Optimal. Leaf size=168 \[ \frac{a^{2/3} e^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 b^{5/3} d}-\frac{a^{2/3} e^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 b^{5/3} d}+\frac{a^{2/3} e^4 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{5/3} d}+\frac{e^4 (c+d x)^2}{2 b d} \]
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Rubi [A] time = 0.134311, antiderivative size = 168, 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, 321, 292, 31, 634, 617, 204, 628} \[ \frac{a^{2/3} e^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 b^{5/3} d}-\frac{a^{2/3} e^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 b^{5/3} d}+\frac{a^{2/3} e^4 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{5/3} d}+\frac{e^4 (c+d x)^2}{2 b d} \]
Antiderivative was successfully verified.
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Rule 372
Rule 321
Rule 292
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{(c e+d e x)^4}{a+b (c+d x)^3} \, dx &=\frac{e^4 \operatorname{Subst}\left (\int \frac{x^4}{a+b x^3} \, dx,x,c+d x\right )}{d}\\ &=\frac{e^4 (c+d x)^2}{2 b d}-\frac{\left (a e^4\right ) \operatorname{Subst}\left (\int \frac{x}{a+b x^3} \, dx,x,c+d x\right )}{b d}\\ &=\frac{e^4 (c+d x)^2}{2 b d}+\frac{\left (a^{2/3} e^4\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,c+d x\right )}{3 b^{4/3} d}-\frac{\left (a^{2/3} e^4\right ) \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 b^{4/3} d}\\ &=\frac{e^4 (c+d x)^2}{2 b d}+\frac{a^{2/3} e^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 b^{5/3} d}-\frac{\left (a^{2/3} e^4\right ) \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^{5/3} d}-\frac{\left (a e^4\right ) \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^{4/3} d}\\ &=\frac{e^4 (c+d x)^2}{2 b d}+\frac{a^{2/3} e^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 b^{5/3} d}-\frac{a^{2/3} e^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 b^{5/3} d}-\frac{\left (a^{2/3} e^4\right ) \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^{5/3} d}\\ &=\frac{e^4 (c+d x)^2}{2 b d}+\frac{a^{2/3} e^4 \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt{3} b^{5/3} d}+\frac{a^{2/3} e^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 b^{5/3} d}-\frac{a^{2/3} e^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 b^{5/3} d}\\ \end{align*}
Mathematica [A] time = 0.0245045, size = 163, normalized size = 0.97 \[ e^4 \left (\frac{a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 b^{5/3} d}-\frac{a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 b^{5/3} d}-\frac{a^{2/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} (c+d x)-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{5/3} d}+\frac{(c+d x)^2}{2 b d}\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.002, size = 102, normalized size = 0.6 \begin{align*}{\frac{{e}^{4}{x}^{2}d}{2\,b}}+{\frac{{e}^{4}xc}{b}}-{\frac{{e}^{4}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{ \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 )} a e^{4}}{b} + \frac{d e^{4} x^{2} + 2 \, c e^{4} x}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56289, size = 420, normalized size = 2.5 \begin{align*} \frac{3 \, d^{2} e^{4} x^{2} + 6 \, c d e^{4} x - 2 \, \sqrt{3} e^{4} \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3}{\left (b d x + b c\right )} \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} - \sqrt{3} a}{3 \, a}\right ) - e^{4} \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a d^{2} x^{2} + 2 \, a c d x + a c^{2} -{\left (b d x + b c\right )} \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}} + a \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}}\right ) + 2 \, e^{4} \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a d x + a c + b \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}\right )}{6 \, b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.543631, size = 66, normalized size = 0.39 \begin{align*} \frac{e^{4} \operatorname{RootSum}{\left (27 t^{3} b^{5} - a^{2}, \left ( t \mapsto t \log{\left (x + \frac{9 t^{2} b^{3} e^{8} + a c e^{8}}{a d e^{8}} \right )} \right )\right )}}{d} + \frac{c e^{4} x}{b} + \frac{d e^{4} x^{2}}{2 b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16982, size = 300, normalized size = 1.79 \begin{align*} -\frac{1}{3} \, \sqrt{3} \left (\frac{a^{2}}{b^{5} d^{3}}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, a b^{2} d^{7} x + 2 \, a b^{2} c d^{6} - \left (a^{2} b\right )^{\frac{2}{3}} b d^{6}\right )}}{3 \, \left (a^{2} b\right )^{\frac{2}{3}} b d^{6}}\right ) e^{4} - \frac{1}{6} \, \left (\frac{a^{2}}{b^{5} d^{3}}\right )^{\frac{1}{3}} e^{4} \log \left (3 \, \left (a^{2} b\right )^{\frac{4}{3}} b^{2} d^{12} +{\left (2 \, a b^{2} d^{7} x + 2 \, a b^{2} c d^{6} - \left (a^{2} b\right )^{\frac{2}{3}} b d^{6}\right )}^{2}\right ) + \frac{1}{3} \, \left (\frac{a^{2}}{b^{5} d^{3}}\right )^{\frac{1}{3}} e^{4} \log \left ({\left | a b^{3} d^{7} x + a b^{3} c d^{6} + \left (a^{2} b\right )^{\frac{2}{3}} b^{2} d^{6} \right |}\right ) + \frac{b d^{7} x^{2} e^{4} + 2 \, b c d^{6} x e^{4}}{2 \, b^{2} d^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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