Optimal. Leaf size=233 \[ -\frac {3 b d^2 e n x^{1+r} (f x)^m}{(1+m+r)^2}-\frac {3 b d e^2 n x^{1+2 r} (f x)^m}{(1+m+2 r)^2}-\frac {b e^3 n x^{1+3 r} (f x)^m}{(1+m+3 r)^2}-\frac {b d^3 n (f x)^{1+m}}{f (1+m)^2}+\frac {3 d^2 e x^{1+r} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{1+m+r}+\frac {3 d e^2 x^{1+2 r} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{1+m+2 r}+\frac {e^3 x^{1+3 r} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{1+m+3 r}+\frac {d^3 (f x)^{1+m} \left (a+b \log \left (c x^n\right )\right )}{f (1+m)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 1.89, antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {276, 20, 30,
2392, 14} \begin {gather*} \frac {d^3 (f x)^{m+1} \left (a+b \log \left (c x^n\right )\right )}{f (m+1)}+\frac {3 d^2 e x^{r+1} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{m+r+1}+\frac {3 d e^2 x^{2 r+1} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{m+2 r+1}+\frac {e^3 x^{3 r+1} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{m+3 r+1}-\frac {b d^3 n (f x)^{m+1}}{f (m+1)^2}-\frac {3 b d^2 e n x^{r+1} (f x)^m}{(m+r+1)^2}-\frac {3 b d e^2 n x^{2 r+1} (f x)^m}{(m+2 r+1)^2}-\frac {b e^3 n x^{3 r+1} (f x)^m}{(m+3 r+1)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 20
Rule 30
Rule 276
Rule 2392
Rubi steps
\begin {align*} \int (f x)^m \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {3 d^2 e x^{1+r} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{1+m+r}+\frac {3 d e^2 x^{1+2 r} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{1+m+2 r}+\frac {e^3 x^{1+3 r} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{1+m+3 r}+\frac {d^3 (f x)^{1+m} \left (a+b \log \left (c x^n\right )\right )}{f (1+m)}-(b n) \int (f x)^m \left (\frac {d^3}{1+m}+\frac {3 d^2 e x^r}{1+m+r}+\frac {3 d e^2 x^{2 r}}{1+m+2 r}+\frac {e^3 x^{3 r}}{1+m+3 r}\right ) \, dx\\ &=\frac {3 d^2 e x^{1+r} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{1+m+r}+\frac {3 d e^2 x^{1+2 r} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{1+m+2 r}+\frac {e^3 x^{1+3 r} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{1+m+3 r}+\frac {d^3 (f x)^{1+m} \left (a+b \log \left (c x^n\right )\right )}{f (1+m)}-(b n) \int \left (\frac {d^3 (f x)^m}{1+m}+\frac {3 d^2 e x^r (f x)^m}{1+m+r}+\frac {3 d e^2 x^{2 r} (f x)^m}{1+m+2 r}+\frac {e^3 x^{3 r} (f x)^m}{1+m+3 r}\right ) \, dx\\ &=-\frac {b d^3 n (f x)^{1+m}}{f (1+m)^2}+\frac {3 d^2 e x^{1+r} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{1+m+r}+\frac {3 d e^2 x^{1+2 r} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{1+m+2 r}+\frac {e^3 x^{1+3 r} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{1+m+3 r}+\frac {d^3 (f x)^{1+m} \left (a+b \log \left (c x^n\right )\right )}{f (1+m)}-\frac {\left (3 b d^2 e n\right ) \int x^r (f x)^m \, dx}{1+m+r}-\frac {\left (3 b d e^2 n\right ) \int x^{2 r} (f x)^m \, dx}{1+m+2 r}-\frac {\left (b e^3 n\right ) \int x^{3 r} (f x)^m \, dx}{1+m+3 r}\\ &=-\frac {b d^3 n (f x)^{1+m}}{f (1+m)^2}+\frac {3 d^2 e x^{1+r} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{1+m+r}+\frac {3 d e^2 x^{1+2 r} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{1+m+2 r}+\frac {e^3 x^{1+3 r} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{1+m+3 