Optimal. Leaf size=346 \[ -\frac {2 b n x^{1-m} (f x)^{m-1} \log \left (\frac {d x^{-m}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^3 e m^2}-\frac {2 b n x (f x)^{m-1} \left (a+b \log \left (c x^n\right )\right )}{3 d^3 m^2 \left (d+e x^m\right )}+\frac {b n x^{1-m} (f x)^{m-1} \left (a+b \log \left (c x^n\right )\right )}{3 d e m^2 \left (d+e x^m\right )^2}-\frac {x^{1-m} (f x)^{m-1} \left (a+b \log \left (c x^n\right )\right )^2}{3 e m \left (d+e x^m\right )^3}+\frac {2 b^2 n^2 x^{1-m} (f x)^{m-1} \text {Li}_2\left (-\frac {d x^{-m}}{e}\right )}{3 d^3 e m^3}+\frac {b^2 n^2 x^{1-m} (f x)^{m-1} \log \left (d+e x^m\right )}{d^3 e m^3}-\frac {b^2 n^2 x^{1-m} \log (x) (f x)^{m-1}}{3 d^3 e m^2}-\frac {b^2 n^2 x^{1-m} (f x)^{m-1}}{3 d^2 e m^3 \left (d+e x^m\right )} \]
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Rubi [A] time = 0.71, antiderivative size = 346, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2339, 2338, 2349, 2345, 2391, 2335, 260, 266, 44} \[ \frac {2 b^2 n^2 x^{1-m} (f x)^{m-1} \text {PolyLog}\left (2,-\frac {d x^{-m}}{e}\right )}{3 d^3 e m^3}-\frac {2 b n x^{1-m} (f x)^{m-1} \log \left (\frac {d x^{-m}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^3 e m^2}-\frac {2 b n x (f x)^{m-1} \left (a+b \log \left (c x^n\right )\right )}{3 d^3 m^2 \left (d+e x^m\right )}+\frac {b n x^{1-m} (f x)^{m-1} \left (a+b \log \left (c x^n\right )\right )}{3 d e m^2 \left (d+e x^m\right )^2}-\frac {x^{1-m} (f x)^{m-1} \left (a+b \log \left (c x^n\right )\right )^2}{3 e m \left (d+e x^m\right )^3}-\frac {b^2 n^2 x^{1-m} (f x)^{m-1}}{3 d^2 e m^3 \left (d+e x^m\right )}-\frac {b^2 n^2 x^{1-m} \log (x) (f x)^{m-1}}{3 d^3 e m^2}+\frac {b^2 n^2 x^{1-m} (f x)^{m-1} \log \left (d+e x^m\right )}{d^3 e m^3} \]
Antiderivative was successfully verified.
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Rule 44
Rule 260
Rule 266
Rule 2335
Rule 2338
Rule 2339
Rule 2345
Rule 2349
Rule 2391
Rubi steps
\begin {align*} \int \frac {(f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^m\right )^4} \, dx &=\left (x^{1-m} (f x)^{-1+m}\right ) \int \frac {x^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^m\right )^4} \, dx\\ &=-\frac {x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{3 e m \left (d+e x^m\right )^3}+\frac {\left (2 b n x^{1-m} (f x)^{-1+m}\right ) \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^m\right )^3} \, dx}{3 e m}\\ &=-\frac {x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{3 e m \left (d+e x^m\right )^3}-\frac {\left (2 b n x^{1-m} (f x)^{-1+m}\right ) \int \frac {x^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^m\right )^3} \, dx}{3 d m}+\frac {\left (2 b n x^{1-m} (f x)^{-1+m}\right ) \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^m\right )^2} \, dx}{3 d e m}\\ &=\frac {b n x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{3 d e m^2 \left (d+e x^m\right )^2}-\frac {x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{3 e m \left (d+e x^m\right )^3}-\frac {\left (2 b n x^{1-m} (f x)^{-1+m}\right ) \int \frac {x^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^m\right )^2} \, dx}{3 d^2 m}+\frac {\left (2 b n x^{1-m} (f x)^{-1+m}\right ) \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^m\right )} \, dx}{3 d^2 e m}-\frac {\left (b^2 n^2 x^{1-m} (f x)^{-1+m}\right ) \int \frac {1}{x \left (d+e x^m\right )^2} \, dx}{3 d e m^2}\\ &=\frac {b n x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{3 d e m^2 \left (d+e x^m\right )^2}-\frac {2 b n x (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{3 