Optimal. Leaf size=94 \[ \frac{1}{3 a^2 d e \left (a+b (c+d x)^3\right )}-\frac{\log \left (a+b (c+d x)^3\right )}{3 a^3 d e}+\frac{\log (c+d x)}{a^3 d e}+\frac{1}{6 a d e \left (a+b (c+d x)^3\right )^2} \]
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Rubi [A] time = 0.0825657, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {372, 266, 44} \[ \frac{1}{3 a^2 d e \left (a+b (c+d x)^3\right )}-\frac{\log \left (a+b (c+d x)^3\right )}{3 a^3 d e}+\frac{\log (c+d x)}{a^3 d e}+\frac{1}{6 a d e \left (a+b (c+d x)^3\right )^2} \]
Antiderivative was successfully verified.
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Rule 372
Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{(c e+d e x) \left (a+b (c+d x)^3\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x \left (a+b x^3\right )^3} \, dx,x,c+d x\right )}{d e}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x (a+b x)^3} \, dx,x,(c+d x)^3\right )}{3 d e}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a^3 x}-\frac{b}{a (a+b x)^3}-\frac{b}{a^2 (a+b x)^2}-\frac{b}{a^3 (a+b x)}\right ) \, dx,x,(c+d x)^3\right )}{3 d e}\\ &=\frac{1}{6 a d e \left (a+b (c+d x)^3\right )^2}+\frac{1}{3 a^2 d e \left (a+b (c+d x)^3\right )}+\frac{\log (c+d x)}{a^3 d e}-\frac{\log \left (a+b (c+d x)^3\right )}{3 a^3 d e}\\ \end{align*}
Mathematica [A] time = 0.0473953, size = 66, normalized size = 0.7 \[ \frac{\frac{a \left (2 \left (a+b (c+d x)^3\right )+a\right )}{\left (a+b (c+d x)^3\right )^2}-2 \log \left (a+b (c+d x)^3\right )+6 \log (c+d x)}{6 a^3 d e} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.023, size = 304, normalized size = 3.2 \begin{align*}{\frac{\ln \left ( dx+c \right ) }{{a}^{3}de}}+{\frac{b{d}^{2}{x}^{3}}{3\,{a}^{2}e \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}}}+{\frac{bcd{x}^{2}}{{a}^{2}e \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}}}+{\frac{b{c}^{2}x}{{a}^{2}e \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}}}+{\frac{b{c}^{3}}{3\,{a}^{2}e \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{1}{2\,ae \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{\ln \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) }{3\,{a}^{3}de}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.03209, size = 348, normalized size = 3.7 \begin{align*} \frac{2 \, b d^{3} x^{3} + 6 \, b c d^{2} x^{2} + 6 \, b c^{2} d x + 2 \, b c^{3} + 3 \, a}{6 \,{\left (a^{2} b^{2} d^{7} e x^{6} + 6 \, a^{2} b^{2} c d^{6} e x^{5} + 15 \, a^{2} b^{2} c^{2} d^{5} e x^{4} + 2 \,{\left (10 \, a^{2} b^{2} c^{3} + a^{3} b\right )} d^{4} e x^{3} + 3 \,{\left (5 \, a^{2} b^{2} c^{4} + 2 \, a^{3} b c\right )} d^{3} e x^{2} + 6 \,{\left (a^{2} b^{2} c^{5} + a^{3} b c^{2}\right )} d^{2} e x +{\left (a^{2} b^{2} c^{6} + 2 \, a^{3} b c^{3} + a^{4}\right )} d e\right )}} - \frac{\log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{3 \, a^{3} d e} + \frac{\log \left (d x + c\right )}{a^{3} d e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.81026, size = 987, normalized size = 10.5 \begin{align*} \frac{2 \, a b d^{3} x^{3} + 6 \, a b c d^{2} x^{2} + 6 \, a b c^{2} d x + 2 \, a b c^{3} + 3 \, a^{2} - 2 \,{\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + b^{2} c^{6} + 2 \,{\left (10 \, b^{2} c^{3} + a b\right )} d^{3} x^{3} + 2 \, a b c^{3} + 3 \,{\left (5 \, b^{2} c^{4} + 2 \, a b c\right )} d^{2} x^{2} + 6 \,{\left (b^{2} c^{5} + a b c^{2}\right )} d x + a^{2}\right )} \log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right ) + 6 \,{\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + b^{2} c^{6} + 2 \,{\left (10 \, b^{2} c^{3} + a b\right )} d^{3} x^{3} + 2 \, a b c^{3} + 3 \,{\left (5 \, b^{2} c^{4} + 2 \, a b c\right )} d^{2} x^{2} + 6 \,{\left (b^{2} c^{5} + a b c^{2}\right )} d x + a^{2}\right )} \log \left (d x + c\right )}{6 \,{\left (a^{3} b^{2} d^{7} e x^{6} + 6 \, a^{3} b^{2} c d^{6} e x^{5} + 15 \, a^{3} b^{2} c^{2} d^{5} e x^{4} + 2 \,{\left (10 \, a^{3} b^{2} c^{3} + a^{4} b\right )} d^{4} e x^{3} + 3 \,{\left (5 \, a^{3} b^{2} c^{4} + 2 \, a^{4} b c\right )} d^{3} e x^{2} + 6 \,{\left (a^{3} b^{2} c^{5} + a^{4} b c^{2}\right )} d^{2} e x +{\left (a^{3} b^{2} c^{6} + 2 \, a^{4} b c^{3} + a^{5}\right )} d e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 166.56, size = 292, normalized size = 3.11 \begin{align*} \frac{3 a + 2 b c^{3} + 6 b c^{2} d x + 6 b c d^{2} x^{2} + 2 b d^{3} x^{3}}{6 a^{4} d e + 12 a^{3} b c^{3} d e + 6 a^{2} b^{2} c^{6} d e + 90 a^{2} b^{2} c^{2} d^{5} e x^{4} + 36 a^{2} b^{2} c d^{6} e x^{5} + 6 a^{2} b^{2} d^{7} e x^{6} + x^{3} \left (12 a^{3} b d^{4} e + 120 a^{2} b^{2} c^{3} d^{4} e\right ) + x^{2} \left (36 a^{3} b c d^{3} e + 90 a^{2} b^{2} c^{4} d^{3} e\right ) + x \left (36 a^{3} b c^{2} d^{2} e + 36 a^{2} b^{2} c^{5} d^{2} e\right )} + \frac{\log{\left (\frac{c}{d} + x \right )}}{a^{3} d e} - \frac{\log{\left (\frac{3 c^{2} x}{d^{2}} + \frac{3 c x^{2}}{d} + x^{3} + \frac{a + b c^{3}}{b d^{3}} \right )}}{3 a^{3} d e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25217, size = 203, normalized size = 2.16 \begin{align*} -\frac{e^{\left (-1\right )} \log \left ({\left | b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a \right |}\right )}{3 \, a^{3} d} + \frac{e^{\left (-1\right )} \log \left ({\left | d x + c \right |}\right )}{a^{3} d} + \frac{{\left (2 \, a b d^{3} x^{3} + 6 \, a b c d^{2} x^{2} + 6 \, a b c^{2} d x + 2 \, a b c^{3} + 3 \, a^{2}\right )} e^{\left (-1\right )}}{6 \,{\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^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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