Optimal. Leaf size=113 \[ \frac {(f h-e i) (a+b \log (c (e+f x)))^3}{3 b d f^2}+\frac {i (e+f x) (a+b \log (c (e+f x)))^2}{d f^2}-\frac {2 a b i x}{d f}-\frac {2 b^2 i (e+f x) \log (c (e+f x))}{d f^2}+\frac {2 b^2 i x}{d f} \]
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Rubi [A] time = 0.20, 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 = {2411, 12, 2346, 2302, 30, 2296, 2295} \[ \frac {(f h-e i) (a+b \log (c (e+f x)))^3}{3 b d f^2}+\frac {i (e+f x) (a+b \log (c (e+f x)))^2}{d f^2}-\frac {2 a b i x}{d f}-\frac {2 b^2 i (e+f x) \log (c (e+f x))}{d f^2}+\frac {2 b^2 i x}{d f} \]
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 2295
Rule 2296
Rule 2302
Rule 2346
Rule 2411
Rubi steps
\begin {align*} \int \frac {(h+186 x) (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx &=\frac {\operatorname {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 {\operatorname {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 \operatorname {Subst}\left (\int (a+b \log (c x))^2 \, dx,x,e+f x\right )}{d f^2}-\frac {(186 e-f h) \operatorname {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) \operatorname {Subst}(\int (a+b \log (c x)) \, dx,x,e+f x)}{d f^2}-\frac {(186 e-f h) \operatorname {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 ) \operatorname {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 = 0.06, size = 89, normalized size = 0.79 \[ \frac {\frac {(f h-e i) (a+b \log (c (e+f x)))^3}{b}+3 i (e+f x) (a+b \log (c (e+f x)))^2-6 b f i x (a-b)-6 b^2 i (e+f x) \log (c (e+f x))}{3 d f^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 141, normalized size = 1.25 \[ \frac {3 \, {\left (a^{2} - 2 \, a b + 2 \, b^{2}\right )} f i x + {\left (b^{2} f h - b^{2} e i\right )} \log \left (c f x + c e\right )^{3} + 3 \, {\left (b^{2} f i x + a b f h - {\left (a b - b^{2}\right )} e i\right )} \log \left (c f x + c e\right )^{2} + 3 \, {\left (a^{2} f h + 2 \, {\left (a b - b^{2}\right )} f i x - {\left (a^{2} - 2 \, a b + 2 \, b^{2}\right )} e i\right )} \log \left (c f x + c e\right )}{3 \, d f^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 240, normalized size = 2.12 \[ \frac {3 \, b^{2} f i x \log \left (c f x + c e\right )^{2} + b^{2} f h \log \left (c f x + c e\right )^{3} - b^{2} i e \log \left (c f x + c e\right )^{3} + 6 \, a b f i x \log \left (c f x + c e\right ) - 6 \, b^{2} f i x \log \left (c f x + c e\right ) + 3 \, a b f h \log \left (c f x + c e\right )^{2} - 3 \, a b i e \log \left (c f x + c e\right )^{2} + 3 \, b^{2} i e \log \left (c f x + c e\right )^{2} + 3 \, a^{2} f i x - 6 \, a b f i x + 6 \, b^{2} f i x + 3 \, a^{2} f h \log \left (f x + e\right ) - 3 \, a^{2} i e \log \left (f x + e\right ) + 6 \, a b i e \log \left (f x + e\right ) - 6 \, b^{2} i e \log \left (f x + e\right )}{3 \, d f^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 341, normalized size = 3.02 \[ -\frac {b^{2} e i \ln \left (c f x +c e \right )^{3}}{3 d \,f^{2}}+\frac {b^{2} h \ln \left (c f x +c e \right )^{3}}{3 d f}+\frac {b^{2} i x \ln \left (c f x +c e \right )^{2}}{d