Optimal. Leaf size=298 \[ \frac {2 b^2 d^2 n^2 x (f x)^{-1+m}}{m^3}+\frac {b^2 d e n^2 x^{1+m} (f x)^{-1+m}}{2 m^3}+\frac {2 b^2 e^2 n^2 x^{1+2 m} (f x)^{-1+m}}{27 m^3}+\frac {b^2 d^3 n^2 x^{1-m} (f x)^{-1+m} \log ^2(x)}{3 e m}-\frac {2 b d^2 n x (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{m^2}-\frac {b d e n x^{1+m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{m^2}-\frac {2 b e^2 n x^{1+2 m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{9 m^2}-\frac {2 b d^3 n x^{1-m} (f x)^{-1+m} \log (x) \left (a+b \log \left (c x^n\right )\right )}{3 e m}+\frac {x^{1-m} (f x)^{-1+m} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 e m} \]
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Rubi [A]
time = 0.30, antiderivative size = 298, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2377, 2376,
272, 45, 2372, 12, 14, 2338} \begin {gather*} -\frac {2 b d^3 n x^{1-m} \log (x) (f x)^{m-1} \left (a+b \log \left (c x^n\right )\right )}{3 e m}-\frac {2 b d^2 n x (f x)^{m-1} \left (a+b \log \left (c x^n\right )\right )}{m^2}-\frac {b d e n x^{m+1} (f x)^{m-1} \left (a+b \log \left (c x^n\right )\right )}{m^2}+\frac {x^{1-m} (f x)^{m-1} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 e m}-\frac {2 b e^2 n x^{2 m+1} (f x)^{m-1} \left (a+b \log \left (c x^n\right )\right )}{9 m^2}+\frac {b^2 d^3 n^2 x^{1-m} \log ^2(x) (f x)^{m-1}}{3 e m}+\frac {2 b^2 d^2 n^2 x (f x)^{m-1}}{m^3}+\frac {b^2 d e n^2 x^{m+1} (f x)^{m-1}}{2 m^3}+\frac {2 b^2 e^2 n^2 x^{2 m+1} (f x)^{m-1}}{27 m^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 45
Rule 272
Rule 2338
Rule 2372
Rule 2376
Rule 2377
Rubi steps
\begin {align*} \int (f x)^{-1+m} \left (d+e x^m\right )^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx &=\left (x^{1-m} (f x)^{-1+m}\right ) \int x^{-1+m} \left (d+e x^m\right )^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx\\ &=\frac {x^{1-m} (f x)^{-1+m} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 e m}-\frac {\left (2 b n x^{1-m} (f x)^{-1+m}\right ) \int \frac {\left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx}{3 e m}\\ &=-\frac {b n x^{1-m} (f x)^{-1+m} \left (\frac {18 d^2 e x^m}{m}+\frac {9 d e^2 x^{2 m}}{m}+\frac {2 e^3 x^{3 m}}{m}+6 d^3 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )}{9 e m}+\frac {x^{1-m} (f x)^{-1+m} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 e m}+\frac {\left (2 b^2 n^2 x^{1-m} (f x)^{-1+m}\right ) \int \frac {e x^m \left (18 d^2+9 d e x^m+2 e^2 x^{2 m}\right )+6 