Optimal. Leaf size=184 \[ \frac{e^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{9 \sqrt [3]{a} b^{5/3} d}-\frac{2 e^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 \sqrt [3]{a} b^{5/3} d}-\frac{2 e^4 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} \sqrt [3]{a} b^{5/3} d}-\frac{e^4 (c+d x)^2}{3 b d \left (a+b (c+d x)^3\right )} \]
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Rubi [A] time = 0.142301, antiderivative size = 184, 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, 292, 31, 634, 617, 204, 628} \[ \frac{e^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{9 \sqrt [3]{a} b^{5/3} d}-\frac{2 e^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 \sqrt [3]{a} b^{5/3} d}-\frac{2 e^4 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} \sqrt [3]{a} b^{5/3} d}-\frac{e^4 (c+d x)^2}{3 b d \left (a+b (c+d x)^3\right )} \]
Antiderivative was successfully verified.
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Rule 372
Rule 288
Rule 292
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{(c e+d e x)^4}{\left (a+b (c+d x)^3\right )^2} \, dx &=\frac{e^4 \operatorname{Subst}\left (\int \frac{x^4}{\left (a+b x^3\right )^2} \, dx,x,c+d x\right )}{d}\\ &=-\frac{e^4 (c+d x)^2}{3 b d \left (a+b (c+d x)^3\right )}+\frac{\left (2 e^4\right ) \operatorname{Subst}\left (\int \frac{x}{a+b x^3} \, dx,x,c+d x\right )}{3 b d}\\ &=-\frac{e^4 (c+d x)^2}{3 b d \left (a+b (c+d x)^3\right )}-\frac{\left (2 e^4\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,c+d x\right )}{9 \sqrt [3]{a} b^{4/3} d}+\frac{\left (2 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 )}{9 \sqrt [3]{a} b^{4/3} d}\\ &=-\frac{e^4 (c+d x)^2}{3 b d \left (a+b (c+d x)^3\right )}-\frac{2 e^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 \sqrt [3]{a} b^{5/3} d}+\frac{e^4 \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 )}{9 \sqrt [3]{a} b^{5/3} d}+\frac{e^4 \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 )}{3 b^{4/3} d}\\ &=-\frac{e^4 (c+d x)^2}{3 b d \left (a+b (c+d x)^3\right )}-\frac{2 e^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 \sqrt [3]{a} b^{5/3} d}+\frac{e^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{9 \sqrt [3]{a} b^{5/3} d}+\frac{\left (2 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 )}{3 \sqrt [3]{a} b^{5/3} d}\\ &=-\frac{e^4 (c+d x)^2}{3 b d \left (a+b (c+d x)^3\right )}-\frac{2 e^4 \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{3 \sqrt{3} \sqrt [3]{a} b^{5/3} d}-\frac{2 e^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 \sqrt [3]{a} b^{5/3} d}+\frac{e^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{9 \sqrt [3]{a} b^{5/3} d}\\ \end{align*}
Mathematica [A] time = 0.0732645, size = 155, normalized size = 0.84 \[ \frac{e^4 \left (\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{\sqrt [3]{a}}-\frac{3 b^{2/3} (c+d x)^2}{a+b (c+d x)^3}-\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{\sqrt [3]{a}}+\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} (c+d x)-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt [3]{a}}\right )}{9 b^{5/3} d} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.011, size = 221, normalized size = 1.2 \begin{align*} -{\frac{{e}^{4}{x}^{2}d}{ \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{2\,{e}^{4}xc}{ \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}^{4}{c}^{2}}{ \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 ) bd}}+{\frac{2\,{e}^{4}}{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{ \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}{3} \,{\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 )} e^{4}}{3 \, b} - \frac{d^{2} e^{4} x^{2} + 2 \, c d e^{4} x + c^{2} e^{4}}{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 = 1.69189, size = 2051, normalized size = 11.15 \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 = 2.6516, size = 131, normalized size = 0.71 \begin{align*} - \frac{c^{2} e^{4} + 2 c d e^{4} x + d^{2} e^{4} x^{2}}{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^{4} \operatorname{RootSum}{\left (729 t^{3} a b^{5} + 8, \left ( t \mapsto t \log{\left (x + \frac{81 t^{2} a b^{3} e^{8} + 4 c e^{8}}{4 d e^{8}} \right )} \right )\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17419, size = 300, normalized size = 1.63 \begin{align*} -\frac{2}{9} \, \sqrt{3} \left (-\frac{e^{12}}{a b^{5} 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 ) - \frac{1}{9} \, \left (-\frac{e^{12}}{a b^{5} 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 ) + \frac{2}{9} \, \left (-\frac{e^{12}}{a b^{5} d^{3}}\right )^{\frac{1}{3}} \log \left ({\left | 3 \, a b^{2} d x + 3 \, a b^{2} c + 3 \, \left (-a^{2} b\right )^{\frac{2}{3}} b \right |}\right ) - \frac{d^{2} x^{2} e^{4} + 2 \, c d x e^{4} + c^{2} e^{4}}{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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