r}+\frac {d^3 (f x)^{1+m} \left (a+b \log \left (c x^n\right )\right )}{f (1+m)}-\frac {\left (3 b d^2 e n x^{-m} (f x)^m\right ) \int x^{m+r} \, dx}{1+m+r}-\frac {\left (3 b d e^2 n x^{-m} (f x)^m\right ) \int x^{m+2 r} \, dx}{1+m+2 r}-\frac {\left (b e^3 n x^{-m} (f x)^m\right ) \int x^{m+3 r} \, dx}{1+m+3 r}\\ &=-\frac {3 b d^2 e n x^{1+r} (f x)^m}{(1+m+r)^2}-\frac {3 b d e^2 n x^{1+2 r} (f x)^m}{(1+m+2 r)^2}-\frac {b e^3 n x^{1+3 r} (f x)^m}{(1+m+3 r)^2}-\frac {b d^3 n (f x)^{1+m}}{f (1+m)^2}+\frac {3 d^2 e x^{1+r} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{1+m+r}+\frac {3 d e^2 x^{1+2 r} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{1+m+2 r}+\frac {e^3 x^{1+3 r} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{1+m+3 r}+\frac {d^3 (f x)^{1+m} \left (a+b \log \left (c x^n\right )\right )}{f (1+m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.35, size = 187, normalized size = 0.80 \begin {gather*} x (f x)^m \left (\frac {b d^3 n \log (x)}{1+m}+\frac {d^3 \left (a+a m-b n-b (1+m) n \log (x)+b (1+m) \log \left (c x^n\right )\right )}{(1+m)^2}+\frac {3 d^2 e x^r \left (-b n+a (1+m+r)+b (1+m+r) \log \left (c x^n\right )\right )}{(1+m+r)^2}+\frac {3 d e^2 x^{2 r} \left (-b n+a (1+m+2 r)+b (1+m+2 r) \log \left (c x^n\right )\right )}{(1+m+2 r)^2}+\frac {e^3 x^{3 r} \left (-b n+a (1+m+3 r)+b (1+m+3 r) \log \left (c x^n\right )\right )}{(1+m+3 r)^2}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 1.37, size = 22706, normalized size = 97.45
method | result | size |
risch | \(\text {Expression too large to display}\) | \(22706\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.32, size = 331, normalized size = 1.42 \begin {gather*} -\frac {b d^{3} f^{m} n x x^{m}}{{\left (m + 1\right )}^{2}} + \frac {3 \, b d^{2} f^{m} x e^{\left (m \log \left (x\right ) + r \log \left (x\right ) + 1\right )} \log \left (c x^{n}\right )}{m + r + 1} + \frac {3 \, a d^{2} f^{m} x e^{\left (m \log \left (x\right ) + r \log \left (x\right ) + 1\right )}}{m + r + 1} - \frac {3 \, b d^{2} f^{m} n x e^{\left (m \log \left (x\right ) + r \log \left (x\right ) + 1\right )}}{{\left (m + r + 1\right )}^{2}} + \frac {3 \, b d f^{m} x e^{\left (m \log \left (x\right ) + 2 \, r \log \left (x\right ) + 2\right )} \log \left (c x^{n}\right )}{m + 2 \, r + 1} + \frac {3 \, a d f^{m} x e^{\left (m \log \left (x\right ) + 2 \, r \log \left (x\right ) + 2\right )}}{m + 2 \, r + 1} - \frac {3 \, b d f^{m} n x e^{\left (m \log \left (x\right ) + 2 \, r \log \left (x\right ) + 2\right )}}{{\left (m + 2 \, r + 1\right )}^{2}} + \frac {\left (f x\right )^{m + 1} b d^{3} \log \left (c x^{n}\right )}{f {\left (m + 1\right )}} + \frac {b f^{m} x e^{\left (m \log \left (x\right ) + 3 \, r \log \left (x\right ) + 3\right )} \log \left (c x^{n}\right )}{m + 3 \, r + 1} + \frac {\left (f x\right )^{m + 1} a d^{3}}{f {\left (m + 1\right )}} + \frac {a f^{m} x e^{\left (m \log \left (x\right ) + 3 \, r \log \left (x\right ) + 3\right )}}{m + 3 \, r + 1} - \frac {b f^{m} n x e^{\left (m \log \left (x\right ) + 3 \, r \log \left (x\right ) + 3\right )}}{{\left (m + 3 \, r + 1\right )}^{2}} \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. 4049 vs.
