3.17 \(\int (e x)^m (a+b x^n) (A+B x^n) (c+d x^n)^3 \, dx\)

Optimal. Leaf size=210 \[ \frac {c^2 x^{n+1} (e x)^m (3 a A d+a B c+A b c)}{m+n+1}+\frac {d^2 x^{4 n+1} (e x)^m (a B d+A b d+3 b B c)}{m+4 n+1}+\frac {c x^{2 n+1} (e x)^m (3 a d (A d+B c)+b c (3 A d+B c))}{m+2 n+1}+\frac {d x^{3 n+1} (e x)^m (a d (A d+3 B c)+3 b c (A d+B c))}{m+3 n+1}+\frac {a A c^3 (e x)^{m+1}}{e (m+1)}+\frac {b B d^3 x^{5 n+1} (e x)^m}{m+5 n+1} \]

[Out]

c^2*(3*A*a*d+A*b*c+B*a*c)*x^(1+n)*(e*x)^m/(1+m+n)+c*(3*a*d*(A*d+B*c)+b*c*(3*A*d+B*c))*x^(1+2*n)*(e*x)^m/(1+m+2
*n)+d*(3*b*c*(A*d+B*c)+a*d*(A*d+3*B*c))*x^(1+3*n)*(e*x)^m/(1+m+3*n)+d^2*(A*b*d+B*a*d+3*B*b*c)*x^(1+4*n)*(e*x)^
m/(1+m+4*n)+b*B*d^3*x^(1+5*n)*(e*x)^m/(1+m+5*n)+a*A*c^3*(e*x)^(1+m)/e/(1+m)

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Rubi [A]  time = 0.26, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {570, 20, 30} \[ \frac {c^2 x^{n+1} (e x)^m (3 a A d+a B c+A b c)}{m+n+1}+\frac {d^2 x^{4 n+1} (e x)^m (a B d+A b d+3 b B c)}{m+4 n+1}+\frac {c x^{2 n+1} (e x)^m (3 a d (A d+B c)+b c (3 A d+B c))}{m+2 n+1}+\frac {d x^{3 n+1} (e x)^m (a d (A d+3 B c)+3 b c (A d+B c))}{m+3 n+1}+\frac {a A c^3 (e x)^{m+1}}{e (m+1)}+\frac {b B d^3 x^{5 n+1} (e x)^m}{m+5 n+1} \]

Antiderivative was successfully verified.

[In]

Int[(e*x)^m*(a + b*x^n)*(A + B*x^n)*(c + d*x^n)^3,x]

[Out]

(c^2*(A*b*c + a*B*c + 3*a*A*d)*x^(1 + n)*(e*x)^m)/(1 + m + n) + (c*(3*a*d*(B*c + A*d) + b*c*(B*c + 3*A*d))*x^(
1 + 2*n)*(e*x)^m)/(1 + m + 2*n) + (d*(3*b*c*(B*c + A*d) + a*d*(3*B*c + A*d))*x^(1 + 3*n)*(e*x)^m)/(1 + m + 3*n
) + (d^2*(3*b*B*c + A*b*d + a*B*d)*x^(1 + 4*n)*(e*x)^m)/(1 + m + 4*n) + (b*B*d^3*x^(1 + 5*n)*(e*x)^m)/(1 + m +
5*n) + (a*A*c^3*(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 570

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^
(r_.), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{a,
b, c, d, e, f, g, m, n}, x] && IGtQ[p, -2] && IGtQ[q, 0] && IGtQ[r, 0]

Rubi steps

\begin {align*} \int (e x)^m \left (a+b x^n\right ) \left (A+B x^n\right ) \left (c+d x^n\right )^3 \, dx &=\int \left (a A c^3 (e x)^m+c^2 (A b c+a B c+3 a A d) x^n (e x)^m+c (3 a d (B c+A d)+b c (B c+3 A d)) x^{2 n} (e x)^m+d (3 b c (B c+A d)+a d (3 B c+A d)) x^{3 n} (e x)^m+d^2 (3 b B c+A b d+a B d) x^{4 n} (e x)^m+b B d^3 x^{5 n} (e x)^m\right ) \, dx\\ &=\frac {a A c^3 (e x)^{1+m}}{e (1+m)}+\left (b B d^3\right ) \int x^{5 n} (e x)^m \, dx+\left (c^2 (A b c+a B c+3 a A d)\right ) \int x^n (e x)^m \, dx+\left (d^2 (3 b B c+A b d+a B d)\right ) \int x^{4 n} (e x)^m \, dx+(d (3 b c (B c+A d)+a d (3 B c+A d))) \int x^{3 n} (e x)^m \, dx+(c (3 a d (B c+A d)+b c (B c+3 A d))) \int x^{2 n} (e x)^m \, dx\\ &=\frac {a A c^3 (e x)^{1+m}}{e (1+m)}+\left (b B d^3 x^{-m} (e x)^m\right ) \int x^{m+5 n} \, dx+\left (c^2 (A b c+a B c+3 a A d) x^{-m} (e x)^m\right ) \int x^{m+n} \, dx+\left (d^2 (3 b B c+A b d+a B d) x^{-m} (e x)^m\right ) \int x^{m+4 n} \, dx+\left (d (3 b c (B c+A d)+a d (3 B c+A d)) x^{-m} (e x)^m\right ) \int x^{m+3 n} \, dx+\left (c (3 a d (B c+A d)+b c (B c+3 A d)) x^{-m} (e x)^m\right ) \int x^{m+2 n} \, dx\\ &=\frac {c^2 (A b c+a B c+3 a A d) x^{1+n} (e x)^m}{1+m+n}+\frac {c (3 a d (B c+A d)+b c (B c+3 A d)) x^{1+2 n} (e x)^m}{1+m+2 n}+\frac {d (3 b c (B c+A d)+a d (3 B c+A d)) x^{1+3 n} (e x)^m}{1+m+3 n}+\frac {d^2 (3 b B c+A b d+a B d) x^{1+4 n} (e x)^m}{1+m+4 n}+\frac {b B d^3 x^{1+5 n} (e x)^m}{1+m+5 n}+\frac {a A c^3 (e x)^{1+m}}{e (1+m)}\\ \end {align*}

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Mathematica [A]  time = 0.75, size = 172, normalized size = 0.82 \[ x (e x)^m \left (\frac {c^2 x^n (3 a A d+a B c+A b c)}{m+n+1}+\frac {d^2 x^{4 n} (a B d+A b d+3 b B c)}{m+4 n+1}+\frac {c x^{2 n} (3 a d (A d+B c)+b c (3 A d+B c))}{m+2 n+1}+\frac {d x^{3 n} (a d (A d+3 B c)+3 b c (A d+B c))}{m+3 n+1}+\frac {a A c^3}{m+1}+\frac {b B d^3 x^{5 n}}{m+5 n+1}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(e*x)^m*(a + b*x^n)*(A + B*x^n)*(c + d*x^n)^3,x]

[Out]

x*(e*x)^m*((a*A*c^3)/(1 + m) + (c^2*(A*b*c + a*B*c + 3*a*A*d)*x^n)/(1 + m + n) + (c*(3*a*d*(B*c + A*d) + b*c*(
B*c + 3*A*d))*x^(2*n))/(1 + m + 2*n) + (d*(3*b*c*(B*c + A*d) + a*d*(3*B*c + A*d))*x^(3*n))/(1 + m + 3*n) + (d^
2*(3*b*B*c + A*b*d + a*B*d)*x^(4*n))/(1 + m + 4*n) + (b*B*d^3*x^(5*n))/(1 + m + 5*n))

