3.1.2 \(\int (e x)^m (a+b x^n)^2 (A+B x^n) (c+d x^n) \, dx\) [2]

Optimal. Leaf size=160 \[ \frac {a (2 A b c+a B c+a A d) x^{1+n} (e x)^m}{1+m+n}+\frac {(a B (2 b c+a d)+A b (b c+2 a d)) x^{1+2 n} (e x)^m}{1+m+2 n}+\frac {b (b B c+A b d+2 a B d) x^{1+3 n} (e x)^m}{1+m+3 n}+\frac {b^2 B d x^{1+4 n} (e x)^m}{1+m+4 n}+\frac {a^2 A c (e x)^{1+m}}{e (1+m)} \]

[Out]

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

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Rubi [A]
time = 0.11, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {584, 20, 30} \begin {gather*} \frac {a^2 A c (e x)^{m+1}}{e (m+1)}+\frac {a x^{n+1} (e x)^m (a A d+a B c+2 A b c)}{m+n+1}+\frac {x^{2 n+1} (e x)^m (A b (2 a d+b c)+a B (a d+2 b c))}{m+2 n+1}+\frac {b x^{3 n+1} (e x)^m (2 a B d+A b d+b B c)}{m+3 n+1}+\frac {b^2 B d x^{4 n+1} (e x)^m}{m+4 n+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(e*x)^m*(a + b*x^n)^2*(A + B*x^n)*(c + d*x^n),x]

[Out]

(a*(2*A*b*c + a*B*c + a*A*d)*x^(1 + n)*(e*x)^m)/(1 + m + n) + ((a*B*(2*b*c + a*d) + A*b*(b*c + 2*a*d))*x^(1 +
2*n)*(e*x)^m)/(1 + m + 2*n) + (b*(b*B*c + A*b*d + 2*a*B*d)*x^(1 + 3*n)*(e*x)^m)/(1 + m + 3*n) + (b^2*B*d*x^(1
+ 4*n)*(e*x)^m)/(1 + m + 4*n) + (a^2*A*c*(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 584

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

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Mathematica [A]
time = 0.32, size = 129, normalized size = 0.81 \begin {gather*} x (e x)^m \left (\frac {a^2 A c}{1+m}+\frac {a (2 A b c+a B c+a A d) x^n}{1+m+n}+\frac {(a B (2 b c+a d)+A b (b c+2 a d)) x^{2 n}}{1+m+2 n}+\frac {b (b B c+A b d+2 a B d) x^{3 n}}{1+m+3 n}+\frac {b^2 B d x^{4 n}}{1+m+4 n}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(e*x)^m*(a + b*x^n)^2*(A + B*x^n)*(c + d*x^n),x]

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.38, size = 2410, normalized size = 15.06

method	result	size
risch	\(\text {Expression too large to display}\)	\(2410\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m*(a+b*x^n)^2*(A+B*x^n)*(c+d*x^n),x,method=_RETURNVERBOSE)

[Out]