\(2 (231) = 462\).
time = 0.44, size = 4049, normalized size = 17.38 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 766 vs.
\(2 (231) = 462\).
time = 4.59, size = 766, normalized size = 3.29 \begin {gather*} \frac {3 \, b d^{2} f^{m} m n x x^{m} x^{r} e \log \left (x\right )}{m^{2} + 2 \, m r + r^{2} + 2 \, m + 2 \, r + 1} + \frac {3 \, b d^{2} f^{m} n r x x^{m} x^{r} e \log \left (x\right )}{m^{2} + 2 \, m r + r^{2} + 2 \, m + 2 \, r + 1} + \frac {b d^{3} f^{m} m n x x^{m} \log \left (x\right )}{m^{2} + 2 \, m + 1} + \frac {3 \, b d f^{m} m n x x^{m} x^{2 \, r} e^{2} \log \left (x\right )}{m^{2} + 4 \, m r + 4 \, r^{2} + 2 \, m + 4 \, r + 1} + \frac {6 \, b d f^{m} n r x x^{m} x^{2 \, r} e^{2} \log \left (x\right )}{m^{2} + 4 \, m r + 4 \, r^{2} + 2 \, m + 4 \, r + 1} + \frac {3 \, b d^{2} f^{m} n x x^{m} x^{r} e \log \left (x\right )}{m^{2} + 2 \, m r + r^{2} + 2 \, m + 2 \, r + 1} - \frac {3 \, b d^{2} f^{m} n x x^{m} x^{r} e}{m^{2} + 2 \, m r + r^{2} + 2 \, m + 2 \, r + 1} + \frac {3 \, b d^{2} f^{m} x x^{m} x^{r} e \log \left (c\right )}{m + r + 1} + \frac {b d^{3} f^{m} n x x^{m} \log \left (x\right )}{m^{2} + 2 \, m + 1} + \frac {b f^{m} m n x x^{m} x^{3 \, r} e^{3} \log \left (x\right )}{m^{2} + 6 \, m r + 9 \, r^{2} + 2 \, m + 6 \, r + 1} + \frac {3 \, b f^{m} n r x x^{m} x^{3 \, r} e^{3} \log \left (x\right )}{m^{2} + 6 \, m r + 9 \, r^{2} + 2 \, m + 6 \, r + 1} + \frac {3 \, b d f^{m} n x x^{m} x^{2 \, r} e^{2} \log \left (x\right )}{m^{2} + 4 \, m r + 4 \, r^{2} + 2 \, m + 4 \, r + 1} - \frac {b d^{3} f^{m} n x x^{m}}{m^{2} + 2 \, m + 1} - \frac {3 \, b d f^{m} n x x^{m} x^{2 \, r} e^{2}}{m^{2} + 4 \, m r + 4 \, r^{2} + 2 \, m + 4 \, r + 1} + \frac {3 \, a d^{2} f^{m} x x^{m} x^{r} e}{m + r + 1} + \frac {3 \, b d f^{m} x x^{m} x^{2 \, r} e^{2} \log \left (c\right )}{m + 2 \, r + 1} + \frac {b f^{m} n x x^{m} x^{3 \, r} e^{3} \log \left (x\right )}{m^{2} + 6 \, m r + 9 \, r^{2} + 2 \, m + 6 \, r + 1} - \frac {b f^{m} n x x^{m} x^{3 \, r} e^{3}}{m^{2} + 6 \, m r + 9 \, r^{2} + 2 \, m + 6 \, r + 1} + \frac {3 \, a d f^{m} x x^{m} x^{2 \, r} e^{2}}{m + 2 \, r + 1} + \frac {\left (f x\right )^{m} b d^{3} x \log \left (c\right )}{m + 1} + \frac {b f^{m} x x^{m} x^{3 \, r} e^{3} \log \left (c\right )}{m + 3 \, r + 1} + \frac {\left (f x\right )^{m} a d^{3} x}{m + 1} + \frac {a f^{m} x x^{m} x^{3 \, r} e^{3}}{m + 3 \, r + 1} \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 {\left (f\,x\right )}^m\,{\left (d+e\,x^r\right )}^3\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________