Optimal. Leaf size=202 \[ -\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{7/3} b^{2/3} d}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{7/3} b^{2/3} d}-\frac{2 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{7/3} b^{2/3} d}+\frac{2 (c+d x)^2}{9 a^2 d \left (a+b (c+d x)^3\right )}+\frac{(c+d x)^2}{6 a d \left (a+b (c+d x)^3\right )^2} \]
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Rubi [A] time = 0.166015, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {372, 290, 292, 31, 634, 617, 204, 628} \[ -\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{7/3} b^{2/3} d}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{7/3} b^{2/3} d}-\frac{2 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{7/3} b^{2/3} d}+\frac{2 (c+d x)^2}{9 a^2 d \left (a+b (c+d x)^3\right )}+\frac{(c+d x)^2}{6 a d \left (a+b (c+d x)^3\right )^2} \]
Antiderivative was successfully verified.
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Rule 372
Rule 290
Rule 292
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{c+d x}{\left (a+b (c+d x)^3\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x}{\left (a+b x^3\right )^3} \, dx,x,c+d x\right )}{d}\\ &=\frac{(c+d x)^2}{6 a d \left (a+b (c+d x)^3\right )^2}+\frac{2 \operatorname{Subst}\left (\int \frac{x}{\left (a+b x^3\right )^2} \, dx,x,c+d x\right )}{3 a d}\\ &=\frac{(c+d x)^2}{6 a d \left (a+b (c+d x)^3\right )^2}+\frac{2 (c+d x)^2}{9 a^2 d \left (a+b (c+d x)^3\right )}+\frac{2 \operatorname{Subst}\left (\int \frac{x}{a+b x^3} \, dx,x,c+d x\right )}{9 a^2 d}\\ &=\frac{(c+d x)^2}{6 a d \left (a+b (c+d x)^3\right )^2}+\frac{2 (c+d x)^2}{9 a^2 d \left (a+b (c+d x)^3\right )}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,c+d x\right )}{27 a^{7/3} \sqrt [3]{b} d}+\frac{2 \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 )}{27 a^{7/3} \sqrt [3]{b} d}\\ &=\frac{(c+d x)^2}{6 a d \left (a+b (c+d x)^3\right )^2}+\frac{2 (c+d x)^2}{9 a^2 d \left (a+b (c+d x)^3\right )}-\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{7/3} b^{2/3} d}+\frac{\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 )}{27 a^{7/3} b^{2/3} d}+\frac{\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 )}{9 a^2 \sqrt [3]{b} d}\\ &=\frac{(c+d x)^2}{6 a d \left (a+b (c+d x)^3\right )^2}+\frac{2 (c+d x)^2}{9 a^2 d \left (a+b (c+d x)^3\right )}-\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{7/3} b^{2/3} d}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{7/3} b^{2/3} d}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}\right )}{9 a^{7/3} b^{2/3} d}\\ &=\frac{(c+d x)^2}{6 a d \left (a+b (c+d x)^3\right )^2}+\frac{2 (c+d x)^2}{9 a^2 d \left (a+b (c+d x)^3\right )}-\frac{2 \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{9 \sqrt{3} a^{7/3} b^{2/3} d}-\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{7/3} b^{2/3} d}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{7/3} b^{2/3} d}\\ \end{align*}