d^3 m^2 \left (d+e x^m\right )}-\frac {x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{3 e m \left (d+e x^m\right )^3}-\frac {2 b n x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x^{-m}}{e}\right )}{3 d^3 e m^2}-\frac {\left (b^2 n^2 x^{1-m} (f x)^{-1+m}\right ) \operatorname {Subst}\left (\int \frac {1}{x (d+e x)^2} \, dx,x,x^m\right )}{3 d e m^3}+\frac {\left (2 b^2 n^2 x^{1-m} (f x)^{-1+m}\right ) \int \frac {x^{-1+m}}{d+e x^m} \, dx}{3 d^3 m^2}+\frac {\left (2 b^2 n^2 x^{1-m} (f x)^{-1+m}\right ) \int \frac {\log \left (1+\frac {d x^{-m}}{e}\right )}{x} \, dx}{3 d^3 e m^2}\\ &=\frac {b n x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{3 d e m^2 \left (d+e x^m\right )^2}-\frac {2 b n x (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{3 d^3 m^2 \left (d+e x^m\right )}-\frac {x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{3 e m \left (d+e x^m\right )^3}-\frac {2 b n x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x^{-m}}{e}\right )}{3 d^3 e m^2}+\frac {2 b^2 n^2 x^{1-m} (f x)^{-1+m} \log \left (d+e x^m\right )}{3 d^3 e m^3}+\frac {2 b^2 n^2 x^{1-m} (f x)^{-1+m} \text {Li}_2\left (-\frac {d x^{-m}}{e}\right )}{3 d^3 e m^3}-\frac {\left (b^2 n^2 x^{1-m} (f x)^{-1+m}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{d^2 x}-\frac {e}{d (d+e x)^2}-\frac {e}{d^2 (d+e x)}\right ) \, dx,x,x^m\right )}{3 d e m^3}\\ &=-\frac {b^2 n^2 x^{1-m} (f x)^{-1+m}}{3 d^2 e m^3 \left (d+e x^m\right )}-\frac {b^2 n^2 x^{1-m} (f x)^{-1+m} \log (x)}{3 d^3 e m^2}+\frac {b n x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{3 d e m^2 \left (d+e x^m\right )^2}-\frac {2 b n x (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{3 d^3 m^2 \left (d+e x^m\right )}-\frac {x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{3 e m \left (d+e x^m\right )^3}-\frac {2 b n x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x^{-m}}{e}\right )}{3 d^3 e m^2}+\frac {b^2 n^2 x^{1-m} (f x)^{-1+m} \log \left (d+e x^m\right )}{d^3 e m^3}+\frac {2 b^2 n^2 x^{1-m} (f x)^{-1+m} \text {Li}_2\left (-\frac {d x^{-m}}{e}\right )}{3 d^3 e m^3}\\ \end {align*}
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Mathematica [A] time = 0.53, size = 240, normalized size = 0.69 \[ \frac {x^{-m} (f x)^m \left (\frac {b n \left (2 a m+2 b m \log \left (c x^n\right )-b n\right )}{d^2 \left (d+e x^m\right )}-\frac {m^2 \left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^m\right )^3}+\frac {b m n \left (a+b \log \left (c x^n\right )\right )}{d \left (d+e x^m\right )^2}-\frac {2 a b m n \log \left (d-d x^m\right )}{d^3}+\frac {2 b^2 m n \left (n \log (x)-\log \left (c x^n\right )\right ) \log \left (d-d x^m\right )}{d^3}+\frac {2 b^2 n^2 \left (\text {Li}_2\left (\frac {e x^m}{d}+1\right )+\left (\log \left (-\frac {e x^m}{d}\right )-m \log (x)\right ) \log \left (d+e x^m\right )+\frac {1}{2} m^2 \log ^2(x)\right )}{d^3}+\frac {3 b^2 n^2 \log \left (d-d x^m\right )}{d^3}\right )}{3 e f m^3} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.49, size = 810, normalized size = 2.34 \[ \frac {{\left (b^{2} e^{3} m^{2} n^{2} \log \relax (x)^{2} + {\left (2 \, b^{2} e^{3} m^{2} n \log \relax (c) + 2 \, a b e^{3} m^{2} n - 3 \, b^{2} e^{3} m n^{2}\right )} \log \relax (x)\right )} f^{m - 1} x^{3 \, m} + {\left (3 \, b^{2} d e^{2} m^{2} n^{2} \log \relax (x)^{2} + 2 \, b^{2} d e^{2} m n \log \relax (c) + 2 \, a b d e^{2} m n - b^{2} d e^{2} n^{2} + {\left (6 \, b^{2} d e^{2} m^{2} n \log \relax (c) + 6 \, a b d e^{2} m^{2} n - 7 \, b^{2} d e^{2} m