\(\int (e x)^m (A+B x^n) (c+d x^n)^2 \, dx\) [11]
Optimal result
Integrand size = 22, antiderivative size = 102 \[
\int (e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )^2 \, dx=\frac {c (B c+2 A d) x^{1+n} (e x)^m}{1+m+n}+\frac {d (2 B c+A d) x^{1+2 n} (e x)^m}{1+m+2 n}+\frac {B d^2 x^{1+3 n} (e x)^m}{1+m+3 n}+\frac {A c^2 (e x)^{1+m}}{e (1+m)}
\]
[Out]
c*(2*A*d+B*c)*x^(1+n)*(e*x)^m/(1+m+n)+d*(A*d+2*B*c)*x^(1+2*n)*(e*x)^m/(1+m+2*n)+B*d^2*x^(1+3*n)*(e*x)^m/(1+m+3
*n)+A*c^2*(e*x)^(1+m)/e/(1+m)
Rubi [A] (verified)
Time = 0.05 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number
of steps used = 8, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {459, 20, 30}
\[
\int (e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )^2 \, dx=\frac {c x^{n+1} (e x)^m (2 A d+B c)}{m+n+1}+\frac {d x^{2 n+1} (e x)^m (A d+2 B c)}{m+2 n+1}+\frac {A c^2 (e x)^{m+1}}{e (m+1)}+\frac {B d^2 x^{3 n+1} (e x)^m}{m+3 n+1}
\]
[In]
Int[(e*x)^m*(A + B*x^n)*(c + d*x^n)^2,x]
[Out]
(c*(B*c + 2*A*d)*x^(1 + n)*(e*x)^m)/(1 + m + n) + (d*(2*B*c + A*d)*x^(1 + 2*n)*(e*x)^m)/(1 + m + 2*n) + (B*d^2
*x^(1 + 3*n)*(e*x)^m)/(1 + m + 3*n) + (A*c^2*(e*x)^(1 + m))/(e*(1 + m))
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 459
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]
Rubi steps \begin{align*}
\text {integral}& = \int \left (A c^2 (e x)^m+c (B c+2 A d) x^n (e x)^m+d (2 B c+A d) x^{2 n} (e x)^m+B d^2 x^{3 n} (e x)^m\right ) \, dx \\ & = \frac {A c^2 (e x)^{1+m}}{e (1+m)}+\left (B d^2\right ) \int x^{3 n} (e x)^m \, dx+(d (2 B c+A d)) \int x^{2 n} (e x)^m \, dx+(c (B c+2 A d)) \int x^n (e x)^m \, dx \\ & = \frac {A c^2 (e x)^{1+m}}{e (1+m)}+\left (B d^2 x^{-m} (e x)^m\right ) \int x^{m+3 n} \, dx+\left (d (2 B c+A d) x^{-m} (e x)^m\right ) \int x^{m+2 n} \, dx+\left (c (B c+2 A d) x^{-m} (e x)^m\right ) \int x^{m+n} \, dx \\ & = \frac {c (B c+2 A d) x^{1+n} (e x)^m}{1+m+n}+\frac {d (2 B c+A d) x^{1+2 n} (e x)^m}{1+m+2 n}+\frac {B d^2 x^{1+3 n} (e x)^m}{1+m+3 n}+\frac {A c^2 (e x)^{1+m}}{e (1+m)} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.18 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.76
\[
\int (e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )^2 \, dx=x (e x)^m \left (\frac {A c^2}{1+m}+\frac {c (B c+2 A d) x^n}{1+m+n}+\frac {d (2 B c+A d) x^{2 n}}{1+m+2 n}+\frac {B d^2 x^{3 n}}{1+m+3 n}\right )
\]
[In]
Integrate[(e*x)^m*(A + B*x^n)*(c + d*x^n)^2,x]
[Out]
x*(e*x)^m*((A*c^2)/(1 + m) + (c*(B*c + 2*A*d)*x^n)/(1 + m + n) + (d*(2*B*c + A*d)*x^(2*n))/(1 + m + 2*n) + (B*
d^2*x^(3*n))/(1 + m + 3*n))
Maple [C] (warning: unable to verify)
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 4.92 (sec) , antiderivative size = 699, normalized size of antiderivative = 6.85
| | |
| method | result | size |
| | |
| risch |
\(\frac {x \left (5 B \,c^{2} x^{n} n +16 B c d m n \,x^{2 n}+6 B c d m \,n^{2} x^{2 n}+A \,c^{2}+B \,c^{2} m^{3} x^{n}+6 B \,d^{2} m n \,x^{3 n}+3 A \,c^{2} m +6 A \,c^{2} n +3 B \,c^{2} m^{2} x^{n}+6 B \,c^{2} n^{2} x^{n}+3 B \,c^{2} x^{n} m +A \,c^{2} m^{3}+12 A \,c^{2} m n +2 A c d \,m^{3} x^{n}+10 A c d \,m^{2} n \,x^{n}+12 A c d m \,n^{2} x^{n}+20 A c d m n \,x^{n}+6 A \,c^{2} n^{3}+3 A \,c^{2} m^{2}+11 A \,c^{2} n^{2}+x^{2 n} A \,d^{2}+x^{n} B \,c^{2}+2 A c d \,x^{n}+6 A \,c^{2} m^{2} n +11 A \,c^{2} m \,n^{2}+5 B \,c^{2} m^{2} n \,x^{n}+6 B \,c^{2} m \,n^{2} x^{n}+6 A c d \,m^{2} x^{n}+8 B c d \,m^{2} n \,x^{2 n}+12 A c d \,n^{2} x^{n}+10 B \,c^{2} m n \,x^{n}+6 A c d \,x^{n} m +10 A c d \,x^{n} n +2 B c d \,x^{2 n}+3 A \,d^{2} x^{2 n} m +3 B \,d^{2} m^{2} x^{3 n}+4 A \,d^{2} x^{2 n} n +3 A \,d^{2} n^{2} x^{2 n}+3 m B \,d^{2} x^{3 n}+3 B \,d^{2} x^{3 n} n +2 B \,d^{2} n^{2} x^{3 n}+B \,d^{2} m^{3} x^{3 n}+3 A \,d^{2} m^{2} x^{2 n}+A \,d^{2} m^{3} x^{2 n}+x^{3 n} B \,d^{2}+8 A \,d^{2} m n \,x^{2 n}+6 B c d \,m^{2} x^{2 n}+3 B \,d^{2} m^{2} n \,x^{3 n}+6 B c d \,n^{2} x^{2 n}+6 B c d \,x^{2 n} m +8 B c d \,x^{2 n} n +3 A \,d^{2} m \,n^{2} x^{2 n}+2 B c d \,m^{3} x^{2 n}+2 B \,d^{2} m \,n^{2} x^{3 n}+4 A \,d^{2} m^{2} n \,x^{2 n}\right ) e^{m} x^{m} {\mathrm e}^{\frac {i \operatorname {csgn}\left (i e x \right ) \pi m \left (\operatorname {csgn}\left (i e x \right )-\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (i e x \right )+\operatorname {csgn}\left (i e \right )\right )}{2}}}{\left (1+m \right ) \left (1+m +n \right ) \left (1+m +2 n \right ) \left (1+m +3 n \right )}\) |