x*(38*B*a^2*d*m*n^2*(x^n)^2+8*B*a*b*c*m^3*(x^n)^2+24*A*b^2*c*m*n*(x^n)^2+27*B*a^2*c*m^2*n*x^n+52*B*a^2*c*m*n^2
*x^n+24*B*a^2*d*m*n*(x^n)^2+19*B*a^2*d*m^2*n^2*(x^n)^2+12*A*a*b*d*m^2*(x^n)^2+38*A*a*b*d*n^2*(x^n)^2+2*x^n*c*a
*b*A+8*B*a^2*d*m^3*n*(x^n)^2+A*a^2*d*x^n+B*a^2*c*x^n+30*A*a^2*c*m^2*n+70*A*a^2*c*m*n^2+30*A*a^2*c*m*n+21*B*b^2
*c*m^2*n*(x^n)^3+28*B*b^2*c*m*n^2*(x^n)^3+18*B*b^2*d*m*n*(x^n)^4+9*A*a^2*d*m^3*n*x^n+26*A*a^2*d*m^2*n^2*x^n+24
*A*a*b*d*m*n^3*(x^n)^2+16*B*a*b*c*m^3*n*(x^n)^2+38*B*a*b*c*m^2*n^2*(x^n)^2+a^2*A*c+48*B*a*b*c*m*n*(x^n)^2+54*A
*a*b*c*m*n*x^n+16*A*a*b*d*m^3*n*(x^n)^2+48*A*a*b*d*m*n*(x^n)^2+(x^n)^2*A*b^2*c+(x^n)^2*B*a^2*d+24*B*a^2*c*m*n^
3*x^n+24*B*a^2*d*m^2*n*(x^n)^2+24*A*a^2*d*n^3*x^n+6*A*b^2*c*m^2*(x^n)^2+19*A*b^2*c*n^2*(x^n)^2+4*A*b^2*d*(x^n)
^3*m+7*A*b^2*d*(x^n)^3*n+4*B*a^2*c*m^3*x^n+56*B*a*b*d*m*n^2*(x^n)^3+18*A*a*b*c*m^3*n*x^n+24*B*a*b*c*m*n^3*(x^n
)^2+42*B*a*b*d*m^2*n*(x^n)^3+(x^n)^3*b^2*B*c+6*A*a^2*c*m^2+14*B*a*b*d*m^3*n*(x^n)^3+28*B*a*b*d*m^2*n^2*(x^n)^3
+16*B*a*b*d*m*n^3*(x^n)^3+b^2*B*d*(x^n)^4+52*A*a*b*c*m^2*n^2*x^n+26*B*a^2*c*n^2*x^n+4*B*a^2*d*(x^n)^2*m+8*B*a^
2*d*(x^n)^2*n+B*b^2*d*m^4*(x^n)^4+38*A*a*b*d*m^2*n^2*(x^n)^2+A*a^2*c*m^4+4*A*a^2*c*m^3+50*A*a^2*c*n^3+35*A*a^2
*c*n^2+11*B*b^2*d*m^2*n^2*(x^n)^4+B*a^2*d*m^4*(x^n)^2+4*B*b^2*c*m^3*(x^n)^3+8*B*b^2*c*n^3*(x^n)^3+6*B*b^2*d*m^
2*(x^n)^4+11*B*b^2*d*n^2*(x^n)^4+A*a^2*d*m^4*x^n+27*A*a^2*d*m*n*x^n+12*A*a*b*c*m^2*x^n+26*B*a^2*c*m^2*n^2*x^n+
104*A*a*b*c*m*n^2*x^n+4*a^2*A*c*m+10*a^2*A*c*n+(x^n)^3*A*b^2*d+54*A*a*b*c*m^2*n*x^n+24*A*a^2*c*n^4+24*B*a*b*c*
n^3*(x^n)^2+12*B*a*b*d*m^2*(x^n)^3+28*B*a*b*d*n^2*(x^n)^3+16*B*a*b*d*n^3*(x^n)^3+12*B*a*b*c*m^2*(x^n)^2+38*B*a
*b*c*n^2*(x^n)^2+8*B*a*b*d*(x^n)^3*m+6*B*b^2*d*m^3*n*(x^n)^4+14*B*a*b*d*(x^n)^3*n+16*A*a*b*d*(x^n)^2*n+27*B*a^
2*c*m*n*x^n+8*B*a*b*c*(x^n)^2*m+16*B*a*b*c*(x^n)^2*n+6*B*b^2*d*m*n^3*(x^n)^4+7*A*b^2*d*m^3*n*(x^n)^3+14*A*b^2*
d*m^2*n^2*(x^n)^3+8*A*b^2*d*m*n^3*(x^n)^3+48*A*a*b*c*m*n^3*x^n+52*A*a*b*c*n^2*x^n+8*A*a*b*d*(x^n)^2*m+28*A*b^2
*d*m*n^2*(x^n)^3+4*B*a^2*d*m^3*(x^n)^2+12*B*a^2*d*n^3*(x^n)^2+6*B*b^2*c*m^2*(x^n)^3+14*B*b^2*c*n^2*(x^n)^3+4*m
*b^2*B*d*(x^n)^4+6*b^2*B*d*(x^n)^4*n+4*A*a^2*d*m^3*x^n+4*A*a^2*d*x^n*m+9*A*a^2*d*x^n*n+4*B*a^2*c*x^n*m+48*A*a*
b*c*n^3*x^n+7*B*b^2*c*m^3*n*(x^n)^3+14*B*b^2*c*m^2*n^2*(x^n)^3+8*B*b^2*c*m*n^3*(x^n)^3+8*A*a*b*c*x^n*m+18*A*a*
b*c*x^n*n+2*B*a*b*d*m^4*(x^n)^3+38*A*b^2*c*m*n^2*(x^n)^2+21*A*b^2*d*m*n*(x^n)^3+9*B*a^2*c*m^3*n*x^n+2*B*a*b*c*
m^4*(x^n)^2+8*B*a*b*d*m^3*(x^n)^3+24*A*a*b*d*n^3*(x^n)^2+24*A*b^2*c*m^2*n*(x^n)^2+2*A*a*b*d*m^4*(x^n)^2+12*B*a
^2*d*m*n^3*(x^n)^2+19*A*b^2*c*m^2*n^2*(x^n)^2+12*A*b^2*c*m*n^3*(x^n)^2+21*A*b^2*d*m^2*n*(x^n)^3+18*B*b^2*d*m^2
*n*(x^n)^4+22*B*b^2*d*m*n^2*(x^n)^4+24*A*a^2*d*m*n^3*x^n+2*A*a*b*c*m^4*x^n+8*A*a*b*d*m^3*(x^n)^2+8*A*b^2*c*m^3
*n*(x^n)^2+76*B*a*b*c*m*n^2*(x^n)^2+42*B*a*b*d*m*n*(x^n)^3+76*A*a*b*d*m*n^2*(x^n)^2+48*B*a*b*c*m^2*n*(x^n)^2+4
8*A*a*b*d*m^2*n*(x^n)^2+21*B*b^2*c*m*n*(x^n)^3+27*A*a^2*d*m^2*n*x^n+52*A*a^2*d*m*n^2*x^n+8*A*a*b*c*m^3*x^n+10*
A*a^2*c*m^3*n+35*A*a^2*c*m^2*n^2+50*A*a^2*c*m*n^3+9*B*a^2*c*x^n*n+24*B*a^2*c*n^3*x^n+6*B*a^2*d*m^2*(x^n)^2+19*
B*a^2*d*n^2*(x^n)^2+A*b^2*d*m^4*(x^n)^3+B*b^2*c*m^4*(x^n)^3+4*B*b^2*d*m^3*(x^n)^4+6*B*b^2*d*n^3*(x^n)^4+A*b^2*
c*m^4*(x^n)^2+4*A*b^2*d*m^3*(x^n)^3+8*A*b^2*d*n^3*(x^n)^3+2*(x^n)^2*A*a*b*d+2*(x^n)^2*B*a*b*c+2*(x^n)^3*B*a*b*
d+4*A*b^2*c*m^3*(x^n)^2+12*A*b^2*c*n^3*(x^n)^2+6*A*b^2*d*m^2*(x^n)^3+14*A*b^2*d*n^2*(x^n)^3+B*a^2*c*m^4*x^n+4*
B*b^2*c*(x^n)^3*m+7*B*b^2*c*(x^n)^3*n+6*A*a^2*d*m^2*x^n+26*A*a^2*d*n^2*x^n+4*A*b^2*c*(x^n)^2*m+8*A*b^2*c*(x^n)
^2*n+6*B*a^2*c*m^2*x^n)/(1+m)/(1+m+n)/(1+m+2*n)/(1+m+3*n)/(1+m+4*n)*exp(1/2*m*(-I*Pi*csgn(I*e*x)^3+I*Pi*csgn(I
*e*x)^2*csgn(I*e)+I*Pi*csgn(I*e*x)^2*csgn(I*x)-I*Pi*csgn(I*e*x)*csgn(I*e)*csgn(I*x)+2*ln(e)+2*ln(x)))

