Optimal. Leaf size=160 \[ -24 a b^3 m^3 n^3 x+24 b^4 m^4 n^4 x-\frac {24 b^4 m^3 n^3 (e+f x) \log \left (c \left (d (e+f x)^m\right )^n\right )}{f}+\frac {12 b^2 m^2 n^2 (e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2}{f}-\frac {4 b m n (e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^3}{f}+\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^4}{f} \]
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Rubi [A]
time = 0.14, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2436, 2333,
2332, 2495} \begin {gather*} -24 a b^3 m^3 n^3 x+\frac {12 b^2 m^2 n^2 (e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2}{f}-\frac {4 b m n (e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^3}{f}+\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^4}{f}-\frac {24 b^4 m^3 n^3 (e+f x) \log \left (c \left (d (e+f x)^m\right )^n\right )}{f}+24 b^4 m^4 n^4 x \end {gather*}
Antiderivative was successfully verified.
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Rule 2332
Rule 2333
Rule 2436
Rule 2495
Rubi steps
\begin {align*} \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^4 \, dx &=\text {Subst}\left (\int \left (a+b \log \left (c d^n (e+f x)^{m n}\right )\right )^4 \, dx,c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right )\\ &=\text {Subst}\left (\frac {\text {Subst}\left (\int \left (a+b \log \left (c d^n x^{m n}\right )\right )^4 \, 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 )^4}{f}-\text {Subst}\left (\frac {(4 b m n) \text {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 {4 b m n (e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^3}{f}+\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^4}{f}+\text {Subst}\left (\frac {\left (12 b^2 m^2 n^2\right ) \text {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 {12 b^2 m^2 n^2 (e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2}{f}-\frac {4 b m n (e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^3}{f}+\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^4}{f}-\text {Subst}\left (\frac {\left (24 b^3 m^3 n^3\right ) \text {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 )\\ &=-24 a b^3 m^3 n^3 x+\frac {12 b^2 m^2 n^2 (e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2}{f}-\frac {4 b m n (e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^3}{f}+\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^4}{f}-\text {Subst}\left (\frac {\left (24 b^4 m^3 n^3\right ) \text {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 )\\ &=-24 a b^3 m^3 n^3 x+24 b^4 m^4 n^4 x-\frac {24 b^4 m^3 n^3 (e+f x) \log \left (c \left (d (e+f x)^m\right )^n\right )}{f}+\frac {12 b^2 m^2 n^2 (e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2}{f}-\frac {4 b m n (e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^3}{f}+\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^4}{f}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 132, normalized size = 0.82 \begin {gather*} \frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^4-4 b m n \left ((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 (a-b m n) x+b (e+f x) \log \left (c \left (d (e+f x)^m\right )^n\right )\right )\right )\right )}{f} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \left (a +b \ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right )\right )^{4}\, dx\]
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. 609 vs.
\(2 (168) = 336\).
time = 0.32, size = 609, normalized size = 3.81 \begin {gather*} b^{4} x \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right )^{4} - 4 \, a^{3} b f m n {\left (\frac {x}{f} - \frac {e \log \left (f x + e\right )}{f^{2}}\right )} + 4 \, a b^{3} x \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right )^{3} + 6 \, a^{2} b^{2} x \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right )^{2} + 4 \, a^{3} b x \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right ) - 6 \, {\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^{2} b^{2} - 4 \, {\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 )} a b^{3} - {\left (4 \, 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 )^{3} + {\left ({\left (\frac {{\left (e \log \left (f x + e\right )^{4} + 4 \, e \log \left (f x + e\right )^{3} + 12 \, e \log \left (f x + e\right )^{2} - 24 \, f x + 24 \, e \log \left (f x + e\right )\right )} m^{2} n^{2}}{f^{3}} - \frac {4 \, {\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 n \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right )}{f^{3}}\right )} f m n + \frac {6 \, {\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 )^{2}}{f^{2}}\right )} f m n\right )} b^{4} + a^{4} x \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. 1438 vs.
\(2 (168) = 336\).
