Optimal. Leaf size=272 \[ -\frac {360 a b^2 n^5 x^m}{m^6}-\frac {9 a^2 b n^3 x^{2 m}}{8 m^4}-\frac {a^3 n x^{3 m}}{9 m^2}+\frac {360 a b^2 n^4 x^m \log \left (c x^n\right )}{m^5}+\frac {9 a^2 b n^2 x^{2 m} \log \left (c x^n\right )}{4 m^3}+\frac {a^3 x^{3 m} \log \left (c x^n\right )}{3 m}-\frac {180 a b^2 n^3 x^m \log ^2\left (c x^n\right )}{m^4}-\frac {9 a^2 b n x^{2 m} \log ^2\left (c x^n\right )}{4 m^2}+\frac {60 a b^2 n^2 x^m \log ^3\left (c x^n\right )}{m^3}+\frac {3 a^2 b x^{2 m} \log ^3\left (c x^n\right )}{2 m}-\frac {15 a b^2 n x^m \log ^4\left (c x^n\right )}{m^2}+\frac {3 a b^2 x^m \log ^5\left (c x^n\right )}{m}+\frac {b^3 \log ^8\left (c x^n\right )}{8 n} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.20, antiderivative size = 272, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {2619, 2341,
2342, 2339, 30} \begin {gather*} \frac {a^3 x^{3 m} \log \left (c x^n\right )}{3 m}-\frac {a^3 n x^{3 m}}{9 m^2}+\frac {9 a^2 b n^2 x^{2 m} \log \left (c x^n\right )}{4 m^3}-\frac {9 a^2 b n x^{2 m} \log ^2\left (c x^n\right )}{4 m^2}+\frac {3 a^2 b x^{2 m} \log ^3\left (c x^n\right )}{2 m}-\frac {9 a^2 b n^3 x^{2 m}}{8 m^4}+\frac {360 a b^2 n^4 x^m \log \left (c x^n\right )}{m^5}-\frac {180 a b^2 n^3 x^m \log ^2\left (c x^n\right )}{m^4}+\frac {60 a b^2 n^2 x^m \log ^3\left (c x^n\right )}{m^3}-\frac {15 a b^2 n x^m \log ^4\left (c x^n\right )}{m^2}+\frac {3 a b^2 x^m \log ^5\left (c x^n\right )}{m}-\frac {360 a b^2 n^5 x^m}{m^6}+\frac {b^3 \log ^8\left (c x^n\right )}{8 n} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 2339
Rule 2341
Rule 2342
Rule 2619
Rubi steps
\begin {align*} \int \frac {\log \left (c x^n\right ) \left (a x^m+b \log ^2\left (c x^n\right )\right )^3}{x} \, dx &=\int \left (a^3 x^{-1+3 m} \log \left (c x^n\right )+3 a^2 b x^{-1+2 m} \log ^3\left (c x^n\right )+3 a b^2 x^{-1+m} \log ^5\left (c x^n\right )+\frac {b^3 \log ^7\left (c x^n\right )}{x}\right ) \, dx\\ &=a^3 \int x^{-1+3 m} \log \left (c x^n\right ) \, dx+\left (3 a^2 b\right ) \int x^{-1+2 m} \log ^3\left (c x^n\right ) \, dx+\left (3 a b^2\right ) \int x^{-1+m} \log ^5\left (c x^n\right ) \, dx+b^3 \int \frac {\log ^7\left (c x^n\right )}{x} \, dx\\ &=-\frac {a^3 n x^{3 m}}{9 m^2}+\frac {a^3 x^{3 m} \log \left (c x^n\right )}{3 m}+\frac {3 a^2 b x^{2 m} \log ^3\left (c x^n\right )}{2 m}+\frac {3 a b^2 x^m \log ^5\left (c x^n\right )}{m}+\frac {b^3 \text {Subst}\left (\int x^7 \, dx,x,\log \left (c x^n\right )\right )}{n}-\frac {\left (9 a^2 b n\right ) \int x^{-1+2 m} \log ^2\left (c x^n\right ) \, dx}{2 m}-\frac {\left (15 a b^2 n\right ) \int x^{-1+m} \log ^4\left (c x^n\right ) \, dx}{m}\\ &=-\frac {a^3 n x^{3 m}}{9 m^2}+\frac {a^3 x^{3 m} \log \left (c x^n\right )}{3 