\(\int (f x)^m (d+e x^r)^3 (a+b \log (c x^n)) \, dx\) [440]

Optimal result

Integrand size = 25, antiderivative size = 233 \[ \int (f x)^m \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {3 b d^2 e n x^{1+r} (f x)^m}{(1+m+r)^2}-\frac {3 b d e^2 n x^{1+2 r} (f x)^m}{(1+m+2 r)^2}-\frac {b e^3 n x^{1+3 r} (f x)^m}{(1+m+3 r)^2}-\frac {b d^3 n (f x)^{1+m}}{f (1+m)^2}+\frac {3 d^2 e x^{1+r} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{1+m+r}+\frac {3 d e^2 x^{1+2 r} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{1+m+2 r}+\frac {e^3 x^{1+3 r} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{1+m+3 r}+\frac {d^3 (f x)^{1+m} \left (a+b \log \left (c x^n\right )\right )}{f (1+m)} \]

[Out]

-3*b*d^2*e*n*x^(1+r)*(f*x)^m/(1+m+r)^2-3*b*d*e^2*n*x^(1+2*r)*(f*x)^m/(1+m+2*r)^2-b*e^3*n*x^(1+3*r)*(f*x)^m/(1+
m+3*r)^2-b*d^3*n*(f*x)^(1+m)/f/(1+m)^2+3*d^2*e*x^(1+r)*(f*x)^m*(a+b*ln(c*x^n))/(1+m+r)+3*d*e^2*x^(1+2*r)*(f*x)
^m*(a+b*ln(c*x^n))/(1+m+2*r)+e^3*x^(1+3*r)*(f*x)^m*(a+b*ln(c*x^n))/(1+m+3*r)+d^3*(f*x)^(1+m)*(a+b*ln(c*x^n))/f
/(1+m)

Rubi [A] (verified)

Time = 1.89 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {276, 20, 30, 2392, 14} \[ \int (f x)^m \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {d^3 (f x)^{m+1} \left (a+b \log \left (c x^n\right )\right )}{f (m+1)}+\frac {3 d^2 e x^{r+1} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{m+r+1}+\frac {3 d e^2 x^{2 r+1} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{m+2 r+1}+\frac {e^3 x^{3 r+1} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{m+3 r+1}-\frac {b d^3 n (f x)^{m+1}}{f (m+1)^2}-\frac {3 b d^2 e n x^{r+1} (f x)^m}{(m+r+1)^2}-\frac {3 b d e^2 n x^{2 r+1} (f x)^m}{(m+2 r+1)^2}-\frac {b e^3 n x^{3 r+1} (f x)^m}{(m+3 r+1)^2} \]

[In]

Int[(f*x)^m*(d + e*x^r)^3*(a + b*Log[c*x^n]),x]

[Out]

(-3*b*d^2*e*n*x^(1 + r)*(f*x)^m)/(1 + m + r)^2 - (3*b*d*e^2*n*x^(1 + 2*r)*(f*x)^m)/(1 + m + 2*r)^2 - (b*e^3*n*
x^(1 + 3*r)*(f*x)^m)/(1 + m + 3*r)^2 - (b*d^3*n*(f*x)^(1 + m))/(f*(1 + m)^2) + (3*d^2*e*x^(1 + r)*(f*x)^m*(a +
b*Log[c*x^n]))/(1 + m + r) + (3*d*e^2*x^(1 + 2*r)*(f*x)^m*(a + b*Log[c*x^n]))/(1 + m + 2*r) + (e^3*x^(1 + 3*r
)*(f*x)^m*(a + b*Log[c*x^n]))/(1 + m + 3*r) + (d^3*(f*x)^(1 + m)*(a + b*Log[c*x^n]))/(f*(1 + m))

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 20

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[b^IntPart[n]*((b*v)^FracPart[n]/(a^IntPart[n]
*(a*v)^FracPart[n])), Int[u*(a*v)^(m + n), x], x] /; FreeQ[{a, b, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n]
&&  !IntegerQ[m + n]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 2392

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = IntHide[(f*x)^m*(d + e*x^r)^q, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[SimplifyIntegrand[u/x,
x], x], x] /; ((EqQ[r, 1] || EqQ[r, 2]) && IntegerQ[m] && IntegerQ[q - 1/2]) || InverseFunctionFreeQ[u, x]] /
; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && IntegerQ[2*q] && ((IntegerQ[m] && IntegerQ[r]) || IGtQ[q, 0])