\(699\) |
| parallelrisch |
\(\frac {12 A x \left (e x \right )^{m} c^{2} m n +3 B x \,x^{n} \left (e x \right )^{m} c^{2} m +5 B x \,x^{n} \left (e x \right )^{m} c^{2} n +2 A x \,x^{n} \left (e x \right )^{m} c d +A x \,x^{2 n} \left (e x \right )^{m} d^{2} m^{3}+3 B x \,x^{3 n} \left (e x \right )^{m} d^{2} m^{2}+2 B x \,x^{3 n} \left (e x \right )^{m} d^{2} n^{2}+3 A x \,x^{2 n} \left (e x \right )^{m} d^{2} m^{2}+3 A x \,x^{2 n} \left (e x \right )^{m} d^{2} n^{2}+3 B x \,x^{3 n} \left (e x \right )^{m} d^{2} m +3 B x \,x^{3 n} \left (e x \right )^{m} d^{2} n +3 A x \,x^{2 n} \left (e x \right )^{m} d^{2} m +4 A x \,x^{2 n} \left (e x \right )^{m} d^{2} n +10 B x \,x^{n} \left (e x \right )^{m} c^{2} m n +6 A x \,x^{n} \left (e x \right )^{m} c d \,m^{2}+10 A x \,x^{n} \left (e x \right )^{m} c d n +6 A x \,x^{n} \left (e x \right )^{m} c d m +20 A x \,x^{n} \left (e x \right )^{m} c d m n +12 A x \,x^{n} \left (e x \right )^{m} c d m \,n^{2}+10 A x \,x^{n} \left (e x \right )^{m} c d \,m^{2} n +12 A x \,x^{n} \left (e x \right )^{m} c d \,n^{2}+2 A x \,x^{n} \left (e x \right )^{m} c d \,m^{3}+2 B x \,x^{2 n} \left (e x \right )^{m} c d +B x \,x^{3 n} \left (e x \right )^{m} d^{2} m^{3}+5 B x \,x^{n} \left (e x \right )^{m} c^{2} m^{2} n +6 B x \,x^{n} \left (e x \right )^{m} c^{2} m \,n^{2}+8 B x \,x^{2 n} \left (e x \right )^{m} c d \,m^{2} n +16 B x \,x^{2 n} \left (e x \right )^{m} c d m n +6 B x \,x^{2 n} \left (e x \right )^{m} c d m \,n^{2}+B x \,x^{n} \left (e x \right )^{m} c^{2} m^{3}+6 A x \left (e x \right )^{m} c^{2} m^{2} n +11 A x \left (e x \right )^{m} c^{2} m \,n^{2}+3 B x \,x^{n} \left (e x \right )^{m} c^{2} m^{2}+6 B x \,x^{n} \left (e x \right )^{m} c^{2} n^{2}+6 B x \,x^{2 n} \left (e x \right )^{m} c d m +6 B x \,x^{2 n} \left (e x \right )^{m} c d \,n^{2}+8 B x \,x^{2 n} \left (e x \right )^{m} c d n +2 B x \,x^{3 n} \left (e x \right )^{m} d^{2} m \,n^{2}+4 A x \,x^{2 n} \left (e x \right )^{m} d^{2} m^{2} n +3 A x \,x^{2 n} \left (e x \right )^{m} d^{2} m \,n^{2}+6 B x \,x^{3 n} \left (e x \right )^{m} d^{2} m n +2 B x \,x^{2 n} \left (e x \right )^{m} c d \,m^{3}+8 A x \,x^{2 n} \left (e x \right )^{m} d^{2} m n +6 B x \,x^{2 n} \left (e x \right )^{m} c d \,m^{2}+3 B x \,x^{3 n} \left (e x \right )^{m} d^{2} m^{2} n +A x \left (e x \right )^{m} c^{2}+A x \left (e x \right )^{m} c^{2} m^{3}+6 A x \left (e x \right )^{m} c^{2} n^{3}+3 A x \left (e x \right )^{m} c^{2} m^{2}+11 A x \left (e x \right )^{m} c^{2} n^{2}+3 A x \left (e x \right )^{m} c^{2} m +6 A x \left (e x \right )^{m} c^{2} n +B x \,x^{n} \left (e x \right )^{m} c^{2}+B x \,x^{3 n} \left (e x \right )^{m} d^{2}+A x \,x^{2 n} \left (e x \right )^{m} d^{2}}{\left (1+m \right ) \left (1+m +n \right ) \left (1+m +2 n \right ) \left (1+m +3 n \right )}\) |
\(982\) |
| | |
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[In]
int((e*x)^m*(A+B*x^n)*(c+d*x^n)^2,x,method=_RETURNVERBOSE)
[Out]
x*(16*B*c*d*m*n*(x^n)^2+5*B*c^2*x^n*n+2*B*c*d*(x^n)^2+3*A*d^2*(x^n)^2*m+6*B*c*d*m*n^2*(x^n)^2+A*c^2+B*c^2*m^3*
x^n+3*B*d^2*m^2*(x^n)^3+8*B*c*d*m^2*n*(x^n)^2+3*A*c^2*m+6*A*c^2*n+4*A*d^2*(x^n)^2*n+3*B*c^2*m^2*x^n+6*B*c^2*n^
2*x^n+3*B*c^2*x^n*m+A*c^2*m^3+12*A*c^2*m*n+6*B*d^2*m*n*(x^n)^3+2*A*c*d*m^3*x^n+8*A*d^2*m*n*(x^n)^2+10*A*c*d*m^
2*n*x^n+12*A*c*d*m*n^2*x^n+20*A*c*d*m*n*x^n+6*A*c^2*n^3+3*A*c^2*m^2+11*A*c^2*n^2+3*A*d^2*n^2*(x^n)^2+3*m*B*d^2
*(x^n)^3+3*B*d^2*(x^n)^3*n+(x^n)^2*A*d^2+x^n*B*c^2+2*B*d^2*n^2*(x^n)^3+(x^n)^3*B*d^2+2*A*c*d*x^n+6*A*c^2*m^2*n
+11*A*c^2*m*n^2+5*B*c^2*m^2*n*x^n+6*B*c^2*m*n^2*x^n+6*B*c*d*m^2*(x^n)^2+3*B*d^2*m^2*n*(x^n)^3+B*d^2*m^3*(x^n)^
3+3*A*d^2*m^2*(x^n)^2+6*B*c*d*n^2*(x^n)^2+6*A*c*d*m^2*x^n+A*d^2*m^3*(x^n)^2+12*A*c*d*n^2*x^n+10*B*c^2*m*n*x^n+
6*B*c*d*(x^n)^2*m+8*B*c*d*(x^n)^2*n+3*A*d^2*m*n^2*(x^n)^2+2*B*c*d*m^3*(x^n)^2+6*A*c*d*x^n*m+10*A*c*d*x^n*n+2*B
*d^2*m*n^2*(x^n)^3+4*A*d^2*m^2*n*(x^n)^2)/(1+m)/(1+m+n)/(1+m+2*n)/(1+m+3*n)*e^m*x^m*exp(1/2*I*csgn(I*e*x)*Pi*m
*(csgn(I*e*x)-csgn(I*x))*(-csgn(I*e*x)+csgn(I*e)))
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 527 vs. \(2 (102) = 204\).
Time = 0.28 (sec) , antiderivative size = 527, normalized size of antiderivative = 5.17
\[
