Optimal. Leaf size=70 \[ -b d^2 n x-\frac {1}{2} b d e n x^2-\frac {1}{9} b e^2 n x^3-\frac {b d^3 n \log (x)}{3 e}+\frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{3 e} \]
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Rubi [A]
time = 0.03, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {32, 2350, 12,
45} \begin {gather*} \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac {b d^3 n \log (x)}{3 e}-b d^2 n x-\frac {1}{2} b d e n x^2-\frac {1}{9} b e^2 n x^3 \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 32
Rule 45
Rule 2350
Rubi steps
\begin {align*} \int (d+e x)^2 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{3 e}-(b n) \int \frac {(d+e x)^3}{3 e x} \, dx\\ &=\frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac {(b n) \int \frac {(d+e x)^3}{x} \, dx}{3 e}\\ &=\frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac {(b n) \int \left (3 d^2 e+\frac {d^3}{x}+3 d e^2 x+e^3 x^2\right ) \, dx}{3 e}\\ &=-b d^2 n x-\frac {1}{2} b d e n x^2-\frac {1}{9} b e^2 n x^3-\frac {b d^3 n \log (x)}{3 e}+\frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{3 e}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 77, normalized size = 1.10 \begin {gather*} \frac {1}{18} x \left (6 a \left (3 d^2+3 d e x+e^2 x^2\right )-b n \left (18 d^2+9 d e x+2 e^2 x^2\right )+6 b \left (3 d^2+3 d e x+e^2 x^2\right ) \log \left (c x^n\right )\right ) \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 = 0.16, size = 414, normalized size = 5.91
method | result | size |
risch | \(\frac {\left (e x +d \right )^{3} b \ln \left (x^{n}\right )}{3 e}-\frac {i e^{2} \pi b \,x^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{6}-\frac {i \pi b \,d^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) x}{2}-\frac {i e \pi b d \,x^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2}+\frac {i e^{2} \pi b \,x^{3} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{6}+\frac {i \pi b \,d^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x}{2}-\frac {i e^{2} \pi b \,x^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{6}-\frac {i \pi b \,d^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3} x}{2}+\frac {i e^{2} \pi b \,x^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{6}-\frac {i e \pi b d \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i e \pi b d \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i \pi b \,d^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x}{2}+\frac {i e \pi b d \,x^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {\ln \left (c \right ) b \,e^{2} x^{3}}{3}-\frac {b \,e^{2} n \,x^{3}}{9}+\ln \left (c \right ) b d e \,x^{2}+\frac {a \,e^{2} x^{3}}{3}-\frac {b d e n \,x^{2}}{2}-\frac {b \,d^{3} n \ln \left (x \right )}{3 e}+\ln \left (c \right ) b \,d^{2} x +a d e \,x^{2}-b \,d^{2} n x +a \,d^{2} x\) | \(414\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 90, normalized size = 1.29 \begin {gather*} -\frac {1}{9} \, b n x^{3} e^{2} - \frac {1}{2} \, b d n x^{2} e + \frac {1}{3} \, b x^{3} e^{2} \log \left (c x^{n}\right ) + b d x^{2} e \log \left (c x^{n}\right ) - b d^{2} n x + \frac {1}{3} \, a x^{3} e^{2} + a d x^{2} e + b d^{2} x \log \left (c x^{n}\right ) + a d^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 106, normalized size = 1.51 \begin {gather*} -\frac {1}{9} \, {\left (b n - 3 \, a\right )} x^{3} e^{2} - \frac {1}{2} \, {\left (b d n - 2 \, a d\right )} x^{2} e - {\left (b d^{2} n - a d^{2}\right )} x + \frac {1}{3} \, {\left (b x^{3} e^{2} + 3 \, b d x^{2} e + 3 \, b d^{2} x\right )} \log \left (c\right ) + \frac {1}{3} \, {\left (b n x^{3} e^{2} + 3 \, b d n x^{2} e + 3 \, b d^{2} n x\right )} \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.19, size = 102, normalized size = 1.46 \begin {gather*} a d^{2} x + a d e x^{2} + \frac {a e^{2} x^{3}}{3} - b d^{2} n x + b d^{2} x \log {\left (c x^{n} \right )} - \frac {b d e n x^{2}}{2} + b d e x^{2} \log {\left (c x^{n} \right )} - \frac {b e^{2} n x^{3}}{9} + \frac {b e^{2} x^{3} \log {\left (c x^{n} \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.24, size = 109, normalized size = 1.56 \begin {gather*} \frac {1}{3} \, b n x^{3} e^{2} \log \left (x\right ) + b d n x^{2} e \log \left (x\right ) - \frac {1}{9} \, b n x^{3} e^{2} - \frac {1}{2} \, b d n x^{2} e + \frac {1}{3} \, b x^{3} e^{2} \log \left (c\right ) + b d x^{2} e \log \left (c\right ) + b d^{2} n x \log \left (x\right ) - b d^{2} n x + \frac {1}{3} \, a x^{3} e^{2} + a d x^{2} e + b d^{2} x \log \left (c\right ) + a d^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.61, size = 73, normalized size = 1.04 \begin {gather*} \ln \left (c\,x^n\right )\,\left (b\,d^2\,x+b\,d\,e\,x^2+\frac {b\,e^2\,x^3}{3}\right )+\frac {e^2\,x^3\,\left (3\,a-b\,n\right )}{9}+d^2\,x\,\left (a-b\,n\right )+\frac {d\,e\,x^2\,\left (2\,a-b\,n\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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