Optimal. Leaf size=154 \[ -\frac {\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{4/3} d}+\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{4/3} d}+\frac {\sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{4/3} d}-\frac {1}{a d (c+d x)} \]
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Rubi [A] time = 0.13, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {372, 325, 292, 31, 634, 617, 204, 628} \begin {gather*} -\frac {\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{4/3} d}+\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{4/3} d}+\frac {\sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{4/3} d}-\frac {1}{a d (c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 292
Rule 325
Rule 372
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {1}{(c+d x)^2 \left (a+b (c+d x)^3\right )} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x^2 \left (a+b x^3\right )} \, dx,x,c+d x\right )}{d}\\ &=-\frac {1}{a d (c+d x)}-\frac {b \operatorname {Subst}\left (\int \frac {x}{a+b x^3} \, dx,x,c+d x\right )}{a d}\\ &=-\frac {1}{a d (c+d x)}+\frac {b^{2/3} \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,c+d x\right )}{3 a^{4/3} d}-\frac {b^{2/3} \operatorname {Subst}\left (\int \frac {\sqrt [3]{a}+\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,c+d x\right )}{3 a^{4/3} d}\\ &=-\frac {1}{a d (c+d x)}+\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{4/3} d}-\frac {\sqrt [3]{b} \operatorname {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^{4/3} d}-\frac {b^{2/3} \operatorname {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 a d}\\ &=-\frac {1}{a d (c+d x)}+\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{4/3} d}-\frac {\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{4/3} d}-\frac {\sqrt [3]{b} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}\right )}{a^{4/3} d}\\ &=-\frac {1}{a d (c+d x)}+\frac {\sqrt [3]{b} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} a^{4/3} d}+\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{4/3} d}-\frac {\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{4/3} d}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 140, normalized size = 0.91 \begin {gather*} \frac {-\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )+2 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )-2 \sqrt {3} \sqrt [3]{b} \tan ^{-1}\left (\frac {2 \sqrt [3]{b} (c+d x)-\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )-\frac {6 \sqrt [3]{a}}{c+d x}}{6 a^{4/3} d} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(c+d x)^2 \left (a+b (c+d x)^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.99, size = 150, normalized size = 0.97 \begin {gather*} -\frac {2 \, \sqrt {3} {\left (d x + c\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (d x + c\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + {\left (d x + c\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} - {\left (a d x + a c\right )} \left (\frac {b}{a}\right )^{\frac {2}{3}} + a \left (\frac {b}{a}\right )^{\frac {1}{3}}\right ) - 2 \, {\left (d x + c\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b d x + b c + a \left (\frac {b}{a}\right )^{\frac {2}{3}}\right ) + 6}{6 \, {\left (a d^{2} x + a c d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 175, normalized size = 1.14 \begin {gather*} \frac {\left (\frac {b}{a d^{3}}\right )^{\frac {1}{3}} \log \left ({\left | -\left (\frac {b}{a d^{3}}\right )^{\frac {1}{3}} - \frac {1}{{\left (d x + c\right )} d} \right |}\right )}{3 \, a} - \frac {\sqrt {3} \left (a^{2} b\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (\left (\frac {b}{a d^{3}}\right )^{\frac {1}{3}} - \frac {2}{{\left (d x + c\right )} d}\right )}}{3 \, \left (\frac {b}{a d^{3}}\right )^{\frac {1}{3}}}\right )}{3 \, a^{2} d} - \frac {\left (a^{2} b\right )^{\frac {1}{3}} \log \left (\left (\frac {b}{a d^{3}}\right )^{\frac {2}{3}} - \frac {\left (\frac {b}{a d^{3}}\right )^{\frac {1}{3}}}{{\left (d x + c\right )} d} + \frac {1}{{\left (d x + c\right )}^{2} d^{2}}\right )}{6 \, a^{2} d} - \frac {1}{{\left (d x + c\right )} a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.00, size = 92, normalized size = 0.60 \begin {gather*} -\frac {\left (\RootOf \left (b \,d^{3} \textit {\_Z}^{3}+3 b \,d^{2} c \,\textit {\_Z}^{2}+3 b d \,c^{2} \textit {\_Z} +b \,c^{3}+a \right ) d +c \right ) \ln \left (-\RootOf \left (b \,d^{3} \textit {\_Z}^{3}+3 b \,d^{2} c \,\textit {\_Z}^{2}+3 b d \,c^{2} \textit {\_Z} +b \,c^{3}+a \right )+x \right )}{3 a d \left (d^{2} \RootOf \left (b \,d^{3} \textit {\_Z}^{3}+3 b \,d^{2} c \,\textit {\_Z}^{2}+3 b d \,c^{2} \textit {\_Z} +b \,c^{3}+a \right )^{2}+2 c d \RootOf \left (b \,d^{3} \textit {\_Z}^{3}+3 b \,d^{2} c \,\textit {\_Z}^{2}+3 b d \,c^{2} \textit {\_Z} +b \,c^{3}+a \right )+c^{2}\right )}-\frac {1}{\left (d x +c \right ) a d} \end {gather*}
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*} -\frac {-\frac {1}{6} \, {\left (2 \, \sqrt {3} \left (-\frac {1}{a b^{2} d^{3}}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, a b d x + 2 \, a b c - \left (-a^{2} b\right )^{\frac {2}{3}}\right )}}{3 \, \left (-a^{2} b\right )^{\frac {2}{3}}}\right ) + \left (-\frac {1}{a b^{2} d^{3}}\right )^{\frac {1}{3}} \log \left ({\left (2 \, a b d x + 2 \, a b c - \left (-a^{2} b\right )^{\frac {2}{3}}\right )}^{2} + 3 \, \left (-a^{2} b\right )^{\frac {4}{3}}\right ) - 2 \, \left (-\frac {1}{a b^{2} d^{3}}\right )^{\frac {1}{3}} \log \left ({\left | a b d x + a b c + \left (-a^{2} b\right )^{\frac {2}{3}} \right |}\right )\right )} b}{a} - \frac {1}{a d^{2} x + a c d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.38, size = 138, normalized size = 0.90 \begin {gather*} \frac {b^{1/3}\,\ln \left (b^{1/3}\,c+a^{1/3}+b^{1/3}\,d\,x\right )}{3\,a^{4/3}\,d}-\frac {1}{a\,d\,\left (c+d\,x\right )}-\frac {b^{1/3}\,\ln \left (2\,b^{1/3}\,c-a^{1/3}+2\,b^{1/3}\,d\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,a^{4/3}\,d}+\frac {b^{1/3}\,\ln \left (2\,b^{1/3}\,c-a^{1/3}+2\,b^{1/3}\,d\,x-\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}{a^{4/3}\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.66, size = 44, normalized size = 0.29 \begin {gather*} - \frac {1}{a c d + a d^{2} x} + \frac {\operatorname {RootSum} {\left (27 t^{3} a^{4} - b, \left (t \mapsto t \log {\left (x + \frac {9 t^{2} a^{3} + b c}{b d} \right )} \right )\right )}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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