\int (e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )^2 \, dx=\frac {{\left (B d^{2} m^{3} + 3 \, B d^{2} m^{2} + 3 \, B d^{2} m + B d^{2} + 2 \, {\left (B d^{2} m + B d^{2}\right )} n^{2} + 3 \, {\left (B d^{2} m^{2} + 2 \, B d^{2} m + B d^{2}\right )} n\right )} x x^{3 \, n} e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )} + {\left ({\left (2 \, B c d + A d^{2}\right )} m^{3} + 2 \, B c d + A d^{2} + 3 \, {\left (2 \, B c d + A d^{2}\right )} m^{2} + 3 \, {\left (2 \, B c d + A d^{2} + {\left (2 \, B c d + A d^{2}\right )} m\right )} n^{2} + 3 \, {\left (2 \, B c d + A d^{2}\right )} m + 4 \, {\left (2 \, B c d + A d^{2} + {\left (2 \, B c d + A d^{2}\right )} m^{2} + 2 \, {\left (2 \, B c d + A d^{2}\right )} m\right )} n\right )} x x^{2 \, n} e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )} + {\left ({\left (B c^{2} + 2 \, A c d\right )} m^{3} + B c^{2} + 2 \, A c d + 3 \, {\left (B c^{2} + 2 \, A c d\right )} m^{2} + 6 \, {\left (B c^{2} + 2 \, A c d + {\left (B c^{2} + 2 \, A c d\right )} m\right )} n^{2} + 3 \, {\left (B c^{2} + 2 \, A c d\right )} m + 5 \, {\left (B c^{2} + 2 \, A c d + {\left (B c^{2} + 2 \, A c d\right )} m^{2} + 2 \, {\left (B c^{2} + 2 \, A c d\right )} m\right )} n\right )} x x^{n} e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )} + {\left (A c^{2} m^{3} + 6 \, A c^{2} n^{3} + 3 \, A c^{2} m^{2} + 3 \, A c^{2} m + A c^{2} + 11 \, {\left (A c^{2} m + A c^{2}\right )} n^{2} + 6 \, {\left (A c^{2} m^{2} + 2 \, A c^{2} m + A c^{2}\right )} n\right )} x e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )}}{m^{4} + 6 \, {\left (m + 1\right )} n^{3} + 4 \, m^{3} + 11 \, {\left (m^{2} + 2 \, m + 1\right )} n^{2} + 6 \, m^{2} + 6 \, {\left (m^{3} + 3 \, m^{2} + 3 \, m + 1\right )} n + 4 \, m + 1}
\]
[In]
integrate((e*x)^m*(A+B*x^n)*(c+d*x^n)^2,x, algorithm="fricas")
[Out]
((B*d^2*m^3 + 3*B*d^2*m^2 + 3*B*d^2*m + B*d^2 + 2*(B*d^2*m + B*d^2)*n^2 + 3*(B*d^2*m^2 + 2*B*d^2*m + B*d^2)*n)
*x*x^(3*n)*e^(m*log(e) + m*log(x)) + ((2*B*c*d + A*d^2)*m^3 + 2*B*c*d + A*d^2 + 3*(2*B*c*d + A*d^2)*m^2 + 3*(2
*B*c*d + A*d^2 + (2*B*c*d + A*d^2)*m)*n^2 + 3*(2*B*c*d + A*d^2)*m + 4*(2*B*c*d + A*d^2 + (2*B*c*d + A*d^2)*m^2
+ 2*(2*B*c*d + A*d^2)*m)*n)*x*x^(2*n)*e^(m*log(e) + m*log(x)) + ((B*c^2 + 2*A*c*d)*m^3 + B*c^2 + 2*A*c*d + 3*
(B*c^2 + 2*A*c*d)*m^2 + 6*(B*c^2 + 2*A*c*d + (B*c^2 + 2*A*c*d)*m)*n^2 + 3*(B*c^2 + 2*A*c*d)*m + 5*(B*c^2 + 2*A
*c*d + (B*c^2 + 2*A*c*d)*m^2 + 2*(B*c^2 + 2*A*c*d)*m)*n)*x*x^n*e^(m*log(e) + m*log(x)) + (A*c^2*m^3 + 6*A*c^2*
n^3 + 3*A*c^2*m^2 + 3*A*c^2*m + A*c^2 + 11*(A*c^2*m + A*c^2)*n^2 + 6*(A*c^2*m^2 + 2*A*c^2*m + A*c^2)*n)*x*e^(m
*log(e) + m*log(x)))/(m^4 + 6*(m + 1)*n^3 + 4*m^3 + 11*(m^2 + 2*m + 1)*n^2 + 6*m^2 + 6*(m^3 + 3*m^2 + 3*m + 1)
*n + 4*m + 1)
Sympy [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 5882 vs. \(2 (94) = 188\).
Time = 3.13 (sec) , antiderivative size = 5882, normalized size of antiderivative = 57.67
\[
\int (e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )^2 \, dx=\text {Too large to display}
\]
[In]
integrate((e*x)**m*(A+B*x**n)*(c+d*x**n)**2,x)
[Out]
Piecewise(((A + B)*(c + d)**2*log(x)/e, Eq(m, -1) & Eq(n, 0)), ((A*c**2*log(x) + 2*A*c*d*x**n/n + A*d**2*x**(2
*n)/(2*n) + B*c**2*x**n/n + B*c*d*x**(2*n)/n + B*d**2*x**(3*n)/(3*n))/e, Eq(m, -1)), (A*c**2*Piecewise((0**(-3
*n - 1)*x, Eq(e, 0)), (Piecewise((-1/(3*n*(e*x)**(3*n)), Ne(n, 0)), (log(e*x), True))/e, True)) + 2*A*c*d*Piec
ewise((-x*x**n*(e*x)**(-3*n - 1)/(2*n), Ne(n, 0)), (x*x**n*(e*x)**(-3*n - 1)*log(x), True)) + A*d**2*Piecewise
((-x*x**(2*n)*(e*x)**(-3*n - 1)/n, Ne(n, 0)), (x*x**(2*n)*(e*x)**(-3*n - 1)*log(x), True)) + B*c**2*Piecewise(
(-x*x**n*(e*x)**(-3*n - 1)/(2*n), Ne(n, 0)), (x*x**n*(e*x)**(-3*n - 1)*log(x), True)) + 2*B*c*d*Piecewise((-x*
x**(2*n)*(e*x)**(-3*n - 1)/n, Ne(n, 0)), (x*x**(2*n)*(e*x)**(-3*n - 1)*log(x), True)) + B*d**2*x*x**(3*n)*(e*x
)**(-3*n - 1)*log(x), Eq(m, -3*n - 1)), (A*c**2*Piecewise((0**(-2*n - 1)*x, Eq(e, 0)), (Piecewise((-1/(2*n*(e*
x)**(2*n)), Ne(n, 0)), (log(e*x), True))/e, True)) + 2*A*c*d*Piecewise((-x*x**n*(e*x)**(-2*n - 1)/n, Ne(n, 0))
, (x*x**n*(e*x)**(-2*n - 1)*log(x), True)) + A*d**2*x*x**(2*n)*(e*x)**(-2*n - 1)*log(x) + B*c**2*Piecewise((-x
*x**n*(e*x)**(-2*n - 1)/n, Ne(n, 0)), (x*x**n*(e*x)**(-2*n - 1)*log(x), True)) + 2*B*c*d*x*x**(2*n)*(e*x)**(-2
*n - 1)*log(x) + B*d**2*Piecewise((x*x**(3*n)*(e*x)**(-2*n - 1)/n, Ne(n, 0)), (x*x**(3*n)*(e*x)**(-2*n - 1)*lo