d^3 m \log (x)}{6 m x} \, dx}{3 e m}\\ &=-\frac {b n x^{1-m} (f x)^{-1+m} \left (\frac {18 d^2 e x^m}{m}+\frac {9 d e^2 x^{2 m}}{m}+\frac {2 e^3 x^{3 m}}{m}+6 d^3 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )}{9 e m}+\frac {x^{1-m} (f x)^{-1+m} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 e m}+\frac {\left (b^2 n^2 x^{1-m} (f x)^{-1+m}\right ) \int \frac {e x^m \left (18 d^2+9 d e x^m+2 e^2 x^{2 m}\right )+6 d^3 m \log (x)}{x} \, dx}{9 e m^2}\\ &=-\frac {b n x^{1-m} (f x)^{-1+m} \left (\frac {18 d^2 e x^m}{m}+\frac {9 d e^2 x^{2 m}}{m}+\frac {2 e^3 x^{3 m}}{m}+6 d^3 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )}{9 e m}+\frac {x^{1-m} (f x)^{-1+m} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 e m}+\frac {\left (b^2 n^2 x^{1-m} (f x)^{-1+m}\right ) \int \left (18 d^2 e x^{-1+m}+9 d e^2 x^{-1+2 m}+2 e^3 x^{-1+3 m}+\frac {6 d^3 m \log (x)}{x}\right ) \, dx}{9 e m^2}\\ &=\frac {2 b^2 d^2 n^2 x (f x)^{-1+m}}{m^3}+\frac {b^2 d e n^2 x^{1+m} (f x)^{-1+m}}{2 m^3}+\frac {2 b^2 e^2 n^2 x^{1+2 m} (f x)^{-1+m}}{27 m^3}-\frac {b n x^{1-m} (f x)^{-1+m} \left (\frac {18 d^2 e x^m}{m}+\frac {9 d e^2 x^{2 m}}{m}+\frac {2 e^3 x^{3 m}}{m}+6 d^3 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )}{9 e m}+\frac {x^{1-m} (f x)^{-1+m} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 e m}+\frac {\left (2 b^2 d^3 n^2 x^{1-m} (f x)^{-1+m}\right ) \int \frac {\log (x)}{x} \, dx}{3 e m}\\ &=\frac {2 b^2 d^2 n^2 x (f x)^{-1+m}}{m^3}+\frac {b^2 d e n^2 x^{1+m} (f x)^{-1+m}}{2 m^3}+\frac {2 b^2 e^2 n^2 x^{1+2 m} (f x)^{-1+m}}{27 m^3}+\frac {b^2 d^3 n^2 x^{1-m} (f x)^{-1+m} \log ^2(x)}{3 e m}-\frac {b n x^{1-m} (f x)^{-1+m} \left (\frac {18 d^2 e x^m}{m}+\frac {9 d e^2 x^{2 m}}{m}+\frac {2 e^3 x^{3 m}}{m}+6 d^3 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )}{9 e m}+\frac {x^{1-m} (f x)^{-1+m} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 e m}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 207, normalized size = 0.69 \begin {gather*} \frac {(f x)^m \left (18 a^2 m^2 \left (3 d^2+3 d e x^m+e^2 x^{2 m}\right )-6 a b m n \left (18 d^2+9 d e x^m+2 e^2 x^{2 m}\right )+b^2 n^2 \left (108 d^2+27 d e x^m+4 e^2 x^{2 m}\right )+6 b m \left (6 a m \left (3 d^2+3 d e x^m+e^2 x^{2 m}\right )-b n \left (18 d^2+9 d e x^m+2 e^2 x^{2 m}\right )\right ) \log \left (c x^n\right )+18 b^2 m^2 \left (3 d^2+3 d e x^m+e^2 x^{2 m}\right ) \log ^2\left (c x^n\right )\right )}{54 f m^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
3.
