Optimal. Leaf size=205 \[ -\frac {2 b^2 e n^2 r}{81 x^3}-\frac {2 b e n (3 a+b n) r}{81 x^3}-\frac {e \left (9 a^2+6 a b n+2 b^2 n^2\right ) r}{81 x^3}-\frac {2 b^2 e n r \log \left (c x^n\right )}{27 x^3}-\frac {2 b e (3 a+b n) r \log \left (c x^n\right )}{27 x^3}-\frac {b^2 e r \log ^2\left (c x^n\right )}{9 x^3}-\frac {2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{27 x^3}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{3 x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.14, antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {2342, 2341,
2413, 12, 14} \begin {gather*} -\frac {e r \left (9 a^2+6 a b n+2 b^2 n^2\right )}{81 x^3}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{3 x^3}-\frac {2 b e r (3 a+b n) \log \left (c x^n\right )}{27 x^3}-\frac {2 b e n r (3 a+b n)}{81 x^3}-\frac {b^2 e r \log ^2\left (c x^n\right )}{9 x^3}-\frac {2 b^2 e n r \log \left (c x^n\right )}{27 x^3}-\frac {2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{27 x^3}-\frac {2 b^2 e n^2 r}{81 x^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 14
Rule 2341
Rule 2342
Rule 2413
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{x^4} \, dx &=-\frac {2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{27 x^3}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{3 x^3}-(e r) \int \frac {-9 a^2 \left (1+\frac {2 b n (3 a+b n)}{9 a^2}\right )-6 b (3 a+b n) \log \left (c x^n\right )-9 b^2 \log ^2\left (c x^n\right )}{27 x^4} \, dx\\ &=-\frac {2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{27 x^3}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{3 x^3}-\frac {1}{27} (e r) \int \frac {-9 a^2 \left (1+\frac {2 b n (3 a+b n)}{9 a^2}\right )-6 b (3 a+b n) \log \left (c x^n\right )-9 b^2 \log ^2\left (c x^n\right )}{x^4} \, dx\\ &=-\frac {2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{27 x^3}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{3 x^3}-\frac {1}{27} (e r) \int \left (\frac {-9 a^2-6 a b n-2 b^2 n^2}{x^4}-\frac {6 b (3 a+b n) \log \left (c x^n\right )}{x^4}-\frac {9 b^2 \log ^2\left (c x^n\right )}{x^4}\right ) \, dx\\ &=-\frac {e \left (9 a^2+6 a b n+2 b^2 n^2\right ) r}{81 x^3}-\frac {2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{27 x^3}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{3 x^3}+\frac {1}{3} \left (b^2 e r\right ) \int \frac {\log ^2\left (c x^n\right )}{x^4} \, dx+\frac {1}{9} (2 b e (3 a+b n) r) \int \frac {\log \left (c x^n\right )}{x^4} \, dx\\ &=-\frac {2 b e n (3 a+b n) r}{81 x^3}-\frac {e \left (9 a^2+6 a b n+2 b^2 n^2\right ) r}{81 x^3}-\frac {2 b e (3 a+b n) r \log \left (c x^n\right )}{27 x^3}-\frac {b^2 e r \log ^2\left (c x^n\right )}{9 x^3}-\frac {2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{27 x^3}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{3 x^3}+\frac {1}{9} \left (2 b^2 e n r\right ) \int \frac {\log \left (c x^n\right )}{x^4} \, dx\\ &=-\frac {2 b^2 e n^2 r}{81 x^3}-\frac {2 b e n (3 a+b n) r}{81 x^3}-\frac {e \left (9 a^2+6 a b n+2 b^2 n^2\right ) r}{81 x^3}-\frac {2 b^2 e n r \log \left (c x^n\right )}{27 x^3}-\frac {2 b e (3 a+b n) r \log \left (c x^n\right )}{27 x^3}-\frac {b^2 e r \log ^2\left (c x^n\right )}{9 x^3}-\frac {2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{27 x^3}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{3 x^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.08, size = 155, normalized size = 0.76 \begin {gather*} -\frac {9 a^2 d+6 a b d n+2 b^2 d n^2+3 a^2 e r+4 a b e n r+2 b^2 e n^2 r+e \left (9 a^2+6 a b n+2 b^2 n^2\right ) \log \left (f x^r\right )+3 b^2 \log ^2\left (c x^n\right ) \left (3 d+e r+3 e \log \left (f x^r\right )\right )+2 b \log \left (c x^n\right ) \left (9 a d+3 b d n+3 a e r+2 b e n r+3 e (3 a+b n) \log \left (f x^r\right )\right )}{27 x^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.45, size = 8407, normalized size = 41.01