g(x), True)), Eq(m, -2*n - 1)), (A*c**2*Piecewise((0**(-n - 1)*x, Eq(e, 0)), (Piecewise((-1/(n*(e*x)**n), Ne(n
, 0)), (log(e*x), True))/e, True)) + 2*A*c*d*x*x**n*(e*x)**(-n - 1)*log(x) + A*d**2*Piecewise((x*x**(2*n)*(e*x
)**(-n - 1)/n, Ne(n, 0)), (x*x**(2*n)*(e*x)**(-n - 1)*log(x), True)) + B*c**2*x*x**n*(e*x)**(-n - 1)*log(x) +
2*B*c*d*Piecewise((x*x**(2*n)*(e*x)**(-n - 1)/n, Ne(n, 0)), (x*x**(2*n)*(e*x)**(-n - 1)*log(x), True)) + B*d**
2*Piecewise((x*x**(3*n)*(e*x)**(-n - 1)/(2*n), Ne(n, 0)), (x*x**(3*n)*(e*x)**(-n - 1)*log(x), True)), Eq(m, -n
- 1)), (A*c**2*m**3*x*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*
m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + 6*A*c**2*m**2*n*x*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 1
1*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + 3*A*c**
2*m**2*x*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*
n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + 11*A*c**2*m*n**2*x*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 +
18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + 12*A*c**2*m*n*x*(e*x
)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n
**3 + 11*n**2 + 6*n + 1) + 3*A*c**2*m*x*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2
+ 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + 6*A*c**2*n**3*x*(e*x)**m/(m**4 + 6*m**3
*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n
+ 1) + 11*A*c**2*n**2*x*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 2
2*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + 6*A*c**2*n*x*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m
**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + A*c**2*x*(
e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m +
6*n**3 + 11*n**2 + 6*n + 1) + 2*A*c*d*m**3*x*x**n*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*
n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + 10*A*c*d*m**2*n*x*x**n*(e*x)*
*m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**
3 + 11*n**2 + 6*n + 1) + 6*A*c*d*m**2*x*x**n*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6
*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + 12*A*c*d*m*n**2*x*x**n*(e*x)**m/(m
**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 1
1*n**2 + 6*n + 1) + 20*A*c*d*m*n*x*x**n*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2
+ 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + 6*A*c*d*m*x*x**n*(e*x)**m/(m**4 + 6*m**
3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*
n + 1) + 12*A*c*d*n**2*x*x**n*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**
3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + 10*A*c*d*n*x*x**n*(e*x)**m/(m**4 + 6*m**3*n + 4*m
**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) +
2*A*c*d*x*x**n*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 +
18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + A*d**2*m**3*x*x**(2*n)*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m
**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + 4*A*d**2*m
**2*n*x*x**(2*n)*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2
+ 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + 3*A*d**2*m**2*x*x**(2*n)*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 +
11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + 3*A*d*
*2*m*n**2*x*x**(2*n)*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*
n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + 8*A*d**2*m*n*x*x**(2*n)*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3
+ 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + 3*A
*d**2*m*x*x**(2*n)*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n*
*2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + 3*A*d**2*n**2*x*x**(2*n)*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3
+ 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + 4*A*
d**2*n*x*x**(2*n)*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**
2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + A*d**2*x*x**(2*n)*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**
2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + B*c**2*m**3*
x*x**n*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n
+ 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + 5*B*c**2*m**2*n*x*x**n*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2
+ 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + 3*B*c**2*m**2*x*x*