m}-\frac {9 a^2 b n x^{2 m} \log ^2\left (c x^n\right )}{4 m^2}+\frac {3 a^2 b x^{2 m} \log ^3\left (c x^n\right )}{2 m}-\frac {15 a b^2 n x^m \log ^4\left (c x^n\right )}{m^2}+\frac {3 a b^2 x^m \log ^5\left (c x^n\right )}{m}+\frac {b^3 \log ^8\left (c x^n\right )}{8 n}+\frac {\left (9 a^2 b n^2\right ) \int x^{-1+2 m} \log \left (c x^n\right ) \, dx}{2 m^2}+\frac {\left (60 a b^2 n^2\right ) \int x^{-1+m} \log ^3\left (c x^n\right ) \, dx}{m^2}\\ &=-\frac {9 a^2 b n^3 x^{2 m}}{8 m^4}-\frac {a^3 n x^{3 m}}{9 m^2}+\frac {9 a^2 b n^2 x^{2 m} \log \left (c x^n\right )}{4 m^3}+\frac {a^3 x^{3 m} \log \left (c x^n\right )}{3 m}-\frac {9 a^2 b n x^{2 m} \log ^2\left (c x^n\right )}{4 m^2}+\frac {60 a b^2 n^2 x^m \log ^3\left (c x^n\right )}{m^3}+\frac {3 a^2 b x^{2 m} \log ^3\left (c x^n\right )}{2 m}-\frac {15 a b^2 n x^m \log ^4\left (c x^n\right )}{m^2}+\frac {3 a b^2 x^m \log ^5\left (c x^n\right )}{m}+\frac {b^3 \log ^8\left (c x^n\right )}{8 n}-\frac {\left (180 a b^2 n^3\right ) \int x^{-1+m} \log ^2\left (c x^n\right ) \, dx}{m^3}\\ &=-\frac {9 a^2 b n^3 x^{2 m}}{8 m^4}-\frac {a^3 n x^{3 m}}{9 m^2}+\frac {9 a^2 b n^2 x^{2 m} \log \left (c x^n\right )}{4 m^3}+\frac {a^3 x^{3 m} \log \left (c x^n\right )}{3 m}-\frac {180 a b^2 n^3 x^m \log ^2\left (c x^n\right )}{m^4}-\frac {9 a^2 b n x^{2 m} \log ^2\left (c x^n\right )}{4 m^2}+\frac {60 a b^2 n^2 x^m \log ^3\left (c x^n\right )}{m^3}+\frac {3 a^2 b x^{2 m} \log ^3\left (c x^n\right )}{2 m}-\frac {15 a b^2 n x^m \log ^4\left (c x^n\right )}{m^2}+\frac {3 a b^2 x^m \log ^5\left (c x^n\right )}{m}+\frac {b^3 \log ^8\left (c x^n\right )}{8 n}+\frac {\left (360 a b^2 n^4\right ) \int x^{-1+m} \log \left (c x^n\right ) \, dx}{m^4}\\ &=-\frac {360 a b^2 n^5 x^m}{m^6}-\frac {9 a^2 b n^3 x^{2 m}}{8 m^4}-\frac {a^3 n x^{3 m}}{9 m^2}+\frac {360 a b^2 n^4 x^m \log \left (c x^n\right )}{m^5}+\frac {9 a^2 b n^2 x^{2 m} \log \left (c x^n\right )}{4 m^3}+\frac {a^3 x^{3 m} \log \left (c x^n\right )}{3 m}-\frac {180 a b^2 n^3 x^m \log ^2\left (c x^n\right )}{m^4}-\frac {9 a^2 b n x^{2 m} \log ^2\left (c x^n\right )}{4 m^2}+\frac {60 a b^2 n^2 x^m \log ^3\left (c x^n\right )}{m^3}+\frac {3 a^2 b x^{2 m} \log ^3\left (c x^n\right )}{2 m}-\frac {15 a b^2 n x^m \log ^4\left (c x^n\right )}{m^2}+\frac {3 a b^2 x^m \log ^5\left (c x^n\right )}{m}+\frac {b^3 \log ^8\left (c x^n\right )}{8 n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.20, size = 336, normalized size = 1.24 \begin {gather*} -\frac {1}{8} b^3 n^7 \log ^8(x)+b^3 n^6 \log ^7(x) \log \left (c x^n\right )-\frac {7}{2} b^3 n^5 \log ^6(x) \log ^2\left (c x^n\right )+7 b^3 n^4 \log ^5(x) \log ^3\left (c x^n\right )-\frac {35}{4} b^3 n^3 \log ^4(x) \log ^4\left (c x^n\right )+7 