Optimal. Leaf size=149 \[ \frac {b (d f-e g)^3 n x}{4 e^3}-\frac {b (d f-e g)^2 n (g+f x)^2}{8 e^2 f}+\frac {b (d f-e g) n (g+f x)^3}{12 e f}-\frac {b n (g+f x)^4}{16 f}-\frac {b (d f-e g)^4 n \log (d+e x)}{4 e^4 f}+\frac {(g+f x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 f} \]
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Rubi [A]
time = 0.09, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2459, 2442, 45}
\begin {gather*} \frac {(f x+g)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 f}-\frac {b n (d f-e g)^4 \log (d+e x)}{4 e^4 f}+\frac {b n x (d f-e g)^3}{4 e^3}-\frac {b n (f x+g)^2 (d f-e g)^2}{8 e^2 f}+\frac {b n (f x+g)^3 (d f-e g)}{12 e f}-\frac {b n (f x+g)^4}{16 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2442
Rule 2459
Rubi steps
\begin {align*} \int \left (f+\frac {g}{x}\right )^3 x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx &=\int (g+f x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx\\ &=\frac {(g+f x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 f}-\frac {(b e n) \int \frac {(g+f x)^4}{d+e x} \, dx}{4 f}\\ &=\frac {(g+f x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 f}-\frac {(b e n) \int \left (\frac {f (-d f+e g)^3}{e^4}+\frac {(-d f+e g)^4}{e^4 (d+e x)}+\frac {f (-d f+e g)^2 (g+f x)}{e^3}+\frac {f (-d f+e g) (g+f x)^2}{e^2}+\frac {f (g+f x)^3}{e}\right ) \, dx}{4 f}\\ &=\frac {b (d f-e g)^3 n x}{4 e^3}-\frac {b (d f-e g)^2 n (g+f x)^2}{8 e^2 f}+\frac {b (d f-e g) n (g+f x)^3}{12 e f}-\frac {b n (g+f x)^4}{16 f}-\frac {b (d f-e g)^4 n \log (d+e x)}{4 e^4 f}+\frac {(g+f x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 f}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 226, normalized size = 1.52 \begin {gather*} \frac {e x \left (12 a e^3 \left (4 g^3+6 f g^2 x+4 f^2 g x^2+f^3 x^3\right )+b n \left (12 d^3 f^3-6 d^2 e f^2 (8 g+f x)+4 d e^2 f \left (18 g^2+6 f g x+f^2 x^2\right )-e^3 \left (48 g^3+36 f g^2 x+16 f^2 g x^2+3 f^3 x^3\right )\right )\right )-12 b d^2 f \left (d^2 f^2-4 d e f g+6 e^2 g^2\right ) n \log (d+e x)+12 b e^3 \left (4 d g^3+e x \left (4 g^3+6 f g^2 x+4 f^2 g x^2+f^3 x^3\right )\right ) \log \left (c (d+e x)^n\right )}{48 e^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.29, size = 836, normalized size = 5.61
method | result | size |
risch | \(\frac {a \,f^{3} x^{4}}{4}+x a \,g^{3}+\frac {b \,g^{3} n d \ln \left (e x +d \right )}{e}-b \,g^{3} n x +\frac {i f^{3} \pi b \,x^{4} \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}}{8}-\frac {i f^{2} \pi b g \,x^{3} \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}}{2}+\frac {i f^{3} \pi b \,x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}}{8}-\frac {3 i f \pi b \,g^{2} x^{2} \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}}{4}+\frac {f^{3} \ln \left (c \right ) b \,x^{4}}{4}+\ln \left (c \right ) b \,g^{3} x +\frac {3 f a \,g^{2} x^{2}}{2}-\frac {f^{3} b n \,x^{4}}{16}+f^{2} a g \,x^{3}+\frac {3 f \ln \left (c \right ) b \,g^{2} x^{2}}{2}-\frac {\ln \left (e x +d \right ) b \,g^{4} n}{4 f}+f^{2} \ln \left (c \right ) b g \,x^{3}+\frac {i \pi b \,g^{3} x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}}{2}+\frac {i \pi b \,g^{3} x \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}}{2}+\frac {f^{2} b d g n \,x^{2}}{2 e}-\frac {f^{2} b \,d^{2} g n x}{e^{2}}+\frac {3 f b d \,g^{2} n x}{2 e}-\frac {3 f \ln \left (e x +d \right ) b \,d^{2} g^{2} n}{2 e^{2}}+\frac {f^{2} \ln \left (e x +d \right ) b \,d^{3} g n}{e^{3}}+\frac {\left (f x +g \right )^{4} b \ln \left (\left (e x +d \right )^{n}\right )}{4 f}-\frac {3 i f \pi b \,g^{2} x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )}{4}-\frac {i f^{2} \pi b g \,x^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )}{2}-\frac {f^{3} \ln \left (e x +d \right ) b \,d^{4} n}{4 e^{4}}+\frac {f^{3} b d n \,x^{3}}{12 e}-\frac {f^{2} b g n \,x^{3}}{3}-\frac {f^{3} b \,d^{2} n \,x^{2}}{8 e^{2}}-\frac {i f^{3} \pi b \,x^{4} \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}}{8}-\frac {i \pi b \,g^{3} x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )}{2}-\frac {3 f b \,g^{2} n \,x^{2}}{4}-\frac {i \pi b \,g^{3} x \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}}{2}+\frac {i f^{2} \pi b g \,x^{3} \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}}{2}+\frac {3 i f \pi b \,g^{2} x^{2} \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}}{4}+\frac {3 i f \pi b \,g^{2} x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}}{4}+\frac {i f^{2} \pi b g \,x^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}}{2}-\frac {i f^{3} \pi b \,x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )}{8}+\frac {f^{3} b \,d^{3} n x}{4 e^{3}}\) | \(836\) |
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. 287 vs.