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fricas [B]  time = 0.81, size = 2833, normalized size = 13.49 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((B*b*d^3*m^5 + 5*B*b*d^3*m^4 + 10*B*b*d^3*m^3 + 10*B*b*d^3*m^2 + 5*B*b*d^3*m + B*b*d^3 + 24*(B*b*d^3*m + B*b*
d^3)*n^4 + 50*(B*b*d^3*m^2 + 2*B*b*d^3*m + B*b*d^3)*n^3 + 35*(B*b*d^3*m^3 + 3*B*b*d^3*m^2 + 3*B*b*d^3*m + B*b*
d^3)*n^2 + 10*(B*b*d^3*m^4 + 4*B*b*d^3*m^3 + 6*B*b*d^3*m^2 + 4*B*b*d^3*m + B*b*d^3)*n)*x*x^(5*n)*e^(m*log(e) +
m*log(x)) + ((3*B*b*c*d^2 + (B*a + A*b)*d^3)*m^5 + 3*B*b*c*d^2 + 5*(3*B*b*c*d^2 + (B*a + A*b)*d^3)*m^4 + 30*(
3*B*b*c*d^2 + (B*a + A*b)*d^3 + (3*B*b*c*d^2 + (B*a + A*b)*d^3)*m)*n^4 + (B*a + A*b)*d^3 + 10*(3*B*b*c*d^2 + (
B*a + A*b)*d^3)*m^3 + 61*(3*B*b*c*d^2 + (B*a + A*b)*d^3 + (3*B*b*c*d^2 + (B*a + A*b)*d^3)*m^2 + 2*(3*B*b*c*d^2
+ (B*a + A*b)*d^3)*m)*n^3 + 10*(3*B*b*c*d^2 + (B*a + A*b)*d^3)*m^2 + 41*(3*B*b*c*d^2 + (B*a + A*b)*d^3 + (3*B
*b*c*d^2 + (B*a + A*b)*d^3)*m^3 + 3*(3*B*b*c*d^2 + (B*a + A*b)*d^3)*m^2 + 3*(3*B*b*c*d^2 + (B*a + A*b)*d^3)*m)
*n^2 + 5*(3*B*b*c*d^2 + (B*a + A*b)*d^3)*m + 11*(3*B*b*c*d^2 + (3*B*b*c*d^2 + (B*a + A*b)*d^3)*m^4 + (B*a + A*
b)*d^3 + 4*(3*B*b*c*d^2 + (B*a + A*b)*d^3)*m^3 + 6*(3*B*b*c*d^2 + (B*a + A*b)*d^3)*m^2 + 4*(3*B*b*c*d^2 + (B*a
+ A*b)*d^3)*m)*n)*x*x^(4*n)*e^(m*log(e) + m*log(x)) + ((3*B*b*c^2*d + A*a*d^3 + 3*(B*a + A*b)*c*d^2)*m^5 + 3*
B*b*c^2*d + A*a*d^3 + 5*(3*B*b*c^2*d + A*a*d^3 + 3*(B*a + A*b)*c*d^2)*m^4 + 40*(3*B*b*c^2*d + A*a*d^3 + 3*(B*a
+ A*b)*c*d^2 + (3*B*b*c^2*d + A*a*d^3 + 3*(B*a + A*b)*c*d^2)*m)*n^4 + 3*(B*a + A*b)*c*d^2 + 10*(3*B*b*c^2*d +
A*a*d^3 + 3*(B*a + A*b)*c*d^2)*m^3 + 78*(3*B*b*c^2*d + A*a*d^3 + 3*(B*a + A*b)*c*d^2 + (3*B*b*c^2*d + A*a*d^3
+ 3*(B*a + A*b)*c*d^2)*m^2 + 2*(3*B*b*c^2*d + A*a*d^3 + 3*(B*a + A*b)*c*d^2)*m)*n^3 + 10*(3*B*b*c^2*d + A*a*d
^3 + 3*(B*a + A*b)*c*d^2)*m^2 + 49*(3*B*b*c^2*d + A*a*d^3 + 3*(B*a + A*b)*c*d^2 + (3*B*b*c^2*d + A*a*d^3 + 3*(
B*a + A*b)*c*d^2)*m^3 + 3*(3*B*b*c^2*d + A*a*d^3 + 3*(B*a + A*b)*c*d^2)*m^2 + 3*(3*B*b*c^2*d + A*a*d^3 + 3*(B*
a + A*b)*c*d^2)*m)*n^2 + 5*(3*B*b*c^2*d + A*a*d^3 + 3*(B*a + A*b)*c*d^2)*m + 12*(3*B*b*c^2*d + A*a*d^3 + (3*B*
b*c^2*d + A*a*d^3 + 3*(B*a + A*b)*c*d^2)*m^4 + 3*(B*a + A*b)*c*d^2 + 4*(3*B*b*c^2*d + A*a*d^3 + 3*(B*a + A*b)*
c*d^2)*m^3 + 6*(3*B*b*c^2*d + A*a*d^3 + 3*(B*a + A*b)*c*d^2)*m^2 + 4*(3*B*b*c^2*d + A*a*d^3 + 3*(B*a + A*b)*c*
d^2)*m)*n)*x*x^(3*n)*e^(m*log(e) + m*log(x)) + ((B*b*c^3 + 3*A*a*c*d^2 + 3*(B*a + A*b)*c^2*d)*m^5 + B*b*c^3 +
3*A*a*c*d^2 + 5*(B*b*c^3 + 3*A*a*c*d^2 + 3*(B*a + A*b)*c^2*d)*m^4 + 60*(B*b*c^3 + 3*A*a*c*d^2 + 3*(B*a + A*b)*
c^2*d + (B*b*c^3 + 3*A*a*c*d^2 + 3*(B*a + A*b)*c^2*d)*m)*n^4 + 3*(B*a + A*b)*c^2*d + 10*(B*b*c^3 + 3*A*a*c*d^2
+ 3*(B*a + A*b)*c^2*d)*m^3 + 107*(B*b*c^3 + 3*A*a*c*d^2 + 3*(B*a + A*b)*c^2*d + (B*b*c^3 + 3*A*a*c*d^2 + 3*(B
*a + A*b)*c^2*d)*m^2 + 2*(B*b*c^3 + 3*A*a*c*d^2 + 3*(B*a + A*b)*c^2*d)*m)*n^3 + 10*(B*b*c^3 + 3*A*a*c*d^2 + 3*
(B*a + A*b)*c^2*d)*m^2 + 59*(B*b*c^3 + 3*A*a*c*d^2 + 3*(B*a + A*b)*c^2*d + (B*b*c^3 + 3*A*a*c*d^2 + 3*(B*a + A
*b)*c^2*d)*m^3 + 3*(B*b*c^3 + 3*A*a*c*d^2 + 3*(B*a + A*b)*c^2*d)*m^2 + 3*(B*b*c^3 + 3*A*a*c*d^2 + 3*(B*a + A*b
)*c^2*d)*m)*n^2 + 5*(B*b*c^3 + 3*A*a*c*d^2 + 3*(B*a + A*b)*c^2*d)*m + 13*(B*b*c^3 + 3*A*a*c*d^2 + (B*b*c^3 + 3
*A*a*c*d^2 + 3*(B*a + A*b)*c^2*d)*m^4 + 3*(B*a + A*b)*c^2*d + 4*(B*b*c^3 + 3*A*a*c*d^2 + 3*(B*a + A*b)*c^2*d)*
m^3 + 6*(B*b*c^3 + 3*A*a*c*d^2 + 3*(B*a + A*b)*c^2*d)*m^2 + 4*(B*b*c^3 + 3*A*a*c*d^2 + 3*(B*a + A*b)*c^2*d)*m)
*n)*x*x^(2*n)*e^(m*log(e) + m*log(x)) + ((3*A*a*c^2*d + (B*a + A*b)*c^3)*m^5 + 3*A*a*c^2*d + 5*(3*A*a*c^2*d +
(B*a + A*b)*c^3)*m^4 + 120*(3*A*a*c^2*d + (B*a + A*b)*c^3 + (3*A*a*c^2*d + (B*a + A*b)*c^3)*m)*n^4 + (B*a + A*
b)*c^3 + 10*(3*A*a*c^2*d + (B*a + A*b)*c^3)*m^3 + 154*(3*A*a*c^2*d + (B*a + A*b)*c^3 + (3*A*a*c^2*d + (B*a + A
*b)*c^3)*m^2 + 2*(3*A*a*c^2*d + (B*a + A*b)*c^3)*m)*n^3 + 10*(3*A*a*c^2*d + (B*a + A*b)*c^3)*m^2 + 71*(3*A*a*c
^2*d + (B*a + A*b)*c^3 + (3*A*a*c^2*d + (B*a + A*b)*c^3)*m^3 + 3*(3*A*a*c^2*d + (B*a + A*b)*c^3)*m^2 + 3*(3*A*
a*c^2*d + (B*a + A*b)*c^3)*m)*n^2 + 5*(3*A*a*c^2*d + (B*a + A*b)*c^3)*m + 14*(3*A*a*c^2*d + (3*A*a*c^2*d + (B*
a + A*b)*c^3)*m^4 + (B*a + A*b)*c^3 + 4*(3*A*a*c^2*d + (B*a + A*b)*c^3)*m^3 + 6*(3*A*a*c^2*d + (B*a + A*b)*c^3
)*m^2 + 4*(3*A*a*c^2*d + (B*a + A*b)*c^3)*m)*n)*x*x^n*e^(m*log(e) + m*log(x)) + (A*a*c^3*m^5 + 120*A*a*c^3*n^5
+ 5*A*a*c^3*m^4 + 10*A*a*c^3*m^3 + 10*A*a*c^3*m^2 + 5*A*a*c^3*m + A*a*c^3 + 274*(A*a*c^3*m + A*a*c^3)*n^4 + 2
25*(A*a*c^3*m^2 + 2*A*a*c^3*m + A*a*c^3)*n^3 + 85*(A*a*c^3*m^3 + 3*A*a*c^3*m^2 + 3*A*a*c^3*m + A*a*c^3)*n^2 +
15*(A*a*c^3*m^4 + 4*A*a*c^3*m^3 + 6*A*a*c^3*m^2 + 4*A*a*c^3*m + A*a*c^3)*n)*x*e^(m*log(e) + m*log(x)))/(m^6 +
120*(m + 1)*n^5 + 6*m^5 + 274*(m^2 + 2*m + 1)*n^4 + 15*m^4 + 225*(m^3 + 3*m^2 + 3*m + 1)*n^3 + 20*m^3 + 85*(m^
4 + 4*m^3 + 6*m^2 + 4*m + 1)*n^2 + 15*m^2 + 15*(m^5 + 5*m^4 + 10*m^3 + 10*m^2 + 5*m + 1)*n + 6*m + 1)