f}-\frac {a b e i \ln \left (c f x +c e \right )^{2}}{d \,f^{2}}+\frac {a b h \ln \left (c f x +c e \right )^{2}}{d f}+\frac {2 a b i x \ln \left (c f x +c e \right )}{d f}+\frac {b^{2} e i \ln \left (c f x +c e \right )^{2}}{d \,f^{2}}-\frac {2 b^{2} i x \ln \left (c f x +c e \right )}{d f}-\frac {a^{2} e i \ln \left (c f x +c e \right )}{d \,f^{2}}+\frac {a^{2} h \ln \left (c f x +c e \right )}{d f}+\frac {a^{2} i x}{d f}+\frac {2 a b e i \ln \left (c f x +c e \right )}{d \,f^{2}}-\frac {2 a b i x}{d f}-\frac {2 b^{2} e i \ln \left (c f x +c e \right )}{d \,f^{2}}+\frac {2 b^{2} i x}{d f}+\frac {a^{2} e i}{d \,f^{2}}-\frac {2 a b e i}{d \,f^{2}}+\frac {2 b^{2} e i}{d \,f^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.52, size = 304, normalized size = 2.69 \[ 2 \, a b i {\left (\frac {x}{d f} - \frac {e \log \left (f x + e\right )}{d f^{2}}\right )} \log \left (c f x + c e\right ) - a b h {\left (\frac {2 \, \log \left (c f x + c e\right ) \log \left (d f x + d e\right )}{d f} - \frac {\log \left (f x + e\right )^{2} + 2 \, \log \left (f x + e\right ) \log \relax (c)}{d f}\right )} + a^{2} i {\left (\frac {x}{d f} - \frac {e \log \left (f x + e\right )}{d f^{2}}\right )} + \frac {b^{2} h \log \left (c f x + c e\right )^{3}}{3 \, d f} + \frac {2 \, a b h \log \left (c f x + c e\right ) \log \left (d f x + d e\right )}{d f} + \frac {a^{2} h \log \left (d f x + d e\right )}{d f} + \frac {{\left (e \log \left (f x + e\right )^{2} - 2 \, f x + 2 \, e \log \left (f x + e\right )\right )} a b i}{d f^{2}} - \frac {{\left (c^{2} e \log \left (c f x + c e\right )^{3} - 3 \, {\left (c f x + c e\right )} {\left (c \log \left (c f x + c e\right )^{2} - 2 \, c \log \left (c f x + c e\right ) + 2 \, c\right )}\right )} b^{2} i}{3 \, c^{2} d f^{2}} \]
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 \[ {\ln \left (c\,\left (e+f\,x\right )\right )}^2\,\left (\frac {b\,\left (a\,f\,h-a\,e\,i+b\,e\,i\right )}{d\,f^2}+\frac {b^2\,i\,x}{d\,f}\right )-\frac {\ln \left (e+f\,x\right )\,\left (a^2\,e\,i-a^2\,f\,h+2\,b^2\,e\,i-2\,a\,b\,e\,i\right )}{d\,f^2}+\frac {i\,x\,\left (a^2-2\,a\,b+2\,b^2\right )}{d\,f}-\frac {b^2\,{\ln \left (c\,\left (e+f\,x\right )\right )}^3\,\left (e\,i-f\,h\right )}{3\,d\,f^2}+\frac {2\,b\,i\,x\,\ln \left (c\,\left (e+f\,x\right )\right )\,\left (a-b\right )}{d\,f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.08, size = 175, normalized size = 1.55 \[ x \left (\frac {a^{2} i}{d f} - \frac {2 a b i}{d f} + \frac {2 b^{2} i}{d f}\right ) + \frac {\left (2 a b i x - 2 b^{2} i x\right ) \log {\left (c \left (e + f x\right ) \right )}}{d f} + \frac {\left (- b^{2} e i + b^{2} f h\right ) \log {\left (c \left (e + f x\right ) \right )}^{3}}{3 d f^{2}} - \frac {\left (a^{2} e i - a^{2} f h - 2 a b e i + 2 b^{2} e i\right ) \log {\left (e + f x \right )}}{d f^{2}} + \frac {\left (- a b e i + a b f h + b^{2} e i + b^{2} f i x\right ) \log {\left (c \left (e + f x\right ) \right )}^{2}}{d f^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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