\(2 (139) = 278\).
time = 0.28, size = 287, normalized size = 1.93 \begin {gather*} \frac {1}{4} \, b f^{3} x^{4} \log \left ({\left (x e + d\right )}^{n} c\right ) + \frac {1}{4} \, a f^{3} x^{4} + b f^{2} g x^{3} \log \left ({\left (x e + d\right )}^{n} c\right ) + a f^{2} g x^{3} - \frac {1}{48} \, {\left (12 \, d^{4} e^{\left (-5\right )} \log \left (x e + d\right ) + {\left (3 \, x^{4} e^{3} - 4 \, d x^{3} e^{2} + 6 \, d^{2} x^{2} e - 12 \, d^{3} x\right )} e^{\left (-4\right )}\right )} b f^{3} n e + \frac {1}{6} \, {\left (6 \, d^{3} e^{\left (-4\right )} \log \left (x e + d\right ) - {\left (2 \, x^{3} e^{2} - 3 \, d x^{2} e + 6 \, d^{2} x\right )} e^{\left (-3\right )}\right )} b f^{2} g n e - \frac {3}{4} \, {\left (2 \, d^{2} e^{\left (-3\right )} \log \left (x e + d\right ) + {\left (x^{2} e - 2 \, d x\right )} e^{\left (-2\right )}\right )} b f g^{2} n e + {\left (d e^{\left (-2\right )} \log \left (x e + d\right ) - x e^{\left (-1\right )}\right )} b g^{3} n e + \frac {3}{2} \, b f g^{2} x^{2} \log \left ({\left (x e + d\right )}^{n} c\right ) + \frac {3}{2} \, a f g^{2} x^{2} + b g^{3} x \log \left ({\left (x e + d\right )}^{n} c\right ) + a g^{3} 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. 304 vs.
\(2 (139) = 278\).
time = 0.36, size = 304, normalized size = 2.04 \begin {gather*} \frac {1}{48} \, {\left (12 \, b d^{3} f^{3} n x e + 12 \, {\left (b f^{3} x^{4} + 4 \, b f^{2} g x^{3} + 6 \, b f g^{2} x^{2} + 4 \, b g^{3} x\right )} e^{4} \log \left (c\right ) - {\left (3 \, {\left (b f^{3} n - 4 \, a f^{3}\right )} x^{4} + 16 \, {\left (b f^{2} g n - 3 \, a f^{2} g\right )} x^{3} + 36 \, {\left (b f g^{2} n - 2 \, a f g^{2}\right )} x^{2} + 48 \, {\left (b g^{3} n - a g^{3}\right )} x\right )} e^{4} + 4 \, {\left (b d f^{3} n x^{3} + 6 \, b d f^{2} g n x^{2} + 18 \, b d f g^{2} n x\right )} e^{3} - 6 \, {\left (b d^{2} f^{3} n x^{2} + 8 \, b d^{2} f^{2} g n x\right )} e^{2} - 12 \, {\left (b d^{4} f^{3} n - 4 \, b d^{3} f^{2} g n e + 6 \, b d^{2} f g^{2} n e^{2} - 4 \, b d g^{3} n e^{3} - {\left (b f^{3} n x^{4} + 4 \, b f^{2} g n x^{3} + 6 \, b f g^{2} n x^{2} + 4 \, b g^{3} n x\right )} e^{4}\right )} \log \left (x e + d\right )\right )} e^{\left (-4\right )} \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. 410 vs.