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Maxima [A]
time = 0.34, size = 310, normalized size = 1.94 \begin {gather*} \frac {\left (x e\right )^{m + 1} A a^{2} c e^{\left (-1\right )}}{m + 1} + \frac {B b^{2} d x e^{\left (m \log \left (x\right ) + 4 \, n \log \left (x\right ) + m\right )}}{m + 4 \, n + 1} + \frac {B b^{2} c x e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right ) + m\right )}}{m + 3 \, n + 1} + \frac {2 \, B a b d x e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right ) + m\right )}}{m + 3 \, n + 1} + \frac {A b^{2} d x e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right ) + m\right )}}{m + 3 \, n + 1} + \frac {2 \, B a b c x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right ) + m\right )}}{m + 2 \, n + 1} + \frac {A b^{2} c x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right ) + m\right )}}{m + 2 \, n + 1} + \frac {B a^{2} d x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right ) + m\right )}}{m + 2 \, n + 1} + \frac {2 \, A a b d x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right ) + m\right )}}{m + 2 \, n + 1} + \frac {B a^{2} c x e^{\left (m \log \left (x\right ) + n \log \left (x\right ) + m\right )}}{m + n + 1} + \frac {2 \, A a b c x e^{\left (m \log \left (x\right ) + n \log \left (x\right ) + m\right )}}{m + n + 1} + \frac {A a^{2} d x e^{\left (m \log \left (x\right ) + n \log \left (x\right ) + m\right )}}{m + n + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*(a+b*x^n)^2*(A+B*x^n)*(c+d*x^n),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1509 vs. \(2 (164) = 328\).
time = 2.33, size = 1509, normalized size = 9.43 \begin {gather*} \frac {{\left (B b^{2} d m^{4} + 4 \, B b^{2} d m^{3} + 6 \, B b^{2} d m^{2} + 4 \, B b^{2} d m + B b^{2} d + 6 \, {\left (B b^{2} d m + B b^{2} d\right )} n^{3} + 11 \, {\left (B b^{2} d m^{2} + 2 \, B b^{2} d m + B b^{2} d\right )} n^{2} + 6 \, {\left (B b^{2} d m^{3} + 3 \, B b^{2} d m^{2} + 3 \, B b^{2} d m + B b^{2} d\right )} n\right )} x x^{4 \, n} e^{\left (m \log \left (x\right ) + m\right )} + {\left ({\left (B b^{2} c + {\left (2 \, B a b + A b^{2}\right )} d\right )} m^{4} + B b^{2} c + 4 \, {\left (B b^{2} c + {\left (2 \, B a b + A b^{2}\right )} d\right )} m^{3} + 8 \, {\left (B b^{2} c + {\left (2 \, B a b + A b^{2}\right )} d + {\left (B b^{2} c + {\left (2 \, B a b + A b^{2}\right )} d\right )} m\right )} n^{3} + 6 \, {\left (B b^{2} c + {\left (2 \, B a b + A b^{2}\right )} d\right )} m^{2} + 14 \, {\left (B b^{2} c + {\left (B b^{2} c + {\left (2 \, B a b + A b^{2}\right )} d\right )} m^{2} + {\left (2 \, B a b + A b^{2}\right )} d + 2 \, {\left (B b^{2} c + {\left (2 \, B a b + A b^{2}\right )} d\right )} m\right )} n^{2} + {\left (2 \, B a b + A b^{2}\right )} d + 4 \, {\left (B b^{2} c + {\left (2 \, B a b + A b^{2}\right )} d\right )} m + 7 \, {\left (B b^{2} c + {\left (B b^{2} c + {\left (2 \, B a b + A b^{2}\right )} d\right )} m^{3} + 3 \, {\left (B b^{2} c + {\left (2 \, B a b + A b^{2}\right )} d\right )} m^{2} + {\left (2 \, B a b + A b^{2}\right )} d + 3 \, {\left (B b^{2} c + {\left (2 \, B a b + A b^{2}\right )} d\right )} m\right )} n\right )} x x^{3 \, n} e^{\left (m \log \left (x\right ) + m\right )} + {\left ({\left ({\left (2 \, B a b + A b^{2}\right )} c + {\left (B a^{2} + 2 \, A a b\right )} d\right )} m^{4} + 4 \, {\left ({\left (2 \, B a b + A b^{2}\right )} c + {\left (B a^{2} + 2 \, A a b\right )} d\right )} m^{3} + 12 \, {\left ({\left (2 \, B a b + A b^{2}\right )} c + {\left (B a^{2} + 2 \, A a b\right )} d + {\left ({\left (2 \, B a b + A b^{2}\right )} c + {\left (B a^{2} + 2 \, A a b\right )} d\right )} m\right )} n^{3} + 6 \, {\left ({\left (2 \, B a b + A b^{2}\right )} c + {\left (B a^{2} + 2 \, A a b\right )} d\right )} m^{2} + 19 \, {\left ({\left ({\left (2 \, B a b + A b^{2}\right )} c + {\left (B a^{2} + 2 \, A a b\right )} d\right )} m^{2} + {\left (2 \, B a b + A b^{2}\right )} c + {\left (B a^{2} + 2 \, A a b\right )} d + 2 \, {\left ({\left (2 \, B a b + A b^{2}\right )} c + {\left (B a^{2} + 2 \, A a b\right )} d\right )} m\right )} n^{2} + {\left (2 \, B a b + A b^{2}\right )} c + {\left (B a^{2} + 2 \, A a b\right )} d + 4 \, {\left ({\left (2 \, B a b + A b^{2}\right )} c + {\left (B a^{2} + 2 \, A a b\right )} d\right )} m + 8 \, {\left ({\left ({\left (2 \, B a b + A b^{2}\right )} c + {\left (B a^{2} + 2 \, A a b\right )} d\right )} m^{3} + 3 \, {\left ({\left (2 \, B a b + A b^{2}\right )} c + {\left (B a^{2} + 2 \, A a b\right )} d\right )} m^{2} + {\left (2 \, B a b + A b^{2}\right )} c + {\left (B a^{2} + 2 \, A a b\right )} d + 3 \, {\left ({\left (2 \, B a b + A b^{2}\right )} c + {\left (B a^{2} + 2 \, A a b\right )} d\right )} m\right )} n\right )} x x^{2 \, n} e^{\left (m \log \left (x\right ) + m\right )} + {\left ({\left (A a^{2} d + {\left (B a^{2} + 2 \, A a b\right )} c\right )} m^{4} + A a^{2} d + 4 \, {\left (A a^{2} d + {\left (B a^{2} + 2 \, A a b\right )} c\right )} m^{3} + 24 \, {\left (A a^{2} d + {\left (B a^{2} + 2 \, A a b\right )} c + {\left (A a^{2} d + {\left (B a^{2} + 2 \, A a b\right )} c\right )} m\right )} n^{3} + 6 \, {\left (A a^{2} d + {\left (B a^{2} + 2 \, A a b\right )} c\right )} m^{2} + 26 \, {\left (A a^{2} d + {\left (A a^{2} d + {\left (B a^{2} + 2 \, A a b\right )} c\right )} m^{2} + {\left (B a^{2} + 2 \, A a b\right )} c + 2 \, {\left (A a^{2} d + {\left (B a^{2} + 2 \, A a b\right )} c\right )} m\right )} n^{2} + {\left (B a^{2} + 2 \, A a b\right )} c + 4 \, {\left (A a^{2} d + {\left (B a^{2} + 2 \, A a b\right )} c\right )} m + 9 \, {\left (A a^{2} d + {\left (A a^{2} d + {\left (B a^{2} + 2 \, A a b\right )} c\right )} m^{3} + 3 \, {\left (A a^{2} d + {\left (B a^{2} + 2 \, A a b\right )} c\right )} m^{2} + {\left (B a^{2} + 2 \, A a b\right )} c + 3 \, {\left (A a^{2} d + {\left (B a^{2} + 2 \, A a b\right )} c\right )} m\right )} n\right )} x x^{n} e^{\left (m \log \left (x\right ) + m\right )} + {\left (A a^{2} c m^{4} + 24 \, A a^{2} c n^{4} + 4 \, A a^{2} c m^{3} + 6 \, A a^{2} c m^{2} + 4 \, A a^{2} c m + A a^{2} c + 50 \, {\left (A a^{2} c m + A a^{2} c\right )} n^{3} + 35 \, {\left (A a^{2} c m^{2} + 2 \, A a^{2} c m + A a^{2} c\right )} n^{2} + 10 \, {\left (A a^{2} c m^{3} + 3 \, A a^{2} c m^{2} + 3 \, A a^{2} c m + A a^{2} c\right )} n\right )} x e^{\left (m \log \left (x\right ) + m\right )}}{m^{5} + 24 \, {\left (m + 1\right )} n^{4} + 5 \, m^{4} + 50 \, {\left (m^{2} + 2 \, m + 1\right )} n^{3} + 10 \, m^{3} + 35 \, {\left (m^{3} + 3 \, m^{2} + 3 \, m + 1\right )} n^{2} + 10 \, m^{2} + 10 \, {\left (m^{4} + 4 \, m^{3} + 6 \, m^{2} + 4 \, m + 1\right )} n + 5 \, m + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*(a+b*x^n)^2*(A+B*x^n)*(c+d*x^n),x, algorithm="fricas")