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 0.30 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.76 \[ \int (f x)^m \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=x (f x)^m \left (-\frac {b d^3 n}{(1+m)^2}-\frac {3 b d^2 e n x^r}{(1+m+r)^2}-\frac {3 b d e^2 n x^{2 r}}{(1+m+2 r)^2}-\frac {b e^3 n x^{3 r}}{(1+m+3 r)^2}+\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{1+m}+\frac {3 d^2 e x^r \left (a+b \log \left (c x^n\right )\right )}{1+m+r}+\frac {3 d e^2 x^{2 r} \left (a+b \log \left (c x^n\right )\right )}{1+m+2 r}+\frac {e^3 x^{3 r} \left (a+b \log \left (c x^n\right )\right )}{1+m+3 r}\right ) \]

[In]

Integrate[(f*x)^m*(d + e*x^r)^3*(a + b*Log[c*x^n]),x]

[Out]

x*(f*x)^m*(-((b*d^3*n)/(1 + m)^2) - (3*b*d^2*e*n*x^r)/(1 + m + r)^2 - (3*b*d*e^2*n*x^(2*r))/(1 + m + 2*r)^2 -
(b*e^3*n*x^(3*r))/(1 + m + 3*r)^2 + (d^3*(a + b*Log[c*x^n]))/(1 + m) + (3*d^2*e*x^r*(a + b*Log[c*x^n]))/(1 + m
+ r) + (3*d*e^2*x^(2*r)*(a + b*Log[c*x^n]))/(1 + m + 2*r) + (e^3*x^(3*r)*(a + b*Log[c*x^n]))/(1 + m + 3*r))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(8182\) vs. \(2(233)=466\).

Time = 57.15 (sec) , antiderivative size = 8183, normalized size of antiderivative = 35.12

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(8183\)
risch	\(\text {Expression too large to display}\)	\(22640\)

[In]

int((f*x)^m*(d+e*x^r)^3*(a+b*ln(c*x^n)),x,method=_RETURNVERBOSE)

[Out]

result too large to display

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4918 vs. \(2 (233) = 466\).

Time = 0.48 (sec) , antiderivative size = 4918, normalized size of antiderivative = 21.11 \[ \int (f x)^m \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Too large to display} \]

[In]

integrate((f*x)^m*(d+e*x^r)^3*(a+b*log(c*x^n)),x, algorithm="fricas")

[Out]