method | result | size |
risch | \(\text {Expression too large to display}\) | \(8407\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.30, size = 236, normalized size = 1.15 \begin {gather*} -\frac {1}{9} \, b^{2} {\left (\frac {r}{x^{3}} + \frac {3 \, \log \left (f x^{r}\right )}{x^{3}}\right )} e \log \left (c x^{n}\right )^{2} - \frac {2}{9} \, a b {\left (\frac {r}{x^{3}} + \frac {3 \, \log \left (f x^{r}\right )}{x^{3}}\right )} e \log \left (c x^{n}\right ) - \frac {2}{27} \, b^{2} d {\left (\frac {n^{2}}{x^{3}} + \frac {3 \, n \log \left (c x^{n}\right )}{x^{3}}\right )} - \frac {2}{27} \, b^{2} {\left (\frac {{\left (r \log \left (x\right ) + r + \log \left (f\right )\right )} n^{2}}{x^{3}} + \frac {n {\left (2 \, r + 3 \, \log \left (f\right ) + 3 \, \log \left (x^{r}\right )\right )} \log \left (c x^{n}\right )}{x^{3}}\right )} e - \frac {2 \, a b n {\left (2 \, r + 3 \, \log \left (f\right ) + 3 \, \log \left (x^{r}\right )\right )} e}{27 \, x^{3}} - \frac {b^{2} d \log \left (c x^{n}\right )^{2}}{3 \, x^{3}} - \frac {2 \, a b d n}{9 \, x^{3}} - \frac {a^{2} r e}{9 \, x^{3}} - \frac {2 \, a b d \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac {a^{2} e \log \left (f x^{r}\right )}{3 \, x^{3}} - \frac {a^{2} d}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.36, size = 337, normalized size = 1.64 \begin {gather*} -\frac {9 \, b^{2} n^{2} r e \log \left (x\right )^{3} + 2 \, b^{2} d n^{2} + 6 \, a b d n + 9 \, a^{2} d + {\left (2 \, b^{2} n^{2} + 4 \, a b n + 3 \, a^{2}\right )} r e + 3 \, {\left (b^{2} r e + 3 \, b^{2} d\right )} \log \left (c\right )^{2} + 9 \, {\left (2 \, b^{2} n r e \log \left (c\right ) + b^{2} n^{2} e \log \left (f\right ) + b^{2} d n^{2} + {\left (b^{2} n^{2} + 2 \, a b n\right )} r e\right )} \log \left (x\right )^{2} + 2 \, {\left (3 \, b^{2} d n + 9 \, a b d + {\left (2 \, b^{2} n + 3 \, a b\right )} r e\right )} \log \left (c\right ) + {\left (9 \, b^{2} e \log \left (c\right )^{2} + 6 \, {\left (b^{2} n + 3 \, a b\right )} e \log \left (c\right ) + {\left (2 \, b^{2} n^{2} + 6 \, a b n + 9 \, a^{2}\right )} e\right )} \log \left (f\right ) + 3 \, {\left (3 \, b^{2} r e \log \left (c\right )^{2} + 2 \, b^{2} d n^{2} + 6 \, a b d n + {\left (2 \, b^{2} n^{2} + 4 \, a b n + 3 \, a^{2}\right )} r e + 2 \, {\left (3 \, b^{2} d n + {\left (2 \, b^{2} n + 3 \, a b\right )} r e\right )} \log \left (c\right ) + 2 \, {\left (3 \, b^{2} n e \log \left (c\right ) + {\left (b^{2} n^{2} + 3 \, a b n\right )} e\right )} \log \left (f\right )\right )} \log \left (x\right )}{27 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 2.72, size = 342, normalized size = 1.67 \begin {gather*} - \frac {a^{2} d}{3 x^{3}} - \frac {a^{2} e r}{9 x^{3}} - \frac {a^{2} e \log {\left (f x^{r} \right )}}{3 x^{3}} - \frac {2 a b d n}{9 x^{3}} - \frac {2 a b d \log {\left (c x^{n} \right )}}{3 x^{3}} - \frac {4 a b e n r}{27 x^{3}} - \frac {2 a b e n \log {\left (f x^{r} \right )}}{9 x^{3}} - \frac {2 a b e r \log {\left (c x^{n} \right )}}{9 x^{3}} - \frac {2 a b e \log {\left (c x^{n} \right )} \log {\left (f x^{r} \right )}}{3 x^{3}} - \frac {2 b^{2} d n^{2}}{27 x^{3}} - \frac {2 b^{2} d n \log {\left (c x^{n} \right )}}{9 x^{3}} - \frac {b^{2} d \log {\left (c x^{n} \right )}^{2}}{3 x^{3}} - \frac {2 b^{2} e n^{2} r}{27 x^{3}} - \frac {2 b^{2} e n^{2} \log {\left (f x^{r} \right )}}{27 x^{3}} - \frac {4 b^{2} e n r \log {\left (c x^{n} \right )}}{27 x^{3}} - \frac {2 b^{2} e n \log {\left (c x^{n} \right )} \log {\left (f x^{r} \right )}}{9 x^{3}} - \frac {b^{2} e r \log {\left (c x^{n} \right )}^{2}}{9 x^{3}} - \frac {b^{2} e \log {\left (c x^{n} \right )}^{2} \log {\left (f x^{r} \right )}}{3 x^{3}} \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. 403 vs.