[Out]

((B*b^2*d*m^4 + 4*B*b^2*d*m^3 + 6*B*b^2*d*m^2 + 4*B*b^2*d*m + B*b^2*d + 6*(B*b^2*d*m + B*b^2*d)*n^3 + 11*(B*b^
2*d*m^2 + 2*B*b^2*d*m + B*b^2*d)*n^2 + 6*(B*b^2*d*m^3 + 3*B*b^2*d*m^2 + 3*B*b^2*d*m + B*b^2*d)*n)*x*x^(4*n)*e^
(m*log(x) + m) + ((B*b^2*c + (2*B*a*b + A*b^2)*d)*m^4 + B*b^2*c + 4*(B*b^2*c + (2*B*a*b + A*b^2)*d)*m^3 + 8*(B
*b^2*c + (2*B*a*b + A*b^2)*d + (B*b^2*c + (2*B*a*b + A*b^2)*d)*m)*n^3 + 6*(B*b^2*c + (2*B*a*b + A*b^2)*d)*m^2
+ 14*(B*b^2*c + (B*b^2*c + (2*B*a*b + A*b^2)*d)*m^2 + (2*B*a*b + A*b^2)*d + 2*(B*b^2*c + (2*B*a*b + A*b^2)*d)*
m)*n^2 + (2*B*a*b + A*b^2)*d + 4*(B*b^2*c + (2*B*a*b + A*b^2)*d)*m + 7*(B*b^2*c + (B*b^2*c + (2*B*a*b + A*b^2)
*d)*m^3 + 3*(B*b^2*c + (2*B*a*b + A*b^2)*d)*m^2 + (2*B*a*b + A*b^2)*d + 3*(B*b^2*c + (2*B*a*b + A*b^2)*d)*m)*n
)*x*x^(3*n)*e^(m*log(x) + m) + (((2*B*a*b + A*b^2)*c + (B*a^2 + 2*A*a*b)*d)*m^4 + 4*((2*B*a*b + A*b^2)*c + (B*
a^2 + 2*A*a*b)*d)*m^3 + 12*((2*B*a*b + A*b^2)*c + (B*a^2 + 2*A*a*b)*d + ((2*B*a*b + A*b^2)*c + (B*a^2 + 2*A*a*
b)*d)*m)*n^3 + 6*((2*B*a*b + A*b^2)*c + (B*a^2 + 2*A*a*b)*d)*m^2 + 19*(((2*B*a*b + A*b^2)*c + (B*a^2 + 2*A*a*b
)*d)*m^2 + (2*B*a*b + A*b^2)*c + (B*a^2 + 2*A*a*b)*d + 2*((2*B*a*b + A*b^2)*c + (B*a^2 + 2*A*a*b)*d)*m)*n^2 +
(2*B*a*b + A*b^2)*c + (B*a^2 + 2*A*a*b)*d + 4*((2*B*a*b + A*b^2)*c + (B*a^2 + 2*A*a*b)*d)*m + 8*(((2*B*a*b + A
*b^2)*c + (B*a^2 + 2*A*a*b)*d)*m^3 + 3*((2*B*a*b + A*b^2)*c + (B*a^2 + 2*A*a*b)*d)*m^2 + (2*B*a*b + A*b^2)*c +
(B*a^2 + 2*A*a*b)*d + 3*((2*B*a*b + A*b^2)*c + (B*a^2 + 2*A*a*b)*d)*m)*n)*x*x^(2*n)*e^(m*log(x) + m) + ((A*a^
2*d + (B*a^2 + 2*A*a*b)*c)*m^4 + A*a^2*d + 4*(A*a^2*d + (B*a^2 + 2*A*a*b)*c)*m^3 + 24*(A*a^2*d + (B*a^2 + 2*A*
a*b)*c + (A*a^2*d + (B*a^2 + 2*A*a*b)*c)*m)*n^3 + 6*(A*a^2*d + (B*a^2 + 2*A*a*b)*c)*m^2 + 26*(A*a^2*d + (A*a^2
*d + (B*a^2 + 2*A*a*b)*c)*m^2 + (B*a^2 + 2*A*a*b)*c + 2*(A*a^2*d + (B*a^2 + 2*A*a*b)*c)*m)*n^2 + (B*a^2 + 2*A*
a*b)*c + 4*(A*a^2*d + (B*a^2 + 2*A*a*b)*c)*m + 9*(A*a^2*d + (A*a^2*d + (B*a^2 + 2*A*a*b)*c)*m^3 + 3*(A*a^2*d +
(B*a^2 + 2*A*a*b)*c)*m^2 + (B*a^2 + 2*A*a*b)*c + 3*(A*a^2*d + (B*a^2 + 2*A*a*b)*c)*m)*n)*x*x^n*e^(m*log(x) +
m) + (A*a^2*c*m^4 + 24*A*a^2*c*n^4 + 4*A*a^2*c*m^3 + 6*A*a^2*c*m^2 + 4*A*a^2*c*m + A*a^2*c + 50*(A*a^2*c*m + A
*a^2*c)*n^3 + 35*(A*a^2*c*m^2 + 2*A*a^2*c*m + A*a^2*c)*n^2 + 10*(A*a^2*c*m^3 + 3*A*a^2*c*m^2 + 3*A*a^2*c*m + A
*a^2*c)*n)*x*e^(m*log(x) + m))/(m^5 + 24*(m + 1)*n^4 + 5*m^4 + 50*(m^2 + 2*m + 1)*n^3 + 10*m^3 + 35*(m^3 + 3*m
^2 + 3*m + 1)*n^2 + 10*m^2 + 10*(m^4 + 4*m^3 + 6*m^2 + 4*m + 1)*n + 5*m + 1)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 24932 vs. \(2 (156) = 312\).
time = 88.12, size = 24932, normalized size = 155.82 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**m*(a+b*x**n)**2*(A+B*x**n)*(c+d*x**n),x)

