Optimal. Leaf size=42 \[ \frac {\log (c+d x)}{a d e}-\frac {\log \left (a+b (c+d x)^3\right )}{3 a d e} \]
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Rubi [A] time = 0.03, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {372, 266, 36, 29, 31} \[ \frac {\log (c+d x)}{a d e}-\frac {\log \left (a+b (c+d x)^3\right )}{3 a d e} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 266
Rule 372
Rubi steps
\begin {align*} \int \frac {1}{(c e+d e x) \left (a+b (c+d x)^3\right )} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x \left (a+b x^3\right )} \, dx,x,c+d x\right )}{d e}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x (a+b x)} \, dx,x,(c+d x)^3\right )}{3 d e}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,(c+d x)^3\right )}{3 a d e}-\frac {b \operatorname {Subst}\left (\int \frac {1}{a+b x} \, dx,x,(c+d x)^3\right )}{3 a d e}\\ &=\frac {\log (c+d x)}{a d e}-\frac {\log \left (a+b (c+d x)^3\right )}{3 a d e}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 42, normalized size = 1.00 \[ \frac {\log (c+d x)}{a d e}-\frac {\log \left (a+b (c+d x)^3\right )}{3 a d e} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.09, size = 54, normalized size = 1.29 \[ -\frac {\log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right ) - 3 \, \log \left (d x + c\right )}{3 \, a d e} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 62, normalized size = 1.48 \[ -\frac {e^{\left (-1\right )} \log \left ({\left | b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a \right |}\right )}{3 \, a d} + \frac {e^{\left (-1\right )} \log \left ({\left | d x + c \right |}\right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 63, normalized size = 1.50 \[ \frac {\ln \left (d x +c \right )}{a d e}-\frac {\ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right )}{3 a d e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 62, normalized size = 1.48 \[ -\frac {\log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{3 \, a d e} + \frac {\log \left (d x + c\right )}{a d e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 62, normalized size = 1.48 \[ \frac {\ln \left (c+d\,x\right )}{a\,d\,e}-\frac {\ln \left (b\,c^3+3\,b\,c^2\,d\,x+3\,b\,c\,d^2\,x^2+b\,d^3\,x^3+a\right )}{3\,a\,d\,e} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.75, size = 53, normalized size = 1.26 \[ \frac {\log {\left (\frac {c}{d} + x \right )}}{a d e} - \frac {\log {\left (\frac {3 c^{2} x}{d^{2}} + \frac {3 c x^{2}}{d} + x^{3} + \frac {a + b c^{3}}{b d^{3}} \right )}}{3 a d e} \]
Verification of antiderivative is not currently implemented for this CAS.
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