*n*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*
m + 6*n**3 + 11*n**2 + 6*n + 1) + 6*B*c**2*m*n**2*x*x**n*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 1
8*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + 10*B*c**2*m*n*x*x**n*(
e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m +
6*n**3 + 11*n**2 + 6*n + 1) + 3*B*c**2*m*x*x**n*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n
+ 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + 6*B*c**2*n**2*x*x**n*(e*x)**m/(
m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 +
11*n**2 + 6*n + 1) + 5*B*c**2*n*x*x**n*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2
+ 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + B*c**2*x*x**n*(e*x)**m/(m**4 + 6*m**3*n
+ 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n +
1) + 2*B*c*d*m**3*x*x**(2*n)*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3
+ 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + 8*B*c*d*m**2*n*x*x**(2*n)*(e*x)**m/(m**4 + 6*m**3*
n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n
+ 1) + 6*B*c*d*m**2*x*x**(2*n)*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n*
*3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + 6*B*c*d*m*n**2*x*x**(2*n)*(e*x)**m/(m**4 + 6*m**
3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*
n + 1) + 16*B*c*d*m*n*x*x**(2*n)*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*
n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + 6*B*c*d*m*x*x**(2*n)*(e*x)**m/(m**4 + 6*m**3*n
+ 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n +
1) + 6*B*c*d*n**2*x*x**(2*n)*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**
3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + 8*B*c*d*n*x*x**(2*n)*(e*x)**m/(m**4 + 6*m**3*n +
4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1)
+ 2*B*c*d*x*x**(2*n)*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m
*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + B*d**2*m**3*x*x**(3*n)*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3
+ 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + 3*B
*d**2*m**2*n*x*x**(3*n)*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22
*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + 3*B*d**2*m**2*x*x**(3*n)*(e*x)**m/(m**4 + 6*m**3*n + 4*
m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) +
2*B*d**2*m*n**2*x*x**(3*n)*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3
+ 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + 6*B*d**2*m*n*x*x**(3*n)*(e*x)**m/(m**4 + 6*m**3*n +
4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1
) + 3*B*d**2*m*x*x**(3*n)*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 +
22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + 2*B*d**2*n**2*x*x**(3*n)*(e*x)**m/(m**4 + 6*m**3*n +
4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1)
+ 3*B*d**2*n*x*x**(3*n)*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 2
2*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + B*d**2*x*x**(3*n)*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 +
11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1), True))
Maxima [A] (verification not implemented)
none
Time = 0.20 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.52
\[
\int (e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )^2 \, dx=\frac {B d^{2} e^{m} x e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right )\right )}}{m + 3 \, n + 1} + \frac {2 \, B c d e^{m} x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right )\right )}}{m + 2 \, n + 1} + \frac {A d^{2} e^{m} x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right )\right )}}{m + 2 \, n + 1} + \frac {B c^{2} e^{m} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{m + n + 1} + \frac {2 \, A c d e^{m} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{m + n + 1} + \frac {\left (e x\right )^{m + 1} A c^{2}}{e {\left (m + 1\right )}}
\]
[In]
integrate((e*x)^m*(A+B*x^n)*(c+d*x^n)^2,x, algorithm="maxima")
[Out]
B*d^2*e^m*x*e^(m*log(x) + 3*n*log(x))/(m + 3*n + 1) + 2*B*c*d*e^m*x*e^(m*log(x) + 2*n*log(x))/(m + 2*n + 1) +
A*d^2*e^m*x*e^(m*log(x) + 2*n*log(x))/(m + 2*n + 1) + B*c^2*e^m*x*e^(m*log(x) + n*log(x))/(m + n + 1) + 2*A*c*
d*e^m*x*e^(m*log(x) + n*log(x))/(m + n + 1) + (e*x)^(m + 1)*A*c^2/(e*(m + 1))
Giac [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 2951 vs. \(2 (102) = 204\).
