Optimal. Leaf size=104 \[ -\frac {2 b d e n x^r}{r^2}-\frac {b e^2 n x^{2 r}}{4 r^2}-\frac {1}{2} b d^2 n \log ^2(x)+\frac {2 d e x^r \left (a+b \log \left (c x^n\right )\right )}{r}+\frac {e^2 x^{2 r} \left (a+b \log \left (c x^n\right )\right )}{2 r}+d^2 \log (x) \left (a+b \log \left (c x^n\right )\right ) \]
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Rubi [A]
time = 0.09, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {272, 45, 2372,
12, 14, 2338} \begin {gather*} d^2 \log (x) \left (a+b \log \left (c x^n\right )\right )+\frac {2 d e x^r \left (a+b \log \left (c x^n\right )\right )}{r}+\frac {e^2 x^{2 r} \left (a+b \log \left (c x^n\right )\right )}{2 r}-\frac {1}{2} b d^2 n \log ^2(x)-\frac {2 b d e n x^r}{r^2}-\frac {b e^2 n x^{2 r}}{4 r^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 45
Rule 272
Rule 2338
Rule 2372
Rubi steps
\begin {align*} \int \frac {\left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {1}{2} \left (\frac {4 d e x^r}{r}+\frac {e^2 x^{2 r}}{r}+2 d^2 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {e x^r \left (4 d+e x^r\right )+2 d^2 r \log (x)}{2 r x} \, dx\\ &=\frac {1}{2} \left (\frac {4 d e x^r}{r}+\frac {e^2 x^{2 r}}{r}+2 d^2 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {(b n) \int \frac {e x^r \left (4 d+e x^r\right )+2 d^2 r \log (x)}{x} \, dx}{2 r}\\ &=\frac {1}{2} \left (\frac {4 d e x^r}{r}+\frac {e^2 x^{2 r}}{r}+2 d^2 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {(b n) \int \left (4 d e x^{-1+r}+e^2 x^{-1+2 r}+\frac {2 d^2 r \log (x)}{x}\right ) \, dx}{2 r}\\ &=-\frac {2 b d e n x^r}{r^2}-\frac {b e^2 n x^{2 r}}{4 r^2}+\frac {1}{2} \left (\frac {4 d e x^r}{r}+\frac {e^2 x^{2 r}}{r}+2 d^2 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-\left (b d^2 n\right ) \int \frac {\log (x)}{x} \, dx\\ &=-\frac {2 b d e n x^r}{r^2}-\frac {b e^2 n x^{2 r}}{4 r^2}-\frac {1}{2} b d^2 n \log ^2(x)+\frac {1}{2} \left (\frac {4 d e x^r}{r}+\frac {e^2 x^{2 r}}{r}+2 d^2 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 87, normalized size = 0.84 \begin {gather*} -\frac {1}{2} b d^2 n \log ^2(x)+d^2 \log (x) \left (a+b \log \left (c x^n\right )\right )+\frac {e x^r \left (2 a r \left (4 d+e x^r\right )-b n \left (8 d+e x^r\right )+2 b r \left (4 d+e x^r\right ) \log \left (c x^n\right )\right )}{4 r^2} \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 = 0.09, size = 487, normalized size = 4.68
method | result | size |
risch | \(\frac {b \left (2 d^{2} \ln \left (x \right ) r +e^{2} x^{2 r}+4 d e \,x^{r}\right ) \ln \left (x^{n}\right )}{2 r}+\frac {i \ln \left (x \right ) \pi b \,d^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i \pi b \,e^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{2 r}}{4 r}-\frac {i \pi b \,e^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) x^{2 r}}{4 r}+\frac {i \pi b d e \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{r}}{r}-\frac {i \pi b \,e^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3} x^{2 r}}{4 r}-\frac {i \pi b d e \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) x^{r}}{r}+\frac {i \pi b \,e^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{2 r}}{4 r}+\frac {i \pi b d e \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{r}}{r}+\frac {i \ln \left (x \right ) \pi b \,d^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i \pi b d e \mathrm {csgn}\left (i c \,x^{n}\right )^{3} x^{r}}{r}-\frac {i \ln \left (x \right ) \pi b \,d^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2}-\frac {i \ln \left (x \right ) \pi b \,d^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2}-\frac {b \,d^{2} n \ln \left (x \right )^{2}}{2}+\ln \left (x \right ) \ln \left (c \right ) b \,d^{2}+\frac {\ln \left (c \right ) b \,e^{2} x^{2 r}}{2 r}+a \,d^{2} \ln \left (x \right )+\frac {a \,x^{2 r} e^{2}}{2 r}-\frac {b \,e^{2} n \,x^{2 r}}{4 r^{2}}+\frac {2 \ln \left (c \right ) b d e \,x^{r}}{r}+\frac {2 a \,x^{r} d e}{r}-\frac {2 b d e n \,x^{r}}{r^{2}}\) | \(487\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 114, normalized size = 1.10 \begin {gather*} \frac {b e^{2} x^{2 \, r} \log \left (c x^{n}\right )}{2 \, r} + \frac {2 \, b d e x^{r} \log \left (c x^{n}\right )}{r} + \frac {b d^{2} \log \left (c x^{n}\right )^{2}}{2 \, n} + a d^{2} \log \left (x\right ) - \frac {b e^{2} n x^{2 \, r}}{4 \, r^{2}} + \frac {a e^{2} x^{2 \, r}}{2 \, r} - \frac {2 \, b d e n x^{r}}{r^{2}} + \frac {2 \, a d e x^{r}}{r} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 116, normalized size = 1.12 \begin {gather*} \frac {2 \, b d^{2} n r^{2} \log \left (x\right )^{2} + {\left (2 \, b n r e^{2} \log \left (x\right ) + 2 \, b r e^{2} \log \left (c\right ) - {\left (b n - 2 \, a r\right )} e^{2}\right )} x^{2 \, r} + 8 \, {\left (b d n r e \log \left (x\right ) + b d r e \log \left (c\right ) - {\left (b d n - a d r\right )} e\right )} x^{r} + 4 \, {\left (b d^{2} r^{2} \log \left (c\right ) + a d^{2} r^{2}\right )} \log \left (x\right )}{4 \, r^{2}} \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. 216 vs.
\(2 (104) = 208\).
time = 5.31, size = 216, normalized size = 2.08 \begin {gather*} \begin {cases} \left (a + b \log {\left (c \right )}\right ) \left (d + e\right )^{2} \log {\left (x \right )} & \text {for}\: n = 0 \wedge r = 0 \\\left (d + e\right )^{2} \left (\begin {cases} a \log {\left (x \right )} & \text {for}\: b = 0 \\- \left (- a - b \log {\left (c \right )}\right ) \log {\left (x \right )} & \text {for}\: n = 0 \\\frac {\left (- a - b \log {\left (c x^{n} \right )}\right )^{2}}{2 b n} & \text {otherwise} \end {cases}\right ) & \text {for}\: r = 0 \\\left (a + b \log {\left (c \right )}\right ) \left (d^{2} \log {\left (x \right )} + \frac {2 d e x^{r}}{r} + \frac {e^{2} x^{2 r}}{2 r}\right ) & \text {for}\: n = 0 \\\frac {a d^{2} \log {\left (c x^{n} \right )}}{n} + \frac {2 a d e x^{r}}{r} + \frac {a e^{2} x^{2 r}}{2 r} + \frac {b d^{2} \log {\left (c x^{n} \right )}^{2}}{2 n} - \frac {2 b d e n x^{r}}{r^{2}} + \frac {2 b d e x^{r} \log {\left (c x^{n} \right )}}{r} - \frac {b e^{2} n x^{2 r}}{4 r^{2}} + \frac {b e^{2} x^{2 r} \log {\left (c x^{n} \right )}}{2 r} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.45, size = 140, normalized size = 1.35 \begin {gather*} \frac {1}{2} \, b d^{2} n \log \left (x\right )^{2} + \frac {2 \, b d n x^{r} e \log \left (x\right )}{r} + b d^{2} \log \left (c\right ) \log \left (x\right ) + \frac {2 \, b d x^{r} e \log \left (c\right )}{r} + a d^{2} \log \left (x\right ) + \frac {b n x^{2 \, r} e^{2} \log \left (x\right )}{2 \, r} - \frac {2 \, b d n x^{r} e}{r^{2}} + \frac {2 \, a d x^{r} e}{r} + \frac {b x^{2 \, r} e^{2} \log \left (c\right )}{2 \, r} - \frac {b n x^{2 \, r} e^{2}}{4 \, r^{2}} + \frac {a x^{2 \, r} e^{2}}{2 \, r} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (d+e\,x^r\right )}^2\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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