(((b*e^3*m^7 + 7*b*e^3*m^6 + 21*b*e^3*m^5 + 35*b*e^3*m^4 + 35*b*e^3*m^3 + 21*b*e^3*m^2 + 12*(b*e^3*m^2 + 2*b*e
^3*m + b*e^3)*r^5 + 7*b*e^3*m + 40*(b*e^3*m^3 + 3*b*e^3*m^2 + 3*b*e^3*m + b*e^3)*r^4 + b*e^3 + 51*(b*e^3*m^4 +
4*b*e^3*m^3 + 6*b*e^3*m^2 + 4*b*e^3*m + b*e^3)*r^3 + 31*(b*e^3*m^5 + 5*b*e^3*m^4 + 10*b*e^3*m^3 + 10*b*e^3*m^
2 + 5*b*e^3*m + b*e^3)*r^2 + 9*(b*e^3*m^6 + 6*b*e^3*m^5 + 15*b*e^3*m^4 + 20*b*e^3*m^3 + 15*b*e^3*m^2 + 6*b*e^3
*m + b*e^3)*r)*x*log(c) + (12*(b*e^3*m^2 + 2*b*e^3*m + b*e^3)*n*r^5 + 40*(b*e^3*m^3 + 3*b*e^3*m^2 + 3*b*e^3*m
+ b*e^3)*n*r^4 + 51*(b*e^3*m^4 + 4*b*e^3*m^3 + 6*b*e^3*m^2 + 4*b*e^3*m + b*e^3)*n*r^3 + 31*(b*e^3*m^5 + 5*b*e^
3*m^4 + 10*b*e^3*m^3 + 10*b*e^3*m^2 + 5*b*e^3*m + b*e^3)*n*r^2 + 9*(b*e^3*m^6 + 6*b*e^3*m^5 + 15*b*e^3*m^4 + 2
0*b*e^3*m^3 + 15*b*e^3*m^2 + 6*b*e^3*m + b*e^3)*n*r + (b*e^3*m^7 + 7*b*e^3*m^6 + 21*b*e^3*m^5 + 35*b*e^3*m^4 +
35*b*e^3*m^3 + 21*b*e^3*m^2 + 7*b*e^3*m + b*e^3)*n)*x*log(x) + (a*e^3*m^7 + 7*a*e^3*m^6 + 21*a*e^3*m^5 + 35*a
*e^3*m^4 + 35*a*e^3*m^3 + 21*a*e^3*m^2 + 12*(a*e^3*m^2 + 2*a*e^3*m + a*e^3)*r^5 + 7*a*e^3*m + 4*(10*a*e^3*m^3
+ 30*a*e^3*m^2 + 30*a*e^3*m + 10*a*e^3 - (b*e^3*m^2 + 2*b*e^3*m + b*e^3)*n)*r^4 + a*e^3 + 3*(17*a*e^3*m^4 + 68
*a*e^3*m^3 + 102*a*e^3*m^2 + 68*a*e^3*m + 17*a*e^3 - 4*(b*e^3*m^3 + 3*b*e^3*m^2 + 3*b*e^3*m + b*e^3)*n)*r^3 +
(31*a*e^3*m^5 + 155*a*e^3*m^4 + 310*a*e^3*m^3 + 310*a*e^3*m^2 + 155*a*e^3*m + 31*a*e^3 - 13*(b*e^3*m^4 + 4*b*e
^3*m^3 + 6*b*e^3*m^2 + 4*b*e^3*m + b*e^3)*n)*r^2 - (b*e^3*m^6 + 6*b*e^3*m^5 + 15*b*e^3*m^4 + 20*b*e^3*m^3 + 15
*b*e^3*m^2 + 6*b*e^3*m + b*e^3)*n + 3*(3*a*e^3*m^6 + 18*a*e^3*m^5 + 45*a*e^3*m^4 + 60*a*e^3*m^3 + 45*a*e^3*m^2
+ 18*a*e^3*m + 3*a*e^3 - 2*(b*e^3*m^5 + 5*b*e^3*m^4 + 10*b*e^3*m^3 + 10*b*e^3*m^2 + 5*b*e^3*m + b*e^3)*n)*r)*
x)*x^(3*r)*e^(m*log(f) + m*log(x)) + 3*((b*d*e^2*m^7 + 7*b*d*e^2*m^6 + 21*b*d*e^2*m^5 + 35*b*d*e^2*m^4 + 35*b*
d*e^2*m^3 + 21*b*d*e^2*m^2 + 18*(b*d*e^2*m^2 + 2*b*d*e^2*m + b*d*e^2)*r^5 + 7*b*d*e^2*m + 57*(b*d*e^2*m^3 + 3*
b*d*e^2*m^2 + 3*b*d*e^2*m + b*d*e^2)*r^4 + b*d*e^2 + 68*(b*d*e^2*m^4 + 4*b*d*e^2*m^3 + 6*b*d*e^2*m^2 + 4*b*d*e
^2*m + b*d*e^2)*r^3 + 38*(b*d*e^2*m^5 + 5*b*d*e^2*m^4 + 10*b*d*e^2*m^3 + 10*b*d*e^2*m^2 + 5*b*d*e^2*m + b*d*e^
2)*r^2 + 10*(b*d*e^2*m^6 + 6*b*d*e^2*m^5 + 15*b*d*e^2*m^4 + 20*b*d*e^2*m^3 + 15*b*d*e^2*m^2 + 6*b*d*e^2*m + b*
d*e^2)*r)*x*log(c) + (18*(b*d*e^2*m^2 + 2*b*d*e^2*m + b*d*e^2)*n*r^5 + 57*(b*d*e^2*m^3 + 3*b*d*e^2*m^2 + 3*b*d
*e^2*m + b*d*e^2)*n*r^4 + 68*(b*d*e^2*m^4 + 4*b*d*e^2*m^3 + 6*b*d*e^2*m^2 + 4*b*d*e^2*m + b*d*e^2)*n*r^3 + 38*
(b*d*e^2*m^5 + 5*b*d*e^2*m^4 + 10*b*d*e^2*m^3 + 10*b*d*e^2*m^2 + 5*b*d*e^2*m + b*d*e^2)*n*r^2 + 10*(b*d*e^2*m^
6 + 6*b*d*e^2*m^5 + 15*b*d*e^2*m^4 + 20*b*d*e^2*m^3 + 15*b*d*e^2*m^2 + 6*b*d*e^2*m + b*d*e^2)*n*r + (b*d*e^2*m