time = 0.32, size = 3038, normalized size = 10.19
method | result | size |
risch | \(\text {Expression too large to display}\) | \(3038\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 425, normalized size = 1.43 \begin {gather*} \frac {b^{2} d f^{m - 1} e^{\left (2 \, m \log \left (x\right ) + 1\right )} \log \left (c x^{n}\right )^{2}}{m} - 2 \, {\left (\frac {f^{m - 1} n x^{m} \log \left (c x^{n}\right )}{m^{2}} - \frac {f^{m - 1} n^{2} x^{m}}{m^{3}}\right )} b^{2} d^{2} - \frac {2 \, a b d^{2} f^{m - 1} n x^{m}}{m^{2}} - \frac {1}{2} \, {\left (\frac {2 \, f^{m - 1} n x^{2 \, m} \log \left (c x^{n}\right )}{m^{2}} - \frac {f^{m - 1} n^{2} x^{2 \, m}}{m^{3}}\right )} b^{2} d e + \frac {2 \, a b d f^{m - 1} e^{\left (2 \, m \log \left (x\right ) + 1\right )} \log \left (c x^{n}\right )}{m} + \frac {\left (f x\right )^{m} b^{2} d^{2} \log \left (c x^{n}\right )^{2}}{f m} + \frac {b^{2} f^{m - 1} e^{\left (3 \, m \log \left (x\right ) + 2\right )} \log \left (c x^{n}\right )^{2}}{3 \, m} - \frac {2}{27} \, {\left (\frac {3 \, f^{m - 1} n x^{3 \, m} \log \left (c x^{n}\right )}{m^{2}} - \frac {f^{m - 1} n^{2} x^{3 \, m}}{m^{3}}\right )} b^{2} e^{2} + \frac {a^{2} d f^{m - 1} e^{\left (2 \, m \log \left (x\right ) + 1\right )}}{m} - \frac {a b d f^{m - 1} n e^{\left (2 \, m \log \left (x\right ) + 1\right )}}{m^{2}} + \frac {2 \, \left (f x\right )^{m} a b d^{2} \log \left (c x^{n}\right )}{f m} + \frac {2 \, a b f^{m - 1} e^{\left (3 \, m \log \left (x\right ) + 2\right )} \log \left (c x^{n}\right )}{3 \, m} + \frac {\left (f x\right )^{m} a^{2} d^{2}}{f m} + \frac {a^{2} f^{m - 1} e^{\left (3 \, m \log \left (x\right ) + 2\right )}}{3 \, m} - \frac {2 \, a b f^{m - 1} n e^{\left (3 \, m \log \left (x\right ) + 2\right )}}{9 \, m^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 411, normalized size = 1.38 \begin {gather*} \frac {2 \, {\left (9 \, b^{2} m^{2} n^{2} e^{2} \log \left (x\right )^{2} + 9 \, b^{2} m^{2} e^{2} \log \left (c\right )^{2} + 6 \, {\left (3 \, a b m^{2} - b^{2} m n\right )} e^{2} \log \left (c\right ) + {\left (9 \, a^{2} m^{2} - 6 \, a b m n + 2 \, b^{2} n^{2}\right )} e^{2} + 6 \, {\left (3 \, b^{2} m^{2} n e^{2} \log \left (c\right ) + {\left (3 \, a b m^{2} n - b^{2} m n^{2}\right )} e^{2}\right )} \log \left (x\right )\right )} f^{m - 1} x^{3 \, m} + 27 \, {\left (2 \, b^{2} d m^{2} n^{2} e \log \left (x\right )^{2} + 2 \, b^{2} d m^{2} e \log \left (c\right )^{2} + 2 \, {\left (2 \, a b d m^{2} - b^{2} d m n\right )} e \log \left (c\right ) + {\left (2 \, a^{2} d m^{2} - 2 \, a b d m n + b^{2} d n^{2}\right )} e + 2 \, {\left (2 \, b^{2} d m^{2} n e \log \left (c\right ) + {\left (2 \, a b d m^{2} n - b^{2} d m n^{2}\right )} e\right )} \log \left (x\right )\right )} f^{m - 1} x^{2 \, m} + 54 \, {\left (b^{2} d^{2} m^{2} n^{2} \log \left (x\right )^{2} + b^{2} d^{2} m^{2} \log \left (c\right )^{2} + a^{2} d^{2} m^{2} - 2 \, a b d^{2} m n + 2 \, b^{2} d^{2} n^{2} + 2 \, {\left (a b d^{2} m^{2} - b^{2} d^{2} m n\right )} \log \left (c\right ) + 2 \, {\left (b^{2} d^{2} m^{2} n \log \left (c\right ) + a b d^{2} m^{2} n - b^{2} d^{2} m n^{2}\right )} \log \left (x\right )\right )} f^{m - 1} x^{m}}{54 \, m^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 715 vs.
