Optimal. Leaf size=121 \[ 6 a b^2 m^2 n^2 x-\frac {3 b m n (e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2}{f}+\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^3}{f}+\frac {6 b^3 m^2 n^2 (e+f x) \log \left (c \left (d (e+f x)^m\right )^n\right )}{f}-6 b^3 m^3 n^3 x \]
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Rubi [A] time = 0.14, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2389, 2296, 2295, 2445} \[ 6 a b^2 m^2 n^2 x-\frac {3 b m n (e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2}{f}+\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^3}{f}+\frac {6 b^3 m^2 n^2 (e+f x) \log \left (c \left (d (e+f x)^m\right )^n\right )}{f}-6 b^3 m^3 n^3 x \]
Antiderivative was successfully verified.
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Rule 2295
Rule 2296
Rule 2389
Rule 2445
Rubi steps
\begin {align*} \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^3 \, dx &=\operatorname {Subst}\left (\int \left (a+b \log \left (c d^n (e+f x)^{m n}\right )\right )^3 \, dx,c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right )\\ &=\operatorname {Subst}\left (\frac {\operatorname {Subst}\left (\int \left (a+b \log \left (c d^n x^{m n}\right )\right )^3 \, dx,x,e+f x\right )}{f},c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right )\\ &=\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^3}{f}-\operatorname {Subst}\left (\frac {(3 b m n) \operatorname {Subst}\left (\int \left (a+b \log \left (c d^n x^{m n}\right )\right )^2 \, dx,x,e+f x\right )}{f},c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right )\\ &=-\frac {3 b m n (e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2}{f}+\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^3}{f}+\operatorname {Subst}\left (\frac {\left (6 b^2 m^2 n^2\right ) \operatorname {Subst}\left (\int \left (a+b \log \left (c d^n x^{m n}\right )\right ) \, dx,x,e+f x\right )}{f},c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right )\\ &=6 a b^2 m^2 n^2 x-\frac {3 b m n (e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2}{f}+\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^3}{f}+\operatorname {Subst}\left (\frac {\left (6 b^3 m^2 n^2\right ) \operatorname {Subst}\left (\int \log \left (c d^n x^{m n}\right ) \, dx,x,e+f x\right )}{f},c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right )\\ &=6 a b^2 m^2 n^2 x-6 b^3 m^3 n^3 x+\frac {6 b^3 m^2 n^2 (e+f x) \log \left (c \left (d (e+f x)^m\right )^n\right )}{f}-\frac {3 b m n (e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2}{f}+\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^3}{f}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 100, normalized size = 0.83 \[ \frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^3-3 b m n \left ((e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2-2 b m n \left (f x (a-b m n)+b (e+f x) \log \left (c \left (d (e+f x)^m\right )^n\right )\right )\right )}{f} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 639, normalized size = 5.28 \[ \frac {b^{3} f n^{3} x \log \relax (d)^{3} + b^{3} f x \log \relax (c)^{3} + {\left (b^{3} f m^{3} n^{3} x + b^{3} e m^{3} n^{3}\right )} \log \left (f x + e\right )^{3} - 3 \, {\left (b^{3} f m n - a b^{2} f\right )} x \log \relax (c)^{2} - 3 \, {\left (b^{3} e m^{3} n^{3} - a b^{2} e m^{2} n^{2} + {\left (b^{3} f m^{3} n^{3} - a b^{2} f m^{2} n^{2}\right )} x - {\left (b^{3} f m^{2} n^{2} x + b^{3} e