^7 + 7*b*d*e^2*m^6 + 21*b*d*e^2*m^5 + 35*b*d*e^2*m^4 + 35*b*d*e^2*m^3 + 21*b*d*e^2*m^2 + 7*b*d*e^2*m + b*d*e^2
)*n)*x*log(x) + (a*d*e^2*m^7 + 7*a*d*e^2*m^6 + 21*a*d*e^2*m^5 + 35*a*d*e^2*m^4 + 35*a*d*e^2*m^3 + 21*a*d*e^2*m
^2 + 18*(a*d*e^2*m^2 + 2*a*d*e^2*m + a*d*e^2)*r^5 + 7*a*d*e^2*m + 3*(19*a*d*e^2*m^3 + 57*a*d*e^2*m^2 + 57*a*d*
e^2*m + 19*a*d*e^2 - 3*(b*d*e^2*m^2 + 2*b*d*e^2*m + b*d*e^2)*n)*r^4 + a*d*e^2 + 4*(17*a*d*e^2*m^4 + 68*a*d*e^2
*m^3 + 102*a*d*e^2*m^2 + 68*a*d*e^2*m + 17*a*d*e^2 - 6*(b*d*e^2*m^3 + 3*b*d*e^2*m^2 + 3*b*d*e^2*m + b*d*e^2)*n
)*r^3 + 2*(19*a*d*e^2*m^5 + 95*a*d*e^2*m^4 + 190*a*d*e^2*m^3 + 190*a*d*e^2*m^2 + 95*a*d*e^2*m + 19*a*d*e^2 - 1
1*(b*d*e^2*m^4 + 4*b*d*e^2*m^3 + 6*b*d*e^2*m^2 + 4*b*d*e^2*m + b*d*e^2)*n)*r^2 - (b*d*e^2*m^6 + 6*b*d*e^2*m^5
+ 15*b*d*e^2*m^4 + 20*b*d*e^2*m^3 + 15*b*d*e^2*m^2 + 6*b*d*e^2*m + b*d*e^2)*n + 2*(5*a*d*e^2*m^6 + 30*a*d*e^2*
m^5 + 75*a*d*e^2*m^4 + 100*a*d*e^2*m^3 + 75*a*d*e^2*m^2 + 30*a*d*e^2*m + 5*a*d*e^2 - 4*(b*d*e^2*m^5 + 5*b*d*e^
2*m^4 + 10*b*d*e^2*m^3 + 10*b*d*e^2*m^2 + 5*b*d*e^2*m + b*d*e^2)*n)*r)*x)*x^(2*r)*e^(m*log(f) + m*log(x)) + 3*
((b*d^2*e*m^7 + 7*b*d^2*e*m^6 + 21*b*d^2*e*m^5 + 35*b*d^2*e*m^4 + 35*b*d^2*e*m^3 + 21*b*d^2*e*m^2 + 36*(b*d^2*
e*m^2 + 2*b*d^2*e*m + b*d^2*e)*r^5 + 7*b*d^2*e*m + 96*(b*d^2*e*m^3 + 3*b*d^2*e*m^2 + 3*b*d^2*e*m + b*d^2*e)*r^
4 + b*d^2*e + 97*(b*d^2*e*m^4 + 4*b*d^2*e*m^3 + 6*b*d^2*e*m^2 + 4*b*d^2*e*m + b*d^2*e)*r^3 + 47*(b*d^2*e*m^5 +
5*b*d^2*e*m^4 + 10*b*d^2*e*m^3 + 10*b*d^2*e*m^2 + 5*b*d^2*e*m + b*d^2*e)*r^2 + 11*(b*d^2*e*m^6 + 6*b*d^2*e*m^
5 + 15*b*d^2*e*m^4 + 20*b*d^2*e*m^3 + 15*b*d^2*e*m^2 + 6*b*d^2*e*m + b*d^2*e)*r)*x*log(c) + (36*(b*d^2*e*m^2 +
2*b*d^2*e*m + b*d^2*e)*n*r^5 + 96*(b*d^2*e*m^3 + 3*b*d^2*e*m^2 + 3*b*d^2*e*m + b*d^2*e)*n*r^4 + 97*(b*d^2*e*m
^4 + 4*b*d^2*e*m^3 + 6*b*d^2*e*m^2 + 4*b*d^2*e*m + b*d^2*e)*n*r^3 + 47*(b*d^2*e*m^5 + 5*b*d^2*e*m^4 + 10*b*d^2
*e*m^3 + 10*b*d^2*e*m^2 + 5*b*d^2*e*m + b*d^2*e)*n*r^2 + 11*(b*d^2*e*m^6 + 6*b*d^2*e*m^5 + 15*b*d^2*e*m^4 + 20
*b*d^2*e*m^3 + 15*b*d^2*e*m^2 + 6*b*d^2*e*m + b*d^2*e)*n*r + (b*d^2*e*m^7 + 7*b*d^2*e*m^6 + 21*b*d^2*e*m^5 + 3
5*b*d^2*e*m^4 + 35*b*d^2*e*m^3 + 21*b*d^2*e*m^2 + 7*b*d^2*e*m + b*d^2*e)*n)*x*log(x) + (a*d^2*e*m^7 + 7*a*d^2*
e*m^6 + 21*a*d^2*e*m^5 + 35*a*d^2*e*m^4 + 35*a*d^2*e*m^3 + 21*a*d^2*e*m^2 + 36*(a*d^2*e*m^2 + 2*a*d^2*e*m + a*
d^2*e)*r^5 + 7*a*d^2*e*m + 12*(8*a*d^2*e*m^3 + 24*a*d^2*e*m^2 + 24*a*d^2*e*m + 8*a*d^2*e - 3*(b*d^2*e*m^2 + 2*
b*d^2*e*m + b*d^2*e)*n)*r^4 + a*d^2*e + (97*a*d^2*e*m^4 + 388*a*d^2*e*m^3 + 582*a*d^2*e*m^2 + 388*a*d^2*e*m +
97*a*d^2*e - 60*(b*d^2*e*m^3 + 3*b*d^2*e*m^2 + 3*b*d^2*e*m + b*d^2*e)*n)*r^3 + (47*a*d^2*e*m^5 + 235*a*d^2*e*m
