Optimal. Leaf size=113 \[ -\frac {2 a b i x}{d f}+\frac {2 b^2 i x}{d f}-\frac {2 b^2 i (e+f x) \log (c (e+f x))}{d f^2}+\frac {i (e+f x) (a+b \log (c (e+f x)))^2}{d f^2}+\frac {(f h-e i) (a+b \log (c (e+f x)))^3}{3 b d f^2} \]
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Rubi [A]
time = 0.14, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {2458, 12, 2388,
2339, 30, 2333, 2332} \begin {gather*} \frac {(f h-e i) (a+b \log (c (e+f x)))^3}{3 b d f^2}+\frac {i (e+f x) (a+b \log (c (e+f x)))^2}{d f^2}-\frac {2 a b i x}{d f}-\frac {2 b^2 i (e+f x) \log (c (e+f x))}{d f^2}+\frac {2 b^2 i x}{d f} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 2332
Rule 2333
Rule 2339
Rule 2388
Rule 2458
Rubi steps
\begin {align*} \int \frac {(h+186 x) (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (\frac {-186 e+f h}{f}+\frac {186 x}{f}\right ) (a+b \log (c x))^2}{d x} \, dx,x,e+f x\right )}{f}\\ &=\frac {\text {Subst}\left (\int \frac {\left (\frac {-186 e+f h}{f}+\frac {186 x}{f}\right ) (a+b \log (c x))^2}{x} \, dx,x,e+f x\right )}{d f}\\ &=\frac {186 \text {Subst}\left (\int (a+b \log (c x))^2 \, dx,x,e+f x\right )}{d f^2}-\frac {(186 e-f h) \text {Subst}\left (\int \frac {(a+b \log (c x))^2}{x} \, dx,x,e+f x\right )}{d f^2}\\ &=\frac {186 (e+f x) (a+b \log (c (e+f x)))^2}{d f^2}-\frac {(372 b) \text {Subst}(\int (a+b \log (c x)) \, dx,x,e+f x)}{d f^2}-\frac {(186 e-f h) \text {Subst}\left (\int x^2 \, dx,x,a+b \log (c (e+f x))\right )}{b d f^2}\\ &=-\frac {372 a b x}{d f}+\frac {186 (e+f x) (a+b \log (c (e+f x)))^2}{d f^2}-\frac {(186 e-f h) (a+b \log (c (e+f x)))^3}{3 b d f^2}-\frac {\left (372 b^2\right ) \text {Subst}(\int \log (c x) \, dx,x,e+f x)}{d f^2}\\ &=-\frac {372 a b x}{d f}+\frac {372 b^2 x}{d f}-\frac {372 b^2 (e+f x) \log (c (e+f x))}{d f^2}+\frac {186 (e+f x) (a+b \log (c (e+f x)))^2}{d f^2}-\frac {(186 e-f h) (a+b \log (c (e+f x)))^3}{3 b d f^2}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 89, normalized size = 0.79 \begin {gather*} \frac {-6 (a-b) b f i x-6 b^2 i (e+f x) \log (c (e+f x))+3 i (e+f x) (a+b \log (c (e+f x)))^2+\frac {(f h-e i) (a+b \log (c (e+f x)))^3}{b}}{3 d f^2} \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. \(256\) vs.
\(2(111)=222\).
