Optimal. Leaf size=655 \[ -\frac {d}{2 a x^2}+\frac {c^{2/3} \left (d+\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3} a \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {c^{2/3} \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b+\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3} a \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}-\frac {c^{2/3} \left (d+\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} a \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}-\frac {c^{2/3} \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \log \left (\sqrt [3]{b+\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} a \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {c^{2/3} \left (d+\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \log \left (\left (b-\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{6 \sqrt [3]{2} a \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {c^{2/3} \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \log \left (\left (b+\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{6 \sqrt [3]{2} a \left (b+\sqrt {b^2-4 a c}\right )^{2/3}} \]
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Rubi [A]
time = 0.73, antiderivative size = 655, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {1518, 1436,
206, 31, 648, 631, 210, 642} \begin {gather*} \frac {c^{2/3} \text {ArcTan}\left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right ) \left (\frac {b d-2 a e}{\sqrt {b^2-4 a c}}+d\right )}{\sqrt [3]{2} \sqrt {3} a \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {c^{2/3} \text {ArcTan}\left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{\sqrt {b^2-4 a c}+b}}}{\sqrt {3}}\right ) \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right )}{\sqrt [3]{2} \sqrt {3} a \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {c^{2/3} \left (\frac {b d-2 a e}{\sqrt {b^2-4 a c}}+d\right ) \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{b-\sqrt {b^2-4 a c}}+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{6 \sqrt [3]{2} a \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {c^{2/3} \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{\sqrt {b^2-4 a c}+b}+\left (\sqrt {b^2-4 a c}+b\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{6 \sqrt [3]{2} a \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}-\frac {c^{2/3} \left (\frac {b d-2 a e}{\sqrt {b^2-4 a c}}+d\right ) \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} a \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}-\frac {c^{2/3} \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} a \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}-\frac {d}{2 a x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 206
Rule 210
Rule 631
Rule 642
Rule 648
Rule 1436
Rule 1518
Rubi steps
\begin {align*} \int \frac {d+e x^3}{x^3 \left (a+b x^3+c x^6\right )} \, dx &=-\frac {d}{2 a x^2}-\frac {\int \frac {2 (b d-a e)+2 c d x^3}{a+b x^3+c x^6} \, dx}{2 a}\\ &=-\frac {d}{2 a x^2}-\frac {\left (c \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right )\right ) \int \frac {1}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^3} \, dx}{2 a}-\frac {\left (c \left (d+\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right )\right ) \int \frac {1}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^3} \, dx}{2 a}\\ &=-\frac {d}{2 a x^2}-\frac {\left (c \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right )\right ) \int \frac {1}{\frac {\sqrt [3]{b+\sqrt {b^2-4 a c}}}{\sqrt [3]{2}}+\sqrt [3]{c} x} \, dx}{3 \sqrt [3]{2} a \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}-\frac {\left (c \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right )\right ) \int \frac {2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-\sqrt [3]{c} x}{\frac {\left (b+\sqrt {b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac {\sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{3 \sqrt [3]{2} a \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}-\frac {\left (c \left (d+\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right )\right ) \int \frac {1}{\frac {\sqrt [3]{b-\sqrt {b^2-4 a c}}}{\sqrt [3]{2}}+\sqrt [3]{c} x} \, dx}{3 \sqrt [3]{2} a \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}-\frac {\left (c \left (d+\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right )\right ) \int \frac {2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-\sqrt [3]{c} x}{\frac {\left (b-\sqrt {b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac {\sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{3 \sqrt [3]{2} a \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\\ &=-\frac {d}{2 a x^2}-\frac {c^{2/3} \left (d+\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} a \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}-\frac {c^{2/3} \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \log \left (\sqrt [3]{b+\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} a \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {\left (c^{2/3} \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right )\right ) \int \frac {-\frac {\sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}}}{\sqrt [3]{2}}+2 c^{2/3} x}{\frac {\left (b+\sqrt {b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac {\sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{6 \sqrt [3]{2} a \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}-\frac {\left (c \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right )\right ) \int \frac {1}{\frac {\left (b+\sqrt {b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac {\sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{2\ 2^{2/3} a \sqrt [3]{b+\sqrt {b^2-4 a c}}}+\frac {\left (c^{2/3} \left (d+\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right )\right ) \int \frac {-\frac {\sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}}}{\sqrt [3]{2}}+2 c^{2/3} x}{\frac {\left (b-\sqrt {b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac {\sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{6 \sqrt [3]{2} a \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}-\frac {\left (c \left (d+\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right )\right ) \int \frac {1}{\frac {\left (b-\sqrt {b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac {\sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{2\ 2^{2/3} a \sqrt [3]{b-\sqrt {b^2-4 a c}}}\\ &=-\frac {d}{2 a x^2}-\frac {c^{2/3} \left (d+\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} a \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}-\frac {c^{2/3} \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \log \left (\sqrt [3]{b+\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} a \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {c^{2/3} \left (d+\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \log \left (\left (b-\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{6 \sqrt [3]{2} a \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {c^{2/3} \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \log \left (\left (b+\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{6 \sqrt [3]{2} a \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}-\frac {\left (c^{2/3} \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right )\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b+\sqrt {b^2-4 a c}}}\right )}{\sqrt [3]{2} a \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}-\frac {\left (c^{2/3} \left (d+\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right )\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}\right )}{\sqrt [3]{2} a \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\\ &=-\frac {d}{2 a x^2}+\frac {c^{2/3} \left (d+\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3} a \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {c^{2/3} \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b+\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3} a \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}-\frac {c^{2/3} \left (d+\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} a \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}-\frac {c^{2/3} \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \log \left (\sqrt [3]{b+\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} a \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {c^{2/3} \left (d+\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \log \left (\left (b-\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{6 \sqrt [3]{2} a \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {c^{2/3} \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \log \left (\left (b+\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{6 \sqrt [3]{2} a \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.03, size = 89, normalized size = 0.14 \begin {gather*} -\frac {d}{2 a x^2}-\frac {\text {RootSum}\left [a+b \text {$\#$1}^3+c \text {$\#$1}^6\&,\frac {b d \log (x-\text {$\#$1})-a e \log (x-\text {$\#$1})+c d \log (x-\text {$\#$1}) \text {$\#$1}^3}{b \text {$\#$1}^2+2 c \text {$\#$1}^5}\&\right ]}{3 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.07, size = 68, normalized size = 0.10
method | result | size |
default | \(\frac {\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{6} c +\textit {\_Z}^{3} b +a \right )}{\sum }\frac {\left (-\textit {\_R}^{3} c d +a e -b d \right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5} c +\textit {\_R}^{2} b}}{3 a}-\frac {d}{2 a \,x^{2}}\) | \(68\) |
risch | \(-\frac {d}{2 a \,x^{2}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (64 c^{3} a^{8}-48 b^{2} c^{2} a^{7}+12 b^{4} c \,a^{6}-b^{6} a^{5}\right ) \textit {\_Z}^{6}+\left (16 a^{5} b \,c^{2} e^{3}+48 a^{5} c^{3} d \,e^{2}-8 a^{4} b^{3} c \,e^{3}-72 a^{4} b^{2} c^{2} d \,e^{2}-96 a^{4} b \,c^{3} d^{2} e -16 a^{4} c^{4} d^{3}+a^{3} b^{5} e^{3}+27 a^{3} b^{4} c d \,e^{2}+96 a^{3} b^{3} c^{2} d^{2} e +56 a^{3} b^{2} c^{3} d^{3}-3 a^{2} b^{6} d \,e^{2}-30 a^{2} b^{5} c \,d^{2} e -41 a^{2} b^{4} c^{2} d^{3}+3 a \,b^{7} d^{2} e +11 a \,b^{6} c \,d^{3}-b^{8} d^{3}\right ) \textit {\_Z}^{3}+a^{3} c^{2} e^{6}-3 a^{2} b \,c^{2} d \,e^{5}+3 a^{2} c^{3} d^{2} e^{4}+3 a \,b^{2} c^{2} d^{2} e^{4}-6 a b \,c^{3} d^{3} e^{3}+3 a \,c^{4} d^{4} e^{2}-b^{3} c^{2} d^{3} e^{3}+3 b^{2} c^{3} d^{4} e^{2}-3 b \,c^{4} d^{5} e +c^{5} d^{6}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (224 c^{3} a^{8}-176 b^{2} c^{2} a^{7}+46 b^{4} c \,a^{6}-4 b^{6} a^{5}\right ) \textit {\_R}^{6}+\left (48 a^{5} b \,c^{2} e^{3}+156 a^{5} c^{3} d \,e^{2}-24 a^{4} b^{3} c \,e^{3}-219 a^{4} b^{2} c^{2} d \,e^{2}-300 a^{4} b \,c^{3} d^{2} e -52 a^{4} c^{4} d^{3}+3 a^{3} b^{5} e^{3}+81 a^{3} b^{4} c d \,e^{2}+291 a^{3} b^{3} c^{2} d^{2} e +173 a^{3} b^{2} c^{3} d^{3}-9 a^{2} b^{6} d \,e^{2}-90 a^{2} b^{5} c \,d^{2} e -124 a^{2} b^{4} c^{2} d^{3}+9 a \,b^{7} d^{2} e +33 a \,b^{6} c \,d^{3}-3 b^{8} d^{3}\right ) \textit {\_R}^{3}+3 a^{3} c^{2} e^{6}-9 a^{2} b \,c^{2} d \,e^{5}+9 a^{2} c^{3} d^{2} e^{4}+9 a \,b^{2} c^{2} d^{2} e^{4}-18 a b \,c^{3} d^{3} e^{3}+9 a \,c^{4} d^{4} e^{2}-3 b^{3} c^{2} d^{3} e^{3}+9 b^{2} c^{3} d^{4} e^{2}-9 b \,c^{4} d^{5} e +3 c^{5} d^{6}\right ) x +\left (-12 a^{6} b \,c^{2} e^{2}-16 a^{6} c^{3} d e +7 a^{5} b^{3} c \,e^{2}+36 a^{5} b^{2} c^{2} d e +20 a^{5} b \,c^{3} d^{2}-a^{4} b^{5} e^{2}-16 a^{4} b^{4} c d e -25 a^{4} b^{3} c^{2} d^{2}+2 a^{3} b^{6} d e +9 a^{3} b^{5} c \,d^{2}-a^{2} b^{7} d^{2}\right ) \textit {\_R}^{4}+\left (-a^{4} c^{2} e^{5}+2 a^{3} b \,c^{2} d \,e^{4}-2 a^{3} c^{3} d^{2} e^{3}-a^{2} b^{2} c^{2} d^{2} e^{3}+2 a^{2} b \,c^{3} d^{3} e^{2}-a^{2} c^{4} d^{4} e \right ) \textit {\_R} \right )\right )}{3}\) | \(970\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 37.90, size = 2500, normalized size = 3.82 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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