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giac [B]  time = 1.06, size = 6927, normalized size = 32.99 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(B*b*d^3*m^5*x*x^m*x^(5*n)*e^m + 10*B*b*d^3*m^4*n*x*x^m*x^(5*n)*e^m + 35*B*b*d^3*m^3*n^2*x*x^m*x^(5*n)*e^m + 5
0*B*b*d^3*m^2*n^3*x*x^m*x^(5*n)*e^m + 24*B*b*d^3*m*n^4*x*x^m*x^(5*n)*e^m + 3*B*b*c*d^2*m^5*x*x^m*x^(4*n)*e^m +
B*a*d^3*m^5*x*x^m*x^(4*n)*e^m + A*b*d^3*m^5*x*x^m*x^(4*n)*e^m + 33*B*b*c*d^2*m^4*n*x*x^m*x^(4*n)*e^m + 11*B*a
*d^3*m^4*n*x*x^m*x^(4*n)*e^m + 11*A*b*d^3*m^4*n*x*x^m*x^(4*n)*e^m + 123*B*b*c*d^2*m^3*n^2*x*x^m*x^(4*n)*e^m +
41*B*a*d^3*m^3*n^2*x*x^m*x^(4*n)*e^m + 41*A*b*d^3*m^3*n^2*x*x^m*x^(4*n)*e^m + 183*B*b*c*d^2*m^2*n^3*x*x^m*x^(4
*n)*e^m + 61*B*a*d^3*m^2*n^3*x*x^m*x^(4*n)*e^m + 61*A*b*d^3*m^2*n^3*x*x^m*x^(4*n)*e^m + 90*B*b*c*d^2*m*n^4*x*x
^m*x^(4*n)*e^m + 30*B*a*d^3*m*n^4*x*x^m*x^(4*n)*e^m + 30*A*b*d^3*m*n^4*x*x^m*x^(4*n)*e^m + 3*B*b*c^2*d*m^5*x*x
^m*x^(3*n)*e^m + 3*B*a*c*d^2*m^5*x*x^m*x^(3*n)*e^m + 3*A*b*c*d^2*m^5*x*x^m*x^(3*n)*e^m + A*a*d^3*m^5*x*x^m*x^(
3*n)*e^m + 36*B*b*c^2*d*m^4*n*x*x^m*x^(3*n)*e^m + 36*B*a*c*d^2*m^4*n*x*x^m*x^(3*n)*e^m + 36*A*b*c*d^2*m^4*n*x*
x^m*x^(3*n)*e^m + 12*A*a*d^3*m^4*n*x*x^m*x^(3*n)*e^m + 147*B*b*c^2*d*m^3*n^2*x*x^m*x^(3*n)*e^m + 147*B*a*c*d^2
*m^3*n^2*x*x^m*x^(3*n)*e^m + 147*A*b*c*d^2*m^3*n^2*x*x^m*x^(3*n)*e^m + 49*A*a*d^3*m^3*n^2*x*x^m*x^(3*n)*e^m +
234*B*b*c^2*d*m^2*n^3*x*x^m*x^(3*n)*e^m + 234*B*a*c*d^2*m^2*n^3*x*x^m*x^(3*n)*e^m + 234*A*b*c*d^2*m^2*n^3*x*x^
m*x^(3*n)*e^m + 78*A*a*d^3*m^2*n^3*x*x^m*x^(3*n)*e^m + 120*B*b*c^2*d*m*n^4*x*x^m*x^(3*n)*e^m + 120*B*a*c*d^2*m
*n^4*x*x^m*x^(3*n)*e^m + 120*A*b*c*d^2*m*n^4*x*x^m*x^(3*n)*e^m + 40*A*a*d^3*m*n^4*x*x^m*x^(3*n)*e^m + B*b*c^3*
m^5*x*x^m*x^(2*n)*e^m + 3*B*a*c^2*d*m^5*x*x^m*x^(2*n)*e^m + 3*A*b*c^2*d*m^5*x*x^m*x^(2*n)*e^m + 3*A*a*c*d^2*m^
5*x*x^m*x^(2*n)*e^m + 13*B*b*c^3*m^4*n*x*x^m*x^(2*n)*e^m + 39*B*a*c^2*d*m^4*n*x*x^m*x^(2*n)*e^m + 39*A*b*c^2*d
*m^4*n*x*x^m*x^(2*n)*e^m + 39*A*a*c*d^2*m^4*n*x*x^m*x^(2*n)*e^m + 59*B*b*c^3*m^3*n^2*x*x^m*x^(2*n)*e^m + 177*B
*a*c^2*d*m^3*n^2*x*x^m*x^(2*n)*e^m + 177*A*b*c^2*d*m^3*n^2*x*x^m*x^(2*n)*e^m + 177*A*a*c*d^2*m^3*n^2*x*x^m*x^(
2*n)*e^m + 107*B*b*c^3*m^2*n^3*x*x^m*x^(2*n)*e^m + 321*B*a*c^2*d*m^2*n^3*x*x^m*x^(2*n)*e^m + 321*A*b*c^2*d*m^2
*n^3*x*x^m*x^(2*n)*e^m + 321*A*a*c*d^2*m^2*n^3*x*x^m*x^(2*n)*e^m + 60*B*b*c^3*m*n^4*x*x^m*x^(2*n)*e^m + 180*B*
a*c^2*d*m*n^4*x*x^m*x^(2*n)*e^m + 180*A*b*c^2*d*m*n^4*x*x^m*x^(2*n)*e^m + 180*A*a*c*d^2*m*n^4*x*x^m*x^(2*n)*e^
m + B*a*c^3*m^5*x*x^m*x^n*e^m + A*b*c^3*m^5*x*x^m*x^n*e^m + 3*A*a*c^2*d*m^5*x*x^m*x^n*e^m + 14*B*a*c^3*m^4*n*x
*x^m*x^n*e^m + 14*A*b*c^3*m^4*n*x*x^m*x^n*e^m + 42*A*a*c^2*d*m^4*n*x*x^m*x^n*e^m + 71*B*a*c^3*m^3*n^2*x*x^m*x^
n*e^m + 71*A*b*c^3*m^3*n^2*x*x^m*x^n*e^m + 213*A*a*c^2*d*m^3*n^2*x*x^m*x^n*e^m + 154*B*a*c^3*m^2*n^3*x*x^m*x^n
*e^m + 154*A*b*c^3*m^2*n^3*x*x^m*x^n*e^m + 462*A*a*c^2*d*m^2*n^3*x*x^m*x^n*e^m + 120*B*a*c^3*m*n^4*x*x^m*x^n*e
^m + 120*A*b*c^3*m*n^4*x*x^m*x^n*e^m + 360*A*a*c^2*d*m*n^4*x*x^m*x^n*e^m + A*a*c^3*m^5*x*x^m*e^m + 15*A*a*c^3*
m^4*n*x*x^m*e^m + 85*A*a*c^3*m^3*n^2*x*x^m*e^m + 225*A*a*c^3*m^2*n^3*x*x^m*e^m + 274*A*a*c^3*m*n^4*x*x^m*e^m +
120*A*a*c^3*n^5*x*x^m*e^m + 5*B*b*d^3*m^4*x*x^m*x^(5*n)*e^m + 40*B*b*d^3*m^3*n*x*x^m*x^(5*n)*e^m + 105*B*b*d^
3*m^2*n^2*x*x^m*x^(5*n)*e^m + 100*B*b*d^3*m*n^3*x*x^m*x^(5*n)*e^m + 24*B*b*d^3*n^4*x*x^m*x^(5*n)*e^m + 15*B*b*
c*d^2*m^4*x*x^m*x^(4*n)*e^m + 5*B*a*d^3*m^4*x*x^m*x^(4*n)*e^m + 5*A*b*d^3*m^4*x*x^m*x^(4*n)*e^m + 132*B*b*c*d^
2*m^3*n*x*x^m*x^(4*n)*e^m + 44*B*a*d^3*m^3*n*x*x^m*x^(4*n)*e^m + 44*A*b*d^3*m^3*n*x*x^m*x^(4*n)*e^m + 369*B*b*
c*d^2*m^2*n^2*x*x^m*x^(4*n)*e^m + 123*B*a*d^3*m^2*n^2*x*x^m*x^(4*n)*e^m + 123*A*b*d^3*m^2*n^2*x*x^m*x^(4*n)*e^
m + 366*B*b*c*d^2*m*n^3*x*x^m*x^(4*n)*e^m + 122*B*a*d^3*m*n^3*x*x^m*x^(4*n)*e^m + 122*A*b*d^3*m*n^3*x*x^m*x^(4
*n)*e^m + 90*B*b*c*d^2*n^4*x*x^m*x^(4*n)*e^m + 30*B*a*d^3*n^4*x*x^m*x^(4*n)*e^m + 30*A*b*d^3*n^4*x*x^m*x^(4*n)
*e^m + 15*B*b*c^2*d*m^4*x*x^m*x^(3*n)*e^m + 15*B*a*c*d^2*m^4*x*x^m*x^(3*n)*e^m + 15*A*b*c*d^2*m^4*x*x^m*x^(3*n
)*e^m + 5*A*a*d^3*m^4*x*x^m*x^(3*n)*e^m + 144*B*b*c^2*d*m^3*n*x*x^m*x^(3*n)*e^m + 144*B*a*c*d^2*m^3*n*x*x^m*x^
(3*n)*e^m + 144*A*b*c*d^2*m^3*n*x*x^m*x^(3*n)*e^m + 48*A*a*d^3*m^3*n*x*x^m*x^(3*n)*e^m + 441*B*b*c^2*d*m^2*n^2
*x*x^m*x^(3*n)*e^m + 441*B*a*c*d^2*m^2*n^2*x*x^m*x^(3*n)*e^m + 441*A*b*c*d^2*m^2*n^2*x*x^m*x^(3*n)*e^m + 147*A
*a*d^3*m^2*n^2*x*x^m*x^(3*n)*e^m + 468*B*b*c^2*d*m*n^3*x*x^m*x^(3*n)*e^m + 468*B*a*c*d^2*m*n^3*x*x^m*x^(3*n)*e
^m + 468*A*b*c*d^2*m*n^3*x*x^m*x^(3*n)*e^m + 156*A*a*d^3*m*n^3*x*x^m*x^(3*n)*e^m + 120*B*b*c^2*d*n^4*x*x^m*x^(
3*n)*e^m + 120*B*a*c*d^2*n^4*x*x^m*x^(3*n)*e^m + 120*A*b*c*d^2*n^4*x*x^m*x^(3*n)*e^m + 40*A*a*d^3*n^4*x*x^m*x^
(3*n)*e^m + 5*B*b*c^3*m^4*x*x^m*x^(2*n)*e^m + 15*B*a*c^2*d*m^4*x*x^m*x^(2*n)*e^m + 15*A*b*c^2*d*m^4*x*x^m*x^(2
*n)*e^m + 15*A*a*c*d^2*m^4*x*x^m*x^(2*n)*e^m + 52*B*b*c^3*m^3*n*x*x^m*x^(2*n)*e^m + 156*B*a*c^2*d*m^3*n*x*x^m*
x^(2*n)*e^m + 156*A*b*c^2*d*m^3*n*x*x^m*x^(2*n)*e^m + 156*A*a*c*d^2*m^3*n*x*x^m*x^(2*n)*e^m + 177*B*b*c^3*m^2*
n^2*x*x^m*x^(2*n)*e^m + 531*B*a*c^2*d*m^2*n^2*x*x^m*x^(2*n)*e^m + 531*A*b*c^2*d*m^2*n^2*x*x^m*x^(2*n)*e^m + 53
1*A*a*c*d^2*m^2*n^2*x*x^m*x^(2*n)*e^m + 214*B*b*c^3*m*n^3*x*x^m*x^(2*n)*e^m + 642*B*a*c^2*d*m*n^3*x*x^m*x^(2*n
)*e^m + 642*A*b*c^2*d*m*n^3*x*x^m*x^(2*n)*e^m + 642*A*a*c*d^2*m*n^3*x*x^m*x^(2*n)*e^m + 60*B*b*c^3*n^4*x*x^m*x
^(2*n)*e^m + 180*B*a*c^2*d*n^4*x*x^m*x^(2*n)*e^m + 180*A*b*c^2*d*n^4*x*x^m*x^(2*n)*e^m + 180*A*a*c*d^2*n^4*x*x
^m*x^(2*n)*e^m + 5*B*a*c^3*m^4*x*x^m*x^n*e^m + 5*A*b*c^3*m^4*x*x^m*x^n*e^m + 15*A*a*c^2*d*m^4*x*x^m*x^n*e^m +
56*B*a*c^3*m^3*n*x*x^m*x^n*e^m + 56*A*b*c^3*m^3*n*x*x^m*x^n*e^m + 168*A*a*c^2*d*m^3*n*x*x^m*x^n*e^m + 213*B*a*
c^3*m^2*n^2*x*x^m*x^n*e^m + 213*A*b*c^3*m^2*n^2*x*x^m*x^n*e^m + 639*A*a*c^2*d*m^2*n^2*x*x^m*x^n*e^m + 308*B*a*
c^3*m*n^3*x*x^m*x^n*e^m + 308*A*b*c^3*m*n^3*x*x^m*x^n*e^m + 924*A*a*c^2*d*m*n^3*x*x^m*x^n*e^m + 120*B*a*c^3*n^
4*x*x^m*x^n*e^m + 120*A*b*c^3*n^4*x*x^m*x^n*e^m + 360*A*a*c^2*d*n^4*x*x^m*x^n*e^m + 5*A*a*c^3*m^4*x*x^m*e^m +