time = 0.36, size = 257, normalized size = 2.27
method | result | size |
norman | \(\frac {i \left (a^{2}-2 b a +2 b^{2}\right ) x}{d f}+\frac {b^{2} i x \ln \left (c \left (f x +e \right )\right )^{2}}{d f}-\frac {\left (a^{2} e i -a^{2} f h -2 a b e i +2 b^{2} e i \right ) \ln \left (c \left (f x +e \right )\right )}{d \,f^{2}}-\frac {b \left (a e i -a f h -b e i \right ) \ln \left (c \left (f x +e \right )\right )^{2}}{d \,f^{2}}-\frac {b^{2} \left (e i -f h \right ) \ln \left (c \left (f x +e \right )\right )^{3}}{3 d \,f^{2}}+\frac {2 b i \left (a -b \right ) x \ln \left (c \left (f x +e \right )\right )}{d f}\) | \(176\) |
risch | \(-\frac {b^{2} \ln \left (c \left (f x +e \right )\right )^{3} e i}{3 d \,f^{2}}+\frac {b^{2} \ln \left (c \left (f x +e \right )\right )^{3} h}{3 d f}-\frac {b \left (-b f i x +a e i -a f h -b e i \right ) \ln \left (c \left (f x +e \right )\right )^{2}}{d \,f^{2}}+\frac {2 b i \left (a -b \right ) x \ln \left (c \left (f x +e \right )\right )}{d f}-\frac {\ln \left (f x +e \right ) a^{2} e i}{d \,f^{2}}+\frac {\ln \left (f x +e \right ) a^{2} h}{d f}+\frac {2 \ln \left (f x +e \right ) a b e i}{d \,f^{2}}-\frac {2 \ln \left (f x +e \right ) b^{2} e i}{d \,f^{2}}+\frac {a^{2} i x}{d f}-\frac {2 a b i x}{d f}+\frac {2 b^{2} i x}{d f}\) | \(221\) |
derivativedivides | \(\frac {-\frac {a^{2} c e i \ln \left (c f x +c e \right )}{f d}+\frac {a^{2} h c \ln \left (c f x +c e \right )}{d}+\frac {a^{2} i \left (c f x +c e \right )}{f d}-\frac {a b c e i \ln \left (c f x +c e \right )^{2}}{f d}+\frac {a b h c \ln \left (c f x +c e \right )^{2}}{d}+\frac {2 a b i \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )-c f x -c e \right )}{f d}-\frac {b^{2} c e i \ln \left (c f x +c e \right )^{3}}{3 f d}+\frac {b^{2} h c \ln \left (c f x +c e \right )^{3}}{3 d}+\frac {b^{2} i \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )^{2}-2 \left (c f x +c e \right ) \ln \left (c f x +c e \right )+2 c f x +2 c e \right )}{f d}}{c f}\) | \(257\) |
default | \(\frac {-\frac {a^{2} c e i \ln \left (c f x +c e \right )}{f d}+\frac {a^{2} h c \ln \left (c f x +c e \right )}{d}+\frac {a^{2} i \left (c f x +c e \right )}{f d}-\frac {a b c e i \ln \left (c f x +c e \right )^{2}}{f d}+\frac {a b h c \ln \left (c f x +c e \right )^{2}}{d}+\frac {2 a b i \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )-c f x -c e \right )}{f d}-\frac {b^{2} c e i \ln \left (c f x +c e \right )^{3}}{3 f d}+\frac {b^{2} h c \ln \left (c f x +c e \right )^{3}}{3 d}+\frac {b^{2} i \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )^{2}-2 \left (c f x +c e \right ) \ln \left (c f x +c e \right )+2 c f x +2 c e \right )}{f d}}{c f}\) | \(257\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 324 vs. \(2 (115) = 230\).
