Optimal. Leaf size=68 \[ -\frac {\log \left (a+b (c+d x)^3\right )}{3 a^2 d e}+\frac {\log (c+d x)}{a^2 d e}+\frac {1}{3 a d e \left (a+b (c+d x)^3\right )} \]
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Rubi [A] time = 0.06, antiderivative size = 68, normalized size of antiderivative = 1.00, 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} \[ -\frac {\log \left (a+b (c+d x)^3\right )}{3 a^2 d e}+\frac {\log (c+d x)}{a^2 d e}+\frac {1}{3 a d e \left (a+b (c+d x)^3\right )} \]
Antiderivative was successfully verified.
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Rule 44
Rule 266
Rule 372
Rubi steps
\begin {align*} \int \frac {1}{(c e+d e x) \left (a+b (c+d x)^3\right )^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x \left (a+b x^3\right )^2} \, dx,x,c+d x\right )}{d e}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x (a+b x)^2} \, dx,x,(c+d x)^3\right )}{3 d e}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{a^2 x}-\frac {b}{a (a+b x)^2}-\frac {b}{a^2 (a+b x)}\right ) \, dx,x,(c+d x)^3\right )}{3 d e}\\ &=\frac {1}{3 a d e \left (a+b (c+d x)^3\right )}+\frac {\log (c+d x)}{a^2 d e}-\frac {\log \left (a+b (c+d x)^3\right )}{3 a^2 d e}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 51, normalized size = 0.75 \[ \frac {\frac {a}{a+b (c+d x)^3}-\log \left (a+b (c+d x)^3\right )+3 \log (c+d x)}{3 a^2 d e} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.79, size = 173, normalized size = 2.54 \[ -\frac {{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )} \log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right ) - 3 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )} \log \left (d x + c\right ) - a}{3 \, {\left (a^{2} b d^{4} e x^{3} + 3 \, a^{2} b c d^{3} e x^{2} + 3 \, a^{2} b c^{2} d^{2} e x + {\left (a^{2} b c^{3} + a^{3}\right )} d e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 107, normalized size = 1.57 \[ -\frac {e^{\left (-1\right )} \log \left ({\left | b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a \right |}\right )}{3 \, a^{2} d} + \frac {e^{\left (-1\right )} \log \left ({\left | d x + c \right |}\right )}{a^{2} d} + \frac {e^{\left (-1\right )}}{3 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )} a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 109, normalized size = 1.60 \[ \frac {1}{3 \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right ) a d e}+\frac {\ln \left (d x +c \right )}{a^{2} d e}-\frac {\ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right )}{3 a^{2} d e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 114, normalized size = 1.68 \[ \frac {1}{3 \, {\left (a b d^{4} e x^{3} + 3 \, a b c d^{3} e x^{2} + 3 \, a b c^{2} d^{2} e x + {\left (a b c^{3} + a^{2}\right )} d e\right )}} - \frac {\log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{3 \, a^{2} d e} + \frac {\log \left (d x + c\right )}{a^{2} d e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.39, size = 116, normalized size = 1.71 \[ \frac {1}{3\,\left (e\,a^2\,d+b\,e\,a\,c^3\,d+3\,b\,e\,a\,c^2\,d^2\,x+3\,b\,e\,a\,c\,d^3\,x^2+b\,e\,a\,d^4\,x^3\right )}+\frac {\ln \left (c+d\,x\right )}{a^2\,d\,e}-\frac {\ln \left (b\,c^3+3\,b\,c^2\,d\,x+3\,b\,c\,d^2\,x^2+b\,d^3\,x^3+a\right )}{3\,a^2\,d\,e} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.17, size = 122, normalized size = 1.79 \[ \frac {1}{3 a^{2} d e + 3 a b c^{3} d e + 9 a b c^{2} d^{2} e x + 9 a b c d^{3} e x^{2} + 3 a b d^{4} e x^{3}} + \frac {\log {\left (\frac {c}{d} + x \right )}}{a^{2} d e} - \frac {\log {\left (\frac {3 c^{2} x}{d^{2}} + \frac {3 c x^{2}}{d} + x^{3} + \frac {a + b c^{3}}{b d^{3}} \right )}}{3 a^{2} d e} \]
Verification of antiderivative is not currently implemented for this CAS.
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