^4 + 470*a*d^2*e*m^3 + 470*a*d^2*e*m^2 + 235*a*d^2*e*m + 47*a*d^2*e - 37*(b*d^2*e*m^4 + 4*b*d^2*e*m^3 + 6*b*d^
2*e*m^2 + 4*b*d^2*e*m + b*d^2*e)*n)*r^2 - (b*d^2*e*m^6 + 6*b*d^2*e*m^5 + 15*b*d^2*e*m^4 + 20*b*d^2*e*m^3 + 15*
b*d^2*e*m^2 + 6*b*d^2*e*m + b*d^2*e)*n + (11*a*d^2*e*m^6 + 66*a*d^2*e*m^5 + 165*a*d^2*e*m^4 + 220*a*d^2*e*m^3
+ 165*a*d^2*e*m^2 + 66*a*d^2*e*m + 11*a*d^2*e - 10*(b*d^2*e*m^5 + 5*b*d^2*e*m^4 + 10*b*d^2*e*m^3 + 10*b*d^2*e*
m^2 + 5*b*d^2*e*m + b*d^2*e)*n)*r)*x)*x^r*e^(m*log(f) + m*log(x)) + ((b*d^3*m^7 + 7*b*d^3*m^6 + 21*b*d^3*m^5 +
35*b*d^3*m^4 + 35*b*d^3*m^3 + 36*(b*d^3*m + b*d^3)*r^6 + 21*b*d^3*m^2 + 132*(b*d^3*m^2 + 2*b*d^3*m + b*d^3)*r
^5 + 7*b*d^3*m + 193*(b*d^3*m^3 + 3*b*d^3*m^2 + 3*b*d^3*m + b*d^3)*r^4 + b*d^3 + 144*(b*d^3*m^4 + 4*b*d^3*m^3
+ 6*b*d^3*m^2 + 4*b*d^3*m + b*d^3)*r^3 + 58*(b*d^3*m^5 + 5*b*d^3*m^4 + 10*b*d^3*m^3 + 10*b*d^3*m^2 + 5*b*d^3*m
+ b*d^3)*r^2 + 12*(b*d^3*m^6 + 6*b*d^3*m^5 + 15*b*d^3*m^4 + 20*b*d^3*m^3 + 15*b*d^3*m^2 + 6*b*d^3*m + b*d^3)*
r)*x*log(c) + (36*(b*d^3*m + b*d^3)*n*r^6 + 132*(b*d^3*m^2 + 2*b*d^3*m + b*d^3)*n*r^5 + 193*(b*d^3*m^3 + 3*b*d
^3*m^2 + 3*b*d^3*m + b*d^3)*n*r^4 + 144*(b*d^3*m^4 + 4*b*d^3*m^3 + 6*b*d^3*m^2 + 4*b*d^3*m + b*d^3)*n*r^3 + 58
*(b*d^3*m^5 + 5*b*d^3*m^4 + 10*b*d^3*m^3 + 10*b*d^3*m^2 + 5*b*d^3*m + b*d^3)*n*r^2 + 12*(b*d^3*m^6 + 6*b*d^3*m
^5 + 15*b*d^3*m^4 + 20*b*d^3*m^3 + 15*b*d^3*m^2 + 6*b*d^3*m + b*d^3)*n*r + (b*d^3*m^7 + 7*b*d^3*m^6 + 21*b*d^3
*m^5 + 35*b*d^3*m^4 + 35*b*d^3*m^3 + 21*b*d^3*m^2 + 7*b*d^3*m + b*d^3)*n)*x*log(x) + (a*d^3*m^7 + 7*a*d^3*m^6
+ 21*a*d^3*m^5 + 35*a*d^3*m^4 + 35*a*d^3*m^3 + 36*(a*d^3*m - b*d^3*n + a*d^3)*r^6 + 21*a*d^3*m^2 + 132*(a*d^3*
m^2 + 2*a*d^3*m + a*d^3 - (b*d^3*m + b*d^3)*n)*r^5 + 7*a*d^3*m + 193*(a*d^3*m^3 + 3*a*d^3*m^2 + 3*a*d^3*m + a*
d^3 - (b*d^3*m^2 + 2*b*d^3*m + b*d^3)*n)*r^4 + a*d^3 + 144*(a*d^3*m^4 + 4*a*d^3*m^3 + 6*a*d^3*m^2 + 4*a*d^3*m
+ a*d^3 - (b*d^3*m^3 + 3*b*d^3*m^2 + 3*b*d^3*m + b*d^3)*n)*r^3 + 58*(a*d^3*m^5 + 5*a*d^3*m^4 + 10*a*d^3*m^3 +
10*a*d^3*m^2 + 5*a*d^3*m + a*d^3 - (b*d^3*m^4 + 4*b*d^3*m^3 + 6*b*d^3*m^2 + 4*b*d^3*m + b*d^3)*n)*r^2 - (b*d^3
*m^6 + 6*b*d^3*m^5 + 15*b*d^3*m^4 + 20*b*d^3*m^3 + 15*b*d^3*m^2 + 6*b*d^3*m + b*d^3)*n + 12*(a*d^3*m^6 + 6*a*d
^3*m^5 + 15*a*d^3*m^4 + 20*a*d^3*m^3 + 15*a*d^3*m^2 + 6*a*d^3*m + a*d^3 - (b*d^3*m^5 + 5*b*d^3*m^4 + 10*b*d^3*
m^3 + 10*b*d^3*m^2 + 5*b*d^3*m + b*d^3)*n)*r)*x)*e^(m*log(f) + m*log(x)))/(m^8 + 8*m^7 + 36*(m^2 + 2*m + 1)*r^
6 + 28*m^6 + 132*(m^3 + 3*m^2 + 3*m + 1)*r^5 + 56*m^5 + 193*(m^4 + 4*m^3 + 6*m^2 + 4*m + 1)*r^4 + 70*m^4 + 144
*(m^5 + 5*m^4 + 10*m^3 + 10*m^2 + 5*m + 1)*r^3 + 56*m^3 + 58*(m^6 + 6*m^5 + 15*m^4 + 20*m^3 + 15*m^2 + 6*m + 1
)*r^2 + 28*m^2 + 12*(m^7 + 7*m^6 + 21*m^5 + 35*m^4 + 35*m^3 + 21*m^2 + 7*m + 1)*r + 8*m + 1)

