Optimal. Leaf size=182 \[ \frac{e^3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 a^{2/3} b^{4/3} d}-\frac{e^3 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{18 a^{2/3} b^{4/3} d}-\frac{e^3 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{2/3} b^{4/3} d}-\frac{e^3 (c+d x)}{3 b d \left (a+b (c+d x)^3\right )} \]
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Rubi [A] time = 0.140213, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {372, 288, 200, 31, 634, 617, 204, 628} \[ \frac{e^3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 a^{2/3} b^{4/3} d}-\frac{e^3 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{18 a^{2/3} b^{4/3} d}-\frac{e^3 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{2/3} b^{4/3} d}-\frac{e^3 (c+d x)}{3 b d \left (a+b (c+d x)^3\right )} \]
Antiderivative was successfully verified.
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Rule 372
Rule 288
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{(c e+d e x)^3}{\left (a+b (c+d x)^3\right )^2} \, dx &=\frac{e^3 \operatorname{Subst}\left (\int \frac{x^3}{\left (a+b x^3\right )^2} \, dx,x,c+d x\right )}{d}\\ &=-\frac{e^3 (c+d x)}{3 b d \left (a+b (c+d x)^3\right )}+\frac{e^3 \operatorname{Subst}\left (\int \frac{1}{a+b x^3} \, dx,x,c+d x\right )}{3 b d}\\ &=-\frac{e^3 (c+d x)}{3 b d \left (a+b (c+d x)^3\right )}+\frac{e^3 \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,c+d x\right )}{9 a^{2/3} b d}+\frac{e^3 \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 )}{9 a^{2/3} b d}\\ &=-\frac{e^3 (c+d x)}{3 b d \left (a+b (c+d x)^3\right )}+\frac{e^3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 a^{2/3} b^{4/3} d}-\frac{e^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 )}{18 a^{2/3} b^{4/3} d}+\frac{e^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 )}{6 \sqrt [3]{a} b d}\\ &=-\frac{e^3 (c+d x)}{3 b d \left (a+b (c+d x)^3\right )}+\frac{e^3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 a^{2/3} b^{4/3} d}-\frac{e^3 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{18 a^{2/3} b^{4/3} d}+\frac{e^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 )}{3 a^{2/3} b^{4/3} d}\\ &=-\frac{e^3 (c+d x)}{3 b d \left (a+b (c+d x)^3\right )}-\frac{e^3 \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{3 \sqrt{3} a^{2/3} b^{4/3} d}+\frac{e^3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 a^{2/3} b^{4/3} d}-\frac{e^3 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{18 a^{2/3} b^{4/3} d}\\ \end{align*}
Mathematica [A] time = 0.0467509, size = 154, normalized size = 0.85 \[ \frac{e^3 \left (-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{a^{2/3}}+\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{a^{2/3}}+\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} (c+d x)-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{a^{2/3}}-\frac{6 \sqrt [3]{b} (c+d x)}{a+b (c+d x)^3}\right )}{18 b^{4/3} d} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.009, size = 166, normalized size = 0.9 \begin{align*} -{\frac{{e}^{3}x}{ \left ( 3\,b{d}^{3}{x}^{3}+9\,bc{d}^{2}{x}^{2}+9\,b{c}^{2}dx+3\,b{c}^{3}+3\,a \right ) b}}-{\frac{{e}^{3}c}{ \left ( 3\,b{d}^{3}{x}^{3}+9\,bc{d}^{2}{x}^{2}+9\,b{c}^{2}dx+3\,b{c}^{3}+3\,a \right ) db}}+{\frac{{e}^{3}}{9\,{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 )} e^{3}}{3 \, b} - \frac{d e^{3} x + c e^{3}}{3 \,{\left (b^{2} d^{4} x^{3} + 3 \, b^{2} c d^{3} x^{2} + 3 \, b^{2} c^{2} d^{2} x +{\left (b^{2} c^{3} + a b\right )} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.6552, size = 2018, normalized size = 11.09 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.19442, size = 110, normalized size = 0.6 \begin{align*} - \frac{c e^{3} + d e^{3} x}{3 a b d + 3 b^{2} c^{3} d + 9 b^{2} c^{2} d^{2} x + 9 b^{2} c d^{3} x^{2} + 3 b^{2} d^{4} x^{3}} + \frac{e^{3} \operatorname{RootSum}{\left (729 t^{3} a^{2} b^{4} - 1, \left ( t \mapsto t \log{\left (x + \frac{9 t a b e^{3} + c e^{3}}{d e^{3}} \right )} \right )\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20993, size = 302, normalized size = 1.66 \begin{align*} \frac{1}{9} \, \sqrt{3} \left (\frac{1}{a^{2} b^{4} 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 ) e^{3} - \frac{1}{18} \, \left (\frac{1}{a^{2} b^{4} d^{3}}\right )^{\frac{1}{3}} e^{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}{9} \, \left (\frac{1}{a^{2} b^{4} d^{3}}\right )^{\frac{1}{3}} e^{3} \log \left ({\left | 3 \, b^{2} d x + 3 \, b^{2} c + 3 \, \left (a b^{2}\right )^{\frac{1}{3}} b \right |}\right ) - \frac{d x e^{3} + c e^{3}}{3 \,{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )} b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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