\(2 (196) = 392\).
time = 6.98, size = 403, normalized size = 1.97 \begin {gather*} -\frac {9 \, b^{2} n^{2} r e \log \left (x\right )^{3} + 9 \, b^{2} n^{2} r e \log \left (x\right )^{2} + 18 \, b^{2} n r e \log \left (c\right ) \log \left (x\right )^{2} + 9 \, b^{2} n^{2} e \log \left (f\right ) \log \left (x\right )^{2} + 6 \, b^{2} n^{2} r e \log \left (x\right ) + 12 \, b^{2} n r e \log \left (c\right ) \log \left (x\right ) + 9 \, b^{2} r e \log \left (c\right )^{2} \log \left (x\right ) + 6 \, b^{2} n^{2} e \log \left (f\right ) \log \left (x\right ) + 18 \, b^{2} n e \log \left (c\right ) \log \left (f\right ) \log \left (x\right ) + 9 \, b^{2} d n^{2} \log \left (x\right )^{2} + 18 \, a b n r e \log \left (x\right )^{2} + 2 \, b^{2} n^{2} r e + 4 \, b^{2} n r e \log \left (c\right ) + 3 \, b^{2} r e \log \left (c\right )^{2} + 2 \, b^{2} n^{2} e \log \left (f\right ) + 6 \, b^{2} n e \log \left (c\right ) \log \left (f\right ) + 9 \, b^{2} e \log \left (c\right )^{2} \log \left (f\right ) + 6 \, b^{2} d n^{2} \log \left (x\right ) + 12 \, a b n r e \log \left (x\right ) + 18 \, b^{2} d n \log \left (c\right ) \log \left (x\right ) + 18 \, a b r e \log \left (c\right ) \log \left (x\right ) + 18 \, a b n e \log \left (f\right ) \log \left (x\right ) + 2 \, b^{2} d n^{2} + 4 \, a b n r e + 6 \, b^{2} d n \log \left (c\right ) + 6 \, a b r e \log \left (c\right ) + 9 \, b^{2} d \log \left (c\right )^{2} + 6 \, a b n e \log \left (f\right ) + 18 \, a b e \log \left (c\right ) \log \left (f\right ) + 18 \, a b d n \log \left (x\right ) + 9 \, a^{2} r e \log \left (x\right ) + 6 \, a b d n + 3 \, a^{2} r e + 18 \, a b d \log \left (c\right ) + 9 \, a^{2} e \log \left (f\right ) + 9 \, a^{2} d}{27 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 4.20, size = 190, normalized size = 0.93 \begin {gather*} -\ln \left (f\,x^r\right )\,\left (\ln \left (c\,x^n\right )\,\left (\frac {2\,a\,b\,e}{3\,x^3}+\frac {2\,b^2\,e\,n}{9\,x^3}\right )+\frac {a^2\,e}{3\,x^3}+\frac {2\,b^2\,e\,n^2}{27\,x^3}+\frac {b^2\,e\,{\ln \left (c\,x^n\right )}^2}{3\,x^3}+\frac {2\,a\,b\,e\,n}{9\,x^3}\right )-\frac {\frac {a^2\,d}{3}+\frac {2\,b^2\,d\,n^2}{27}+\frac {a^2\,e\,r}{9}+\frac {2\,b^2\,e\,n^2\,r}{27}+\frac {2\,a\,b\,d\,n}{9}+\frac {4\,a\,b\,e\,n\,r}{27}}{x^3}-\frac {b^2\,{\ln \left (c\,x^n\right )}^2\,\left (3\,d+e\,r\right )}{9\,x^3}-\frac {2\,b\,\ln \left (c\,x^n\right )\,\left (9\,a\,d+3\,b\,d\,n+3\,a\,e\,r+2\,b\,e\,n\,r\right )}{27\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________