Optimal. Leaf size=107 \[ -\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e (d+e x)^2}-\frac {x \left (2 a+b n+2 b \log \left (c x^n\right )\right )}{2 e^2 (d+e x)}+\frac {\left (2 a+3 b n+2 b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{2 e^3}+\frac {b n \text {Li}_2\left (-\frac {e x}{d}\right )}{e^3} \]
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Rubi [A]
time = 0.09, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2384, 2354,
2438} \begin {gather*} \frac {b n \text {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^3}+\frac {\log \left (\frac {e x}{d}+1\right ) \left (2 a+2 b \log \left (c x^n\right )+3 b n\right )}{2 e^3}-\frac {x \left (2 a+2 b \log \left (c x^n\right )+b n\right )}{2 e^2 (d+e x)}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e (d+e x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2354
Rule 2384
Rule 2438
Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^3} \, dx &=\int \left (\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{e^2 (d+e x)^3}-\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{e^2 (d+e x)^2}+\frac {a+b \log \left (c x^n\right )}{e^2 (d+e x)}\right ) \, dx\\ &=\frac {\int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{e^2}-\frac {(2 d) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{e^2}+\frac {d^2 \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^3} \, dx}{e^2}\\ &=-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^3 (d+e x)^2}-\frac {2 x \left (a+b \log \left (c x^n\right )\right )}{e^2 (d+e x)}+\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^3}-\frac {(b n) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{e^3}+\frac {\left (b d^2 n\right ) \int \frac {1}{x (d+e x)^2} \, dx}{2 e^3}+\frac {(2 b n) \int \frac {1}{d+e x} \, dx}{e^2}\\ &=-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^3 (d+e x)^2}-\frac {2 x \left (a+b \log \left (c x^n\right )\right )}{e^2 (d+e x)}+\frac {2 b n \log (d+e x)}{e^3}+\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^3}+\frac {b n \text {Li}_2\left (-\frac {e x}{d}\right )}{e^3}+\frac {\left (b d^2 n\right ) \int \left (\frac {1}{d^2 x}-\frac {e}{d (d+e x)^2}-\frac {e}{d^2 (d+e x)}\right ) \, dx}{2 e^3}\\ &=\frac {b d n}{2 e^3 (d+e x)}+\frac {b n \log (x)}{2 e^3}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^3 (d+e x)^2}-\frac {2 x \left (a+b \log \left (c x^n\right )\right )}{e^2 (d+e x)}+\frac {3 b n \log (d+e x)}{2 e^3}+\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^3}+\frac {b n \text {Li}_2\left (-\frac {e x}{d}\right )}{e^3}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 122, normalized size = 1.14 \begin {gather*} \frac {-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2}+\frac {4 d \left (a+b \log \left (c x^n\right )\right )}{d+e x}-4 b n (\log (x)-\log (d+e x))+b n \left (\frac {d}{d+e x}+\log (x)-\log (d+e x)\right )+2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )+2 b n \text {Li}_2\left (-\frac {e x}{d}\right )}{2 e^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.11, size = 596, normalized size = 5.57
method | result | size |
risch | \(\frac {a \ln \left (e x +d \right )}{e^{3}}+\frac {b \ln \left (c \right ) \ln \left (e x +d \right )}{e^{3}}+\frac {b \ln \left (x^{n}\right ) \ln \left (e x +d \right )}{e^{3}}+\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (e x +d \right )}{2 e^{3}}+\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} d^{2}}{4 e^{3} \left (e x +d \right )^{2}}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (e x +d \right )}{2 e^{3}}-\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} d}{e^{3} \left (e x +d \right )}-\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \left (e x +d \right )}{2 e^{3}}+\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} d}{e^{3} \left (e x +d \right )}-\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} d^{2}}{4 e^{3} \left (e x +d \right )^{2}}-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) \ln \left (e x +d \right )}{2 e^{3}}-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} d^{2}}{4 e^{3} \left (e x +d \right )^{2}}+\frac {2 a d}{e^{3} \left (e x +d \right )}-\frac {a \,d^{2}}{2 e^{3} \left (e x +d \right )^{2}}-\frac {3 b n \ln \left (e x \right )}{2 e^{3}}+\frac {3 b n \ln \left (e x +d \right )}{2 e^{3}}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} d}{e^{3} \left (e x +d \right )}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) d^{2}}{4 e^{3} \left (e x +d \right )^{2}}-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) d}{e^{3} \left (e x +d \right )}-\frac {b \ln \left (c \right ) d^{2}}{2 e^{3} \left (e x +d \right )^{2}}+\frac {2 b \ln \left (c \right ) d}{e^{3} \left (e x +d \right )}+\frac {b n d}{2 e^{3} \left (e x +d \right )}-\frac {b n \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e^{3}}-\frac {b \ln \left (x^{n}\right ) d^{2}}{2 e^{3} \left (e x +d \right )^{2}}+\frac {2 b \ln \left (x^{n}\right ) d}{e^{3} \left (e x +d \right )}-\frac {b n \dilog \left (-\frac {e x}{d}\right )}{e^{3}}\) | \(596\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 20.36, size = 347, normalized size = 3.24 \begin {gather*} \frac {a d^{2} \left (\begin {cases} \frac {x}{d^{3}} & \text {for}\: e = 0 \\- \frac {1}{2 e \left (d + e x\right )^{2}} & \text {otherwise} \end {cases}\right )}{e^{2}} - \frac {2 a d \left (\begin {cases} \frac {x}{d^{2}} & \text {for}\: e = 0 \\- \frac {1}{d e + e^{2} x} & \text {otherwise} \end {cases}\right )}{e^{2}} + \frac {a \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x \right )}}{e} & \text {otherwise} \end {cases}\right )}{e^{2}} - \frac {b d^{2} n \left (\begin {cases} \frac {x}{d^{3}} & \text {for}\: e = 0 \\- \frac {1}{2 d^{2} e + 2 d e^{2} x} - \frac {\log {\left (x \right )}}{2 d^{2} e} + \frac {\log {\left (\frac {d}{e} + x \right )}}{2 d^{2} e} & \text {otherwise} \end {cases}\right )}{e^{2}} + \frac {b d^{2} \left (\begin {cases} \frac {x}{d^{3}} & \text {for}\: e = 0 \\- \frac {1}{2 e \left (d + e x\right )^{2}} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{e^{2}} + \frac {2 b d n \left (\begin {cases} \frac {x}{d^{2}} & \text {for}\: e = 0 \\- \frac {\log {\left (x \right )}}{d e} + \frac {\log {\left (\frac {d}{e} + x \right )}}{d e} & \text {otherwise} \end {cases}\right )}{e^{2}} - \frac {2 b d \left (\begin {cases} \frac {x}{d^{2}} & \text {for}\: e = 0 \\- \frac {1}{d e + e^{2} x} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{e^{2}} - \frac {b n \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\begin {cases} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (d \right )} \log {\left (x \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (d \right )} \log {\left (\frac {1}{x} \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\left (d \right )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (d \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {otherwise} \end {cases}}{e} & \text {otherwise} \end {cases}\right )}{e^{2}} + \frac {b \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x \right )}}{e} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{e^{2}} \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 {x^2\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (d+e\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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