60*A*a*c^3*m^3*n*x*x^m*e^m + 255*A*a*c^3*m^2*n^2*x*x^m*e^m + 450*A*a*c^3*m*n^3*x*x^m*e^m + 274*A*a*c^3*n^4*x*x
^m*e^m + 10*B*b*d^3*m^3*x*x^m*x^(5*n)*e^m + 60*B*b*d^3*m^2*n*x*x^m*x^(5*n)*e^m + 105*B*b*d^3*m*n^2*x*x^m*x^(5*
n)*e^m + 50*B*b*d^3*n^3*x*x^m*x^(5*n)*e^m + 30*B*b*c*d^2*m^3*x*x^m*x^(4*n)*e^m + 10*B*a*d^3*m^3*x*x^m*x^(4*n)*
e^m + 10*A*b*d^3*m^3*x*x^m*x^(4*n)*e^m + 198*B*b*c*d^2*m^2*n*x*x^m*x^(4*n)*e^m + 66*B*a*d^3*m^2*n*x*x^m*x^(4*n
)*e^m + 66*A*b*d^3*m^2*n*x*x^m*x^(4*n)*e^m + 369*B*b*c*d^2*m*n^2*x*x^m*x^(4*n)*e^m + 123*B*a*d^3*m*n^2*x*x^m*x
^(4*n)*e^m + 123*A*b*d^3*m*n^2*x*x^m*x^(4*n)*e^m + 183*B*b*c*d^2*n^3*x*x^m*x^(4*n)*e^m + 61*B*a*d^3*n^3*x*x^m*
x^(4*n)*e^m + 61*A*b*d^3*n^3*x*x^m*x^(4*n)*e^m + 30*B*b*c^2*d*m^3*x*x^m*x^(3*n)*e^m + 30*B*a*c*d^2*m^3*x*x^m*x
^(3*n)*e^m + 30*A*b*c*d^2*m^3*x*x^m*x^(3*n)*e^m + 10*A*a*d^3*m^3*x*x^m*x^(3*n)*e^m + 216*B*b*c^2*d*m^2*n*x*x^m
*x^(3*n)*e^m + 216*B*a*c*d^2*m^2*n*x*x^m*x^(3*n)*e^m + 216*A*b*c*d^2*m^2*n*x*x^m*x^(3*n)*e^m + 72*A*a*d^3*m^2*
n*x*x^m*x^(3*n)*e^m + 441*B*b*c^2*d*m*n^2*x*x^m*x^(3*n)*e^m + 441*B*a*c*d^2*m*n^2*x*x^m*x^(3*n)*e^m + 441*A*b*
c*d^2*m*n^2*x*x^m*x^(3*n)*e^m + 147*A*a*d^3*m*n^2*x*x^m*x^(3*n)*e^m + 234*B*b*c^2*d*n^3*x*x^m*x^(3*n)*e^m + 23
4*B*a*c*d^2*n^3*x*x^m*x^(3*n)*e^m + 234*A*b*c*d^2*n^3*x*x^m*x^(3*n)*e^m + 78*A*a*d^3*n^3*x*x^m*x^(3*n)*e^m + 1
0*B*b*c^3*m^3*x*x^m*x^(2*n)*e^m + 30*B*a*c^2*d*m^3*x*x^m*x^(2*n)*e^m + 30*A*b*c^2*d*m^3*x*x^m*x^(2*n)*e^m + 30
*A*a*c*d^2*m^3*x*x^m*x^(2*n)*e^m + 78*B*b*c^3*m^2*n*x*x^m*x^(2*n)*e^m + 234*B*a*c^2*d*m^2*n*x*x^m*x^(2*n)*e^m
+ 234*A*b*c^2*d*m^2*n*x*x^m*x^(2*n)*e^m + 234*A*a*c*d^2*m^2*n*x*x^m*x^(2*n)*e^m + 177*B*b*c^3*m*n^2*x*x^m*x^(2
*n)*e^m + 531*B*a*c^2*d*m*n^2*x*x^m*x^(2*n)*e^m + 531*A*b*c^2*d*m*n^2*x*x^m*x^(2*n)*e^m + 531*A*a*c*d^2*m*n^2*
x*x^m*x^(2*n)*e^m + 107*B*b*c^3*n^3*x*x^m*x^(2*n)*e^m + 321*B*a*c^2*d*n^3*x*x^m*x^(2*n)*e^m + 321*A*b*c^2*d*n^
3*x*x^m*x^(2*n)*e^m + 321*A*a*c*d^2*n^3*x*x^m*x^(2*n)*e^m + 10*B*a*c^3*m^3*x*x^m*x^n*e^m + 10*A*b*c^3*m^3*x*x^
m*x^n*e^m + 30*A*a*c^2*d*m^3*x*x^m*x^n*e^m + 84*B*a*c^3*m^2*n*x*x^m*x^n*e^m + 84*A*b*c^3*m^2*n*x*x^m*x^n*e^m +
252*A*a*c^2*d*m^2*n*x*x^m*x^n*e^m + 213*B*a*c^3*m*n^2*x*x^m*x^n*e^m + 213*A*b*c^3*m*n^2*x*x^m*x^n*e^m + 639*A
*a*c^2*d*m*n^2*x*x^m*x^n*e^m + 154*B*a*c^3*n^3*x*x^m*x^n*e^m + 154*A*b*c^3*n^3*x*x^m*x^n*e^m + 462*A*a*c^2*d*n
^3*x*x^m*x^n*e^m + 10*A*a*c^3*m^3*x*x^m*e^m + 90*A*a*c^3*m^2*n*x*x^m*e^m + 255*A*a*c^3*m*n^2*x*x^m*e^m + 225*A
*a*c^3*n^3*x*x^m*e^m + 10*B*b*d^3*m^2*x*x^m*x^(5*n)*e^m + 40*B*b*d^3*m*n*x*x^m*x^(5*n)*e^m + 35*B*b*d^3*n^2*x*
x^m*x^(5*n)*e^m + 30*B*b*c*d^2*m^2*x*x^m*x^(4*n)*e^m + 10*B*a*d^3*m^2*x*x^m*x^(4*n)*e^m + 10*A*b*d^3*m^2*x*x^m
*x^(4*n)*e^m + 132*B*b*c*d^2*m*n*x*x^m*x^(4*n)*e^m + 44*B*a*d^3*m*n*x*x^m*x^(4*n)*e^m + 44*A*b*d^3*m*n*x*x^m*x
^(4*n)*e^m + 123*B*b*c*d^2*n^2*x*x^m*x^(4*n)*e^m + 41*B*a*d^3*n^2*x*x^m*x^(4*n)*e^m + 41*A*b*d^3*n^2*x*x^m*x^(
4*n)*e^m + 30*B*b*c^2*d*m^2*x*x^m*x^(3*n)*e^m + 30*B*a*c*d^2*m^2*x*x^m*x^(3*n)*e^m + 30*A*b*c*d^2*m^2*x*x^m*x^
(3*n)*e^m + 10*A*a*d^3*m^2*x*x^m*x^(3*n)*e^m + 144*B*b*c^2*d*m*n*x*x^m*x^(3*n)*e^m + 144*B*a*c*d^2*m*n*x*x^m*x
^(3*n)*e^m + 144*A*b*c*d^2*m*n*x*x^m*x^(3*n)*e^m + 48*A*a*d^3*m*n*x*x^m*x^(3*n)*e^m + 147*B*b*c^2*d*n^2*x*x^m*
x^(3*n)*e^m + 147*B*a*c*d^2*n^2*x*x^m*x^(3*n)*e^m + 147*A*b*c*d^2*n^2*x*x^m*x^(3*n)*e^m + 49*A*a*d^3*n^2*x*x^m
*x^(3*n)*e^m + 10*B*b*c^3*m^2*x*x^m*x^(2*n)*e^m + 30*B*a*c^2*d*m^2*x*x^m*x^(2*n)*e^m + 30*A*b*c^2*d*m^2*x*x^m*
x^(2*n)*e^m + 30*A*a*c*d^2*m^2*x*x^m*x^(2*n)*e^m + 52*B*b*c^3*m*n*x*x^m*x^(2*n)*e^m + 156*B*a*c^2*d*m*n*x*x^m*
x^(2*n)*e^m + 156*A*b*c^2*d*m*n*x*x^m*x^(2*n)*e^m + 156*A*a*c*d^2*m*n*x*x^m*x^(2*n)*e^m + 59*B*b*c^3*n^2*x*x^m
*x^(2*n)*e^m + 177*B*a*c^2*d*n^2*x*x^m*x^(2*n)*e^m + 177*A*b*c^2*d*n^2*x*x^m*x^(2*n)*e^m + 177*A*a*c*d^2*n^2*x
*x^m*x^(2*n)*e^m + 10*B*a*c^3*m^2*x*x^m*x^n*e^m + 10*A*b*c^3*m^2*x*x^m*x^n*e^m + 30*A*a*c^2*d*m^2*x*x^m*x^n*e^
m + 56*B*a*c^3*m*n*x*x^m*x^n*e^m + 56*A*b*c^3*m*n*x*x^m*x^n*e^m + 168*A*a*c^2*d*m*n*x*x^m*x^n*e^m + 71*B*a*c^3
*n^2*x*x^m*x^n*e^m + 71*A*b*c^3*n^2*x*x^m*x^n*e^m + 213*A*a*c^2*d*n^2*x*x^m*x^n*e^m + 10*A*a*c^3*m^2*x*x^m*e^m
+ 60*A*a*c^3*m*n*x*x^m*e^m + 85*A*a*c^3*n^2*x*x^m*e^m + 5*B*b*d^3*m*x*x^m*x^(5*n)*e^m + 10*B*b*d^3*n*x*x^m*x^
(5*n)*e^m + 15*B*b*c*d^2*m*x*x^m*x^(4*n)*e^m + 5*B*a*d^3*m*x*x^m*x^(4*n)*e^m + 5*A*b*d^3*m*x*x^m*x^(4*n)*e^m +
33*B*b*c*d^2*n*x*x^m*x^(4*n)*e^m + 11*B*a*d^3*n*x*x^m*x^(4*n)*e^m + 11*A*b*d^3*n*x*x^m*x^(4*n)*e^m + 15*B*b*c
^2*d*m*x*x^m*x^(3*n)*e^m + 15*B*a*c*d^2*m*x*x^m*x^(3*n)*e^m + 15*A*b*c*d^2*m*x*x^m*x^(3*n)*e^m + 5*A*a*d^3*m*x
*x^m*x^(3*n)*e^m + 36*B*b*c^2*d*n*x*x^m*x^(3*n)*e^m + 36*B*a*c*d^2*n*x*x^m*x^(3*n)*e^m + 36*A*b*c*d^2*n*x*x^m*
x^(3*n)*e^m + 12*A*a*d^3*n*x*x^m*x^(3*n)*e^m + 5*B*b*c^3*m*x*x^m*x^(2*n)*e^m + 15*B*a*c^2*d*m*x*x^m*x^(2*n)*e^
m + 15*A*b*c^2*d*m*x*x^m*x^(2*n)*e^m + 15*A*a*c*d^2*m*x*x^m*x^(2*n)*e^m + 13*B*b*c^3*n*x*x^m*x^(2*n)*e^m + 39*
B*a*c^2*d*n*x*x^m*x^(2*n)*e^m + 39*A*b*c^2*d*n*x*x^m*x^(2*n)*e^m + 39*A*a*c*d^2*n*x*x^m*x^(2*n)*e^m + 5*B*a*c^
3*m*x*x^m*x^n*e^m + 5*A*b*c^3*m*x*x^m*x^n*e^m + 15*A*a*c^2*d*m*x*x^m*x^n*e^m + 14*B*a*c^3*n*x*x^m*x^n*e^m + 14
*A*b*c^3*n*x*x^m*x^n*e^m + 42*A*a*c^2*d*n*x*x^m*x^n*e^m + 5*A*a*c^3*m*x*x^m*e^m + 15*A*a*c^3*n*x*x^m*e^m + B*b
*d^3*x*x^m*x^(5*n)*e^m + 3*B*b*c*d^2*x*x^m*x^(4*n)*e^m + B*a*d^3*x*x^m*x^(4*n)*e^m + A*b*d^3*x*x^m*x^(4*n)*e^m
+ 3*B*b*c^2*d*x*x^m*x^(3*n)*e^m + 3*B*a*c*d^2*x*x^m*x^(3*n)*e^m + 3*A*b*c*d^2*x*x^m*x^(3*n)*e^m + A*a*d^3*x*x
^m*x^(3*n)*e^m + B*b*c^3*x*x^m*x^(2*n)*e^m + 3*B*a*c^2*d*x*x^m*x^(2*n)*e^m + 3*A*b*c^2*d*x*x^m*x^(2*n)*e^m + 3
*A*a*c*d^2*x*x^m*x^(2*n)*e^m + B*a*c^3*x*x^m*x^n*e^m + A*b*c^3*x*x^m*x^n*e^m + 3*A*a*c^2*d*x*x^m*x^n*e^m + A*a
*c^3*x*x^m*e^m)/(m^6 + 15*m^5*n + 85*m^4*n^2 + 225*m^3*n^3 + 274*m^2*n^4 + 120*m*n^5 + 6*m^5 + 75*m^4*n + 340*
m^3*n^2 + 675*m^2*n^3 + 548*m*n^4 + 120*n^5 + 15*m^4 + 150*m^3*n + 510*m^2*n^2 + 675*m*n^3 + 274*n^4 + 20*m^3
+ 150*m^2*n + 340*m*n^2 + 225*n^3 + 15*m^2 + 75*m*n + 85*n^2 + 6*m + 15*n + 1)

