Optimal. Leaf size=127 \[ -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {2 d e x^{r-3} \left (a+b \log \left (c x^n\right )\right )}{3-r}-\frac {e^2 x^{2 r-3} \left (a+b \log \left (c x^n\right )\right )}{3-2 r}-\frac {b d^2 n}{9 x^3}-\frac {2 b d e n x^{r-3}}{(3-r)^2}-\frac {b e^2 n x^{2 r-3}}{(3-2 r)^2} \]
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Rubi [A] time = 0.17, antiderivative size = 109, normalized size of antiderivative = 0.86, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {270, 2334, 12, 14} \[ -\frac {1}{3} \left (\frac {d^2}{x^3}+\frac {6 d e x^{r-3}}{3-r}+\frac {3 e^2 x^{2 r-3}}{3-2 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b d^2 n}{9 x^3}-\frac {2 b d e n x^{r-3}}{(3-r)^2}-\frac {b e^2 n x^{2 r-3}}{(3-2 r)^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 270
Rule 2334
Rubi steps
\begin {align*} \int \frac {\left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx &=-\frac {1}{3} \left (\frac {d^2}{x^3}+\frac {6 d e x^{-3+r}}{3-r}+\frac {3 e^2 x^{-3+2 r}}{3-2 r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {-d^2+\frac {6 d e x^r}{-3+r}+\frac {3 e^2 x^{2 r}}{-3+2 r}}{3 x^4} \, dx\\ &=-\frac {1}{3} \left (\frac {d^2}{x^3}+\frac {6 d e x^{-3+r}}{3-r}+\frac {3 e^2 x^{-3+2 r}}{3-2 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{3} (b n) \int \frac {-d^2+\frac {6 d e x^r}{-3+r}+\frac {3 e^2 x^{2 r}}{-3+2 r}}{x^4} \, dx\\ &=-\frac {1}{3} \left (\frac {d^2}{x^3}+\frac {6 d e x^{-3+r}}{3-r}+\frac {3 e^2 x^{-3+2 r}}{3-2 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{3} (b n) \int \left (-\frac {d^2}{x^4}+\frac {6 d e x^{-4+r}}{-3+r}+\frac {3 e^2 x^{2 (-2+r)}}{-3+2 r}\right ) \, dx\\ &=-\frac {b d^2 n}{9 x^3}-\frac {2 b d e n x^{-3+r}}{(3-r)^2}-\frac {b e^2 n x^{-3+2 r}}{(3-2 r)^2}-\frac {1}{3} \left (\frac {d^2}{x^3}+\frac {6 d e x^{-3+r}}{3-r}+\frac {3 e^2 x^{-3+2 r}}{3-2 r}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}
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Mathematica [A] time = 0.31, size = 127, normalized size = 1.00 \[ \frac {a \left (-3 d^2+\frac {18 d e x^r}{r-3}+\frac {9 e^2 x^{2 r}}{2 r-3}\right )+3 b \log \left (c x^n\right ) \left (-d^2+\frac {6 d e x^r}{r-3}+\frac {3 e^2 x^{2 r}}{2 r-3}\right )+b n \left (-d^2-\frac {18 d e x^r}{(r-3)^2}-\frac {9 e^2 x^{2 r}}{(3-2 r)^2}\right )}{9 x^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 466, normalized size = 3.67 \[ -\frac {4 \, {\left (b d^{2} n + 3 \, a d^{2}\right )} r^{4} + 81 \, b d^{2} n - 36 \, {\left (b d^{2} n + 3 \, a d^{2}\right )} r^{3} + 243 \, a d^{2} + 117 \, {\left (b d^{2} n + 3 \, a d^{2}\right )} r^{2} - 162 \, {\left (b d^{2} n + 3 \, a d^{2}\right )} r - 9 \, {\left (2 \, a e^{2} r^{3} - 9 \, b e^{2} n - 27 \, a e^{2} - {\left (b e^{2} n + 15 \, a e^{2}\right )} r^{2} + 6 \, {\left (b e^{2} n + 6 \, a e^{2}\right )} r + {\left (2 \, b e^{2} r^{3} - 15 \, b e^{2} r^{2} + 36 \, b e^{2} r - 27 \, b e^{2}\right )} \log \relax (c) + {\left (2 \, b e^{2} n r^{3} - 15 \, b e^{2} n r^{2} + 36 \, b e^{2} n r - 27 \, b e^{2} n\right )} \log \relax (x)\right )} x^{2 \, r} - 18 \, {\left (4 \, a d e r^{3} - 9 \, b d e n - 27 \, a d e - 4 \, {\left (b d e n + 6 \, a d e\right )} r^{2} + 3 \, {\left (4 \, b d e n + 15 \, a d e\right )} r + {\left (4 \, b d e r^{3} - 24 \, b d e r^{2} + 45 \, b d e r - 27 \, b d e\right )} \log \relax (c) + {\left (4 \, b d e n r^{3} - 24 \, b d e n r^{2} + 45 \, b d e n r - 27 \, b d e n\right )} \log \relax (x)\right )} x^{r} + 3 \, {\left (4 \, b d^{2} r^{4} - 36 \, b d^{2} r^{3} + 117 \, b d^{2} r^{2} - 162 \, b d^{2} r + 81 \, b d^{2}\right )} \log \relax (c) + 3 \, {\left (4 \, b d^{2} n r^{4} - 36 \, b d^{2} n r^{3} + 117 \, b d^{2} n r^{2} - 162 \, b d^{2} n r + 81 \, b d^{2} n\right )} \log \relax (x)}{9 \, {\left (4 \, r^{4} - 36 \, r^{3} + 117 \, r^{2} - 162 \, r + 81\right )} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x^{r} + d\right )}^{2} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.35, size = 1930, normalized size = 15.20 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {{\left (d+e\,x^r\right )}^2\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 93.83, size = 235, normalized size = 1.85 \[ - \frac {a d^{2}}{3 x^{3}} + 2 a d e \left (\begin {cases} \frac {x^{r}}{r x^{3} - 3 x^{3}} & \text {for}\: r \neq 3 \\\log {\relax (x )} & \text {otherwise} \end {cases}\right ) + a e^{2} \left (\begin {cases} \frac {x^{2 r}}{2 r x^{3} - 3 x^{3}} & \text {for}\: r \neq \frac {3}{2} \\\log {\relax (x )} & \text {otherwise} \end {cases}\right ) - \frac {b d^{2} n}{9 x^{3}} - \frac {b d^{2} \log {\left (c x^{n} \right )}}{3 x^{3}} - 2 b d e n \left (\begin {cases} \frac {\begin {cases} \frac {x^{r}}{r x^{3} - 3 x^{3}} & \text {for}\: r \neq 3 \\\log {\relax (x )} & \text {otherwise} \end {cases}}{r - 3} & \text {for}\: r > -\infty \wedge r < \infty \wedge r \neq 3 \\\frac {\log {\relax (x )}^{2}}{2} & \text {otherwise} \end {cases}\right ) + 2 b d e \left (\begin {cases} \frac {x^{r - 3}}{r - 3} & \text {for}\: r - 4 \neq -1 \\\log {\relax (x )} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} - b e^{2} n \left (\begin {cases} \frac {\begin {cases} \frac {x^{2 r}}{2 r x^{3} - 3 x^{3}} & \text {for}\: r \neq \frac {3}{2} \\\log {\relax (x )} & \text {otherwise} \end {cases}}{2 r - 3} & \text {for}\: r > -\infty \wedge r < \infty \wedge r \neq \frac {3}{2} \\\frac {\log {\relax (x )}^{2}}{2} & \text {otherwise} \end {cases}\right ) + b e^{2} \left (\begin {cases} \frac {x^{2 r - 3}}{2 r - 3} & \text {for}\: 2 r - 4 \neq -1 \\\log {\relax (x )} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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