[Out]

Piecewise(((A + B)*(a + b)**2*(c + d)*log(x)/e, Eq(m, -1) & Eq(n, 0)), ((A*a**2*c*log(x) + A*a**2*d*x**n/n + 2
*A*a*b*c*x**n/n + A*a*b*d*x**(2*n)/n + A*b**2*c*x**(2*n)/(2*n) + A*b**2*d*x**(3*n)/(3*n) + B*a**2*c*x**n/n + B
*a**2*d*x**(2*n)/(2*n) + B*a*b*c*x**(2*n)/n + 2*B*a*b*d*x**(3*n)/(3*n) + B*b**2*c*x**(3*n)/(3*n) + B*b**2*d*x*
*(4*n)/(4*n))/e, Eq(m, -1)), (A*a**2*c*Piecewise((-1/(4*n*(e*x)**(4*n)), Ne(n, 0)), (log(x), True))/e + A*a**2
*d*Piecewise((-x**n/(3*n*(e*x)**(4*n)), Ne(n, 0)), (log(x), True))/e + 2*A*a*b*c*Piecewise((-x**n/(3*n*(e*x)**
(4*n)), Ne(n, 0)), (log(x), True))/e + 2*A*a*b*d*Piecewise((-x**(2*n)/(2*n*(e*x)**(4*n)), Ne(n, 0)), (log(x),
True))/e + A*b**2*c*Piecewise((-x**(2*n)/(2*n*(e*x)**(4*n)), Ne(n, 0)), (log(x), True))/e + A*b**2*d*Piecewise
((-x**(3*n)/(n*(e*x)**(4*n)), Ne(n, 0)), (log(x), True))/e + B*a**2*c*Piecewise((-x**n/(3*n*(e*x)**(4*n)), Ne(
n, 0)), (log(x), True))/e + B*a**2*d*Piecewise((-x**(2*n)/(2*n*(e*x)**(4*n)), Ne(n, 0)), (log(x), True))/e + 2
*B*a*b*c*Piecewise((-x**(2*n)/(2*n*(e*x)**(4*n)), Ne(n, 0)), (log(x), True))/e + 2*B*a*b*d*Piecewise((-x**(3*n
)/(n*(e*x)**(4*n)), Ne(n, 0)), (log(x), True))/e + B*b**2*c*Piecewise((-x**(3*n)/(n*(e*x)**(4*n)), Ne(n, 0)),
(log(x), True))/e + B*b**2*d*Piecewise((0, (Abs(x) < 1) & (1/Abs(x) < 1)), (log(x)/e**(4*n), Abs(x) < 1), (-lo
g(1/x)/e**(4*n), 1/Abs(x) < 1), (-meijerg(((), (1, 1)), ((0, 0), ()), x)/e**(4*n) + meijerg(((1, 1), ()), ((),
(0, 0)), x)/e**(4*n), True))/e, Eq(m, -4*n - 1)), (A*a**2*c*Piecewise((-1/(3*n*(e*x)**(3*n)), Ne(n, 0)), (log
(x), True))/e + A*a**2*d*Piecewise((-x**n/(2*n*(e*x)**(3*n)), Ne(n, 0)), (log(x), True))/e + 2*A*a*b*c*Piecewi
se((-x**n/(2*n*(e*x)**(3*n)), Ne(n, 0)), (log(x), True))/e + 2*A*a*b*d*Piecewise((-x**(2*n)/(n*(e*x)**(3*n)),
Ne(n, 0)), (log(x), True))/e + A*b**2*c*Piecewise((-x**(2*n)/(n*(e*x)**(3*n)), Ne(n, 0)), (log(x), True))/e +
A*b**2*d*Piecewise((0, (Abs(x) < 1) & (1/Abs(x) < 1)), (log(x)/e**(3*n), Abs(x) < 1), (-log(1/x)/e**(3*n), 1/A
bs(x) < 1), (-meijerg(((), (1, 1)), ((0, 0), ()), x)/e**(3*n) + meijerg(((1, 1), ()), ((), (0, 0)), x)/e**(3*n
), True))/e + B*a**2*c*Piecewise((-x**n/(2*n*(e*x)**(3*n)), Ne(n, 0)), (log(x), True))/e + B*a**2*d*Piecewise(
(-x**(2*n)/(n*(e*x)**(3*n)), Ne(n, 0)), (log(x), True))/e + 2*B*a*b*c*Piecewise((-x**(2*n)/(n*(e*x)**(3*n)), N
e(n, 0)), (log(x), True))/e + 2*B*a*b*d*Piecewise((0, (Abs(x) < 1) & (1/Abs(x) < 1)), (log(x)/e**(3*n), Abs(x)
< 1), (-log(1/x)/e**(3*n), 1/Abs(x) < 1), (-meijerg(((), (1, 1)), ((0, 0), ()), x)/e**(3*n) + meijerg(((1, 1)
, ()), ((), (0, 0)), x)/e**(3*n), True))/e + B*b**2*c*Piecewise((0, (Abs(x) < 1) & (1/Abs(x) < 1)), (log(x)/e*
*(3*n), Abs(x) < 1), (-log(1/x)/e**(3*n), 1/Abs(x) < 1), (-meijerg(((), (1, 1)), ((0, 0), ()), x)/e**(3*n) + m
eijerg(((1, 1), ()), ((), (0, 0)), x)/e**(3*n), True))/e + B*b**2*d*Piecewise((x**(4*n)/(n*(e*x)**(3*n)), Ne(n
, 0)), (log(x), True))/e, Eq(m, -3*n - 1)), (A*a**2*c*Piecewise((-1/(2*n*(e*x)**(2*n)), Ne(n, 0)), (log(x), Tr
ue))/e + A*a**2*d*Piecewise((-x**n/(n*(e*x)**(2*n)), Ne(n, 0)), (log(x), True))/e + 2*A*a*b*c*Piecewise((-x**n
/(n*(e*x)**(2*n)), Ne(n, 0)), (log(x), True))/e + 2*A*a*b*d*Piecewise((0, (Abs(x) < 1) & (1/Abs(x) < 1)), (log
(x)/e**(2*n), Abs(x) < 1), (-log(1/x)/e**(2*n), 1/Abs(x) < 1), (-meijerg(((), (1, 1)), ((0, 0), ()), x)/e**(2*
n) + meijerg(((1, 1), ()), ((), (0, 0)), x)/e**(2*n), True))/e + A*b**2*c*Piecewise((0, (Abs(x) < 1) & (1/Abs(
x) < 1)), (log(x)/e**(2*n), Abs(x) < 1), (-log(1/x)/e**(2*n), 1/Abs(x) < 1), (-meijerg(((), (1, 1)), ((0, 0),
()), x)/e**(2*n) + meijerg(((1, 1), ()), ((), (0, 0)), x)/e**(2*n), True))/e + A*b**2*d*Piecewise((x**(3*n)/(n
*(e*x)**(2*n)), Ne(n, 0)), (log(x), True))/e + B*a**2*c*Piecewise((-x**n/(n*(e*x)**(2*n)), Ne(n, 0)), (log(x),
True))/e + B*a**2*d*Piecewise((0, (Abs(x) < 1) & (1/Abs(x) < 1)), (log(x)/e**(2*n), Abs(x) < 1), (-log(1/x)/e
**(2*n), 1/Abs(x) < 1), (-meijerg(((), (1, 1)), ((0, 0), ()), x)/e**(2*n) + meijerg(((1, 1), ()), ((), (0, 0))
, x)/e**(2*n), True))/e + 2*B*a*b*c*Piecewise((0, (Abs(x) < 1) & (1/Abs(x) < 1)), (log(x)/e**(2*n), Abs(x) < 1
), (-log(1/x)/e**(2*n), 1/Abs(x) < 1), (-meijerg(((), (1, 1)), ((0, 0), ()), x)/e**(2*n) + meijerg(((1, 1), ()
), ((), (0, 0)), x)/e**(2*n), True))/e + 2*B*a*b*d*Piecewise((x**(3*n)/(n*(e*x)**(2*n)), Ne(n, 0)), (log(x), T
rue))/e + B*b**2*c*Piecewise((x**(3*n)/(n*(e*x)**(2*n)), Ne(n, 0)), (log(x), True))/e + B*b**2*d*Piecewise((x*
*(4*n)/(2*n*(e*x)**(2*n)), Ne(n, 0)), (log(x), True))/e, Eq(m, -2*n - 1)), (A*a**2*c*Piecewise((-1/(n*(e*x)**n
), Ne(n, 0)), (log(x), True))/e + A*a**2*d*Piecewise((0, (Abs(x) < 1) & (1/Abs(x) < 1)), (log(x)/e**n, Abs(x)
< 1), (-log(1/x)/e**n, 1/Abs(x) < 1), (-meijerg(((), (1, 1)), ((0, 0), ()), x)/e**n + meijerg(((1, 1), ()), ((
), (0, 0)), x)/e**n, True))/e + 2*A*a*b*c*Piecewise((0, (Abs(x) < 1) & (1/Abs(x) < 1)), (log(x)/e**n, Abs(x) <
1), (-log(1/x)/e**n, 1/Abs(x) < 1), (-meijerg(((), (1, 1)), ((0, 0), ()), x)/e**n + meijerg(((1, 1), ()), (()
, (0, 0)), x)/e**n, True))/e + 2*A*a*b*d*Piecew...

