3.1.1 \(\int (e x)^m (a+b x^n)^3 (A+B x^n) (c+d x^n) \, dx\) [1]

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

[Out]

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

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Rubi [A]
time = 0.18, 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 = {584, 20, 30} \begin {gather*} \frac {a^3 A c (e x)^{m+1}}{e (m+1)}+\frac {a^2 x^{n+1} (e x)^m (a A d+a B c+3 A b c)}{m+n+1}+\frac {b^2 x^{4 n+1} (e x)^m (3 a B d+A b d+b B c)}{m+4 n+1}+\frac {a x^{2 n+1} (e x)^m (3 A b (a d+b c)+a B (a d+3 b c))}{m+2 n+1}+\frac {b x^{3 n+1} (e x)^m (A b (3 a d+b c)+3 a B (a d+b c))}{m+3 n+1}+\frac {b^3 B d x^{5 n+1} (e x)^m}{m+5 n+1} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]
time = 0.69, size = 172, normalized size = 0.82 \begin {gather*} x (e x)^m \left (\frac {a^3 A c}{1+m}+\frac {a^2 (3 A b c+a B c+a A d) x^n}{1+m+n}+\frac {a (3 A b (b c+a d)+a B (3 b c+a d)) x^{2 n}}{1+m+2 n}+\frac {b (3 a B (b c+a d)+A b (b c+3 a d)) x^{3 n}}{1+m+3 n}+\frac {b^2 (b B c+A b d+3 a B d) x^{4 n}}{1+m+4 n}+\frac {b^3 B d x^{5 n}}{1+m+5 n}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.40, size = 4972, normalized size = 23.68