m^{2} n^{2}\right )} \log \relax (c) - {\left (b^{3} f m^{2} n^{3} x + b^{3} e m^{2} n^{3}\right )} \log \relax (d)\right )} \log \left (f x + e\right )^{2} + 3 \, {\left (2 \, b^{3} f m^{2} n^{2} - 2 \, a b^{2} f m n + a^{2} b f\right )} x \log \relax (c) + 3 \, {\left (b^{3} f n^{2} x \log \relax (c) - {\left (b^{3} f m n^{3} - a b^{2} f n^{2}\right )} x\right )} \log \relax (d)^{2} - {\left (6 \, b^{3} f m^{3} n^{3} - 6 \, a b^{2} f m^{2} n^{2} + 3 \, a^{2} b f m n - a^{3} f\right )} x + 3 \, {\left (2 \, b^{3} e m^{3} n^{3} - 2 \, a b^{2} e m^{2} n^{2} + a^{2} b e m n + {\left (b^{3} f m n x + b^{3} e m n\right )} \log \relax (c)^{2} + {\left (b^{3} f m n^{3} x + b^{3} e m n^{3}\right )} \log \relax (d)^{2} + {\left (2 \, b^{3} f m^{3} n^{3} - 2 \, a b^{2} f m^{2} n^{2} + a^{2} b f m n\right )} x - 2 \, {\left (b^{3} e m^{2} n^{2} - a b^{2} e m n + {\left (b^{3} f m^{2} n^{2} - a b^{2} f m n\right )} x\right )} \log \relax (c) - 2 \, {\left (b^{3} e m^{2} n^{3} - a b^{2} e m n^{2} + {\left (b^{3} f m^{2} n^{3} - a b^{2} f m n^{2}\right )} x - {\left (b^{3} f m n^{2} x + b^{3} e m n^{2}\right )} \log \relax (c)\right )} \log \relax (d)\right )} \log \left (f x + e\right ) + 3 \, {\left (b^{3} f n x \log \relax (c)^{2} - 2 \, {\left (b^{3} f m n^{2} - a b^{2} f n\right )} x \log \relax (c) + {\left (2 \, b^{3} f m^{2} n^{3} - 2 \, a b^{2} f m n^{2} + a^{2} b f n\right )} x\right )} \log \relax (d)}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.28, size = 822, normalized size = 6.79 \[ \frac {{\left (f x + e\right )} b^{3} m^{3} n^{3} \log \left (f x + e\right )^{3}}{f} - \frac {3 \, {\left (f x + e\right )} b^{3} m^{3} n^{3} \log \left (f x + e\right )^{2}}{f} + \frac {3 \, {\left (f x + e\right )} b^{3} m^{2} n^{3} \log \left (f x + e\right )^{2} \log \relax (d)}{f} + \frac {6 \, {\left (f x + e\right )} b^{3} m^{3} n^{3} \log \left (f x + e\right )}{f} + \frac {3 \, {\left (f x + e\right )} b^{3} m^{2} n^{2} \log \left (f x + e\right )^{2} \log \relax (c)}{f} - \frac {6 \, {\left (f x + e\right )} b^{3} m^{2} n^{3} \log \left (f x + e\right ) \log \relax (d)}{f} + \frac {3 \, {\left (f x + e\right )} b^{3} m n^{3} \log \left (f x + e\right ) \log \relax (d)^{2}}{f} - \frac {6 \, {\left (f x + e\right )} b^{3} m^{3} n^{3}}{f} + \frac {3 \, {\left (f x + e\right )} a b^{2} m^{2} n^{2} \log \left (f x + e\right )^{2}}{f} - \frac {6 \, {\left (f x + e\right )} b^{3} m^{2} n^{2} \log \left (f x + e\right ) \log \relax (c)}{f} + \frac {6 \, {\left (f x + e\right )} b^{3} m^{2} n^{3} \log \relax (d)}{f} + \frac {6 \, {\left (f x + e\right )} b^{3} m n^{2} \log \left (f x + e\right ) \log \relax (c) \log \relax (d)}{f} - \frac {3 \, {\left (f x + e\right )} b^{3} m n^{3} \log \relax (d)^{2}}{f} + \frac {{\left (f x + e\right )} b^{3} n^{3} \log \relax (d)^{3}}{f} - \frac {6 \, {\left (f x + e\right )} a b^{2} m^{2} n^{2} \log \left (f x + e\right )}{f} + \frac {6 \, {\left (f x + e\right )} b^{3} m^{2} n^{2} \log \relax (c)}{f} + \frac {3 \, {\left (f x + e\right )} b^{3} m n \log \left (f x + e\right ) \log \relax (c)^{2}}{f} + \frac {6 \, {\left (f x + e\right )} a b^{2} m n^{2} \log \left (f x + e\right ) \log \relax (d)}{f} - \frac {6 \, {\left (f x + e\right )} b^{3} m n^{2} \log \relax (c) \log \relax (d)}{f} + \frac {3 \, {\left (f x + e\right )} b^{3} n^{2} \log \relax (c) \log \relax (d)^{2}}{f} + \frac {6 \, {\left (f x + e\right )} a