Sympy [F(-2)]

Exception generated. \[ \int (f x)^m \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((f*x)**m*(d+e*x**r)**3*(a+b*ln(c*x**n)),x)

[Out]

Exception raised: TypeError >> Invalid comparison of non-real zoo

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.47 \[ \int (f x)^m \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {b e^{3} f^{m} x e^{\left (m \log \left (x\right ) + 3 \, r \log \left (x\right )\right )} \log \left (c x^{n}\right )}{m + 3 \, r + 1} + \frac {3 \, b d e^{2} f^{m} x e^{\left (m \log \left (x\right ) + 2 \, r \log \left (x\right )\right )} \log \left (c x^{n}\right )}{m + 2 \, r + 1} + \frac {3 \, b d^{2} e f^{m} x e^{\left (m \log \left (x\right ) + r \log \left (x\right )\right )} \log \left (c x^{n}\right )}{m + r + 1} - \frac {b d^{3} f^{m} n x x^{m}}{{\left (m + 1\right )}^{2}} + \frac {a e^{3} f^{m} x e^{\left (m \log \left (x\right ) + 3 \, r \log \left (x\right )\right )}}{m + 3 \, r + 1} - \frac {b e^{3} f^{m} n x e^{\left (m \log \left (x\right ) + 3 \, r \log \left (x\right )\right )}}{{\left (m + 3 \, r + 1\right )}^{2}} + \frac {3 \, a d e^{2} f^{m} x e^{\left (m \log \left (x\right ) + 2 \, r \log \left (x\right )\right )}}{m + 2 \, r + 1} - \frac {3 \, b d e^{2} f^{m} n x e^{\left (m \log \left (x\right ) + 2 \, r \log \left (x\right )\right )}}{{\left (m + 2 \, r + 1\right )}^{2}} + \frac {3 \, a d^{2} e f^{m} x e^{\left (m \log \left (x\right ) + r \log \left (x\right )\right )}}{m + r + 1} - \frac {3 \, b d^{2} e f^{m} n x e^{\left (m \log \left (x\right ) + r \log \left (x\right )\right )}}{{\left (m + r + 1\right )}^{2}} + \frac {\left (f x\right )^{m + 1} b d^{3} \log \left (c x^{n}\right )}{f {\left (m + 1\right )}} + \frac {\left (f x\right )^{m + 1} a d^{3}}{f {\left (m + 1\right )}} \]

