Optimal. Leaf size=144 \[ \frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 b^{4/3} d}-\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 b^{4/3} d}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{4/3} d}+\frac{x}{b} \]
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Rubi [A] time = 0.134834, antiderivative size = 144, 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, 321, 200, 31, 634, 617, 204, 628} \[ \frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 b^{4/3} d}-\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 b^{4/3} d}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{4/3} d}+\frac{x}{b} \]
Antiderivative was successfully verified.
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Rule 372
Rule 321
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{(c+d x)^3}{a+b (c+d x)^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^3}{a+b x^3} \, dx,x,c+d x\right )}{d}\\ &=\frac{x}{b}-\frac{a \operatorname{Subst}\left (\int \frac{1}{a+b x^3} \, dx,x,c+d x\right )}{b d}\\ &=\frac{x}{b}-\frac{\sqrt [3]{a} \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,c+d x\right )}{3 b d}-\frac{\sqrt [3]{a} \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 b d}\\ &=\frac{x}{b}-\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 b^{4/3} d}+\frac{\sqrt [3]{a} \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 b^{4/3} d}-\frac{a^{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 b d}\\ &=\frac{x}{b}-\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 b^{4/3} d}+\frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 b^{4/3} d}-\frac{\sqrt [3]{a} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}\right )}{b^{4/3} d}\\ &=\frac{x}{b}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt{3} b^{4/3} d}-\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 b^{4/3} d}+\frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 b^{4/3} d}\\ \end{align*}
Mathematica [A] time = 0.0196716, size = 142, normalized size = 0.99 \[ \frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )-2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )-2 \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} (c+d x)-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )+6 \sqrt [3]{b} c+6 \sqrt [3]{b} d x}{6 b^{4/3} d} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.001, size = 78, normalized size = 0.5 \begin{align*}{\frac{x}{b}}-{\frac{a}{3\,{b}^{2}d}\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}}}} \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 )} a}{b} + \frac{x}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49415, size = 321, normalized size = 2.23 \begin{align*} \frac{6 \, d x + 2 \, \sqrt{3} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3}{\left (b d x + b c\right )} \left (-\frac{a}{b}\right )^{\frac{2}{3}} - \sqrt{3} a}{3 \, a}\right ) - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} +{\left (d x + c\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right ) + 2 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left (d x + c - \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}{6 \, b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.454441, size = 27, normalized size = 0.19 \begin{align*} \frac{\operatorname{RootSum}{\left (27 t^{3} b^{4} + a, \left ( t \mapsto t \log{\left (x + \frac{- 3 t b + c}{d} \right )} \right )\right )}}{d} + \frac{x}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1674, size = 288, normalized size = 2. \begin{align*} \frac{1}{3} \, \sqrt{3} \left (-\frac{a}{b^{4} d^{3}}\right )^{\frac{1}{3}} \arctan \left (-\frac{b d^{4} x + b c d^{3} - \left (-a b^{2}\right )^{\frac{1}{3}} d^{3}}{\sqrt{3} b d^{4} x + \sqrt{3} b c d^{3} + \sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}} d^{3}}\right ) - \frac{1}{6} \, \left (-\frac{a}{b^{4} d^{3}}\right )^{\frac{1}{3}} \log \left ({\left (\sqrt{3} b d^{4} x + \sqrt{3} b c d^{3} + \sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}} d^{3}\right )}^{2} +{\left (b d^{4} x + b c d^{3} - \left (-a b^{2}\right )^{\frac{1}{3}} d^{3}\right )}^{2}\right ) + \frac{1}{3} \, \left (-\frac{a}{b^{4} d^{3}}\right )^{\frac{1}{3}} \log \left ({\left | -b^{2} d^{4} x - b^{2} c d^{3} + \left (-a b^{2}\right )^{\frac{1}{3}} b d^{3} \right |}\right ) + \frac{x}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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