b^{2} m^{2} n^{2}}{f} + \frac {6 \, {\left (f x + e\right )} a b^{2} m n \log \left (f x + e\right ) \log \relax (c)}{f} - \frac {3 \, {\left (f x + e\right )} b^{3} m n \log \relax (c)^{2}}{f} - \frac {6 \, {\left (f x + e\right )} a b^{2} m n^{2} \log \relax (d)}{f} + \frac {3 \, {\left (f x + e\right )} b^{3} n \log \relax (c)^{2} \log \relax (d)}{f} + \frac {3 \, {\left (f x + e\right )} a b^{2} n^{2} \log \relax (d)^{2}}{f} + \frac {3 \, {\left (f x + e\right )} a^{2} b m n \log \left (f x + e\right )}{f} - \frac {6 \, {\left (f x + e\right )} a b^{2} m n \log \relax (c)}{f} + \frac {{\left (f x + e\right )} b^{3} \log \relax (c)^{3}}{f} + \frac {6 \, {\left (f x + e\right )} a b^{2} n \log \relax (c) \log \relax (d)}{f} - \frac {3 \, {\left (f x + e\right )} a^{2} b m n}{f} + \frac {3 \, {\left (f x + e\right )} a b^{2} \log \relax (c)^{2}}{f} + \frac {3 \, {\left (f x + e\right )} a^{2} b n \log \relax (d)}{f} + \frac {3 \, {\left (f x + e\right )} a^{2} b \log \relax (c)}{f} + \frac {{\left (f x + e\right )} a^{3}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right )+a \right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.79, size = 317, normalized size = 2.62 \[ -3 \, a^{2} b f m n {\left (\frac {x}{f} - \frac {e \log \left (f x + e\right )}{f^{2}}\right )} + b^{3} x \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right )^{3} + 3 \, a b^{2} x \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right )^{2} + 3 \, a^{2} b x \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right ) - 3 \, {\left (2 \, f m n {\left (\frac {x}{f} - \frac {e \log \left (f x + e\right )}{f^{2}}\right )} \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right ) + \frac {{\left (e \log \left (f x + e\right )^{2} - 2 \, f x + 2 \, e \log \left (f x + e\right )\right )} m^{2} n^{2}}{f}\right )} a b^{2} - {\left (3 \, f m n {\left (\frac {x}{f} - \frac {e \log \left (f x + e\right )}{f^{2}}\right )} \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right )^{2} - {\left (\frac {{\left (e \log \left (f x + e\right )^{3} + 3 \, e \log \left (f x + e\right )^{2} - 6 \, f x + 6 \, e \log \left (f x + e\right )\right )} m^{2} n^{2}}{f^{2}} - \frac {3 \, {\left (e \log \left (f x + e\right )^{2} - 2 \, f x + 2 \, e \log \left (f x + e\right )\right )} m n \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right )}{f^{2}}\right )} f m n\right )} b^{3} + a^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.41, size = 242, normalized size = 2.00 \[ x\,\left (a^3-3\,a^2\,b\,m\,n+6\,a\,b^2\,m^2\,n^2-6\,b^3\,m^3\,n^3\right )+{\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^m\right )}^n\right )}^2\,\left (\frac {3\,\left (a\,b^2\,e-b^3\,e\,m\,n\right )}{f}+3\,b^2\,x\,\left (a-b\,m\,n\right )\right )+{\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^m\right )}^n\right )}^3\,\left (b^3\,x+\frac {b^3\,e}{f}\right )+\frac {\ln \left (e+f\,x\right )\,\left (3\,e\,a^2\,b\,m\,n-6\,e\,a\,b^2\,m^2\,n^2+6\,e\,b^3\,m^3\,n^3\right )}{f}+\frac {\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^m\right )}^n\right )\,\left (3\,b\,f\,\left (a^2-2\,a\,b\,m\,n+2\,b^2\,m^2\,n^2\right )\,x^2+3\,b\,e\,\left (a^2-2\,a\,b\,m\,n+2\,b^2\,m^2\,n^2\right )\,x\right )}{e+f\,x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 9.84, size = 1023, normalized size = 8.45 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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