[In]

integrate((f*x)^m*(d+e*x^r)^3*(a+b*log(c*x^n)),x, algorithm="maxima")

[Out]

b*e^3*f^m*x*e^(m*log(x) + 3*r*log(x))*log(c*x^n)/(m + 3*r + 1) + 3*b*d*e^2*f^m*x*e^(m*log(x) + 2*r*log(x))*log
(c*x^n)/(m + 2*r + 1) + 3*b*d^2*e*f^m*x*e^(m*log(x) + r*log(x))*log(c*x^n)/(m + r + 1) - b*d^3*f^m*n*x*x^m/(m
+ 1)^2 + a*e^3*f^m*x*e^(m*log(x) + 3*r*log(x))/(m + 3*r + 1) - b*e^3*f^m*n*x*e^(m*log(x) + 3*r*log(x))/(m + 3*
r + 1)^2 + 3*a*d*e^2*f^m*x*e^(m*log(x) + 2*r*log(x))/(m + 2*r + 1) - 3*b*d*e^2*f^m*n*x*e^(m*log(x) + 2*r*log(x
))/(m + 2*r + 1)^2 + 3*a*d^2*e*f^m*x*e^(m*log(x) + r*log(x))/(m + r + 1) - 3*b*d^2*e*f^m*n*x*e^(m*log(x) + r*l
og(x))/(m + r + 1)^2 + (f*x)^(m + 1)*b*d^3*log(c*x^n)/(f*(m + 1)) + (f*x)^(m + 1)*a*d^3/(f*(m + 1))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 772 vs. \(2 (233) = 466\).

Time = 0.37 (sec) , antiderivative size = 772, normalized size of antiderivative = 3.31 \[ \int (f x)^m \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {b e^{3} f^{m} m n x x^{m} x^{3 \, r} \log \left (x\right )}{m^{2} + 6 \, m r + 9 \, r^{2} + 2 \, m + 6 \, r + 1} + \frac {3 \, b e^{3} f^{m} n r x x^{m} x^{3 \, r} \log \left (x\right )}{m^{2} + 6 \, m r + 9 \, r^{2} + 2 \, m + 6 \, r + 1} + \frac {3 \, b d e^{2} f^{m} m n x x^{m} x^{2 \, r} \log \left (x\right )}{m^{2} + 4 \, m r + 4 \, r^{2} + 2 \, m + 4 \, r + 1} + \frac {6 \, b d e^{2} f^{m} n r x x^{m} x^{2 \, r} \log \left (x\right )}{m^{2} + 4 \, m r + 4 \, r^{2} + 2 \, m + 4 \, r + 1} + \frac {3 \, b d^{2} e f^{m} m n x x^{m} x^{r} \log \left (x\right )}{m^{2} + 2 \, m r + r^{2} + 2 \, m + 2 \, r + 1} + \frac {3 \, b d^{2} e f^{m} n r x x^{m} x^{r} \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 {b e^{3} f^{m} n x x^{m} x^{3 \, r} \log \left (x\right )}{m^{2} + 6 \, m r + 9 \, r^{2} + 2 \, m + 6 \, r + 1} + \frac {3 \, b d e^{2} f^{m} n x x^{m} x^{2 \, r} \log \left (x\right )}{m^{2} + 4 \, m r + 4 \, r^{2} + 2 \, m + 4 \, r + 1} + \frac {3 \, b d^{2} e f^{m} n x x^{m} x^{r} \log \left (x\right )}{m^{2} + 2 \, m r + r^{2} + 2 \, m + 2 \, r + 1} - \frac {b e^{3} f^{m} n x x^{m} x^{3 \, r}}{m^{2} + 6 \, m r + 9 \, r^{2} + 2 \, m + 6 \, r + 1} - \frac {3 \, b d e^{2} f^{m} n x x^{m} x^{2 \, r}}{m^{2} + 4 \, m r + 4 \, r^{2} + 2 \, m + 4 \, r + 1} - \frac {3 \, b d^{2} e f^{m} n x x^{m} x^{r}}{m^{2} + 2 \, m r + r^{2} + 2 \, m + 2 \, r + 1} + \frac {b e^{3} f^{m} x x^{m} x^{3 \, r} \log \left (c\right )}{m + 3 \, r + 1} + \frac {3 \, b d e^{2} f^{m} x x^{m} x^{2 \, r} \log \left (c\right )}{m + 2 \, r + 1} + \frac {3 \, b d^{2} e f^{m} x x^{m} x^{r} \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 d^{3} f^{m} n x x^{m}}{m^{2} + 2 \, m + 1} + \frac {a e^{3} f^{m} x x^{m} x^{3 \, r}}{m + 3 \, r + 1} + \frac {3 \, a d e^{2} f^{m} x x^{m} x^{2 \, r}}{m + 2 \, r + 1} + \frac {3 \, a d^{2} e f^{m} x x^{m} x^{r}}{m + r + 1} + \frac {\left (f x\right )^{m} b d^{3} x \log \left (c\right )}{m + 1} + \frac {\left (f x\right )^{m} a d^{3} x}{m + 1} \]