b^3 n^2 \log ^3(x) \log ^5\left (c x^n\right )-\frac {7}{2} b^3 n \log ^2(x) \log ^6\left (c x^n\right )+b^3 \log (x) \log ^7\left (c x^n\right )+\frac {a x^m \left (-n \left (25920 b^2 n^4+81 a b m^2 n^2 x^m+8 a^2 m^4 x^{2 m}\right )+6 m \left (4320 b^2 n^4+27 a b m^2 n^2 x^m+4 a^2 m^4 x^{2 m}\right ) \log \left (c x^n\right )-162 b m^2 n \left (80 b n^2+a m^2 x^m\right ) \log ^2\left (c x^n\right )+108 b m^3 \left (40 b n^2+a m^2 x^m\right ) \log ^3\left (c x^n\right )-1080 b^2 m^4 n \log ^4\left (c x^n\right )+216 b^2 m^5 \log ^5\left (c x^n\right )\right )}{72 m^6} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 3.89, size = 61910, normalized size = 227.61
method | result | size |
risch | \(\text {Expression too large to display}\) | \(61910\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1115 vs.
\(2 (258) = 516\).
time = 0.32, size = 1115, normalized size = 4.10 \begin {gather*} \frac {1}{84} \, {\left (\frac {12 \, b^{3} \log \left (c x^{n}\right )^{7}}{n} + \frac {252 \, a b^{2} x^{m} \log \left (c x^{n}\right )^{4}}{m} + \frac {126 \, a^{2} b x^{2 \, m} \log \left (c x^{n}\right )^{2}}{m} - 1008 \, {\left (\frac {n x^{m} \log \left (c x^{n}\right )^{3}}{m^{2}} - \frac {3 \, {\left (\frac {n x^{m} \log \left (c x^{n}\right )^{2}}{m^{2}} - \frac {2 \, n {\left (\frac {n x^{m} \log \left (c x^{n}\right )}{m^{2}} - \frac {n^{2} x^{m}}{m^{3}}\right )}}{m}\right )} n}{m}\right )} a b^{2} - 63 \, a^{2} b {\left (\frac {2 \, n x^{2 \, m} \log \left (c x^{n}\right )}{m^{2}} - \frac {n^{2} x^{2 \, m}}{m^{3}}\right )} + \frac {28 \, a^{3} x^{3 \, m}}{m}\right )} \log \left (c x^{n}\right ) + \frac {9 \, b^{3} m^{6} n^{7} \log \left (x\right )^{8} - 72 \, b^{3} m^{6} n^{6} \log \left (c\right ) \log \left (x\right )^{7} + 252 \, b^{3} m^{6} n^{5} \log \left (c\right )^{2} \log \left (x\right )^{6} - 504 \, b^{3} m^{6} n^{4} \log \left (c\right )^{3} \log \left (x\right )^{5} + 630 \, b^{3} m^{6} n^{3} \log \left (c\right )^{4} \log \left (x\right )^{4} - 504 \, b^{3} m^{6} n^{2} \log \left (c\right )^{5} \log \left (x\right )^{3} + 252 \, b^{3} m^{6} n \log \left (c\right )^{6} \log \left (x\right )^{2} - 72 \, b^{3} m^{6} \log \left (c\right )^{7} \log \left (x\right ) - 72 \, b^{3} m^{6} \log \left (x\right ) \log \left (x^{n}\right )^{7} - 56 \, a^{3} m^{4} n x^{3 \, m} + 252 \, {\left (b^{3} m^{6} n \log \left (x\right )^{2} - 2 \, b^{3} m^{6} \log \left (c\right ) \log \left (x\right )\right )} \log \left (x^{n}\right )^{6} - 504 \, {\left (b^{3} m^{6} n^{2} \log \left (x\right )^{3} - 3 \, b^{3} m^{6} n \log \left (c\right ) \log \left (x\right )^{2} + 3 \, b^{3} m^{6} \log \left (c\right )^{2} \log \left (x\right )\right )} \log \left (x^{n}\right )^{5} - 189 \, {\left (2 \, m^{4} n \log \left (c\right )^{2} - 4 \, m^{3} n^{2} \log \left (c\right ) + 3 \, m^{2} n^{3}\right )} a^{2} b x^{2 \, m} - 1512 \, {\left (m^{4} n \log \left (c\right )^{4} - 8 \, m^{3} n^{2} \log \left (c\right )^{3} + 36 \, m^{2} n^{3} \log \left (c\right )^{2} - 96 \, m n^{4} \log \left (c\right ) + 120 \, n^{5}\right )} a b^{2} x^{m} + 126 \, {\left (5 \, b^{3} m^{6} n^{3} \log \left (x\right )^{4} - 20 \, b^{3} m^{6} n^{2} \log \left (c\right ) \log \left (x\right )^{3} + 30 \, b^{3} m^{6} n \log \left (c\right )^{2} \log \left (x\right )^{2} - 20 \, b^{3} m^{6} \log \left (c\right )^{3} \log \left (x\right ) - 12 \, a b^{2} m^{4} n x^{m}\right )} \log \left (x^{n}\right )^{4} - 504 \, {\left (b^{3} m^{6} n^{4} \log \left (x\right )^{5} - 5 \, b^{3} m^{6} n^{3} \log \left (c\right ) \log \left (x\right )^{4} + 10 \, b^{3} m^{6} n^{2} \log \left (c\right )^{2} \log \left (x\right )^{3} - 10 \, b^{3} m^{6} n \log \left (c\right )^{3} \log \left (x\right )^{2} + 5 \, b^{3} m^{6} \log \left (c\right )^{4} \log \left (x\right ) + 12 \, {\left (m^{4} n \log \left (c\right ) - 2 \, m^{3} n^{2}\right )} a b^{2} x^{m}\right )} \log \left (x^{n}\right )^{3} + 126 \, {\left (2 \, b^{3} m^{6} n^{5} \log \left (x\right )^{6} - 12 \, b^{3} m^{6} n^{4} \log \left (c\right ) \log \left (x\right )^{5} + 30 \, b^{3} m^{6} n^{3} \log \left (c\right )^{2} \log \left (x\right )^{4} - 40 \, b^{3} m^{6} n^{2} \log \left (c\right )^{3} \log \left (x\right )^{3} + 30 \, b^{3} m^{6} n \log \left (c\right )^{4} \log \left (x\right )^{2} - 12 \, b^{3} m^{6} \log \left (c\right )^{5} \log \left (x\right ) - 3 \, a^{2} b m^{4} n x^{2 \, m} - 72 \, {\left (m^{4} n \log \left (c\right )^{2} - 4 \, m^{3} n^{2} \log \left (c\right ) + 6 \, m^{2} n^{3}\right )} a b^{2} x^{m}\right )} \log \left (x^{n}\right )^{2} - 36 \, {\left (2 \, b^{3} m^{6} n^{6} \log \left (x\right )^{7} - 14 \, b^{3} m^{6} n^{5} \log \left (c\right ) \log \left (x\right )^{6} + 42 \, b^{3} m^{6} n^{4} \log \left (c\right )^{2} \log \left (x\right )^{5} - 70 \, b^{3} m^{6} n^{3} \log \left (c\right )^{3} \log \left (x\right )^{4} + 70 \, b^{3} m^{6} n^{2} \log \left (c\right )^{4} \log \left (x\right )^{3} - 42 \, b^{3} m^{6} n \log \left (c\right )^{5} \log \left (x\right )^{2} + 14 \, b^{3} m^{6} \log \left (c\right )^{6} \log \left (x\right ) + 21 \, {\left (m^{4} n \log \left (c\right ) - m^{3} n^{2}\right )} a^{2} b x^{2 \, m} + 168 \, {\left (m^{4} n \log \left (c\right )^{3} - 6 \, m^{3} n^{2} \log \left (c\right )^{2} + 18 \, m^{2} n^{3} \log \left (c\right ) - 24 \, m n^{4}\right )} a b^{2} x^{m}\right )} \log \left (x^{n}\right )}{504 \, m^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 655 vs.
