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 = 0.125382, antiderivative size = 154, normalized size of antiderivative = 1., 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} \[ -\frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{4/3} d}+\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{4/3} d}+\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{4/3} d}-\frac{1}{a d (c+d x)} \]
Antiderivative was successfully verified.
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Rule 372
Rule 325
Rule 292
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
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*}
Mathematica [A] time = 0.0424312, size = 140, normalized size = 0.91 \[ \frac{-\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )+2 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )-2 \sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} (c+d x)-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )-\frac{6 \sqrt [3]{a}}{c+d x}}{6 a^{4/3} d} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.004, size = 92, normalized size = 0.6 \begin{align*} -{\frac{1}{ad \left ( dx+c \right ) }}-{\frac{1}{3\,ad}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{3}b{d}^{3}+3\,{{\it \_Z}}^{2}bc{d}^{2}+3\,{\it \_Z}\,b{c}^{2}d+b{c}^{3}+a \right ) }{\frac{ \left ({\it \_R}\,d+c \right ) \ln \left ( x-{\it \_R} \right ) }{{d}^{2}{{\it \_R}}^{2}+2\,cd{\it \_R}+{c}^{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49883, size = 378, normalized size = 2.45 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.766905, size = 44, normalized size = 0.29 \begin{align*} - \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14125, size = 236, normalized size = 1.53 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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