n^{2}\right )} \log \relax (x)\right )} f^{m - 1} x^{2 \, m} + {\left (3 \, b^{2} d^{2} e m^{2} n^{2} \log \relax (x)^{2} + 5 \, b^{2} d^{2} e m n \log \relax (c) + 5 \, a b d^{2} e m n - 2 \, b^{2} d^{2} e n^{2} + 2 \, {\left (3 \, b^{2} d^{2} e m^{2} n \log \relax (c) + 3 \, a b d^{2} e m^{2} n - 2 \, b^{2} d^{2} e m n^{2}\right )} \log \relax (x)\right )} f^{m - 1} x^{m} - {\left (b^{2} d^{3} m^{2} \log \relax (c)^{2} + a^{2} d^{3} m^{2} - 3 \, a b d^{3} m n + b^{2} d^{3} n^{2} + {\left (2 \, a b d^{3} m^{2} - 3 \, b^{2} d^{3} m n\right )} \log \relax (c)\right )} f^{m - 1} - 2 \, {\left (b^{2} e^{3} f^{m - 1} n^{2} x^{3 \, m} + 3 \, b^{2} d e^{2} f^{m - 1} n^{2} x^{2 \, m} + 3 \, b^{2} d^{2} e f^{m - 1} n^{2} x^{m} + b^{2} d^{3} f^{m - 1} n^{2}\right )} {\rm Li}_2\left (-\frac {e x^{m} + d}{d} + 1\right ) - {\left ({\left (2 \, b^{2} e^{3} m n \log \relax (c) + 2 \, a b e^{3} m n - 3 \, b^{2} e^{3} n^{2}\right )} f^{m - 1} x^{3 \, m} + 3 \, {\left (2 \, b^{2} d e^{2} m n \log \relax (c) + 2 \, a b d e^{2} m n - 3 \, b^{2} d e^{2} n^{2}\right )} f^{m - 1} x^{2 \, m} + 3 \, {\left (2 \, b^{2} d^{2} e m n \log \relax (c) + 2 \, a b d^{2} e m n - 3 \, b^{2} d^{2} e n^{2}\right )} f^{m - 1} x^{m} + {\left (2 \, b^{2} d^{3} m n \log \relax (c) + 2 \, a b d^{3} m n - 3 \, b^{2} d^{3} n^{2}\right )} f^{m - 1}\right )} \log \left (e x^{m} + d\right ) - 2 \, {\left (b^{2} e^{3} f^{m - 1} m n^{2} x^{3 \, m} \log \relax (x) + 3 \, b^{2} d e^{2} f^{m - 1} m n^{2} x^{2 \, m} \log \relax (x) + 3 \, b^{2} d^{2} e f^{m - 1} m n^{2} x^{m} \log \relax (x) + b^{2} d^{3} f^{m - 1} m n^{2} \log \relax (x)\right )} \log \left (\frac {e x^{m} + d}{d}\right )}{3 \, {\left (d^{3} e^{4} m^{3} x^{3 \, m} + 3 \, d^{4} e^{3} m^{3} x^{2 \, m} + 3 \, d^{5} e^{2} m^{3} x^{m} + d^{6} e m^{3}\right )}} \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \left (f x\right )^{m - 1}}{{\left (e x^{m} + d\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.98, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right )^{2} \left (f x \right )^{m -1}}{\left (e \,x^{m}+d \right )^{4}}\, dx \]
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 \[ \frac {1}{3} \, a b f^{m} n {\left (\frac {2 \, e x^{m} + 3 \, d}{{\left (d^{2} e^{3} f m x^{2 \, m} + 2 \, d^{3} e^{2} f m x^{m} + d^{4} e f m\right )} m} + \frac {2 \, \log \relax (x)}{d^{3} e f m} - \frac {2 \, \log \left (e x^{m} + d\right )}{d^{3} e f m^{2}}\right )} - \frac {1}{3} \, {\left (\frac {f^{m} \log \left (x^{n}\right )^{2}}{e^{4} f m x^{3 \, m} + 3 \, d e^{3} f m x^{2 \, m} + 3 \, d^{2} e^{2} f m x^{m} + d^{3} e f m} - 3 \, \int \frac {3 \, e f^{m} m x^{m} \log \relax (c)^{2} + 2 \, {\left (d f^{m} n + {\left (3 \, e f^{m} m \log \relax (c) + e f^{m} n\right )} x^{m}\right )} \log \left (x^{n}\right )}{3 \, {\left (e^{5} f m x x^{4 \, m} + 4 \, d e^{4} f m x x^{3 \, m} + 6 \, d^{2} e^{3} f m x x^{2 \, m} + 4 \, d^{3} e^{2} f m x x^{m} + d^{4} e f m x\right )}}\,{d x}\right )} b^{2} - \frac {2 \, a b f^{m} \log \left (c x^{n}\right )}{3 \, {\left (e^{4} f m x^{3 \, m} + 3 \, d e^{3} f m x^{2 \, m} + 3 \, d^{2} e^{2} f m x^{m} + d^{3} e f m\right )}} - \frac {a^{2} f^{m}}{3 \, {\left (e^{4} f m x^{3 \, m} + 3 \, d e^{3} f m x^{2 \, m} + 3 \, d^{2} e^{2} f m x^{m} + d^{3} e f m\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (f\,x\right )}^{m-1}\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{{\left (d+e\,x^m\right )}^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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