\(2 (258) = 516\).
time = 0.38, size = 655, normalized size = 2.41 \begin {gather*} \frac {9 \, b^{3} m^{6} n^{7} \log \left (x\right )^{8} + 72 \, b^{3} m^{6} n^{6} \log \left (c\right ) \log \left (x\right )^{7} + 252 \, b^{3} m^{6} n^{5} \log \left (c\right )^{2} \log \left (x\right )^{6} + 504 \, b^{3} m^{6} n^{4} \log \left (c\right )^{3} \log \left (x\right )^{5} + 630 \, b^{3} m^{6} n^{3} \log \left (c\right )^{4} \log \left (x\right )^{4} + 504 \, b^{3} m^{6} n^{2} \log \left (c\right )^{5} \log \left (x\right )^{3} + 252 \, b^{3} m^{6} n \log \left (c\right )^{6} \log \left (x\right )^{2} + 72 \, b^{3} m^{6} \log \left (c\right )^{7} \log \left (x\right ) + 8 \, {\left (3 \, a^{3} m^{5} n \log \left (x\right ) + 3 \, a^{3} m^{5} \log \left (c\right ) - a^{3} m^{4} n\right )} x^{3 \, m} + 27 \, {\left (4 \, a^{2} b m^{5} n^{3} \log \left (x\right )^{3} + 4 \, a^{2} b m^{5} \log \left (c\right )^{3} - 6 \, a^{2} b m^{4} n \log \left (c\right )^{2} + 6 \, a^{2} b m^{3} n^{2} \log \left (c\right ) - 3 \, a^{2} b m^{2} n^{3} + 6 \, {\left (2 \, a^{2} b m^{5} n^{2} \log \left (c\right ) - a^{2} b m^{4} n^{3}\right )} \log \left (x\right )^{2} + 6 \, {\left (2 \, a^{2} b m^{5} n \log \left (c\right )^{2} - 2 \, a^{2} b m^{4} n^{2} \log \left (c\right ) + a^{2} b m^{3} n^{3}\right )} \log \left (x\right )\right )} x^{2 \, m} + 216 \, {\left (a b^{2} m^{5} n^{5} \log \left (x\right )^{5} + a b^{2} m^{5} \log \left (c\right )^{5} - 5 \, a b^{2} m^{4} n \log \left (c\right )^{4} + 20 \, a b^{2} m^{3} n^{2} \log \left (c\right )^{3} - 60 \, a b^{2} m^{2} n^{3} \log \left (c\right )^{2} + 120 \, a b^{2} m n^{4} \log \left (c\right ) - 120 \, a b^{2} n^{5} + 5 \, {\left (a b^{2} m^{5} n^{4} \log \left (c\right ) - a b^{2} m^{4} n^{5}\right )} \log \left (x\right )^{4} + 10 \, {\left (a b^{2} m^{5} n^{3} \log \left (c\right )^{2} - 2 \, a b^{2} m^{4} n^{4} \log \left (c\right ) + 2 \, a b^{2} m^{3} n^{5}\right )} \log \left (x\right )^{3} + 10 \, {\left (a b^{2} m^{5} n^{2} \log \left (c\right )^{3} - 3 \, a b^{2} m^{4} n^{3} \log \left (c\right )^{2} + 6 \, a b^{2} m^{3} n^{4} \log \left (c\right ) - 6 \, a b^{2} m^{2} n^{5}\right )} \log \left (x\right )^{2} + 5 \, {\left (a b^{2} m^{5} n \log \left (c\right )^{4} - 4 \, a b^{2} m^{4} n^{2} \log \left (c\right )^{3} + 12 \, a b^{2} m^{3} n^{3} \log \left (c\right )^{2} - 24 \, a b^{2} m^{2} n^{4} \log \left (c\right ) + 24 \, a b^{2} m n^{5}\right )} \log \left (x\right )\right )} x^{m}}{72 \, m^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 30.90, size = 411, normalized size = 1.51 \begin {gather*} - a^{3} n \left (\begin {cases} \frac {\begin {cases} \frac {x^{3 m}}{3 