time = 0.38, size = 1438, normalized size = 8.99 \begin {gather*} \frac {b^{4} f n^{4} x \log \left (d\right )^{4} + b^{4} f x \log \left (c\right )^{4} + {\left (b^{4} f m^{4} n^{4} x + b^{4} m^{4} n^{4} e\right )} \log \left (f x + e\right )^{4} - 4 \, {\left (b^{4} f m n - a b^{3} f\right )} x \log \left (c\right )^{3} - 4 \, {\left ({\left (b^{4} f m^{4} n^{4} - a b^{3} f m^{3} n^{3}\right )} x + {\left (b^{4} m^{4} n^{4} - a b^{3} m^{3} n^{3}\right )} e - {\left (b^{4} f m^{3} n^{3} x + b^{4} m^{3} n^{3} e\right )} \log \left (c\right ) - {\left (b^{4} f m^{3} n^{4} x + b^{4} m^{3} n^{4} e\right )} \log \left (d\right )\right )} \log \left (f x + e\right )^{3} + 6 \, {\left (2 \, b^{4} f m^{2} n^{2} - 2 \, a b^{3} f m n + a^{2} b^{2} f\right )} x \log \left (c\right )^{2} + 4 \, {\left (b^{4} f n^{3} x \log \left (c\right ) - {\left (b^{4} f m n^{4} - a b^{3} f n^{3}\right )} x\right )} \log \left (d\right )^{3} + 6 \, {\left ({\left (b^{4} f m^{2} n^{2} x + b^{4} m^{2} n^{2} e\right )} \log \left (c\right )^{2} + {\left (b^{4} f m^{2} n^{4} x + b^{4} m^{2} n^{4} e\right )} \log \left (d\right )^{2} + {\left (2 \, b^{4} f m^{4} n^{4} - 2 \, a b^{3} f m^{3} n^{3} + a^{2} b^{2} f m^{2} n^{2}\right )} x + {\left (2 \, b^{4} m^{4} n^{4} - 2 \, a b^{3} m^{3} n^{3} + a^{2} b^{2} m^{2} n^{2}\right )} e - 2 \, {\left ({\left (b^{4} f m^{3} n^{3} - a b^{3} f m^{2} n^{2}\right )} x + {\left (b^{4} m^{3} n^{3} - a b^{3} m^{2} n^{2}\right )} e\right )} \log \left (c\right ) - 2 \, {\left ({\left (b^{4} f m^{3} n^{4} - a b^{3} f m^{2} n^{3}\right )} x + {\left (b^{4} m^{3} n^{4} - a b^{3} m^{2} n^{3}\right )} e - {\left (b^{4} f m^{2} n^{3} x + b^{4} m^{2} n^{3} e\right )} \log \left (c\right )\right )} \log \left (d\right )\right )} \log \left (f x + e\right )^{2} - 4 \, {\left (6 \, b^{4} f m^{3} n^{3} - 6 \, a b^{3} f m^{2} n^{2} + 3 \, a^{2} b^{2} f m n - a^{3} b f\right )} x \log \left (c\right ) + 6 \, {\left (b^{4} f n^{2} x \log \left (c\right )^{2} - 2 \, {\left (b^{4} f m n^{3} - a b^{3} f n^{2}\right )} x \log \left (c\right ) + {\left (2 \, b^{4} f m^{2} n^{4} - 2 \, a b^{3} f m n^{3} + a^{2} b^{2} f n^{2}\right )} x\right )} \log \left (d\right )^{2} + {\left (24 \, b^{4} f m^{4} n^{4} - 24 \, a b^{3} f m^{3} n^{3} + 12 \, a^{2} b^{2} f m^{2} n^{2} - 4 \, a^{3} b f m n + a^{4} f\right )} x + 4 \, {\left ({\left (b^{4} f m n x + b^{4} m n e\right )} \log \left (c\right )^{3} + {\left (b^{4} f m n^{4} x + b^{4} m n^{4} e\right )} \log \left (d\right )^{3} - 3 \, {\left ({\left (b^{4} f m^{2} n^{2} - a b^{3} f m n\right )} x + {\left (b^{4} m^{2} n^{2} - a b^{3} m n\right )} e\right )} \log \left (c\right )^{2} - 3 \, {\left ({\left (b^{4} f m^{2} n^{4} - a b^{3} f m n^{3}\right )} x + {\left (b^{4} m^{2} n^{4} - a b^{3} m n^{3}\right )} e - {\left (b^{4} f m n^{3} x + b^{4} m n^{3} e\right )} \log \left (c\right )\right )} \log \left (d\right )^{2} - {\left (6 \, b^{4} f m^{4} n^{4} - 6 \, a b^{3} f m^{3} n^{3} + 3 \, a^{2} b^{2} f m^{2} n^{2} - a^{3} b f m n\right )} x - {\left (6 \, b^{4} m^{4} n^{4} - 6 \, a b^{3} m^{3} n^{3} + 3 \, a^{2} b^{2} m^{2} n^{2} - a^{3} b m n\right )} e + 3 \, {\left ({\left (2 \, b^{4} f m^{3} n^{3} - 2 \, a b^{3} f m^{2} n^{2} + a^{2} b^{2} f m n\right )} x + {\left (2 \, b^{4} m^{3} n^{3} - 2 \, a b^{3} m^{2} n^{2} + a^{2} b^{2} m n\right )} e\right )} \log \left (c\right ) + 3 \, {\left ({\left (b^{4} f m n^{2} x + b^{4} m n^{2} e\right )} \log \left (c\right )^{2} + {\left (2 \, b^{4} f m^{3} n^{4} - 2 \, a b^{3} f m^{2} n^{3} + a^{2} b^{2} f m n^{2}\right )} x + {\left (2 \, b^{4} m^{3} n^{4} - 2 \, a b^{3} m^{2} n^{3} + a^{2} b^{2} m n^{2}\right )} e - 2 \, {\left ({\left (b^{4} f m^{2} n^{3} - a b^{3} f m n^{2}\right )} x + {\left (b^{4} m^{2} n^{3} - a b^{3} m n^{2}\right )} e\right )} \log \left (c\right )\right )} \log \left (d\right )\right )} \log \left (f x + e\right ) + 4 \, {\left (b^{4} f n x \log \left (c\right )^{3} - 3 \, {\left (b^{4} f m n^{2} - a b^{3} f n\right )} x \log \left (c\right )^{2} + 3 \, {\left (2 \, b^{4} f m^{2} n^{3} - 2 \, a b^{3} f m n^{2} + a^{2} b^{2} f n\right )} x \log \left (c\right ) - {\left (6 \, b^{4} f m^{3} n^{4} - 6 \, a b^{3} f m^{2} n^{3} + 3 \, a^{2} b^{2} f m n^{2} - a^{3} b f n\right )} x\right )} \log \left (d\right )}{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. 609 vs.