time = 0.30, size = 324, normalized size = 2.87 \begin {gather*} -a b h {\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 )} + 2 i \, a b {\left (\frac {x}{d f} - \frac {e \log \left (f x + e\right )}{d f^{2}}\right )} \log \left (c f x + c e\right ) + \frac {b^{2} h \log \left (c f x + c e\right )^{3}}{3 \, d f} + i \, a^{2} {\left (\frac {x}{d f} - \frac {e \log \left (f x + e\right )}{d f^{2}}\right )} + \frac {2 \, a b h \log \left (c f x + c e\right ) \log \left (d f x + d e\right )}{d f} + \frac {a^{2} h \log \left (d f x + d e\right )}{d f} + \frac {i \, {\left (e \log \left (f x + e\right )^{2} - 2 \, f x + 2 \, e \log \left (f x + e\right )\right )} a b}{d f^{2}} - \frac {i \, {\left (c^{2} e \log \left (c f x + c e\right )^{3} - 3 \, {\left (c f x + c e\right )} {\left (c \log \left (c f x + c e\right )^{2} - 2 \, c \log \left (c f x + c e\right ) + 2 \, c\right )}\right )} b^{2}}{3 \, c^{2} d f^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 148, normalized size = 1.31 \begin {gather*} \frac {{\left (b^{2} f h - i \, b^{2} e\right )} \log \left (c f x + c e\right )^{3} - 3 \, {\left (-i \, a^{2} + 2 i \, a b - 2 i \, b^{2}\right )} f x + 3 \, {\left (a b f h + i \, b^{2} f x - {\left (i \, a b - i \, b^{2}\right )} e\right )} \log \left (c f x + c e\right )^{2} + 3 \, {\left (a^{2} f h - 2 \, {\left (-i \, a b + i \, b^{2}\right )} f x - {\left (i \, a^{2} - 2 i \, a b + 2 i \, b^{2}\right )} e\right )} \log \left (c f x + c e\right )}{3 \, d f^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.36, size = 175, normalized size = 1.55 \begin {gather*} x \left (\frac {a^{2} i}{d f} - \frac {2 a b i}{d f} + \frac {2 b^{2} i}{d f}\right ) + \frac {\left (2 a b i x - 2 b^{2} i x\right ) \log {\left (c \left (e + f x\right ) \right )}}{d f} + \frac {\left (- b^{2} e i + b^{2} f h\right ) \log {\left (c \left (e + f x\right ) \right )}^{3}}{3 d f^{2}} - \frac {\left (a^{2} e i - a^{2} f h - 2 a b e i + 2 b^{2} e i\right ) \log {\left (e + f x \right )}}{d f^{2}} + \frac {\left (- a b e i + a b f h + b^{2} e i + b^{2} f i x\right ) \log {\left (c \left (e + f x\right ) \right )}^{2}}{d f^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.88, size = 228, normalized size = 2.02 \begin {gather*} \frac {b^{2} f h \log \left (c f x + c e\right )^{3} + 3 \, a b f h \log \left (c f x + c e\right )^{2} + 3 i \, b^{2} f x \log \left (c f x + c e\right )^{2} - i \, b^{2} e \log \left (c f x + c e\right )^{3} + 6 i \, a b f x \log \left (c f x + c e\right ) - 6 i \, b^{2} f x \log \left (c f x + c e\right ) - 3 i \, a b e \log \left (c f x + c e\right )^{2} + 3 i \, b^{2} e \log \left (c f x + c e\right )^{2} + 3 \, a^{2} f h \log \left (f x + e\right ) + 3 i \, a^{2} f x - 6 i \, a b f x + 6 i \, b^{2} f x - 3 i \, a^{2} e \log \left (f x + e\right ) + 6 i \, a b e \log \left (f x + e\right ) - 6 i \, b^{2} e \log \left (f x + e\right )}{3 \, d f^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.34, size = 163, normalized size = 1.44 \begin {gather*} {\ln \left (c\,\left (e+f\,x\right )\right )}^2\,\left (\frac {b\,\left (a\,f\,h-a\,e\,i+b\,e\,i\right )}{d\,f^2}+\frac {b^2\,i\,x}{d\,f}\right )-\frac {\ln \left (e+f\,x\right )\,\left (a^2\,e\,i-a^2\,f\,h+2\,b^2\,e\,i-2\,a\,b\,e\,i\right )}{d\,f^2}+\frac {i\,x\,\left (a^2-2\,a\,b+2\,b^2\right )}{d\,f}-\frac {b^2\,{\ln \left (c\,\left (e+f\,x\right )\right )}^3\,\left (e\,i-f\,h\right )}{3\,d\,f^2}+\frac {2\,b\,i\,x\,\ln \left (c\,\left (e+f\,x\right )\right )\,\left (a-b\right )}{d\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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