b e) x}{e^2}+\frac {c x^4}{4 e}-\frac {\left (c d^2-b d e+a e^2\right ) \tan ^{-1}\left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{\sqrt {3} d^{2/3} e^{7/3}}+\frac {\left (c d^2-b d e+a e^2\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{2/3} e^{7/3}}-\frac {\left (c d^2-b d e+a e^2\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{2/3} e^{7/3}}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 176, normalized size = 0.94 \begin {gather*} \frac {12 \sqrt [3]{e} (-c d+b e) x+3 c e^{4/3} x^4-\frac {4 \sqrt {3} \left (c d^2+e (-b d+a e)\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}}{\sqrt {3}}\right )}{d^{2/3}}+\frac {4 \left (c d^2+e (-b d+a e)\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{d^{2/3}}-\frac {2 \left (c d^2+e (-b d+a e)\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{d^{2/3}}}{12 e^{7/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 133, normalized size = 0.71
method | result | size |
risch | \(\frac {c \,x^{4}}{4 e}+\frac {b x}{e}-\frac {c d x}{e^{2}}+\frac {\munderset {\textit {\_R} =\RootOf \left (e \,\textit {\_Z}^{3}+d \right )}{\sum }\frac {\left (a \,e^{2}-d e b +c \,d^{2}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{3 e^{3}}\) | \(67\) |
default | \(\frac {\frac {1}{4} c e \,x^{4}+e b x -c d x}{e^{2}}+\frac {\left (\frac {\ln \left (x +\left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}{3 e \left (\frac {d}{e}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {d}{e}\right )^{\frac {1}{3}} x +\left (\frac {d}{e}\right )^{\frac {2}{3}}\right )}{6 e \left (\frac {d}{e}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {d}{e}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 e \left (\frac {d}{e}\right )^{\frac {2}{3}}}\right ) \left (a \,e^{2}-d e b +c \,d^{2}\right )}{e^{2}}\) | \(133\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 146, normalized size = 0.78 \begin {gather*} \frac {\sqrt {3} {\left (c d^{2} - b d e + a e^{2}\right )} \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 {7}{3}\right )}}{3 \, d^{\frac {2}{3}}} - \frac {{\left (c d^{2} - b d e + a e^{2}\right )} e^{\left (-\frac {7}{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 {{\left (c d^{2} - b d e + a e^{2}\right )} e^{\left (-\frac {7}{3}\right )} \log \left (d^{\frac {1}{3}} e^{\left (-\frac {1}{3}\right )} + x\right )}{3 \, d^{\frac {2}{3}}} + \frac {1}{4} \, {\left (c x^{4} e - 4 \, {\left (c d - b e\right )} x\right )} e^{\left (-2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 199, normalized size = 1.06 \begin {gather*} -\frac {{\left (12 \, c d^{3} x e - 12 \, \sqrt {\frac {1}{3}} {\left (c d^{3} e - b d^{2} e^{2} + a d e^{3}\right )} {\left (d^{2}\right )}^{\frac {1}{6}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, {\left (d^{2}\right )}^{\frac {2}{3}} x e^{\frac {2}{3}} - {\left (d^{2}\right )}^{\frac {1}{3}} d e^{\frac {1}{3}}\right )} {\left (d^{2}\right )}^{\frac {1}{6}} e^{\left (-\frac {1}{3}\right )}}{d^{2}}\right ) e^{\left (-\frac {1}{3}\right )} + 2 \, {\left (c d^{2} - b d e + a e^{2}\right )} {\left (d^{2}\right )}^{\frac {2}{3}} e^{\frac {2}{3}} \log \left (d x^{2} e - {\left (d^{2}\right )}^{\frac {2}{3}} x e^{\frac {2}{3}} + {\left (d^{2}\right )}^{\frac {1}{3}} d e^{\frac {1}{3}}\right ) - 4 \, {\left (c d^{2} - b d e + a e^{2}\right )} {\left (d^{2}\right )}^{\frac {2}{3}} e^{\frac {2}{3}} \log \left (d x e + {\left (d^{2}\right )}^{\frac {2}{3}} e^{\frac {2}{3}}\right ) - 3 \, {\left (c d^{2} x^{4} + 4 \, b d^{2} x\right )} e^{2}\right )} e^{\left (-3\right )}}{12 \, d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.45, size = 175, normalized size = 0.93 \begin {gather*} \frac {c x^{4}}{4 e} + x \left (\frac {b}{e} - \frac {c d}{e^{2}}\right ) + \operatorname {RootSum} {\left (27 t^{3} d^{2} e^{7} - a^{3} e^{6} + 3 a^{2} b d e^{5} - 3 a^{2} c d^{2} e^{4} - 3 a b^{2} d^{2} e^{4} + 6 a b c d^{3} e^{3} - 3 a c^{2} d^{4} e^{2} + b^{3} d^{3} e^{3} - 3 b^{2} c d^{4} e^{2} + 3 b c^{2} d^{5} e - c^{3} d^{6}, \left ( t \mapsto t \log {\left (\frac {3 t d e^{2}}{a e^{2} - b d e + c d^{2}} + x \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.15, size = 173, normalized size = 0.92 \begin {gather*} -\frac {\sqrt {3} {\left (c d^{2} - b d e + 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 )}}{3 \, \left (-d e^{2}\right )^{\frac {2}{3}}} - \frac {{\left (c d^{2} - b d e + 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 )}{6 \, \left (-d e^{2}\right )^{\frac {2}{3}}} - \frac {{\left (c d^{2} e^{2} - b d e^{3} + a e^{4}\right )} \left (-d e^{\left (-1\right )}\right )^{\frac {1}{3}} e^{\left (-4\right )} \log \left ({\left | x - \left (-d e^{\left (-1\right )}\right )^{\frac {1}{3}} \right |}\right )}{3 \, d} + \frac {1}{4} \, {\left (c x^{4} e^{3} - 4 \, c d x e^{2} + 4 \, b x e^{3}\right )} e^{\left (-4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.27, size = 165, normalized size = 0.88 \begin {gather*} x\,\left (\frac {b}{e}-\frac {c\,d}{e^2}\right )+\frac {c\,x^4}{4\,e}+\frac {\ln \left (e^{1/3}\,x+d^{1/3}\right )\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{3\,d^{2/3}\,e^{7/3}}+\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 (c\,d^2-b\,d\,e+a\,e^2\right )}{3\,d^{2/3}\,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 (c\,d^2-b\,d\,e+a\,e^2\right )}{3\,d^{2/3}\,e^{7/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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