[In]

integrate((f*x)^m*(d+e*x^r)^3*(a+b*log(c*x^n)),x, algorithm="giac")

[Out]

b*e^3*f^m*m*n*x*x^m*x^(3*r)*log(x)/(m^2 + 6*m*r + 9*r^2 + 2*m + 6*r + 1) + 3*b*e^3*f^m*n*r*x*x^m*x^(3*r)*log(x
)/(m^2 + 6*m*r + 9*r^2 + 2*m + 6*r + 1) + 3*b*d*e^2*f^m*m*n*x*x^m*x^(2*r)*log(x)/(m^2 + 4*m*r + 4*r^2 + 2*m +
4*r + 1) + 6*b*d*e^2*f^m*n*r*x*x^m*x^(2*r)*log(x)/(m^2 + 4*m*r + 4*r^2 + 2*m + 4*r + 1) + 3*b*d^2*e*f^m*m*n*x*
x^m*x^r*log(x)/(m^2 + 2*m*r + r^2 + 2*m + 2*r + 1) + 3*b*d^2*e*f^m*n*r*x*x^m*x^r*log(x)/(m^2 + 2*m*r + r^2 + 2
*m + 2*r + 1) + b*d^3*f^m*m*n*x*x^m*log(x)/(m^2 + 2*m + 1) + b*e^3*f^m*n*x*x^m*x^(3*r)*log(x)/(m^2 + 6*m*r + 9
*r^2 + 2*m + 6*r + 1) + 3*b*d*e^2*f^m*n*x*x^m*x^(2*r)*log(x)/(m^2 + 4*m*r + 4*r^2 + 2*m + 4*r + 1) + 3*b*d^2*e
*f^m*n*x*x^m*x^r*log(x)/(m^2 + 2*m*r + r^2 + 2*m + 2*r + 1) - b*e^3*f^m*n*x*x^m*x^(3*r)/(m^2 + 6*m*r + 9*r^2 +
2*m + 6*r + 1) - 3*b*d*e^2*f^m*n*x*x^m*x^(2*r)/(m^2 + 4*m*r + 4*r^2 + 2*m + 4*r + 1) - 3*b*d^2*e*f^m*n*x*x^m*
x^r/(m^2 + 2*m*r + r^2 + 2*m + 2*r + 1) + b*e^3*f^m*x*x^m*x^(3*r)*log(c)/(m + 3*r + 1) + 3*b*d*e^2*f^m*x*x^m*x
^(2*r)*log(c)/(m + 2*r + 1) + 3*b*d^2*e*f^m*x*x^m*x^r*log(c)/(m + r + 1) + b*d^3*f^m*n*x*x^m*log(x)/(m^2 + 2*m
+ 1) - b*d^3*f^m*n*x*x^m/(m^2 + 2*m + 1) + a*e^3*f^m*x*x^m*x^(3*r)/(m + 3*r + 1) + 3*a*d*e^2*f^m*x*x^m*x^(2*r
)/(m + 2*r + 1) + 3*a*d^2*e*f^m*x*x^m*x^r/(m + r + 1) + (f*x)^m*b*d^3*x*log(c)/(m + 1) + (f*x)^m*a*d^3*x/(m +
1)

Mupad [F(-1)]

Timed out. \[ \int (f x)^m \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\int {\left (f\,x\right )}^m\,{\left (d+e\,x^r\right )}^3\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]

[In]

int((f*x)^m*(d + e*x^r)^3*(a + b*log(c*x^n)),x)

[Out]

int((f*x)^m*(d + e*x^r)^3*(a + b*log(c*x^n)), x)