________________________________________________________________________________________

maple [C]  time = 0.18, size = 4972, normalized size = 23.68 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m*(b*x^n+a)*(B*x^n+A)*(d*x^n+c)^3,x)

[Out]

x*(44*B*a*d^3*m^3*n*(x^n)^4+123*B*a*d^3*m^2*n^2*(x^n)^4+122*B*a*d^3*m*n^3*(x^n)^4+3*B*b*c^2*d*m^5*(x^n)^3+15*B
*b*c*d^2*m^4*(x^n)^4+A*a*d^3*(x^n)^3+B*b*c^3*(x^n)^2+A*b*c^3*x^n+B*a*c^3*x^n+b*B*d^3*(x^n)^5+A*b*d^3*(x^n)^4+B
*a*d^3*(x^n)^4+10*A*a*c^3*m^2+85*A*a*c^3*n^2+120*A*a*c^3*n^5+A*a*c^3*m^5+5*A*a*c^3*m^4+274*A*a*c^3*n^4+10*A*a*
c^3*m^3+225*A*a*c^3*n^3+a*A*c^3+5*a*A*c^3*m+15*a*A*c^3*n+40*A*a*d^3*m*n^4*(x^n)^3+3*A*b*c*d^2*m^5*(x^n)^3+44*A
*b*d^3*m^3*n*(x^n)^4+123*A*b*d^3*m^2*n^2*(x^n)^4+122*A*b*d^3*m*n^3*(x^n)^4+3*B*a*c*d^2*m^5*(x^n)^3+84*B*a*c^3*
m^2*n*x^n+213*B*a*c^3*m*n^2*x^n+30*B*a*c^2*d*m^2*(x^n)^2+177*B*a*c^2*d*n^2*(x^n)^2+15*B*a*c*d^2*(x^n)^3*m+36*B
*a*c*d^2*(x^n)^3*n+52*B*b*c^3*m*n*(x^n)^2+15*B*b*c^2*d*(x^n)^3*m+36*B*b*c^2*d*(x^n)^3*n+30*A*a*c^2*d*m^2*x^n+2
13*A*a*c^2*d*n^2*x^n+15*A*a*c*d^2*(x^n)^2*m+90*A*a*c^3*m^2*n+66*B*a*d^3*m^2*n*(x^n)^4+123*B*a*d^3*m*n^2*(x^n)^
4+13*B*b*c^3*m^4*n*(x^n)^2+59*B*b*c^3*m^3*n^2*(x^n)^2+107*B*b*c^3*m^2*n^3*(x^n)^2+60*B*b*c^3*m*n^4*(x^n)^2+15*
B*b*c^2*d*m^4*(x^n)^3+120*B*b*c^2*d*n^4*(x^n)^3+30*B*b*c*d^2*m^3*(x^n)^4+11*B*a*d^3*m^4*n*(x^n)^4+41*B*a*d^3*m
^3*n^2*(x^n)^4+39*A*a*c*d^2*(x^n)^2*n+56*A*b*c^3*m*n*x^n+15*A*b*c^2*d*(x^n)^2*m+39*A*b*c^2*d*(x^n)^2*n+56*B*a*
c^3*m*n*x^n+15*B*a*c^2*d*(x^n)^2*m+39*B*a*c^2*d*(x^n)^2*n+213*B*a*c^3*m^2*n^2*x^n+308*B*a*c^3*m*n^3*x^n+30*B*a
*c^2*d*m^3*(x^n)^2+321*B*a*c^2*d*n^3*(x^n)^2+30*B*a*c*d^2*m^2*(x^n)^3+44*B*a*d^3*m*n*(x^n)^4+52*B*b*c^3*m^3*n*
(x^n)^2+177*B*b*c^3*m^2*n^2*(x^n)^2+214*B*b*c^3*m*n^3*(x^n)^2+30*B*b*c^2*d*m^3*(x^n)^3+234*B*b*c^2*d*n^3*(x^n)
^3+30*B*b*c*d^2*m^2*(x^n)^4+123*B*b*c*d^2*n^2*(x^n)^4+15*A*a*c^2*d*m^4*x^n+360*A*a*c^2*d*n^4*x^n+30*A*a*c*d^2*
m^3*(x^n)^2+15*A*a*c^2*d*x^n*m+42*A*a*c^2*d*x^n*n+15*A*b*c*d^2*(x^n)^3*m+36*A*b*c*d^2*(x^n)^3*n+3*A*b*c^2*d*m^
5*(x^n)^2+15*A*b*c*d^2*m^4*(x^n)^3+120*A*b*c*d^2*n^4*(x^n)^3+66*A*b*d^3*m^2*n*(x^n)^4+123*A*b*d^3*m*n^2*(x^n)^
4+3*B*a*c^2*d*m^5*(x^n)^2+15*B*a*c*d^2*m^4*(x^n)^3+120*B*a*c*d^2*n^4*(x^n)^3+90*B*b*c*d^2*n^4*(x^n)^4+60*B*b*d
^3*m^2*n*(x^n)^5+105*B*b*d^3*m*n^2*(x^n)^5+3*A*a*c*d^2*m^5*(x^n)^2+48*A*a*d^3*m^3*n*(x^n)^3+10*B*b*d^3*m^4*n*(
x^n)^5+35*B*b*d^3*m^3*n^2*(x^n)^5+147*B*a*c*d^2*n^2*(x^n)^3+78*B*b*c^3*m^2*n*(x^n)^2+177*B*b*c^3*m*n^2*(x^n)^2
+30*B*b*c^2*d*m^2*(x^n)^3+147*B*b*c^2*d*n^2*(x^n)^3+15*B*b*c*d^2*(x^n)^4*m+33*B*b*c*d^2*(x^n)^4*n+30*A*a*c^2*d
*m^3*x^n+462*A*a*c^2*d*n^3*x^n+321*A*a*c*d^2*n^3*(x^n)^2+48*A*a*d^3*m*n*(x^n)^3+56*A*b*c^3*m^3*n*x^n+213*A*b*c
^3*m^2*n^2*x^n+308*A*b*c^3*m*n^3*x^n+30*A*b*c^2*d*m^3*(x^n)^2+321*A*b*c^2*d*n^3*(x^n)^2+30*A*b*c*d^2*m^2*(x^n)
^3+255*A*a*c^3*m*n^2+60*A*a*c^3*m*n+61*B*a*d^3*m^2*n^3*(x^n)^4+30*B*a*d^3*m*n^4*(x^n)^4+3*B*b*c*d^2*m^5*(x^n)^
4+40*B*b*d^3*m^3*n*(x^n)^5+105*B*b*d^3*m^2*n^2*(x^n)^5+100*B*b*d^3*m*n^3*(x^n)^5+12*A*a*d^3*m^4*n*(x^n)^3+49*A
*a*d^3*m^3*n^2*(x^n)^3+78*A*a*d^3*m^2*n^3*(x^n)^3+50*B*b*d^3*m^2*n^3*(x^n)^5+24*B*b*d^3*m*n^4*(x^n)^5+11*A*b*d
^3*m^4*n*(x^n)^4+41*A*b*d^3*m^3*n^2*(x^n)^4+61*A*b*d^3*m^2*n^3*(x^n)^4+30*A*a*c*d^2*m^2*(x^n)^2+177*A*a*c*d^2*
n^2*(x^n)^2+84*A*b*c^3*m^2*n*x^n+213*A*b*c^3*m*n^2*x^n+30*A*b*c^2*d*m^2*(x^n)^2+177*A*b*c^2*d*n^2*(x^n)^2+15*B
*a*c^2*d*m^4*(x^n)^2+180*B*a*c^2*d*n^4*(x^n)^2+30*B*a*c*d^2*m^3*(x^n)^3+234*B*a*c*d^2*n^3*(x^n)^3+24*B*b*d^3*n
^4*(x^n)^5+A*b*c^3*m^5*x^n+10*A*b*d^3*m^2*(x^n)^4+41*A*b*d^3*n^2*(x^n)^4+B*a*c^3*m^5*x^n+10*B*a*d^3*m^2*(x^n)^
4+41*B*a*d^3*n^2*(x^n)^4+5*B*b*c^3*m^4*(x^n)^2+60*B*b*c^3*n^4*(x^n)^2+5*m*b*B*d^3*(x^n)^5+147*A*b*c*d^2*n^2*(x
^n)^3+56*B*a*c^3*m^3*n*x^n+183*B*b*c*d^2*n^3*(x^n)^4+40*B*b*d^3*m*n*(x^n)^5+3*A*a*c^2*d*m^5*x^n+15*A*a*c*d^2*m
^4*(x^n)^2+180*A*a*c*d^2*n^4*(x^n)^2+72*A*a*d^3*m^2*n*(x^n)^3+147*A*a*d^3*m*n^2*(x^n)^3+14*A*b*c^3*m^4*n*x^n+7
1*A*b*c^3*m^3*n^2*x^n+154*A*b*c^3*m^2*n^3*x^n+120*A*b*c^3*m*n^4*x^n+15*A*b*c^2*d*m^4*(x^n)^2+180*A*b*c^2*d*n^4
