Optimal. Leaf size=201 \[ \frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{5/3} b^{4/3} d}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{54 a^{5/3} b^{4/3} d}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{5/3} b^{4/3} d}+\frac{c+d x}{18 a b d \left (a+b (c+d x)^3\right )}-\frac{c+d x}{6 b d \left (a+b (c+d x)^3\right )^2} \]
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Rubi [A] time = 0.163744, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {372, 288, 199, 200, 31, 634, 617, 204, 628} \[ \frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{5/3} b^{4/3} d}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{54 a^{5/3} b^{4/3} d}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{5/3} b^{4/3} d}+\frac{c+d x}{18 a b d \left (a+b (c+d x)^3\right )}-\frac{c+d x}{6 b d \left (a+b (c+d x)^3\right )^2} \]
Antiderivative was successfully verified.
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Rule 372
Rule 288
Rule 199
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{(c+d x)^3}{\left (a+b (c+d x)^3\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^3}{\left (a+b x^3\right )^3} \, dx,x,c+d x\right )}{d}\\ &=-\frac{c+d x}{6 b d \left (a+b (c+d x)^3\right )^2}+\frac{\operatorname{Subst}\left (\int \frac{1}{\left (a+b x^3\right )^2} \, dx,x,c+d x\right )}{6 b d}\\ &=-\frac{c+d x}{6 b d \left (a+b (c+d x)^3\right )^2}+\frac{c+d x}{18 a b d \left (a+b (c+d x)^3\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{a+b x^3} \, dx,x,c+d x\right )}{9 a b d}\\ &=-\frac{c+d x}{6 b d \left (a+b (c+d x)^3\right )^2}+\frac{c+d x}{18 a b d \left (a+b (c+d x)^3\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,c+d x\right )}{27 a^{5/3} b d}+\frac{\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 )}{27 a^{5/3} b d}\\ &=-\frac{c+d x}{6 b d \left (a+b (c+d x)^3\right )^2}+\frac{c+d x}{18 a b d \left (a+b (c+d x)^3\right )}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{5/3} b^{4/3} 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 )}{54 a^{5/3} b^{4/3} 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 )}{18 a^{4/3} b d}\\ &=-\frac{c+d x}{6 b d \left (a+b (c+d x)^3\right )^2}+\frac{c+d x}{18 a b d \left (a+b (c+d x)^3\right )}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{5/3} b^{4/3} d}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{54 a^{5/3} b^{4/3} d}+\frac{\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^{5/3} b^{4/3} d}\\ &=-\frac{c+d x}{6 b d \left (a+b (c+d x)^3\right )^2}+\frac{c+d x}{18 a b d \left (a+b (c+d x)^3\right )}-\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{9 \sqrt{3} a^{5/3} b^{4/3} d}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{5/3} b^{4/3} d}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{54 a^{5/3} b^{4/3} d}\\ \end{align*}
Mathematica [A] time = 0.113399, size = 179, normalized size = 0.89 \[ \frac{-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{a^{5/3}}+\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{a^{5/3}}+\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} (c+d x)-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{a^{5/3}}+\frac{3 \sqrt [3]{b} (c+d x)}{a \left (a+b (c+d x)^3\right )}-\frac{9 \sqrt [3]{b} (c+d x)}{\left (a+b (c+d x)^3\right )^2}}{54 b^{4/3} d} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.016, size = 186, normalized size = 0.9 \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{{d}^{3}{x}^{4}}{18\,a}}+{\frac{2\,c{d}^{2}{x}^{3}}{9\,a}}+{\frac{{c}^{2}d{x}^{2}}{3\,a}}-{\frac{ \left ( -2\,b{c}^{3}+a \right ) x}{9\,ab}}-{\frac{c \left ( -b{c}^{3}+2\,a \right ) }{18\,abd}} \right ) }+{\frac{1}{27\,a{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{b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + b c^{4} + 2 \,{\left (2 \, b c^{3} - a\right )} d x - 2 \, a c}{18 \,{\left (a b^{3} d^{7} x^{6} + 6 \, a b^{3} c d^{6} x^{5} + 15 \, a b^{3} c^{2} d^{5} x^{4} + 2 \,{\left (10 \, a b^{3} c^{3} + a^{2} b^{2}\right )} d^{4} x^{3} + 3 \,{\left (5 \, a b^{3} c^{4} + 2 \, a^{2} b^{2} c\right )} d^{3} x^{2} + 6 \,{\left (a b^{3} c^{5} + a^{2} b^{2} c^{2}\right )} d^{2} x +{\left (a b^{3} c^{6} + 2 \, a^{2} b^{2} c^{3} + a^{3} b\right )} d\right )}} + \frac{\frac{1}{3} \, \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 ) - \frac{1}{6} \, \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 ) + \frac{1}{3} \, \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 )}{9 \, a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.99109, size = 3501, normalized size = 17.42 \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 = 24.4362, size = 260, normalized size = 1.29 \begin{align*} \frac{- 2 a c + b c^{4} + 6 b c^{2} d^{2} x^{2} + 4 b c d^{3} x^{3} + b d^{4} x^{4} + x \left (- 2 a d + 4 b c^{3} d\right )}{18 a^{3} b d + 36 a^{2} b^{2} c^{3} d + 18 a b^{3} c^{6} d + 270 a b^{3} c^{2} d^{5} x^{4} + 108 a b^{3} c d^{6} x^{5} + 18 a b^{3} d^{7} x^{6} + x^{3} \left (36 a^{2} b^{2} d^{4} + 360 a b^{3} c^{3} d^{4}\right ) + x^{2} \left (108 a^{2} b^{2} c d^{3} + 270 a b^{3} c^{4} d^{3}\right ) + x \left (108 a^{2} b^{2} c^{2} d^{2} + 108 a b^{3} c^{5} d^{2}\right )} + \frac{\operatorname{RootSum}{\left (19683 t^{3} a^{5} b^{4} - 1, \left ( t \mapsto t \log{\left (x + \frac{27 t a^{2} b + c}{d} \right )} \right )\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18688, size = 360, normalized size = 1.79 \begin{align*} \frac{1}{27} \, \sqrt{3} \left (\frac{1}{a^{5} 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 ) - \frac{1}{54} \, \left (\frac{1}{a^{5} b^{4} 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 ) + \frac{1}{27} \, \left (\frac{1}{a^{5} b^{4} d^{3}}\right )^{\frac{1}{3}} \log \left ({\left | 9 \, a b^{2} d x + 9 \, a b^{2} c + 9 \, \left (a b^{2}\right )^{\frac{1}{3}} a b \right |}\right ) + \frac{b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4} - 2 \, a d x - 2 \, a c}{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 b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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