Optimal. Leaf size=237 \[ -\frac{20 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{11/3} d e^3}+\frac{10 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{11/3} d e^3}+\frac{20 b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{11/3} d e^3}+\frac{4}{9 a^2 d e^3 (c+d x)^2 \left (a+b (c+d x)^3\right )}-\frac{10}{9 a^3 d e^3 (c+d x)^2}+\frac{1}{6 a d e^3 (c+d x)^2 \left (a+b (c+d x)^3\right )^2} \]
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Rubi [A] time = 0.183631, antiderivative size = 237, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {372, 290, 325, 200, 31, 634, 617, 204, 628} \[ -\frac{20 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{11/3} d e^3}+\frac{10 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{11/3} d e^3}+\frac{20 b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{11/3} d e^3}+\frac{4}{9 a^2 d e^3 (c+d x)^2 \left (a+b (c+d x)^3\right )}-\frac{10}{9 a^3 d e^3 (c+d x)^2}+\frac{1}{6 a d e^3 (c+d x)^2 \left (a+b (c+d x)^3\right )^2} \]
Antiderivative was successfully verified.
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Rule 372
Rule 290
Rule 325
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{(c e+d e x)^3 \left (a+b (c+d x)^3\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^3 \left (a+b x^3\right )^3} \, dx,x,c+d x\right )}{d e^3}\\ &=\frac{1}{6 a d e^3 (c+d x)^2 \left (a+b (c+d x)^3\right )^2}+\frac{4 \operatorname{Subst}\left (\int \frac{1}{x^3 \left (a+b x^3\right )^2} \, dx,x,c+d x\right )}{3 a d e^3}\\ &=\frac{1}{6 a d e^3 (c+d x)^2 \left (a+b (c+d x)^3\right )^2}+\frac{4}{9 a^2 d e^3 (c+d x)^2 \left (a+b (c+d x)^3\right )}+\frac{20 \operatorname{Subst}\left (\int \frac{1}{x^3 \left (a+b x^3\right )} \, dx,x,c+d x\right )}{9 a^2 d e^3}\\ &=-\frac{10}{9 a^3 d e^3 (c+d x)^2}+\frac{1}{6 a d e^3 (c+d x)^2 \left (a+b (c+d x)^3\right )^2}+\frac{4}{9 a^2 d e^3 (c+d x)^2 \left (a+b (c+d x)^3\right )}-\frac{(20 b) \operatorname{Subst}\left (\int \frac{1}{a+b x^3} \, dx,x,c+d x\right )}{9 a^3 d e^3}\\ &=-\frac{10}{9 a^3 d e^3 (c+d x)^2}+\frac{1}{6 a d e^3 (c+d x)^2 \left (a+b (c+d x)^3\right )^2}+\frac{4}{9 a^2 d e^3 (c+d x)^2 \left (a+b (c+d x)^3\right )}-\frac{(20 b) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,c+d x\right )}{27 a^{11/3} d e^3}-\frac{(20 b) \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^{11/3} d e^3}\\ &=-\frac{10}{9 a^3 d e^3 (c+d x)^2}+\frac{1}{6 a d e^3 (c+d x)^2 \left (a+b (c+d x)^3\right )^2}+\frac{4}{9 a^2 d e^3 (c+d x)^2 \left (a+b (c+d x)^3\right )}-\frac{20 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{11/3} d e^3}+\frac{\left (10 b^{2/3}\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 )}{27 a^{11/3} d e^3}-\frac{(10 b) \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^{10/3} d e^3}\\ &=-\frac{10}{9 a^3 d e^3 (c+d x)^2}+\frac{1}{6 a d e^3 (c+d x)^2 \left (a+b (c+d x)^3\right )^2}+\frac{4}{9 a^2 d e^3 (c+d x)^2 \left (a+b (c+d x)^3\right )}-\frac{20 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{11/3} d e^3}+\frac{10 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{11/3} d e^3}-\frac{\left (20 b^{2/3}\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 )}{9 a^{11/3} d e^3}\\ &=-\frac{10}{9 a^3 d e^3 (c+d x)^2}+\frac{1}{6 a d e^3 (c+d x)^2 \left (a+b (c+d x)^3\right )^2}+\frac{4}{9 a^2 d e^3 (c+d x)^2 \left (a+b (c+d x)^3\right )}+\frac{20 b^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{9 \sqrt{3} a^{11/3} d e^3}-\frac{20 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{11/3} d e^3}+\frac{10 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{11/3} d e^3}\\ \end{align*}