*(x^n)^2+30*A*b*c*d^2*m^3*(x^n)^3+234*A*b*c*d^2*n^3*(x^n)^3+44*A*b*d^3*m*n*(x^n)^4+14*B*a*c^3*m^4*n*x^n+71*B*a
*c^3*m^3*n^2*x^n+154*B*a*c^3*m^2*n^3*x^n+120*B*a*c^3*m*n^4*x^n+147*A*a*d^3*m^2*n^2*(x^n)^3+156*A*a*d^3*m*n^3*(
x^n)^3+50*B*b*d^3*n^3*(x^n)^5+5*A*a*d^3*m^4*(x^n)^3+154*B*a*c^3*n^3*x^n+10*B*b*c^3*m^2*(x^n)^2+59*B*b*c^3*n^2*
(x^n)^2+10*A*b*c^3*m^2*x^n+71*A*b*c^3*n^2*x^n+10*B*a*c^3*m^2*x^n+30*A*b*d^3*m*n^4*(x^n)^4+A*a*d^3*m^5*(x^n)^3+
5*A*b*d^3*m^4*(x^n)^4+30*A*b*d^3*n^4*(x^n)^4+5*B*a*d^3*m^4*(x^n)^4+30*B*a*d^3*n^4*(x^n)^4+10*B*b*d^3*m^3*(x^n)
^5+B*b*d^3*m^5*(x^n)^5+A*b*d^3*m^5*(x^n)^4+B*a*d^3*m^5*(x^n)^4+5*B*b*d^3*m^4*(x^n)^5+5*A*a*d^3*(x^n)^3*m+12*A*
a*d^3*(x^n)^3*n+10*A*b*c^3*m^3*x^n+154*A*b*c^3*n^3*x^n+10*B*a*c^3*m^3*x^n+10*B*b*d^3*m^2*(x^n)^5+35*B*b*d^3*n^
2*(x^n)^5+10*A*a*d^3*m^3*(x^n)^3+78*A*a*d^3*n^3*(x^n)^3+40*A*a*d^3*n^4*(x^n)^3+10*A*b*d^3*m^3*(x^n)^4+61*A*b*d
^3*n^3*(x^n)^4+60*A*a*c^3*m^3*n+255*A*a*c^3*m^2*n^2+450*A*a*c^3*m*n^3+15*A*a*c^3*m^4*n+85*A*a*c^3*m^3*n^2+225*
A*a*c^3*m^2*n^3+274*A*a*c^3*m*n^4+10*B*a*d^3*m^3*(x^n)^4+61*B*a*d^3*n^3*(x^n)^4+B*b*c^3*m^5*(x^n)^2+3*(x^n)^2*
d*c^2*A*b+3*(x^n)^2*d*c^2*B*a+3*(x^n)^4*b*B*c*d^2+10*b*B*d^3*(x^n)^5*n+10*A*a*d^3*m^2*(x^n)^3+49*A*a*d^3*n^2*(
x^n)^3+5*A*b*c^3*m^4*x^n+120*A*b*c^3*n^4*x^n+5*A*b*d^3*(x^n)^4*m+11*A*b*d^3*(x^n)^4*n+5*B*a*c^3*m^4*x^n+120*B*
a*c^3*n^4*x^n+5*B*a*d^3*(x^n)^4*m+11*B*a*d^3*(x^n)^4*n+10*B*b*c^3*m^3*(x^n)^2+107*B*b*c^3*n^3*(x^n)^2+3*x^n*a*
A*c^2*d+3*(x^n)^3*A*b*c*d^2+3*(x^n)^3*B*a*c*d^2+3*(x^n)^3*b*B*c^2*d+3*(x^n)^2*a*A*c*d^2+71*B*a*c^3*n^2*x^n+5*B
*b*c^3*(x^n)^2*m+13*B*b*c^3*(x^n)^2*n+5*A*b*c^3*x^n*m+14*A*b*c^3*x^n*n+5*B*a*c^3*x^n*m+14*B*a*c^3*x^n*n+441*A*
b*c*d^2*m^2*n^2*(x^n)^3+39*A*b*c^2*d*m^4*n*(x^n)^2+177*A*b*c^2*d*m^3*n^2*(x^n)^2+321*A*b*c^2*d*m^2*n^3*(x^n)^2
+180*A*b*c^2*d*m*n^4*(x^n)^2+144*A*b*c*d^2*m^3*n*(x^n)^3+639*A*a*c^2*d*m^2*n^2*x^n+924*A*a*c^2*d*m*n^3*x^n+234
*A*a*c*d^2*m^2*n*(x^n)^2+531*A*a*c*d^2*m*n^2*(x^n)^2+177*A*a*c*d^2*m^3*n^2*(x^n)^2+321*A*a*c*d^2*m^2*n^3*(x^n)
^2+180*A*a*c*d^2*m*n^4*(x^n)^2+144*B*a*c*d^2*m*n*(x^n)^3+144*B*b*c^2*d*m*n*(x^n)^3+252*A*a*c^2*d*m^2*n*x^n+639
*A*a*c^2*d*m*n^2*x^n+441*B*b*c^2*d*m*n^2*(x^n)^3+132*B*b*c*d^2*m*n*(x^n)^4+168*A*a*c^2*d*m^3*n*x^n+468*B*b*c^2
*d*m*n^3*(x^n)^3+234*A*b*c^2*d*m^2*n*(x^n)^2+531*A*b*c^2*d*m*n^2*(x^n)^2+144*A*b*c*d^2*m*n*(x^n)^3+234*B*a*c^2
*d*m^2*n*(x^n)^2+531*B*a*c^2*d*m*n^2*(x^n)^2+180*B*a*c^2*d*m*n^4*(x^n)^2+144*B*a*c*d^2*m^3*n*(x^n)^3+441*B*a*c
*d^2*m^2*n^2*(x^n)^3+468*B*a*c*d^2*m*n^3*(x^n)^3+144*B*b*c^2*d*m^3*n*(x^n)^3+441*B*b*c^2*d*m^2*n^2*(x^n)^3+360
*A*a*c^2*d*m*n^4*x^n+156*A*a*c*d^2*m^3*n*(x^n)^2+468*A*b*c*d^2*m*n^3*(x^n)^3+39*B*a*c^2*d*m^4*n*(x^n)^2+177*B*
a*c^2*d*m^3*n^2*(x^n)^2+321*B*a*c^2*d*m^2*n^3*(x^n)^2+366*B*b*c*d^2*m*n^3*(x^n)^4+39*A*a*c*d^2*m^4*n*(x^n)^2+1
20*B*a*c*d^2*m*n^4*(x^n)^3+36*B*b*c^2*d*m^4*n*(x^n)^3+147*B*b*c^2*d*m^3*n^2*(x^n)^3+234*B*b*c^2*d*m^2*n^3*(x^n
)^3+120*B*b*c^2*d*m*n^4*(x^n)^3+132*B*b*c*d^2*m^3*n*(x^n)^4+369*B*b*c*d^2*m^2*n^2*(x^n)^4+168*A*a*c^2*d*m*n*x^
n+147*A*b*c*d^2*m^3*n^2*(x^n)^3+234*A*b*c*d^2*m^2*n^3*(x^n)^3+120*A*b*c*d^2*m*n^4*(x^n)^3+36*B*a*c*d^2*m^4*n*(
x^n)^3+147*B*a*c*d^2*m^3*n^2*(x^n)^3+234*B*a*c*d^2*m^2*n^3*(x^n)^3+216*B*a*c*d^2*m^2*n*(x^n)^3+441*B*a*c*d^2*m
*n^2*(x^n)^3+216*B*b*c^2*d*m^2*n*(x^n)^3+531*A*a*c*d^2*m^2*n^2*(x^n)^2+156*A*a*c*d^2*m*n*(x^n)^2+156*A*b*c^2*d
*m*n*(x^n)^2+156*B*a*c^2*d*m*n*(x^n)^2+531*A*b*c^2*d*m^2*n^2*(x^n)^2+642*A*b*c^2*d*m*n^3*(x^n)^2+216*A*b*c*d^2
*m^2*n*(x^n)^3+441*A*b*c*d^2*m*n^2*(x^n)^3+156*B*a*c^2*d*m^3*n*(x^n)^2+531*B*a*c^2*d*m^2*n^2*(x^n)^2+642*B*a*c
^2*d*m*n^3*(x^n)^2+198*B*b*c*d^2*m^2*n*(x^n)^4+369*B*b*c*d^2*m*n^2*(x^n)^4+42*A*a*c^2*d*m^4*n*x^n+213*A*a*c^2*
d*m^3*n^2*x^n+462*A*a*c^2*d*m^2*n^3*x^n+642*A*a*c*d^2*m*n^3*(x^n)^2+156*A*b*c^2*d*m^3*n*(x^n)^2+33*B*b*c*d^2*m
^4*n*(x^n)^4+123*B*b*c*d^2*m^3*n^2*(x^n)^4+183*B*b*c*d^2*m^2*n^3*(x^n)^4+90*B*b*c*d^2*m*n^4*(x^n)^4+36*A*b*c*d
^2*m^4*n*(x^n)^3)/(m+1)/(m+n+1)/(m+2*n+1)/(m+3*n+1)/(m+4*n+1)/(1+m+5*n)*exp(1/2*(-I*Pi*csgn(I*e)*csgn(I*x)*csg
n(I*e*x)+I*Pi*csgn(I*e)*csgn(I*e*x)^2+I*Pi*csgn(I*x)*csgn(I*e*x)^2-I*Pi*csgn(I*e*x)^3+2*ln(e)+2*ln(x))*m)

