Optimal. Leaf size=156 \[ -\frac{b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{5/3} d}+\frac{b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{5/3} d}+\frac{b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{5/3} d}-\frac{1}{2 a d (c+d x)^2} \]
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Rubi [A] time = 0.123323, antiderivative size = 156, 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, 200, 31, 634, 617, 204, 628} \[ -\frac{b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{5/3} d}+\frac{b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{5/3} d}+\frac{b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{5/3} d}-\frac{1}{2 a d (c+d x)^2} \]
Antiderivative was successfully verified.
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Rule 372
Rule 325
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{(c+d x)^3 \left (a+b (c+d x)^3\right )} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^3 \left (a+b x^3\right )} \, dx,x,c+d x\right )}{d}\\ &=-\frac{1}{2 a d (c+d x)^2}-\frac{b \operatorname{Subst}\left (\int \frac{1}{a+b x^3} \, dx,x,c+d x\right )}{a d}\\ &=-\frac{1}{2 a d (c+d x)^2}-\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,c+d x\right )}{3 a^{5/3} d}-\frac{b \operatorname{Subst}\left (\int \frac{2 \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^{5/3} d}\\ &=-\frac{1}{2 a d (c+d x)^2}-\frac{b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{5/3} d}+\frac{b^{2/3} \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^{5/3} d}-\frac{b \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^{4/3} d}\\ &=-\frac{1}{2 a d (c+d x)^2}-\frac{b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{5/3} d}+\frac{b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{5/3} d}-\frac{b^{2/3} \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^{5/3} d}\\ &=-\frac{1}{2 a d (c+d x)^2}+\frac{b^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt{3} a^{5/3} d}-\frac{b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{5/3} d}+\frac{b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{5/3} d}\\ \end{align*}
Mathematica [A] time = 0.0359816, size = 139, normalized size = 0.89 \[ \frac{b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )-\frac{3 a^{2/3}}{(c+d x)^2}-2 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )-2 \sqrt{3} b^{2/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} (c+d x)-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{6 a^{5/3} d} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.003, size = 87, normalized size = 0.6 \begin{align*} -{\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{\ln \left ( x-{\it \_R} \right ) }{{d}^{2}{{\it \_R}}^{2}+2\,cd{\it \_R}+{c}^{2}}}}-{\frac{1}{2\,ad \left ( dx+c \right ) ^{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^{2} b d^{3}}\right )^{\frac{1}{3}} \arctan \left (-\frac{b d x + b c + \left (a b^{2}\right )^{\frac{1}{3}}}{\sqrt{3} b d x + \sqrt{3} b c - \sqrt{3} \left (a b^{2}\right )^{\frac{1}{3}}}\right ) - \left (\frac{1}{a^{2} b d^{3}}\right )^{\frac{1}{3}} \log \left ({\left (\sqrt{3} b d x + \sqrt{3} b c - \sqrt{3} \left (a b^{2}\right )^{\frac{1}{3}}\right )}^{2} +{\left (b d x + b c + \left (a b^{2}\right )^{\frac{1}{3}}\right )}^{2}\right ) + 2 \, \left (\frac{1}{a^{2} b d^{3}}\right )^{\frac{1}{3}} \log \left ({\left | b d x + b c + \left (a b^{2}\right )^{\frac{1}{3}} \right |}\right )\right )} b}{a} - \frac{1}{2 \,{\left (a d^{3} x^{2} + 2 \, a c d^{2} x + a c^{2} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56277, size = 540, normalized size = 3.46 \begin{align*} \frac{2 \, \sqrt{3}{\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3}{\left (a d x + a c\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}} - \sqrt{3} b}{3 \, b}\right ) -{\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} + a^{2} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}} +{\left (a b d x + a b c\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}\right ) + 2 \,{\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b d x + b c - a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}\right ) - 3}{6 \,{\left (a d^{3} x^{2} + 2 \, a c d^{2} x + a c^{2} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.17084, size = 61, normalized size = 0.39 \begin{align*} - \frac{1}{2 a c^{2} d + 4 a c d^{2} x + 2 a d^{3} x^{2}} + \frac{\operatorname{RootSum}{\left (27 t^{3} a^{5} + b^{2}, \left ( t \mapsto t \log{\left (x + \frac{- 3 t a^{2} + 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 = 1.14308, size = 252, normalized size = 1.62 \begin{align*} \frac{1}{3} \, \sqrt{3} \left (-\frac{b^{2}}{a^{5} d^{3}}\right )^{\frac{1}{3}} \arctan \left (-\frac{b d x + b c - \left (-a b^{2}\right )^{\frac{1}{3}}}{\sqrt{3} b d x + \sqrt{3} b c + \sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}}}\right ) - \frac{1}{6} \, \left (-\frac{b^{2}}{a^{5} d^{3}}\right )^{\frac{1}{3}} \log \left ({\left (\sqrt{3} b d x + \sqrt{3} b c + \sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}}\right )}^{2} +{\left (b d x + b c - \left (-a b^{2}\right )^{\frac{1}{3}}\right )}^{2}\right ) + \frac{1}{3} \, \left (-\frac{b^{2}}{a^{5} d^{3}}\right )^{\frac{1}{3}} \log \left ({\left | -a b d x - a b c + \left (-a b^{2}\right )^{\frac{1}{3}} a \right |}\right ) - \frac{1}{2 \,{\left (d x + c\right )}^{2} a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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