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 3415 vs. \(2 (164) = 328\).
time = 1.35, size = 3415, normalized size = 21.34 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*(a+b*x^n)^2*(A+B*x^n)*(c+d*x^n),x, algorithm="giac")

[Out]

(B*b^2*d*m^4*x*x^m*x^(4*n)*e^m + 6*B*b^2*d*m^3*n*x*x^m*x^(4*n)*e^m + 11*B*b^2*d*m^2*n^2*x*x^m*x^(4*n)*e^m + 6*
B*b^2*d*m*n^3*x*x^m*x^(4*n)*e^m + B*b^2*c*m^4*x*x^m*x^(3*n)*e^m + 2*B*a*b*d*m^4*x*x^m*x^(3*n)*e^m + A*b^2*d*m^
4*x*x^m*x^(3*n)*e^m + 7*B*b^2*c*m^3*n*x*x^m*x^(3*n)*e^m + 14*B*a*b*d*m^3*n*x*x^m*x^(3*n)*e^m + 7*A*b^2*d*m^3*n
*x*x^m*x^(3*n)*e^m + 14*B*b^2*c*m^2*n^2*x*x^m*x^(3*n)*e^m + 28*B*a*b*d*m^2*n^2*x*x^m*x^(3*n)*e^m + 14*A*b^2*d*
m^2*n^2*x*x^m*x^(3*n)*e^m + 8*B*b^2*c*m*n^3*x*x^m*x^(3*n)*e^m + 16*B*a*b*d*m*n^3*x*x^m*x^(3*n)*e^m + 8*A*b^2*d
*m*n^3*x*x^m*x^(3*n)*e^m + 2*B*a*b*c*m^4*x*x^m*x^(2*n)*e^m + A*b^2*c*m^4*x*x^m*x^(2*n)*e^m + B*a^2*d*m^4*x*x^m
*x^(2*n)*e^m + 2*A*a*b*d*m^4*x*x^m*x^(2*n)*e^m + 16*B*a*b*c*m^3*n*x*x^m*x^(2*n)*e^m + 8*A*b^2*c*m^3*n*x*x^m*x^
(2*n)*e^m + 8*B*a^2*d*m^3*n*x*x^m*x^(2*n)*e^m + 16*A*a*b*d*m^3*n*x*x^m*x^(2*n)*e^m + 38*B*a*b*c*m^2*n^2*x*x^m*
x^(2*n)*e^m + 19*A*b^2*c*m^2*n^2*x*x^m*x^(2*n)*e^m + 19*B*a^2*d*m^2*n^2*x*x^m*x^(2*n)*e^m + 38*A*a*b*d*m^2*n^2
*x*x^m*x^(2*n)*e^m + 24*B*a*b*c*m*n^3*x*x^m*x^(2*n)*e^m + 12*A*b^2*c*m*n^3*x*x^m*x^(2*n)*e^m + 12*B*a^2*d*m*n^
3*x*x^m*x^(2*n)*e^m + 24*A*a*b*d*m*n^3*x*x^m*x^(2*n)*e^m + B*a^2*c*m^4*x*x^m*x^n*e^m + 2*A*a*b*c*m^4*x*x^m*x^n
*e^m + A*a^2*d*m^4*x*x^m*x^n*e^m + 9*B*a^2*c*m^3*n*x*x^m*x^n*e^m + 18*A*a*b*c*m^3*n*x*x^m*x^n*e^m + 9*A*a^2*d*
m^3*n*x*x^m*x^n*e^m + 26*B*a^2*c*m^2*n^2*x*x^m*x^n*e^m + 52*A*a*b*c*m^2*n^2*x*x^m*x^n*e^m + 26*A*a^2*d*m^2*n^2
*x*x^m*x^n*e^m + 24*B*a^2*c*m*n^3*x*x^m*x^n*e^m + 48*A*a*b*c*m*n^3*x*x^m*x^n*e^m + 24*A*a^2*d*m*n^3*x*x^m*x^n*
e^m + A*a^2*c*m^4*x*x^m*e^m + 10*A*a^2*c*m^3*n*x*x^m*e^m + 35*A*a^2*c*m^2*n^2*x*x^m*e^m + 50*A*a^2*c*m*n^3*x*x
^m*e^m + 24*A*a^2*c*n^4*x*x^m*e^m + 4*B*b^2*d*m^3*x*x^m*x^(4*n)*e^m + 18*B*b^2*d*m^2*n*x*x^m*x^(4*n)*e^m + 22*
B*b^2*d*m*n^2*x*x^m*x^(4*n)*e^m + 6*B*b^2*d*n^3*x*x^m*x^(4*n)*e^m + 4*B*b^2*c*m^3*x*x^m*x^(3*n)*e^m + 8*B*a*b*
d*m^3*x*x^m*x^(3*n)*e^m + 4*A*b^2*d*m^3*x*x^m*x^(3*n)*e^m + 21*B*b^2*c*m^2*n*x*x^m*x^(3*n)*e^m + 42*B*a*b*d*m^
2*n*x*x^m*x^(3*n)*e^m + 21*A*b^2*d*m^2*n*x*x^m*x^(3*n)*e^m + 28*B*b^2*c*m*n^2*x*x^m*x^(3*n)*e^m + 56*B*a*b*d*m
*n^2*x*x^m*x^(3*n)*e^m + 28*A*b^2*d*m*n^2*x*x^m*x^(3*n)*e^m + 8*B*b^2*c*n^3*x*x^m*x^(3*n)*e^m + 16*B*a*b*d*n^3
*x*x^m*x^(3*n)*e^m + 8*A*b^2*d*n^3*x*x^m*x^(3*n)*e^m + 8*B*a*b*c*m^3*x*x^m*x^(2*n)*e^m + 4*A*b^2*c*m^3*x*x^m*x
^(2*n)*e^m + 4*B*a^2*d*m^3*x*x^m*x^(2*n)*e^m + 8*A*a*b*d*m^3*x*x^m*x^(2*n)*e^m + 48*B*a*b*c*m^2*n*x*x^m*x^(2*n