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maxima [B]  time = 1.01, size = 464, normalized size = 2.21 \[ \frac {B b d^{3} e^{m} x e^{\left (m \log \relax (x) + 5 \, n \log \relax (x)\right )}}{m + 5 \, n + 1} + \frac {3 \, B b c d^{2} e^{m} x e^{\left (m \log \relax (x) + 4 \, n \log \relax (x)\right )}}{m + 4 \, n + 1} + \frac {B a d^{3} e^{m} x e^{\left (m \log \relax (x) + 4 \, n \log \relax (x)\right )}}{m + 4 \, n + 1} + \frac {A b d^{3} e^{m} x e^{\left (m \log \relax (x) + 4 \, n \log \relax (x)\right )}}{m + 4 \, n + 1} + \frac {3 \, B b c^{2} d e^{m} x e^{\left (m \log \relax (x) + 3 \, n \log \relax (x)\right )}}{m + 3 \, n + 1} + \frac {3 \, B a c d^{2} e^{m} x e^{\left (m \log \relax (x) + 3 \, n \log \relax (x)\right )}}{m + 3 \, n + 1} + \frac {3 \, A b c d^{2} e^{m} x e^{\left (m \log \relax (x) + 3 \, n \log \relax (x)\right )}}{m + 3 \, n + 1} + \frac {A a d^{3} e^{m} x e^{\left (m \log \relax (x) + 3 \, n \log \relax (x)\right )}}{m + 3 \, n + 1} + \frac {B b c^{3} e^{m} x e^{\left (m \log \relax (x) + 2 \, n \log \relax (x)\right )}}{m + 2 \, n + 1} + \frac {3 \, B a c^{2} d e^{m} x e^{\left (m \log \relax (x) + 2 \, n \log \relax (x)\right )}}{m + 2 \, n + 1} + \frac {3 \, A b c^{2} d e^{m} x e^{\left (m \log \relax (x) + 2 \, n \log \relax (x)\right )}}{m + 2 \, n + 1} + \frac {3 \, A a c d^{2} e^{m} x e^{\left (m \log \relax (x) + 2 \, n \log \relax (x)\right )}}{m + 2 \, n + 1} + \frac {B a c^{3} e^{m} x e^{\left (m \log \relax (x) + n \log \relax (x)\right )}}{m + n + 1} + \frac {A b c^{3} e^{m} x e^{\left (m \log \relax (x) + n \log \relax (x)\right )}}{m + n + 1} + \frac {3 \, A a c^{2} d e^{m} x e^{\left (m \log \relax (x) + n \log \relax (x)\right )}}{m + n + 1} + \frac {\left (e x\right )^{m + 1} A a c^{3}}{e {\left (m + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*b*d^3*e^m*x*e^(m*log(x) + 5*n*log(x))/(m + 5*n + 1) + 3*B*b*c*d^2*e^m*x*e^(m*log(x) + 4*n*log(x))/(m + 4*n +
1) + B*a*d^3*e^m*x*e^(m*log(x) + 4*n*log(x))/(m + 4*n + 1) + A*b*d^3*e^m*x*e^(m*log(x) + 4*n*log(x))/(m + 4*n
+ 1) + 3*B*b*c^2*d*e^m*x*e^(m*log(x) + 3*n*log(x))/(m + 3*n + 1) + 3*B*a*c*d^2*e^m*x*e^(m*log(x) + 3*n*log(x)
)/(m + 3*n + 1) + 3*A*b*c*d^2*e^m*x*e^(m*log(x) + 3*n*log(x))/(m + 3*n + 1) + A*a*d^3*e^m*x*e^(m*log(x) + 3*n*
log(x))/(m + 3*n + 1) + B*b*c^3*e^m*x*e^(m*log(x) + 2*n*log(x))/(m + 2*n + 1) + 3*B*a*c^2*d*e^m*x*e^(m*log(x)
+ 2*n*log(x))/(m + 2*n + 1) + 3*A*b*c^2*d*e^m*x*e^(m*log(x) + 2*n*log(x))/(m + 2*n + 1) + 3*A*a*c*d^2*e^m*x*e^
(m*log(x) + 2*n*log(x))/(m + 2*n + 1) + B*a*c^3*e^m*x*e^(m*log(x) + n*log(x))/(m + n + 1) + A*b*c^3*e^m*x*e^(m
*log(x) + n*log(x))/(m + n + 1) + 3*A*a*c^2*d*e^m*x*e^(m*log(x) + n*log(x))/(m + n + 1) + (e*x)^(m + 1)*A*a*c^
3/(e*(m + 1))