\(2 (155) = 310\).
time = 3.13, size = 609, normalized size = 3.81 \begin {gather*} \begin {cases} a^{4} x + \frac {4 a^{3} b e \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}}{f} - 4 a^{3} b m n x + 4 a^{3} b x \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )} - \frac {12 a^{2} b^{2} e m n \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}}{f} + \frac {6 a^{2} b^{2} e \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}^{2}}{f} + 12 a^{2} b^{2} m^{2} n^{2} x - 12 a^{2} b^{2} m n x \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )} + 6 a^{2} b^{2} x \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}^{2} + \frac {24 a b^{3} e m^{2} n^{2} \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}}{f} - \frac {12 a b^{3} e m n \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}^{2}}{f} + \frac {4 a b^{3} e \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}^{3}}{f} - 24 a b^{3} m^{3} n^{3} x + 24 a b^{3} m^{2} n^{2} x \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )} - 12 a b^{3} m n x \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}^{2} + 4 a b^{3} x \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}^{3} - \frac {24 b^{4} e m^{3} n^{3} \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}}{f} + \frac {12 b^{4} e m^{2} n^{2} \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}^{2}}{f} - \frac {4 b^{4} e m n \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}^{3}}{f} + \frac {b^{4} e \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}^{4}}{f} + 24 b^{4} m^{4} n^{4} x - 24 b^{4} m^{3} n^{3} x \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )} + 12 b^{4} m^{2} n^{2} x \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}^{2} - 4 b^{4} m n x \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}^{3} + b^{4} x \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}^{4} & \text {for}\: f \neq 0 \\x \left (a + b \log {\left (c \left (d e^{m}\right )^{n} \right )}\right )^{4} & \text {otherwise} \end {cases} \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. 1802 vs.
\(2 (168) = 336\).
time = 6.32, size = 1802, normalized size = 11.26 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.58, size = 380, normalized size = 2.38 \begin {gather*} {\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^m\right )}^n\right )}^3\,\left (\frac {4\,\left (a\,b^3\,e-b^4\,e\,m\,n\right )}{f}+4\,b^3\,x\,\left (a-b\,m\,n\right )\right )+{\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^m\right )}^n\right )}^4\,\left (b^4\,x+\frac {b^4\,e}{f}\right )+x\,\left (a^4-4\,a^3\,b\,m\,n+12\,a^2\,b^2\,m^2\,n^2-24\,a\,b^3\,m^3\,n^3+24\,b^4\,m^4\,n^4\right )+{\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^m\right )}^n\right )}^2\,\left (\frac {6\,\left (e\,a^2\,b^2-2\,e\,a\,b^3\,m\,n+2\,e\,b^4\,m^2\,n^2\right )}{f}+6\,b^2\,x\,\left (a^2-2\,a\,b\,m\,n+2\,b^2\,m^2\,n^2\right )\right )-\frac {\ln \left (e+f\,x\right )\,\left (-4\,e\,a^3\,b\,m\,n+12\,e\,a^2\,b^2\,m^2\,n^2-24\,e\,a\,b^3\,m^3\,n^3+24\,e\,b^4\,m^4\,n^4\right )}{f}+\frac {\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^m\right )}^n\right )\,\left (4\,b\,f\,\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 )\,x^2+4\,b\,e\,\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 )\,x\right )}{e+f\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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