Mathematica [A] time = 0.119765, size = 180, normalized size = 0.89 \[ \frac{\frac{2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{b^{2/3}}+\frac{9 a^{4/3} (c+d x)^2}{\left (a+b (c+d x)^3\right )^2}-\frac{4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{b^{2/3}}+\frac{4 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} (c+d x)-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{b^{2/3}}+\frac{12 \sqrt [3]{a} (c+d x)^2}{a+b (c+d x)^3}}{54 a^{7/3} d} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.015, size = 214, normalized size = 1.1 \begin{align*}{\frac{1}{ \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}} \left ({\frac{2\,b{d}^{4}{x}^{5}}{9\,{a}^{2}}}+{\frac{10\,bc{d}^{3}{x}^{4}}{9\,{a}^{2}}}+{\frac{20\,{c}^{2}{d}^{2}b{x}^{3}}{9\,{a}^{2}}}+{\frac{d \left ( 40\,b{c}^{3}+7\,a \right ){x}^{2}}{18\,{a}^{2}}}+{\frac{c \left ( 10\,b{c}^{3}+7\,a \right ) x}{9\,{a}^{2}}}+{\frac{{c}^{2} \left ( 4\,b{c}^{3}+7\,a \right ) }{18\,{a}^{2}d}} \right ) }+{\frac{2}{27\,b{a}^{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{4 \, b d^{5} x^{5} + 20 \, b c d^{4} x^{4} + 40 \, b c^{2} d^{3} x^{3} + 4 \, b c^{5} +{\left (40 \, b c^{3} + 7 \, a\right )} d^{2} x^{2} + 7 \, a c^{2} + 2 \,{\left (10 \, b c^{4} + 7 \, a c\right )} d x}{18 \,{\left (a^{2} b^{2} d^{7} x^{6} + 6 \, a^{2} b^{2} c d^{6} x^{5} + 15 \, a^{2} b^{2} c^{2} d^{5} x^{4} + 2 \,{\left (10 \, a^{2} b^{2} c^{3} + a^{3} b\right )} d^{4} x^{3} + 3 \,{\left (5 \, a^{2} b^{2} c^{4} + 2 \, a^{3} b c\right )} d^{3} x^{2} + 6 \,{\left (a^{2} b^{2} c^{5} + a^{3} b c^{2}\right )} d^{2} x +{\left (a^{2} b^{2} c^{6} + 2 \, a^{3} b c^{3} + a^{4}\right )} d\right )}} + \frac{2 \,{\left (-\frac{1}{3} \, \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 ) - \frac{1}{6} \, \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 ) + \frac{1}{3} \, \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 )}}{9 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.9354, size = 3613, normalized size = 17.89 \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 = 17.3841, size = 296, normalized size = 1.47 \begin{align*} \frac{7 a c^{2} + 4 b c^{5} + 40 b c^{2} d^{3} x^{3} + 20 b c d^{4} x^{4} + 4 b d^{5} x^{5} + x^{2} \left (7 a d^{2} + 40 b c^{3} d^{2}\right ) + x \left (14 a c d + 20 b c^{4} d\right )}{18 a^{4} d + 36 a^{3} b c^{3} d + 18 a^{2} b^{2} c^{6} d + 270 a^{2} b^{2} c^{2} d^{5} x^{4} + 108 a^{2} b^{2} c d^{6} x^{5} + 18 a^{2} b^{2} d^{7} x^{6} + x^{3} \left (36 a^{3} b d^{4} + 360 a^{2} b^{2} c^{3} d^{4}\right ) + x^{2} \left (108 a^{3} b c d^{3} + 270 a^{2} b^{2} c^{4} d^{3}\right ) + x \left (108 a^{3} b c^{2} d^{2} + 108 a^{2} b^{2} c^{5} d^{2}\right )} + \frac{\operatorname{RootSum}{\left (19683 t^{3} a^{7} b^{2} + 8, \left ( t \mapsto t \log{\left (x + \frac{729 t^{2} a^{5} b + 4 c}{4 d} \right )} \right )\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19586, size = 370, normalized size = 1.83 \begin{align*} -\frac{2}{27} \, \sqrt{3} \left (-\frac{1}{a^{7} 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 ) - \frac{1}{27} \, \left (-\frac{1}{a^{7} 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 ) + \frac{2}{27} \, \left (-\frac{1}{a^{7} b^{2} d^{3}}\right )^{\frac{1}{3}} \log \left ({\left | 9 \, a^{3} b d x + 9 \, a^{3} b c + 9 \, \left (-a^{2} b\right )^{\frac{2}{3}} a^{2} \right |}\right ) + \frac{4 \, b d^{5} x^{5} + 20 \, b c d^{4} x^{4} + 40 \, b c^{2} d^{3} x^{3} + 40 \, b c^{3} d^{2} x^{2} + 20 \, b c^{4} d x + 4 \, b c^{5} + 7 \, a d^{2} x^{2} + 14 \, a c d x + 7 \, a c^{2}}{18 \,{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}^{2} a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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