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mupad [B]  time = 5.65, size = 1089, normalized size = 5.19 \[ \frac {A\,a\,c^3\,x\,{\left (e\,x\right )}^m}{m+1}+\frac {d^2\,x\,x^{4\,n}\,{\left (e\,x\right )}^m\,\left (A\,b\,d+B\,a\,d+3\,B\,b\,c\right )\,\left (m^4+11\,m^3\,n+4\,m^3+41\,m^2\,n^2+33\,m^2\,n+6\,m^2+61\,m\,n^3+82\,m\,n^2+33\,m\,n+4\,m+30\,n^4+61\,n^3+41\,n^2+11\,n+1\right )}{m^5+15\,m^4\,n+5\,m^4+85\,m^3\,n^2+60\,m^3\,n+10\,m^3+225\,m^2\,n^3+255\,m^2\,n^2+90\,m^2\,n+10\,m^2+274\,m\,n^4+450\,m\,n^3+255\,m\,n^2+60\,m\,n+5\,m+120\,n^5+274\,n^4+225\,n^3+85\,n^2+15\,n+1}+\frac {c\,x\,x^{2\,n}\,{\left (e\,x\right )}^m\,\left (3\,A\,a\,d^2+B\,b\,c^2+3\,A\,b\,c\,d+3\,B\,a\,c\,d\right )\,\left (m^4+13\,m^3\,n+4\,m^3+59\,m^2\,n^2+39\,m^2\,n+6\,m^2+107\,m\,n^3+118\,m\,n^2+39\,m\,n+4\,m+60\,n^4+107\,n^3+59\,n^2+13\,n+1\right )}{m^5+15\,m^4\,n+5\,m^4+85\,m^3\,n^2+60\,m^3\,n+10\,m^3+225\,m^2\,n^3+255\,m^2\,n^2+90\,m^2\,n+10\,m^2+274\,m\,n^4+450\,m\,n^3+255\,m\,n^2+60\,m\,n+5\,m+120\,n^5+274\,n^4+225\,n^3+85\,n^2+15\,n+1}+\frac {d\,x\,x^{3\,n}\,{\left (e\,x\right )}^m\,\left (A\,a\,d^2+3\,B\,b\,c^2+3\,A\,b\,c\,d+3\,B\,a\,c\,d\right )\,\left (m^4+12\,m^3\,n+4\,m^3+49\,m^2\,n^2+36\,m^2\,n+6\,m^2+78\,m\,n^3+98\,m\,n^2+36\,m\,n+4\,m+40\,n^4+78\,n^3+49\,n^2+12\,n+1\right )}{m^5+15\,m^4\,n+5\,m^4+85\,m^3\,n^2+60\,m^3\,n+10\,m^3+225\,m^2\,n^3+255\,m^2\,n^2+90\,m^2\,n+10\,m^2+274\,m\,n^4+450\,m\,n^3+255\,m\,n^2+60\,m\,n+5\,m+120\,n^5+274\,n^4+225\,n^3+85\,n^2+15\,n+1}+\frac {c^2\,x\,x^n\,{\left (e\,x\right )}^m\,\left (3\,A\,a\,d+A\,b\,c+B\,a\,c\right )\,\left (m^4+14\,m^3\,n+4\,m^3+71\,m^2\,n^2+42\,m^2\,n+6\,m^2+154\,m\,n^3+142\,m\,n^2+42\,m\,n+4\,m+120\,n^4+154\,n^3+71\,n^2+14\,n+1\right )}{m^5+15\,m^4\,n+5\,m^4+85\,m^3\,n^2+60\,m^3\,n+10\,m^3+225\,m^2\,n^3+255\,m^2\,n^2+90\,m^2\,n+10\,m^2+274\,m\,n^4+450\,m\,n^3+255\,m\,n^2+60\,m\,n+5\,m+120\,n^5+274\,n^4+225\,n^3+85\,n^2+15\,n+1}+\frac {B\,b\,d^3\,x\,x^{5\,n}\,{\left (e\,x\right )}^m\,\left (m^4+10\,m^3\,n+4\,m^3+35\,m^2\,n^2+30\,m^2\,n+6\,m^2+50\,m\,n^3+70\,m\,n^2+30\,m\,n+4\,m+24\,n^4+50\,n^3+35\,n^2+10\,n+1\right )}{m^5+15\,m^4\,n+5\,m^4+85\,m^3\,n^2+60\,m^3\,n+10\,m^3+225\,m^2\,n^3+255\,m^2\,n^2+90\,m^2\,n+10\,m^2+274\,m\,n^4+450\,m\,n^3+255\,m\,n^2+60\,m\,n+5\,m+120\,n^5+274\,n^4+225\,n^3+85\,n^2+15\,n+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m*(A + B*x^n)*(a + b*x^n)*(c + d*x^n)^3,x)

[Out]

(A*a*c^3*x*(e*x)^m)/(m + 1) + (d^2*x*x^(4*n)*(e*x)^m*(A*b*d + B*a*d + 3*B*b*c)*(4*m + 11*n + 33*m*n + 82*m*n^2
+ 33*m^2*n + 61*m*n^3 + 11*m^3*n + 6*m^2 + 4*m^3 + m^4 + 41*n^2 + 61*n^3 + 30*n^4 + 41*m^2*n^2 + 1))/(5*m + 1
5*n + 60*m*n + 255*m*n^2 + 90*m^2*n + 450*m*n^3 + 60*m^3*n + 274*m*n^4 + 15*m^4*n + 10*m^2 + 10*m^3 + 5*m^4 +
m^5 + 85*n^2 + 225*n^3 + 274*n^4 + 120*n^5 + 255*m^2*n^2 + 225*m^2*n^3 + 85*m^3*n^2 + 1) + (c*x*x^(2*n)*(e*x)^
m*(3*A*a*d^2 + B*b*c^2 + 3*A*b*c*d + 3*B*a*c*d)*(4*m + 13*n + 39*m*n + 118*m*n^2 + 39*m^2*n + 107*m*n^3 + 13*m
^3*n + 6*m^2 + 4*m^3 + m^4 + 59*n^2 + 107*n^3 + 60*n^4 + 59*m^2*n^2 + 1))/(5*m + 15*n + 60*m*n + 255*m*n^2 + 9
0*m^2*n + 450*m*n^3 + 60*m^3*n + 274*m*n^4 + 15*m^4*n + 10*m^2 + 10*m^3 + 5*m^4 + m^5 + 85*n^2 + 225*n^3 + 274
*n^4 + 120*n^5 + 255*m^2*n^2 + 225*m^2*n^3 + 85*m^3*n^2 + 1) + (d*x*x^(3*n)*(e*x)^m*(A*a*d^2 + 3*B*b*c^2 + 3*A
*b*c*d + 3*B*a*c*d)*(4*m + 12*n + 36*m*n + 98*m*n^2 + 36*m^2*n + 78*m*n^3 + 12*m^3*n + 6*m^2 + 4*m^3 + m^4 + 4
9*n^2 + 78*n^3 + 40*n^4 + 49*m^2*n^2 + 1))/(5*m + 15*n + 60*m*n + 255*m*n^2 + 90*m^2*n + 450*m*n^3 + 60*m^3*n
+ 274*m*n^4 + 15*m^4*n + 10*m^2 + 10*m^3 + 5*m^4 + m^5 + 85*n^2 + 225*n^3 + 274*n^4 + 120*n^5 + 255*m^2*n^2 +
225*m^2*n^3 + 85*m^3*n^2 + 1) + (c^2*x*x^n*(e*x)^m*(3*A*a*d + A*b*c + B*a*c)*(4*m + 14*n + 42*m*n + 142*m*n^2
+ 42*m^2*n + 154*m*n^3 + 14*m^3*n + 6*m^2 + 4*m^3 + m^4 + 71*n^2 + 154*n^3 + 120*n^4 + 71*m^2*n^2 + 1))/(5*m +
15*n + 60*m*n + 255*m*n^2 + 90*m^2*n + 450*m*n^3 + 60*m^3*n + 274*m*n^4 + 15*m^4*n + 10*m^2 + 10*m^3 + 5*m^4
+ m^5 + 85*n^2 + 225*n^3 + 274*n^4 + 120*n^5 + 255*m^2*n^2 + 225*m^2*n^3 + 85*m^3*n^2 + 1) + (B*b*d^3*x*x^(5*n
)*(e*x)^m*(4*m + 10*n + 30*m*n + 70*m*n^2 + 30*m^2*n + 50*m*n^3 + 10*m^3*n + 6*m^2 + 4*m^3 + m^4 + 35*n^2 + 50
*n^3 + 24*n^4 + 35*m^2*n^2 + 1))/(5*m + 15*n + 60*m*n + 255*m*n^2 + 90*m^2*n + 450*m*n^3 + 60*m^3*n + 274*m*n^
4 + 15*m^4*n + 10*m^2 + 10*m^3 + 5*m^4 + m^5 + 85*n^2 + 225*n^3 + 274*n^4 + 120*n^5 + 255*m^2*n^2 + 225*m^2*n^
3 + 85*m^3*n^2 + 1)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**m*(a+b*x**n)*(A+B*x**n)*(c+d*x**n)**3,x)

[Out]

Timed out

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