\(2 (128) = 256\).
time = 26.99, size = 410, normalized size = 2.75 \begin {gather*} \begin {cases} \frac {a f^{3} x^{4}}{4} + a f^{2} g x^{3} + \frac {3 a f g^{2} x^{2}}{2} + a g^{3} x - \frac {b d^{4} f^{3} \log {\left (c \left (d + e x\right )^{n} \right )}}{4 e^{4}} + \frac {b d^{3} f^{3} n x}{4 e^{3}} + \frac {b d^{3} f^{2} g \log {\left (c \left (d + e x\right )^{n} \right )}}{e^{3}} - \frac {b d^{2} f^{3} n x^{2}}{8 e^{2}} - \frac {b d^{2} f^{2} g n x}{e^{2}} - \frac {3 b d^{2} f g^{2} \log {\left (c \left (d + e x\right )^{n} \right )}}{2 e^{2}} + \frac {b d f^{3} n x^{3}}{12 e} + \frac {b d f^{2} g n x^{2}}{2 e} + \frac {3 b d f g^{2} n x}{2 e} + \frac {b d g^{3} \log {\left (c \left (d + e x\right )^{n} \right )}}{e} - \frac {b f^{3} n x^{4}}{16} + \frac {b f^{3} x^{4} \log {\left (c \left (d + e x\right )^{n} \right )}}{4} - \frac {b f^{2} g n x^{3}}{3} + b f^{2} g x^{3} \log {\left (c \left (d + e x\right )^{n} \right )} - \frac {3 b f g^{2} n x^{2}}{4} + \frac {3 b f g^{2} x^{2} \log {\left (c \left (d + e x\right )^{n} \right )}}{2} - b g^{3} n x + b g^{3} x \log {\left (c \left (d + e x\right )^{n} \right )} & \text {for}\: e \neq 0 \\\left (a + b \log {\left (c d^{n} \right )}\right ) \left (\frac {f^{3} x^{4}}{4} + f^{2} g x^{3} + \frac {3 f g^{2} x^{2}}{2} + g^{3} x\right ) & \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. 780 vs.
\(2 (139) = 278\).
time = 4.80, size = 780, normalized size = 5.23 \begin {gather*} \frac {1}{4} \, {\left (x e + d\right )}^{4} b f^{3} n e^{\left (-4\right )} \log \left (x e + d\right ) - {\left (x e + d\right )}^{3} b d f^{3} n e^{\left (-4\right )} \log \left (x e + d\right ) + \frac {3}{2} \, {\left (x e + d\right )}^{2} b d^{2} f^{3} n e^{\left (-4\right )} \log \left (x e + d\right ) - {\left (x e + d\right )} b d^{3} f^{3} n e^{\left (-4\right )} \log \left (x e + d\right ) - \frac {1}{16} \, {\left (x e + d\right )}^{4} b f^{3} n e^{\left (-4\right )} + \frac {1}{3} \, {\left (x e + d\right )}^{3} b d f^{3} n e^{\left (-4\right )} - \frac {3}{4} \, {\left (x e + d\right )}^{2} b d^{2} f^{3} n e^{\left (-4\right )} + {\left (x e + d\right )} b d^{3} f^{3} n e^{\left (-4\right )} + {\left (x e + d\right )}^{3} b f^{2} g n e^{\left (-3\right )} \log \left (x e + d\right ) - 3 \, {\left (x e + d\right )}^{2} b d f^{2} g n e^{\left (-3\right )} \log \left (x e + d\right ) + 3 \, {\left (x e + d\right )} b d^{2} f^{2} g n e^{\left (-3\right )} \log \left (x e + d\right ) + \frac {1}{4} \, {\left (x e + d\right )}^{4} b f^{3} e^{\left (-4\right )} \log \left (c\right ) - {\left (x e + d\right )}^{3} b d f^{3} e^{\left (-4\right )} \log \left (c\right ) + \frac {3}{2} \, {\left (x e + d\right )}^{2} b d^{2} f^{3} e^{\left (-4\right )} \log \left (c\right ) - {\left (x e + d\right )} b d^{3} f^{3} e^{\left (-4\right )} \log \left (c\right ) - \frac {1}{3} \, {\left (x e + d\right )}^{3} b f^{2} g n e^{\left (-3\right )} + \frac {3}{2} \, {\left (x e + d\right )}^{2} b d f^{2} g n e^{\left (-3\right )} - 3 \, {\left (x e + d\right )} b d^{2} f^{2} g n e^{\left (-3\right )} + \frac {1}{4} \, {\left (x e + d\right )}^{4} a f^{3} e^{\left (-4\right )} - {\left (x e + d\right )}^{3} a d f^{3} e^{\left (-4\right )} + \frac {3}{2} \, {\left (x e + d\right )}^{2} a d^{2} f^{3} e^{\left (-4\right )} - {\left (x e + d\right )} a d^{3} f^{3} e^{\left (-4\right )} + \frac {3}{2} \, {\left (x e + d\right )}^{2} b f g^{2} n e^{\left (-2\right )} \log \left (x e + d\right ) - 3 \, {\left (x e + d\right )} b d f g^{2} n e^{\left (-2\right )} \log \left (x e + d\right ) + {\left (x e + d\right )}^{3} b f^{2} g e^{\left (-3\right )} \log \left (c\right ) - 3 \, {\left (x e + d\right )}^{2} b d f^{2} g e^{\left (-3\right )} \log \left (c\right ) + 3 \, {\left (x e + d\right )} b d^{2} f^{2} g e^{\left (-3\right )} \log \left (c\right ) - \frac {3}{4} \, {\left (x e + d\right )}^{2} b f g^{2} n e^{\left (-2\right )} + 3 \, {\left (x e + d\right )} b d f g^{2} n e^{\left (-2\right )} + {\left (x e + d\right )}^{3} a f^{2} g e^{\left (-3\right )} - 3 \, {\left (x e + d\right )}^{2} a d f^{2} g e^{\left (-3\right )} + 3 \, {\left (x e + d\right )} a d^{2} f^{2} g e^{\left (-3\right )} + {\left (x e + d\right )} b g^{3} n e^{\left (-1\right )} \log \left (x e + d\right ) + \frac {3}{2} \, {\left (x e + d\right )}^{2} b f g^{2} e^{\left (-2\right )} \log \left (c\right ) - 3 \, {\left (x e + d\right )} b d f g^{2} e^{\left (-2\right )} \log \left (c\right ) - {\left (x e + d\right )} b g^{3} n e^{\left (-1\right )} + \frac {3}{2} \, {\left (x e + d\right )}^{2} a f g^{2} e^{\left (-2\right )} - 3 \, {\left (x e + d\right )} a d f g^{2} e^{\left (-2\right )} + {\left (x e + d\right )} b g^{3} e^{\left (-1\right )} \log \left (c\right ) + {\left (x e + d\right )} a g^{3} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.37, size = 352, normalized size = 2.36 \begin {gather*} x\,\left (\frac {4\,a\,e\,g^3+12\,a\,d\,f\,g^2-4\,b\,e\,g^3\,n}{4\,e}+\frac {d\,\left (\frac {d\,\left (\frac {f^2\,\left (a\,d\,f+3\,a\,e\,g-b\,e\,g\,n\right )}{e}-\frac {d\,f^3\,\left (4\,a-b\,n\right )}{4\,e}\right )}{e}-\frac {3\,f\,g\,\left (2\,a\,d\,f+2\,a\,e\,g-b\,e\,g\,n\right )}{2\,e}\right )}{e}\right )+x^3\,\left (\frac {f^2\,\left (a\,d\,f+3\,a\,e\,g-b\,e\,g\,n\right )}{3\,e}-\frac {d\,f^3\,\left (4\,a-b\,n\right )}{12\,e}\right )+\ln \left (c\,{\left (d+e\,x\right )}^n\right )\,\left (\frac {b\,f^3\,x^4}{4}+b\,f^2\,g\,x^3+\frac {3\,b\,f\,g^2\,x^2}{2}+b\,g^3\,x\right )-x^2\,\left (\frac {d\,\left (\frac {f^2\,\left (a\,d\,f+3\,a\,e\,g-b\,e\,g\,n\right )}{e}-\frac {d\,f^3\,\left (4\,a-b\,n\right )}{4\,e}\right )}{2\,e}-\frac {3\,f\,g\,\left (2\,a\,d\,f+2\,a\,e\,g-b\,e\,g\,n\right )}{4\,e}\right )-\frac {\ln \left (d+e\,x\right )\,\left (b\,n\,d^4\,f^3-4\,b\,n\,d^3\,e\,f^2\,g+6\,b\,n\,d^2\,e^2\,f\,g^2-4\,b\,n\,d\,e^3\,g^3\right )}{4\,e^4}+\frac {f^3\,x^4\,\left (4\,a-b\,n\right )}{16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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