m} & \text {for}\: m \neq 0 \\\log {\left (x \right )} & \text {otherwise} \end {cases}}{3 m} & \text {for}\: m > -\infty \wedge m < \infty \wedge m \neq 0 \\\frac {\log {\left (x \right )}^{2}}{2} & \text {otherwise} \end {cases}\right ) + a^{3} \left (\begin {cases} \frac {x^{3 m}}{3 m} & \text {for}\: m \neq 0 \\\log {\left (x \right )} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} + 3 a^{2} b \left (\begin {cases} \frac {x^{2 m} \log {\left (c x^{n} \right )}^{3}}{2 m} - \frac {3 n x^{2 m} \log {\left (c x^{n} \right )}^{2}}{4 m^{2}} + \frac {3 n^{2} x^{2 m} \log {\left (c x^{n} \right )}}{4 m^{3}} - \frac {3 n^{3} x^{2 m}}{8 m^{4}} & \text {for}\: m \neq 0 \\\begin {cases} 0 & \text {for}\: \frac {1}{\left |{c x^{n}}\right |} < 1 \wedge \left |{c x^{n}}\right | < 1 \\\frac {\log {\left (c x^{n} \right )}^{4}}{4 n} & \text {for}\: \left |{c x^{n}}\right | < 1 \\\frac {\log {\left (\frac {x^{- n}}{c} \right )}^{4}}{4 n} & \text {for}\: \frac {1}{\left |{c x^{n}}\right |} < 1 \\\frac {6 {G_{5, 5}^{5, 0}\left (\begin {matrix} & 1, 1, 1, 1, 1 \\0, 0, 0, 0, 0 & \end {matrix} \middle | {c x^{n}} \right )}}{n} + \frac {6 {G_{5, 5}^{0, 5}\left (\begin {matrix} 1, 1, 1, 1, 1 & \\ & 0, 0, 0, 0, 0 \end {matrix} \middle | {c x^{n}} \right )}}{n} & \text {otherwise} \end {cases} & \text {otherwise} \end {cases}\right ) + 3 a b^{2} \left (\begin {cases} \frac {x^{m} \log {\left (c x^{n} \right )}^{5}}{m} - \frac {5 n x^{m} \log {\left (c x^{n} \right )}^{4}}{m^{2}} + \frac {20 n^{2} x^{m} \log {\left (c x^{n} \right )}^{3}}{m^{3}} - \frac {60 n^{3} x^{m} \log {\left (c x^{n} \right )}^{2}}{m^{4}} + \frac {120 n^{4} x^{m} \log {\left (c x^{n} \right )}}{m^{5}} - \frac {120 n^{5} x^{m}}{m^{6}} & \text {for}\: m \neq 0 \\\begin {cases} 0 & \text {for}\: \frac {1}{\left |{c x^{n}}\right |} < 1 \wedge \left |{c x^{n}}\right | < 1 \\\frac {\log {\left (c x^{n} \right )}^{6}}{6 n} & \text {for}\: \left |{c x^{n}}\right | < 1 \\\frac {\log {\left (\frac {x^{- n}}{c} \right )}^{6}}{6 n} & \text {for}\: \frac {1}{\left |{c x^{n}}\right |} < 1 \\\frac {120 {G_{7, 7}^{7, 0}\left (\begin {matrix} & 1, 1, 1, 1, 1, 1, 1 \\0, 0, 0, 0, 0, 0, 0 & \end {matrix} \middle | {c x^{n}} \right )}}{n} + \frac {120 {G_{7, 7}^{0, 7}\left (\begin {matrix} 1, 1, 1, 1, 1, 1, 1 & \\ & 0, 0, 0, 0, 0, 0, 0 \end {matrix} \middle | {c x^{n}} \right )}}{n} & \text {otherwise} \end {cases} & \text {otherwise} \end {cases}\right ) - b^{3} \left (\begin {cases} - \log {\left (c \right )}^{7} \log {\left (x \right )} & \text {for}\: n = 0 \\- \frac {\log {\left (c x^{n} \right )}^{8}}{8 n} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 766 vs.
\(2 (258) = 516\).