Time = 0.30 (sec) , antiderivative size = 2951, normalized size of antiderivative = 28.93
\[
\int (e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )^2 \, dx=\text {Too large to display}
\]
[In]
integrate((e*x)^m*(A+B*x^n)*(c+d*x^n)^2,x, algorithm="giac")
[Out]
(B*d^2*m^3*x*x^(3*n)*e^(m*log(e) + m*log(x)) + 3*B*d^2*m^2*n*x*x^(3*n)*e^(m*log(e) + m*log(x)) + 2*B*d^2*m*n^2
*x*x^(3*n)*e^(m*log(e) + m*log(x)) + 2*B*c*d*m^3*x*x^(2*n)*e^(m*log(e) + m*log(x)) + A*d^2*m^3*x*x^(2*n)*e^(m*
log(e) + m*log(x)) + B*d^2*m^3*x*x^(2*n)*e^(m*log(e) + m*log(x)) + 8*B*c*d*m^2*n*x*x^(2*n)*e^(m*log(e) + m*log
(x)) + 4*A*d^2*m^2*n*x*x^(2*n)*e^(m*log(e) + m*log(x)) + 3*B*d^2*m^2*n*x*x^(2*n)*e^(m*log(e) + m*log(x)) + 6*B
*c*d*m*n^2*x*x^(2*n)*e^(m*log(e) + m*log(x)) + 3*A*d^2*m*n^2*x*x^(2*n)*e^(m*log(e) + m*log(x)) + 2*B*d^2*m*n^2
*x*x^(2*n)*e^(m*log(e) + m*log(x)) + B*c^2*m^3*x*x^n*e^(m*log(e) + m*log(x)) + 2*A*c*d*m^3*x*x^n*e^(m*log(e) +
m*log(x)) + 2*B*c*d*m^3*x*x^n*e^(m*log(e) + m*log(x)) + A*d^2*m^3*x*x^n*e^(m*log(e) + m*log(x)) + B*d^2*m^3*x
*x^n*e^(m*log(e) + m*log(x)) + 5*B*c^2*m^2*n*x*x^n*e^(m*log(e) + m*log(x)) + 10*A*c*d*m^2*n*x*x^n*e^(m*log(e)
+ m*log(x)) + 8*B*c*d*m^2*n*x*x^n*e^(m*log(e) + m*log(x)) + 4*A*d^2*m^2*n*x*x^n*e^(m*log(e) + m*log(x)) + 3*B*
d^2*m^2*n*x*x^n*e^(m*log(e) + m*log(x)) + 6*B*c^2*m*n^2*x*x^n*e^(m*log(e) + m*log(x)) + 12*A*c*d*m*n^2*x*x^n*e
^(m*log(e) + m*log(x)) + 6*B*c*d*m*n^2*x*x^n*e^(m*log(e) + m*log(x)) + 3*A*d^2*m*n^2*x*x^n*e^(m*log(e) + m*log
(x)) + 2*B*d^2*m*n^2*x*x^n*e^(m*log(e) + m*log(x)) + A*c^2*m^3*x*e^(m*log(e) + m*log(x)) + B*c^2*m^3*x*e^(m*lo
g(e) + m*log(x)) + 2*A*c*d*m^3*x*e^(m*log(e) + m*log(x)) + 2*B*c*d*m^3*x*e^(m*log(e) + m*log(x)) + A*d^2*m^3*x
*e^(m*log(e) + m*log(x)) + B*d^2*m^3*x*e^(m*log(e) + m*log(x)) + 6*A*c^2*m^2*n*x*e^(m*log(e) + m*log(x)) + 5*B
*c^2*m^2*n*x*e^(m*log(e) + m*log(x)) + 10*A*c*d*m^2*n*x*e^(m*log(e) + m*log(x)) + 8*B*c*d*m^2*n*x*e^(m*log(e)
+ m*log(x)) + 4*A*d^2*m^2*n*x*e^(m*log(e) + m*log(x)) + 3*B*d^2*m^2*n*x*e^(m*log(e) + m*log(x)) + 11*A*c^2*m*n
^2*x*e^(m*log(e) + m*log(x)) + 6*B*c^2*m*n^2*x*e^(m*log(e) + m*log(x)) + 12*A*c*d*m*n^2*x*e^(m*log(e) + m*log(
x)) + 6*B*c*d*m*n^2*x*e^(m*log(e) + m*log(x)) + 3*A*d^2*m*n^2*x*e^(m*log(e) + m*log(x)) + 2*B*d^2*m*n^2*x*e^(m
*log(e) + m*log(x)) + 6*A*c^2*n^3*x*e^(m*log(e) + m*log(x)) + 3*B*d^2*m^2*x*x^(3*n)*e^(m*log(e) + m*log(x)) +
6*B*d^2*m*n*x*x^(3*n)*e^(m*log(e) + m*log(x)) + 2*B*d^2*n^2*x*x^(3*n)*e^(m*log(e) + m*log(x)) + 6*B*c*d*m^2*x*