)*e^m + 24*A*b^2*c*m^2*n*x*x^m*x^(2*n)*e^m + 24*B*a^2*d*m^2*n*x*x^m*x^(2*n)*e^m + 48*A*a*b*d*m^2*n*x*x^m*x^(2*
n)*e^m + 76*B*a*b*c*m*n^2*x*x^m*x^(2*n)*e^m + 38*A*b^2*c*m*n^2*x*x^m*x^(2*n)*e^m + 38*B*a^2*d*m*n^2*x*x^m*x^(2
*n)*e^m + 76*A*a*b*d*m*n^2*x*x^m*x^(2*n)*e^m + 24*B*a*b*c*n^3*x*x^m*x^(2*n)*e^m + 12*A*b^2*c*n^3*x*x^m*x^(2*n)
*e^m + 12*B*a^2*d*n^3*x*x^m*x^(2*n)*e^m + 24*A*a*b*d*n^3*x*x^m*x^(2*n)*e^m + 4*B*a^2*c*m^3*x*x^m*x^n*e^m + 8*A
*a*b*c*m^3*x*x^m*x^n*e^m + 4*A*a^2*d*m^3*x*x^m*x^n*e^m + 27*B*a^2*c*m^2*n*x*x^m*x^n*e^m + 54*A*a*b*c*m^2*n*x*x
^m*x^n*e^m + 27*A*a^2*d*m^2*n*x*x^m*x^n*e^m + 52*B*a^2*c*m*n^2*x*x^m*x^n*e^m + 104*A*a*b*c*m*n^2*x*x^m*x^n*e^m
+ 52*A*a^2*d*m*n^2*x*x^m*x^n*e^m + 24*B*a^2*c*n^3*x*x^m*x^n*e^m + 48*A*a*b*c*n^3*x*x^m*x^n*e^m + 24*A*a^2*d*n
^3*x*x^m*x^n*e^m + 4*A*a^2*c*m^3*x*x^m*e^m + 30*A*a^2*c*m^2*n*x*x^m*e^m + 70*A*a^2*c*m*n^2*x*x^m*e^m + 50*A*a^
2*c*n^3*x*x^m*e^m + 6*B*b^2*d*m^2*x*x^m*x^(4*n)*e^m + 18*B*b^2*d*m*n*x*x^m*x^(4*n)*e^m + 11*B*b^2*d*n^2*x*x^m*
x^(4*n)*e^m + 6*B*b^2*c*m^2*x*x^m*x^(3*n)*e^m + 12*B*a*b*d*m^2*x*x^m*x^(3*n)*e^m + 6*A*b^2*d*m^2*x*x^m*x^(3*n)
*e^m + 21*B*b^2*c*m*n*x*x^m*x^(3*n)*e^m + 42*B*a*b*d*m*n*x*x^m*x^(3*n)*e^m + 21*A*b^2*d*m*n*x*x^m*x^(3*n)*e^m
+ 14*B*b^2*c*n^2*x*x^m*x^(3*n)*e^m + 28*B*a*b*d*n^2*x*x^m*x^(3*n)*e^m + 14*A*b^2*d*n^2*x*x^m*x^(3*n)*e^m + 12*
B*a*b*c*m^2*x*x^m*x^(2*n)*e^m + 6*A*b^2*c*m^2*x*x^m*x^(2*n)*e^m + 6*B*a^2*d*m^2*x*x^m*x^(2*n)*e^m + 12*A*a*b*d
*m^2*x*x^m*x^(2*n)*e^m + 48*B*a*b*c*m*n*x*x^m*x^(2*n)*e^m + 24*A*b^2*c*m*n*x*x^m*x^(2*n)*e^m + 24*B*a^2*d*m*n*
x*x^m*x^(2*n)*e^m + 48*A*a*b*d*m*n*x*x^m*x^(2*n)*e^m + 38*B*a*b*c*n^2*x*x^m*x^(2*n)*e^m + 19*A*b^2*c*n^2*x*x^m
*x^(2*n)*e^m + 19*B*a^2*d*n^2*x*x^m*x^(2*n)*e^m + 38*A*a*b*d*n^2*x*x^m*x^(2*n)*e^m + 6*B*a^2*c*m^2*x*x^m*x^n*e
^m + 12*A*a*b*c*m^2*x*x^m*x^n*e^m + 6*A*a^2*d*m^2*x*x^m*x^n*e^m + 27*B*a^2*c*m*n*x*x^m*x^n*e^m + 54*A*a*b*c*m*
n*x*x^m*x^n*e^m + 27*A*a^2*d*m*n*x*x^m*x^n*e^m + 26*B*a^2*c*n^2*x*x^m*x^n*e^m + 52*A*a*b*c*n^2*x*x^m*x^n*e^m +
26*A*a^2*d*n^2*x*x^m*x^n*e^m + 6*A*a^2*c*m^2*x*x^m*e^m + 30*A*a^2*c*m*n*x*x^m*e^m + 35*A*a^2*c*n^2*x*x^m*e^m
+ 4*B*b^2*d*m*x*x^m*x^(4*n)*e^m + 6*B*b^2*d*n*x*x^m*x^(4*n)*e^m + 4*B*b^2*c*m*x*x^m*x^(3*n)*e^m + 8*B*a*b*d*m*
x*x^m*x^(3*n)*e^m + 4*A*b^2*d*m*x*x^m*x^(3*n)*e^m + 7*B*b^2*c*n*x*x^m*x^(3*n)*e^m + 14*B*a*b*d*n*x*x^m*x^(3*n)
*e^m + 7*A*b^2*d*n*x*x^m*x^(3*n)*e^m + 8*B*a*b*c*m*x*x^m*x^(2*n)*e^m + 4*A*b^2*c*m*x*x^m*x^(2*n)*e^m + 4*B*a^2
*d*m*x*x^m*x^(2*n)*e^m + 8*A*a*b*d*m*x*x^m*x^(2*n)*e^m + 16*B*a*b*c*n*x*x^m*x^(2*n)*e^m + 8*A*b^2*c*n*x*x^m*x^
(2*n)*e^m + 8*B*a^2*d*n*x*x^m*x^(2*n)*e^m + 16*...