Mathematica [A] time = 0.0981259, size = 195, normalized size = 0.82 \[ \frac{20 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )-\frac{9 a^{5/3} b (c+d x)}{\left (a+b (c+d x)^3\right )^2}-\frac{33 a^{2/3} b (c+d x)}{a+b (c+d x)^3}-\frac{27 a^{2/3}}{(c+d x)^2}-40 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )-40 \sqrt{3} b^{2/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} (c+d x)-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{54 a^{11/3} d e^3} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.021, size = 446, normalized size = 1.9 \begin{align*} -{\frac{1}{2\,{a}^{3}d{e}^{3} \left ( dx+c \right ) ^{2}}}-{\frac{11\,{b}^{2}{d}^{3}{x}^{4}}{18\,{e}^{3}{a}^{3} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}}}-{\frac{22\,c{d}^{2}{b}^{2}{x}^{3}}{9\,{e}^{3}{a}^{3} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}}}-{\frac{11\,{b}^{2}{c}^{2}d{x}^{2}}{3\,{e}^{3}{a}^{3} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}}}-{\frac{22\,{b}^{2}x{c}^{3}}{9\,{e}^{3}{a}^{3} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}}}-{\frac{7\,bx}{9\,{e}^{3}{a}^{2} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}}}-{\frac{11\,{b}^{2}{c}^{4}}{18\,{e}^{3}{a}^{3} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}d}}-{\frac{7\,bc}{9\,{e}^{3}{a}^{2} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}d}}-{\frac{20}{27\,{a}^{3}d{e}^{3}}\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{20 \, b^{2} d^{6} x^{6} + 120 \, b^{2} c d^{5} x^{5} + 300 \, b^{2} c^{2} d^{4} x^{4} + 20 \, b^{2} c^{6} + 16 \,{\left (25 \, b^{2} c^{3} + 2 \, a b\right )} d^{3} x^{3} + 32 \, a b c^{3} + 12 \,{\left (25 \, b^{2} c^{4} + 8 \, a b c\right )} d^{2} x^{2} + 24 \,{\left (5 \, b^{2} c^{5} + 4 \, a b c^{2}\right )} d x + 9 \, a^{2}}{18 \,{\left (a^{3} b^{2} d^{9} e^{3} x^{8} + 8 \, a^{3} b^{2} c d^{8} e^{3} x^{7} + 28 \, a^{3} b^{2} c^{2} d^{7} e^{3} x^{6} + 2 \,{\left (28 \, a^{3} b^{2} c^{3} + a^{4} b\right )} d^{6} e^{3} x^{5} + 10 \,{\left (7 \, a^{3} b^{2} c^{4} + a^{4} b c\right )} d^{5} e^{3} x^{4} + 4 \,{\left (14 \, a^{3} b^{2} c^{5} + 5 \, a^{4} b c^{2}\right )} d^{4} e^{3} x^{3} +{\left (28 \, a^{3} b^{2} c^{6} + 20 \, a^{4} b c^{3} + a^{5}\right )} d^{3} e^{3} x^{2} + 2 \,{\left (4 \, a^{3} b^{2} c^{7} + 5 \, a^{4} b c^{4} + a^{5} c\right )} d^{2} e^{3} x +{\left (a^{3} b^{2} c^{8} + 2 \, a^{4} b c^{5} + a^{5} c^{2}\right )} d e^{3}\right )}} - \frac{\frac{10}{3} \,{\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 )} b}{9 \, a^{3} e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.04842, size = 2225, normalized size = 9.39 \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 [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26768, size = 513, normalized size = 2.16 \begin{align*} \frac{20}{27} \, \sqrt{3} \left (-\frac{b^{2} e^{\left (-9\right )}}{a^{11} 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{10}{27} \, \left (-\frac{b^{2} e^{\left (-9\right )}}{a^{11} 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{20}{27} \, \left (-\frac{b^{2} e^{\left (-9\right )}}{a^{11} d^{3}}\right )^{\frac{1}{3}} \log \left ({\left | -9 \, a^{3} b d x e^{3} - 9 \, a^{3} b c e^{3} + 9 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} e^{3} \right |}\right ) - \frac{{\left (20 \, b^{2} d^{6} x^{6} + 120 \, b^{2} c d^{5} x^{5} + 300 \, b^{2} c^{2} d^{4} x^{4} + 400 \, b^{2} c^{3} d^{3} x^{3} + 300 \, b^{2} c^{4} d^{2} x^{2} + 120 \, b^{2} c^{5} d x + 20 \, b^{2} c^{6} + 32 \, a b d^{3} x^{3} + 96 \, a b c d^{2} x^{2} + 96 \, a b c^{2} d x + 32 \, a b c^{3} + 9 \, a^{2}\right )} e^{\left (-3\right )}}{18 \,{\left (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} + a d x + a c\right )}^{2} a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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