x^(2*n)*e^(m*log(e) + m*log(x)) + 3*A*d^2*m^2*x*x^(2*n)*e^(m*log(e) + m*log(x)) + 3*B*d^2*m^2*x*x^(2*n)*e^(m*l
og(e) + m*log(x)) + 16*B*c*d*m*n*x*x^(2*n)*e^(m*log(e) + m*log(x)) + 8*A*d^2*m*n*x*x^(2*n)*e^(m*log(e) + m*log
(x)) + 6*B*d^2*m*n*x*x^(2*n)*e^(m*log(e) + m*log(x)) + 6*B*c*d*n^2*x*x^(2*n)*e^(m*log(e) + m*log(x)) + 3*A*d^2
*n^2*x*x^(2*n)*e^(m*log(e) + m*log(x)) + 2*B*d^2*n^2*x*x^(2*n)*e^(m*log(e) + m*log(x)) + 3*B*c^2*m^2*x*x^n*e^(
m*log(e) + m*log(x)) + 6*A*c*d*m^2*x*x^n*e^(m*log(e) + m*log(x)) + 6*B*c*d*m^2*x*x^n*e^(m*log(e) + m*log(x)) +
3*A*d^2*m^2*x*x^n*e^(m*log(e) + m*log(x)) + 3*B*d^2*m^2*x*x^n*e^(m*log(e) + m*log(x)) + 10*B*c^2*m*n*x*x^n*e^
(m*log(e) + m*log(x)) + 20*A*c*d*m*n*x*x^n*e^(m*log(e) + m*log(x)) + 16*B*c*d*m*n*x*x^n*e^(m*log(e) + m*log(x)
) + 8*A*d^2*m*n*x*x^n*e^(m*log(e) + m*log(x)) + 6*B*d^2*m*n*x*x^n*e^(m*log(e) + m*log(x)) + 6*B*c^2*n^2*x*x^n*
e^(m*log(e) + m*log(x)) + 12*A*c*d*n^2*x*x^n*e^(m*log(e) + m*log(x)) + 6*B*c*d*n^2*x*x^n*e^(m*log(e) + m*log(x
)) + 3*A*d^2*n^2*x*x^n*e^(m*log(e) + m*log(x)) + 2*B*d^2*n^2*x*x^n*e^(m*log(e) + m*log(x)) + 3*A*c^2*m^2*x*e^(
m*log(e) + m*log(x)) + 3*B*c^2*m^2*x*e^(m*log(e) + m*log(x)) + 6*A*c*d*m^2*x*e^(m*log(e) + m*log(x)) + 6*B*c*d
*m^2*x*e^(m*log(e) + m*log(x)) + 3*A*d^2*m^2*x*e^(m*log(e) + m*log(x)) + 3*B*d^2*m^2*x*e^(m*log(e) + m*log(x))
+ 12*A*c^2*m*n*x*e^(m*log(e) + m*log(x)) + 10*B*c^2*m*n*x*e^(m*log(e) + m*log(x)) + 20*A*c*d*m*n*x*e^(m*log(e
) + m*log(x)) + 16*B*c*d*m*n*x*e^(m*log(e) + m*log(x)) + 8*A*d^2*m*n*x*e^(m*log(e) + m*log(x)) + 6*B*d^2*m*n*x
*e^(m*log(e) + m*log(x)) + 11*A*c^2*n^2*x*e^(m*log(e) + m*log(x)) + 6*B*c^2*n^2*x*e^(m*log(e) + m*log(x)) + 12
*A*c*d*n^2*x*e^(m*log(e) + m*log(x)) + 6*B*c*d*n^2*x*e^(m*log(e) + m*log(x)) + 3*A*d^2*n^2*x*e^(m*log(e) + m*l
og(x)) + 2*B*d^2*n^2*x*e^(m*log(e) + m*log(x)) + 3*B*d^2*m*x*x^(3*n)*e^(m*log(e) + m*log(x)) + 3*B*d^2*n*x*x^(
3*n)*e^(m*log(e) + m*log(x)) + 6*B*c*d*m*x*x^(2*n)*e^(m*log(e) + m*log(x)) + 3*A*d^2*m*x*x^(2*n)*e^(m*log(e) +
m*log(x)) + 3*B*d^2*m*x*x^(2*n)*e^(m*log(e) + m*log(x)) + 8*B*c*d*n*x*x^(2*n)*e^(m*log(e) + m*log(x)) + 4*A*d
^2*n*x*x^(2*n)*e^(m*log(e) + m*log(x)) + 3*B*d^2*n*x*x^(2*n)*e^(m*log(e) + m*log(x)) + 3*B*c^2*m*x*x^n*e^(m*lo
g(e) + m*log(x)) + 6*A*c*d*m*x*x^n*e^(m*log(e) + m*log(x)) + 6*B*c*d*m*x*x^n*e^(m*log(e) + m*log(x)) + 3*A*d^2
*m*x*x^n*e^(m*log(e) + m*log(x)) + 3*B*d^2*m*x*x^n*e^(m*log(e) + m*log(x)) + 5*B*c^2*n*x*x^n*e^(m*log(e) + m*l
og(x)) + 10*A*c*d*n*x*x^n*e^(m*log(e) + m*log(x)) + 8*B*c*d*n*x*x^n*e^(m*log(e) + m*log(x)) + 4*A*d^2*n*x*x^n*
