Optimal. Leaf size=27 \[ \frac {(a+b \log (c (e+f x)))^3}{3 b d f} \]
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Rubi [A]
time = 0.04, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2437, 12, 2339,
30} \begin {gather*} \frac {(a+b \log (c (e+f x)))^3}{3 b d f} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 2339
Rule 2437
Rubi steps
\begin {align*} \int \frac {(a+b \log (c (e+f x)))^2}{d e+d f x} \, dx &=\frac {\text {Subst}\left (\int \frac {(a+b \log (c x))^2}{d x} \, dx,x,e+f x\right )}{f}\\ &=\frac {\text {Subst}\left (\int \frac {(a+b \log (c x))^2}{x} \, dx,x,e+f x\right )}{d f}\\ &=\frac {\text {Subst}\left (\int x^2 \, dx,x,a+b \log (c (e+f x))\right )}{b d f}\\ &=\frac {(a+b \log (c (e+f x)))^3}{3 b d f}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 27, normalized size = 1.00 \begin {gather*} \frac {(a+b \log (c (e+f x)))^3}{3 b d f} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(63\) vs.
\(2(25)=50\).
time = 0.28, size = 64, normalized size = 2.37
method | result | size |
risch | \(\frac {b^{2} \ln \left (c \left (f x +e \right )\right )^{3}}{3 d f}+\frac {b a \ln \left (c \left (f x +e \right )\right )^{2}}{d f}+\frac {a^{2} \ln \left (f x +e \right )}{d f}\) | \(58\) |
norman | \(\frac {a^{2} \ln \left (c \left (f x +e \right )\right )}{d f}+\frac {b a \ln \left (c \left (f x +e \right )\right )^{2}}{d f}+\frac {b^{2} \ln \left (c \left (f x +e \right )\right )^{3}}{3 d f}\) | \(60\) |
derivativedivides | \(\frac {\frac {c \,a^{2} \ln \left (c f x +c e \right )}{d}+\frac {c a b \ln \left (c f x +c e \right )^{2}}{d}+\frac {c \,b^{2} \ln \left (c f x +c e \right )^{3}}{3 d}}{c f}\) | \(64\) |
default | \(\frac {\frac {c \,a^{2} \ln \left (c f x +c e \right )}{d}+\frac {c a b \ln \left (c f x +c e \right )^{2}}{d}+\frac {c \,b^{2} \ln \left (c f x +c e \right )^{3}}{3 d}}{c f}\) | \(64\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 136 vs.
\(2 (26) = 52\).
time = 0.29, size = 136, normalized size = 5.04 \begin {gather*} -a b {\left (\frac {2 \, \log \left (c f x + c e\right ) \log \left (d f x + d e\right )}{d f} - \frac {\log \left (f x + e\right )^{2} + 2 \, \log \left (f x + e\right ) \log \left (c\right )}{d f}\right )} + \frac {b^{2} \log \left (c f x + c e\right )^{3}}{3 \, d f} + \frac {2 \, a b \log \left (c f x + c e\right ) \log \left (d f x + d e\right )}{d f} + \frac {a^{2} \log \left (d f x + d e\right )}{d f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 56 vs.
\(2 (26) = 52\).
time = 0.37, size = 56, normalized size = 2.07 \begin {gather*} \frac {b^{2} \log \left (c f x + c e\right )^{3} + 3 \, a b \log \left (c f x + c e\right )^{2} + 3 \, a^{2} \log \left (c f x + c e\right )}{3 \, d f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 51 vs.
\(2 (19) = 38\).
time = 0.09, size = 51, normalized size = 1.89 \begin {gather*} \frac {a^{2} \log {\left (d e + d f x \right )}}{d f} + \frac {a b \log {\left (c \left (e + f x\right ) \right )}^{2}}{d f} + \frac {b^{2} \log {\left (c \left (e + f x\right ) \right )}^{3}}{3 d f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 53 vs.
\(2 (26) = 52\).
time = 4.75, size = 53, normalized size = 1.96 \begin {gather*} \frac {b^{2} \log \left (c f x + c e\right )^{3} + 3 \, a b \log \left (c f x + c e\right )^{2} + 3 \, a^{2} \log \left (f x + e\right )}{3 \, d f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.48, size = 50, normalized size = 1.85 \begin {gather*} \frac {3\,\ln \left (e+f\,x\right )\,a^2+3\,a\,b\,{\ln \left (c\,e+c\,f\,x\right )}^2+b^2\,{\ln \left (c\,e+c\,f\,x\right )}^3}{3\,d\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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