time = 5.48, size = 766, normalized size = 2.82 \begin {gather*} \frac {1}{8} \, b^{3} n^{7} \log \left (x\right )^{8} + b^{3} n^{6} \log \left (c\right ) \log \left (x\right )^{7} + \frac {7}{2} \, b^{3} n^{5} \log \left (c\right )^{2} \log \left (x\right )^{6} + 7 \, b^{3} n^{4} \log \left (c\right )^{3} \log \left (x\right )^{5} + \frac {35}{4} \, b^{3} n^{3} \log \left (c\right )^{4} \log \left (x\right )^{4} + 7 \, b^{3} n^{2} \log \left (c\right )^{5} \log \left (x\right )^{3} + \frac {3 \, a b^{2} n^{5} x^{m} \log \left (x\right )^{5}}{m} + \frac {7}{2} \, b^{3} n \log \left (c\right )^{6} \log \left (x\right )^{2} + \frac {15 \, a b^{2} n^{4} x^{m} \log \left (c\right ) \log \left (x\right )^{4}}{m} + b^{3} \log \left (c\right )^{7} \log \left (x\right ) + \frac {30 \, a b^{2} n^{3} x^{m} \log \left (c\right )^{2} \log \left (x\right )^{3}}{m} - \frac {15 \, a b^{2} n^{5} x^{m} \log \left (x\right )^{4}}{m^{2}} + \frac {30 \, a b^{2} n^{2} x^{m} \log \left (c\right )^{3} \log \left (x\right )^{2}}{m} - \frac {60 \, a b^{2} n^{4} x^{m} \log \left (c\right ) \log \left (x\right )^{3}}{m^{2}} + \frac {15 \, a b^{2} n x^{m} \log \left (c\right )^{4} \log \left (x\right )}{m} - \frac {90 \, a b^{2} n^{3} x^{m} \log \left (c\right )^{2} \log \left (x\right )^{2}}{m^{2}} + \frac {3 \, a^{2} b n^{3} x^{2 \, m} \log \left (x\right )^{3}}{2 \, m} + \frac {60 \, a b^{2} n^{5} x^{m} \log \left (x\right )^{3}}{m^{3}} + \frac {3 \, a b^{2} x^{m} \log \left (c\right )^{5}}{m} - \frac {60 \, a b^{2} n^{2} x^{m} \log \left (c\right )^{3} \log \left (x\right )}{m^{2}} + \frac {9 \, a^{2} b n^{2} x^{2 \, m} \log \left (c\right ) \log \left (x\right )^{2}}{2 \, m} + \frac {180 \, a b^{2} n^{4} x^{m} \log \left (c\right ) \log \left (x\right )^{2}}{m^{3}} - \frac {15 \, a b^{2} n x^{m} \log \left (c\right )^{4}}{m^{2}} + \frac {9 \, a^{2} b n x^{2 \, m} \log \left (c\right )^{2} \log \left (x\right )}{2 \, m} + \frac {180 \, a b^{2} n^{3} x^{m} \log \left (c\right )^{2} \log \left (x\right )}{m^{3}} - \frac {9 \, a^{2} b n^{3} x^{2 \, m} \log \left (x\right )^{2}}{4 \, m^{2}} - \frac {180 \, a b^{2} n^{5} x^{m} \log \left (x\right )^{2}}{m^{4}} + \frac {3 \, a^{2} b x^{2 \, m} \log \left (c\right )^{3}}{2 \, m} + \frac {60 \, a b^{2} n^{2} x^{m} \log \left (c\right )^{3}}{m^{3}} - \frac {9 \, a^{2} b n^{2} x^{2 \, m} \log \left (c\right ) \log \left (x\right )}{2 \, m^{2}} - \frac {360 \, a b^{2} n^{4} x^{m} \log \left (c\right ) \log \left (x\right )}{m^{4}} - \frac {9 \, a^{2} b n x^{2 \, m} \log \left (c\right )^{2}}{4 \, m^{2}} - \frac {180 \, a b^{2} n^{3} x^{m} \log \left (c\right )^{2}}{m^{4}} + \frac {a^{3} n x^{3 \, m} \log \left (x\right )}{3 \, m} + \frac {9 \, a^{2} b n^{3} x^{2 \, m} \log \left (x\right )}{4 \, m^{3}} + \frac {360 \, a b^{2} n^{5} x^{m} \log \left (x\right )}{m^{5}} + \frac {a^{3} x^{3 \, m} \log \left (c\right )}{3 \, m} + \frac {9 \, a^{2} b n^{2} x^{2 \, m} \log \left (c\right )}{4 \, m^{3}} + \frac {360 \, a b^{2} n^{4} x^{m} \log \left (c\right )}{m^{5}} - \frac {a^{3} n x^{3 \, m}}{9 \, m^{2}} - \frac {9 \, a^{2} b n^{3} x^{2 \, m}}{8 \, m^{4}} - \frac {360 \, a b^{2} n^{5} x^{m}}{m^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\ln \left (c\,x^n\right )\,{\left (a\,x^m+b\,{\ln \left (c\,x^n\right )}^2\right )}^3}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________