\(2 (284) = 568\).
time = 7.72, size = 715, normalized size = 2.40 \begin {gather*} \frac {b^{2} d^{2} f^{m} n^{2} x^{m} \log \left (x\right )^{2}}{f m} + \frac {b^{2} d f^{m} n^{2} x^{2 \, m} e \log \left (x\right )^{2}}{f m} + \frac {2 \, b^{2} d^{2} f^{m} n x^{m} \log \left (c\right ) \log \left (x\right )}{f m} + \frac {2 \, b^{2} d f^{m} n x^{2 \, m} e \log \left (c\right ) \log \left (x\right )}{f m} + \frac {b^{2} f^{m} n^{2} x^{3 \, m} e^{2} \log \left (x\right )^{2}}{3 \, f m} + \frac {b^{2} d^{2} f^{m} x^{m} \log \left (c\right )^{2}}{f m} + \frac {b^{2} d f^{m} x^{2 \, m} e \log \left (c\right )^{2}}{f m} + \frac {2 \, a b d^{2} f^{m} n x^{m} \log \left (x\right )}{f m} - \frac {2 \, b^{2} d^{2} f^{m} n^{2} x^{m} \log \left (x\right )}{f m^{2}} + \frac {2 \, a b d f^{m} n x^{2 \, m} e \log \left (x\right )}{f m} - \frac {b^{2} d f^{m} n^{2} x^{2 \, m} e \log \left (x\right )}{f m^{2}} + \frac {2 \, b^{2} f^{m} n x^{3 \, m} e^{2} \log \left (c\right ) \log \left (x\right )}{3 \, f m} + \frac {2 \, a b d^{2} f^{m} x^{m} \log \left (c\right )}{f m} - \frac {2 \, b^{2} d^{2} f^{m} n x^{m} \log \left (c\right )}{f m^{2}} + \frac {2 \, a b d f^{m} x^{2 \, m} e \log \left (c\right )}{f m} - \frac {b^{2} d f^{m} n x^{2 \, m} e \log \left (c\right )}{f m^{2}} + \frac {b^{2} f^{m} x^{3 \, m} e^{2} \log \left (c\right )^{2}}{3 \, f m} + \frac {2 \, a b f^{m} n x^{3 \, m} e^{2} \log \left (x\right )}{3 \, f m} - \frac {2 \, b^{2} f^{m} n^{2} x^{3 \, m} e^{2} \log \left (x\right )}{9 \, f m^{2}} + \frac {a^{2} d^{2} f^{m} x^{m}}{f m} - \frac {2 \, a b d^{2} f^{m} n x^{m}}{f m^{2}} + \frac {2 \, b^{2} d^{2} f^{m} n^{2} x^{m}}{f m^{3}} + \frac {a^{2} d f^{m} x^{2 \, m} e}{f m} - \frac {a b d f^{m} n x^{2 \, m} e}{f m^{2}} + \frac {b^{2} d f^{m} n^{2} x^{2 \, m} e}{2 \, f m^{3}} + \frac {2 \, a b f^{m} x^{3 \, m} e^{2} \log \left (c\right )}{3 \, f m} - \frac {2 \, b^{2} f^{m} n x^{3 \, m} e^{2} \log \left (c\right )}{9 \, f m^{2}} + \frac {a^{2} f^{m} x^{3 \, m} e^{2}}{3 \, f m} - \frac {2 \, a b f^{m} n x^{3 \, m} e^{2}}{9 \, f m^{2}} + \frac {2 \, b^{2} f^{m} n^{2} x^{3 \, m} e^{2}}{27 \, f m^{3}} \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 = -0.00 \begin {gather*} \int {\left (f\,x\right )}^{m-1}\,{\left (d+e\,x^m\right )}^2\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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