Optimal. Leaf size=224 \[ \frac {\sqrt [3]{b} c \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} c+b^{2/3} c^2\right )}+\frac {\log (x)}{a+b c^3}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} c\right )}-\frac {\left (2 \sqrt [3]{a}-\sqrt [3]{b} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{2/3} \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} c+b^{2/3} c^2\right )} \]
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Rubi [A]
time = 0.23, antiderivative size = 238, normalized size of antiderivative = 1.06, number of steps
used = 11, number of rules used = 10, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.588, Rules used = {378, 6857,
1885, 1874, 31, 648, 631, 210, 642, 266} \begin {gather*} \frac {\sqrt [3]{b} c \left (\sqrt [3]{a}+\sqrt [3]{b} c\right ) \text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} \left (a+b c^3\right )}-\frac {\sqrt [3]{b} c \left (\sqrt [3]{a}-\sqrt [3]{b} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{2/3} \left (a+b c^3\right )}+\frac {\sqrt [3]{b} c \left (\sqrt [3]{a}-\sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} \left (a+b c^3\right )}-\frac {\log \left (a+b (c+d x)^3\right )}{3 \left (a+b c^3\right )}+\frac {\log (x)}{a+b c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 210
Rule 266
Rule 378
Rule 631
Rule 642
Rule 648
Rule 1874
Rule 1885
Rule 6857
Rubi steps
\begin {align*} \int \frac {1}{x \left (a+b (c+d x)^3\right )} \, dx &=\text {Subst}\left (\int \frac {1}{(-c+x) \left (a+b x^3\right )} \, dx,x,c+d x\right )\\ &=\text {Subst}\left (\int \left (-\frac {1}{\left (a+b c^3\right ) (c-x)}-\frac {b \left (c^2+c x+x^2\right )}{\left (a+b c^3\right ) \left (a+b x^3\right )}\right ) \, dx,x,c+d x\right )\\ &=\frac {\log (x)}{a+b c^3}-\frac {b \text {Subst}\left (\int \frac {c^2+c x+x^2}{a+b x^3} \, dx,x,c+d x\right )}{a+b c^3}\\ &=\frac {\log (x)}{a+b c^3}-\frac {b \text {Subst}\left (\int \frac {x^2}{a+b x^3} \, dx,x,c+d x\right )}{a+b c^3}-\frac {b \text {Subst}\left (\int \frac {c^2+c x}{a+b x^3} \, dx,x,c+d x\right )}{a+b c^3}\\ &=\frac {\log (x)}{a+b c^3}-\frac {\log \left (a+b (c+d x)^3\right )}{3 \left (a+b c^3\right )}-\frac {b^{2/3} \text {Subst}\left (\int \frac {\sqrt [3]{a} \left (\sqrt [3]{a} c+2 \sqrt [3]{b} c^2\right )+\sqrt [3]{b} \left (\sqrt [3]{a} c-\sqrt [3]{b} c^2\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,c+d x\right )}{3 a^{2/3} \left (a+b c^3\right )}+\frac {\left (b^{2/3} c \left (\sqrt [3]{a}-\sqrt [3]{b} c\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,c+d x\right )}{3 a^{2/3} \left (a+b c^3\right )}\\ &=\frac {\log (x)}{a+b c^3}+\frac {\sqrt [3]{b} c \left (\sqrt [3]{a}-\sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} \left (a+b c^3\right )}-\frac {\log \left (a+b (c+d x)^3\right )}{3 \left (a+b c^3\right )}-\frac {\left (\sqrt [3]{b} c \left (\sqrt [3]{a}-\sqrt [3]{b} c\right )\right ) \text {Subst}\left (\int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,c+d x\right )}{6 a^{2/3} \left (a+b c^3\right )}-\frac {\left (b^{2/3} c \left (\sqrt [3]{a}+\sqrt [3]{b} c\right )\right ) \text {Subst}\left (\int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,c+d x\right )}{2 \sqrt [3]{a} \left (a+b c^3\right )}\\ &=\frac {\log (x)}{a+b c^3}+\frac {\sqrt [3]{b} c \left (\sqrt [3]{a}-\sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} \left (a+b c^3\right )}-\frac {\sqrt [3]{b} c \left (\sqrt [3]{a}-\sqrt [3]{b} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{2/3} \left (a+b c^3\right )}-\frac {\log \left (a+b (c+d x)^3\right )}{3 \left (a+b c^3\right )}-\frac {\left (\sqrt [3]{b} c \left (\sqrt [3]{a}+\sqrt [3]{b} c\right )\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}\right )}{a^{2/3} \left (a+b c^3\right )}\\ &=\frac {\sqrt [3]{b} c \left (\sqrt [3]{a}+\sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} a^{2/3} \left (a+b c^3\right )}+\frac {\log (x)}{a+b c^3}+\frac {\sqrt [3]{b} c \left (\sqrt [3]{a}-\sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} \left (a+b c^3\right )}-\frac {\sqrt [3]{b} c \left (\sqrt [3]{a}-\sqrt [3]{b} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{2/3} \left (a+b c^3\right )}-\frac {\log \left (a+b (c+d x)^3\right )}{3 \left (a+b c^3\right )}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.04, size = 119, normalized size = 0.53 \begin {gather*} -\frac {-3 \log (x)+\text {RootSum}\left [a+b c^3+3 b c^2 d \text {$\#$1}+3 b c d^2 \text {$\#$1}^2+b d^3 \text {$\#$1}^3\&,\frac {3 c^2 \log (x-\text {$\#$1})+3 c d \log (x-\text {$\#$1}) \text {$\#$1}+d^2 \log (x-\text {$\#$1}) \text {$\#$1}^2}{c^2+2 c d \text {$\#$1}+d^2 \text {$\#$1}^2}\&\right ]}{3 \left (a+b c^3\right )} \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 = 0.04, size = 105, normalized size = 0.47
method | result | size |
default | \(-\frac {\munderset {\textit {\_R} =\RootOf \left (d^{3} b \,\textit {\_Z}^{3}+3 b c \,d^{2} \textit {\_Z}^{2}+3 b \,c^{2} d \textit {\_Z} +b \,c^{3}+a \right )}{\sum }\frac {\left (d^{2} \textit {\_R}^{2}+3 c d \textit {\_R} +3 c^{2}\right ) \ln \left (x -\textit {\_R} \right )}{d^{2} \textit {\_R}^{2}+2 c d \textit {\_R} +c^{2}}}{3 \left (b \,c^{3}+a \right )}+\frac {\ln \left (x \right )}{b \,c^{3}+a}\) | \(105\) |
risch | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (1+\left (a^{2} b \,c^{3}+a^{3}\right ) \textit {\_Z}^{3}+3 a^{2} \textit {\_Z}^{2}+3 a \textit {\_Z} \right )}{\sum }\textit {\_R} \ln \left (\left (\left (2 a b \,c^{3} d -4 d \,a^{2}\right ) \textit {\_R}^{2}+\left (b \,c^{3} d -8 a d \right ) \textit {\_R} -4 d \right ) x +\left (a b \,c^{4}+a^{2} c \right ) \textit {\_R}^{2}+\left (b \,c^{4}-2 a c \right ) \textit {\_R} -3 c \right )\right )}{3}+\frac {\ln \left (-x \right )}{b \,c^{3}+a}\) | \(125\) |
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 [C] Result contains complex when optimal does not.
time = 1.18, size = 4370, normalized size = 19.51 \begin {gather*} \text {Too large to display} \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 = 0.12, size = 553, normalized size = 2.47 \begin {gather*} \frac {\ln \left (x\right )}{b\,c^3+a}+\left (\sum _{k=1}^3\ln \left ({\mathrm {root}\left (27\,a^2\,b\,c^3\,z^3+27\,a^3\,z^3+27\,a^2\,z^2+9\,a\,z+1,z,k\right )}^2\,b^4\,c^4\,d^8\,3-\mathrm {root}\left (27\,a^2\,b\,c^3\,z^3+27\,a^3\,z^3+27\,a^2\,z^2+9\,a\,z+1,z,k\right )\,b^3\,c\,d^8\,3-\mathrm {root}\left (27\,a^2\,b\,c^3\,z^3+27\,a^3\,z^3+27\,a^2\,z^2+9\,a\,z+1,z,k\right )\,b^3\,d^9\,x\,4-{\mathrm {root}\left (27\,a^2\,b\,c^3\,z^3+27\,a^3\,z^3+27\,a^2\,z^2+9\,a\,z+1,z,k\right )}^2\,a\,b^3\,c\,d^8\,6-{\mathrm {root}\left (27\,a^2\,b\,c^3\,z^3+27\,a^3\,z^3+27\,a^2\,z^2+9\,a\,z+1,z,k\right )}^2\,a\,b^3\,d^9\,x\,24+{\mathrm {root}\left (27\,a^2\,b\,c^3\,z^3+27\,a^3\,z^3+27\,a^2\,z^2+9\,a\,z+1,z,k\right )}^3\,a^2\,b^3\,c\,d^8\,9+{\mathrm {root}\left (27\,a^2\,b\,c^3\,z^3+27\,a^3\,z^3+27\,a^2\,z^2+9\,a\,z+1,z,k\right )}^3\,a\,b^4\,c^4\,d^8\,9-{\mathrm {root}\left (27\,a^2\,b\,c^3\,z^3+27\,a^3\,z^3+27\,a^2\,z^2+9\,a\,z+1,z,k\right )}^3\,a^2\,b^3\,d^9\,x\,36+{\mathrm {root}\left (27\,a^2\,b\,c^3\,z^3+27\,a^3\,z^3+27\,a^2\,z^2+9\,a\,z+1,z,k\right )}^2\,b^4\,c^3\,d^9\,x\,3+{\mathrm {root}\left (27\,a^2\,b\,c^3\,z^3+27\,a^3\,z^3+27\,a^2\,z^2+9\,a\,z+1,z,k\right )}^3\,a\,b^4\,c^3\,d^9\,x\,18\right )\,\mathrm {root}\left (27\,a^2\,b\,c^3\,z^3+27\,a^3\,z^3+27\,a^2\,z^2+9\,a\,z+1,z,k\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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