e^(m*log(e) + m*log(x)) + 3*B*d^2*n*x*x^n*e^(m*log(e) + m*log(x)) + 3*A*c^2*m*x*e^(m*log(e) + m*log(x)) + 3*B*
c^2*m*x*e^(m*log(e) + m*log(x)) + 6*A*c*d*m*x*e^(m*log(e) + m*log(x)) + 6*B*c*d*m*x*e^(m*log(e) + m*log(x)) +
3*A*d^2*m*x*e^(m*log(e) + m*log(x)) + 3*B*d^2*m*x*e^(m*log(e) + m*log(x)) + 6*A*c^2*n*x*e^(m*log(e) + m*log(x)
) + 5*B*c^2*n*x*e^(m*log(e) + m*log(x)) + 10*A*c*d*n*x*e^(m*log(e) + m*log(x)) + 8*B*c*d*n*x*e^(m*log(e) + m*l
og(x)) + 4*A*d^2*n*x*e^(m*log(e) + m*log(x)) + 3*B*d^2*n*x*e^(m*log(e) + m*log(x)) + B*d^2*x*x^(3*n)*e^(m*log(
e) + m*log(x)) + 2*B*c*d*x*x^(2*n)*e^(m*log(e) + m*log(x)) + A*d^2*x*x^(2*n)*e^(m*log(e) + m*log(x)) + B*d^2*x
*x^(2*n)*e^(m*log(e) + m*log(x)) + B*c^2*x*x^n*e^(m*log(e) + m*log(x)) + 2*A*c*d*x*x^n*e^(m*log(e) + m*log(x))
+ 2*B*c*d*x*x^n*e^(m*log(e) + m*log(x)) + A*d^2*x*x^n*e^(m*log(e) + m*log(x)) + B*d^2*x*x^n*e^(m*log(e) + m*l
og(x)) + A*c^2*x*e^(m*log(e) + m*log(x)) + B*c^2*x*e^(m*log(e) + m*log(x)) + 2*A*c*d*x*e^(m*log(e) + m*log(x))
+ 2*B*c*d*x*e^(m*log(e) + m*log(x)) + A*d^2*x*e^(m*log(e) + m*log(x)) + B*d^2*x*e^(m*log(e) + m*log(x)))/(m^4
+ 6*m^3*n + 11*m^2*n^2 + 6*m*n^3 + 4*m^3 + 18*m^2*n + 22*m*n^2 + 6*n^3 + 6*m^2 + 18*m*n + 11*n^2 + 4*m + 6*n
+ 1)
Mupad [B] (verification not implemented)
Time = 9.12 (sec) , antiderivative size = 265, normalized size of antiderivative = 2.60
\[
\int (e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )^2 \, dx=\frac {A\,c^2\,x\,{\left (e\,x\right )}^m}{m+1}+\frac {c\,x\,x^n\,{\left (e\,x\right )}^m\,\left (2\,A\,d+B\,c\right )\,\left (m^2+5\,m\,n+2\,m+6\,n^2+5\,n+1\right )}{m^3+6\,m^2\,n+3\,m^2+11\,m\,n^2+12\,m\,n+3\,m+6\,n^3+11\,n^2+6\,n+1}+\frac {d\,x\,x^{2\,n}\,{\left (e\,x\right )}^m\,\left (A\,d+2\,B\,c\right )\,\left (m^2+4\,m\,n+2\,m+3\,n^2+4\,n+1\right )}{m^3+6\,m^2\,n+3\,m^2+11\,m\,n^2+12\,m\,n+3\,m+6\,n^3+11\,n^2+6\,n+1}+\frac {B\,d^2\,x\,x^{3\,n}\,{\left (e\,x\right )}^m\,\left (m^2+3\,m\,n+2\,m+2\,n^2+3\,n+1\right )}{m^3+6\,m^2\,n+3\,m^2+11\,m\,n^2+12\,m\,n+3\,m+6\,n^3+11\,n^2+6\,n+1}
\]
[In]
int((e*x)^m*(A + B*x^n)*(c + d*x^n)^2,x)
[Out]
(A*c^2*x*(e*x)^m)/(m + 1) + (c*x*x^n*(e*x)^m*(2*A*d + B*c)*(2*m + 5*n + 5*m*n + m^2 + 6*n^2 + 1))/(3*m + 6*n +
12*m*n + 11*m*n^2 + 6*m^2*n + 3*m^2 + m^3 + 11*n^2 + 6*n^3 + 1) + (d*x*x^(2*n)*(e*x)^m*(A*d + 2*B*c)*(2*m + 4
*n + 4*m*n + m^2 + 3*n^2 + 1))/(3*m + 6*n + 12*m*n + 11*m*n^2 + 6*m^2*n + 3*m^2 + m^3 + 11*n^2 + 6*n^3 + 1) +
(B*d^2*x*x^(3*n)*(e*x)^m*(2*m + 3*n + 3*m*n + m^2 + 2*n^2 + 1))/(3*m + 6*n + 12*m*n + 11*m*n^2 + 6*m^2*n + 3*m
^2 + m^3 + 11*n^2 + 6*n^3 + 1)