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Mupad [B]
time = 5.23, size = 588, normalized size = 3.68 \begin {gather*} \frac {x\,x^{2\,n}\,{\left (e\,x\right )}^m\,\left (A\,b^2\,c+B\,a^2\,d+2\,A\,a\,b\,d+2\,B\,a\,b\,c\right )\,\left (m^3+8\,m^2\,n+3\,m^2+19\,m\,n^2+16\,m\,n+3\,m+12\,n^3+19\,n^2+8\,n+1\right )}{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}+\frac {A\,a^2\,c\,x\,{\left (e\,x\right )}^m}{m+1}+\frac {a\,x\,x^n\,{\left (e\,x\right )}^m\,\left (A\,a\,d+2\,A\,b\,c+B\,a\,c\right )\,\left (m^3+9\,m^2\,n+3\,m^2+26\,m\,n^2+18\,m\,n+3\,m+24\,n^3+26\,n^2+9\,n+1\right )}{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}+\frac {b\,x\,x^{3\,n}\,{\left (e\,x\right )}^m\,\left (A\,b\,d+2\,B\,a\,d+B\,b\,c\right )\,\left (m^3+7\,m^2\,n+3\,m^2+14\,m\,n^2+14\,m\,n+3\,m+8\,n^3+14\,n^2+7\,n+1\right )}{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}+\frac {B\,b^2\,d\,x\,x^{4\,n}\,{\left (e\,x\right )}^m\,\left (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\right )}{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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m*(A + B*x^n)*(a + b*x^n)^2*(c + d*x^n),x)

[Out]

(x*x^(2*n)*(e*x)^m*(A*b^2*c + B*a^2*d + 2*A*a*b*d + 2*B*a*b*c)*(3*m + 8*n + 16*m*n + 19*m*n^2 + 8*m^2*n + 3*m^
2 + m^3 + 19*n^2 + 12*n^3 + 1))/(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) + (A*a^2*c*x*(e*x)^m)/(m + 1) + (a*x*x^n*(e*x)^m*(A*a*d
+ 2*A*b*c + B*a*c)*(3*m + 9*n + 18*m*n + 26*m*n^2 + 9*m^2*n + 3*m^2 + m^3 + 26*n^2 + 24*n^3 + 1))/(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) + (b*x*x^(3*n)*(e*x)^m*(A*b*d + 2*B*a*d + B*b*c)*(3*m + 7*n + 14*m*n + 14*m*n^2 + 7*m^2*n + 3*m^2 +
m^3 + 14*n^2 + 8*n^3 + 1))/(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) + (B*b^2*d*x*x^(4*n)*(e*x)^m*(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))/(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)

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