method	result	size
risch	\(\text {Expression too large to display}\)	\(4972\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*(147*A*a*b^2*d*n^2*(x^n)^3+48*A*b^3*c*m*n*(x^n)^3+56*B*a^3*c*m^3*n*x^n+3*(x^n)^3*B*a*b^2*c+3*(x^n)^2*d*A*a^2
*b+B*a^3*c*x^n+105*B*b^3*d*m^2*n^2*(x^n)^5+100*B*b^3*d*m*n^3*(x^n)^5+60*B*a^3*d*m*n^4*(x^n)^2+3*B*a^2*b*c*m^5*
(x^n)^2+13*B*a^3*d*m^4*n*(x^n)^2+59*B*a^3*d*m^3*n^2*(x^n)^2+39*A*a*b^2*c*(x^n)^2*n+56*B*a^3*c*m*n*x^n+15*B*a^2
*b*c*(x^n)^2*m+30*B*a*b^2*c*m^3*(x^n)^3+234*B*a*b^2*c*n^3*(x^n)^3+30*B*a*b^2*d*m^2*(x^n)^4+234*A*a*b^2*d*m^2*n
^3*(x^n)^3+120*A*a*b^2*d*m*n^4*(x^n)^3+36*B*a^2*b*d*m^4*n*(x^n)^3+147*B*a^2*b*d*m^3*n^2*(x^n)^3+234*B*a^2*b*d*
m^2*n^3*(x^n)^3+A*a^3*c*m^5+36*B*a*b^2*c*(x^n)^3*n+56*A*a^3*d*m*n*x^n+30*A*a^2*b*c*m^2*x^n+213*A*a^2*b*c*n^2*x
^n+15*A*a^2*b*d*(x^n)^2*m+123*B*a*b^2*d*n^2*(x^n)^4+44*B*b^3*c*m*n*(x^n)^4+56*A*a^3*d*m^3*n*x^n+213*A*a^3*d*m^
2*n^2*x^n+308*A*a^3*d*m*n^3*x^n+A*a^3*c+90*B*a*b^2*d*m*n^4*(x^n)^4+36*A*a*b^2*d*m^4*n*(x^n)^3+B*a^3*d*(x^n)^2+
A*a^3*d*x^n+321*A*a*b^2*c*n^3*(x^n)^2+30*A*a*b^2*d*m^2*(x^n)^3+120*B*a^2*b*d*m*n^4*(x^n)^3+36*B*a*b^2*c*m^4*n*
(x^n)^3+147*B*a*b^2*c*m^3*n^2*(x^n)^3+10*B*b^3*d*m^4*n*(x^n)^5+39*A*a^2*b*d*(x^n)^2*n+30*A*a*b^2*c*m^2*(x^n)^2
+15*A*a*b^2*c*(x^n)^2*m+5*A*a^3*c*m^4+274*A*a^3*c*n^4+10*A*a^3*c*m^3+225*A*a^3*c*n^3+10*A*a^3*c*m^2+85*A*a^3*c
*n^2+B*b^3*c*(x^n)^4+A*b^3*c*(x^n)^3+122*A*b^3*d*m*n^3*(x^n)^4+147*B*a*b^2*c*n^2*(x^n)^3+15*B*a*b^2*d*(x^n)^4*
m+33*B*a*b^2*d*(x^n)^4*n+84*A*a^3*d*m^2*n*x^n+213*A*a^3*d*m*n^2*x^n+321*A*a^2*b*d*n^3*(x^n)^2+30*A*a*b^2*c*m^3
*(x^n)^2+33*B*a*b^2*d*m^4*n*(x^n)^4+123*B*a*b^2*d*m^3*n^2*(x^n)^4+183*B*a*b^2*d*m^2*n^3*(x^n)^4+107*B*a^3*d*m^
2*n^3*(x^n)^2+120*A*a^3*c*n^5+5*A*a^3*c*m+15*A*a^3*c*n+213*B*a^3*c*m^2*n^2*x^n+308*B*a^3*c*m*n^3*x^n+78*B*a^3*
d*m^2*n*(x^n)^2+177*B*a^3*d*m*n^2*(x^n)^2+30*B*a^2*b*c*m^3*(x^n)^2+15*B*a*b^2*c*(x^n)^3*m+180*B*a^2*b*c*m*n^4*
(x^n)^2+144*B*a^2*b*d*m^3*n*(x^n)^3+441*B*a^2*b*d*m^2*n^2*(x^n)^3+5*A*b^3*d*(x^n)^4*m+11*A*b^3*d*(x^n)^4*n+5*B
*a^3*c*m^4*x^n+120*B*a^3*c*n^4*x^n+10*B*a^3*d*m^3*(x^n)^2+107*B*a^3*d*n^3*(x^n)^2+5*B*b^3*c*(x^n)^4*m+11*B*b^3
*c*(x^n)^4*n+35*B*b^3*d*n^2*(x^n)^5+A*a^3*d*m^5*x^n+10*A*b^3*c*m^3*(x^n)^3+78*A*b^3*c*n^3*(x^n)^3+10*A*b^3*d*m
^2*(x^n)^4+41*A*b^3*d*n^2*(x^n)^4+10*A*a^3*d*m^2*x^n+71*A*a^3*d*n^2*x^n+5*A*b^3*d*m^4*(x^n)^4+30*A*b^3*d*n^4*(
x^n)^4+5*B*a^3*d*(x^n)^2*m+13*B*a^3*d*(x^n)^2*n+5*A*a^3*d*x^n*m+14*A*a^3*d*x^n*n+3*(x^n)^2*c*B*a^2*b+3*x^n*c*A
*a^2*b+5*B*a^3*c*x^n*m+B*a^3*c*m^5*x^n+5*B*a^3*d*m^4*(x^n)^2+60*B*a^3*d*n^4*(x^n)^2+10*B*b^3*c*m^2*(x^n)^4+41*
B*b^3*c*n^2*(x^n)^4+5*m*b^3*B*d*(x^n)^5+3*(x^n)^3*B*a^2*b*d+3*(x^n)^2*c*A*a*b^2+B*a^3*d*m^5*(x^n)^2+10*B*b^3*c
*m^3*(x^n)^4+61*B*b^3*c*n^3*(x^n)^4+5*B*b^3*c*m^4*(x^n)^4+10*B*a^3*c*m^2*x^n+71*B*a^3*c*n^2*x^n+10*b^3*B*d*(x^
n)^5*n+5*A*a^3*d*m^4*x^n+120*A*a^3*d*n^4*x^n+10*A*b^3*c*m^2*(x^n)^3+49*A*b^3*c*n^2*(x^n)^3+14*B*a^3*c*x^n*n+15
4*B*a^3*c*m^2*n^3*x^n+120*B*a^3*c*m*n^4*x^n+52*B*a^3*d*m^3*n*(x^n)^2+177*B*a^3*d*m^2*n^2*(x^n)^2+214*B*a^3*d*m
*n^3*(x^n)^2+321*B*a^2*b*c*n^3*(x^n)^2+30*B*a^2*b*d*m^2*(x^n)^3+147*B*a^2*b*d*n^2*(x^n)^3+30*B*a*b^2*c*m^2*(x^
n)^3+90*B*a*b^2*d*n^4*(x^n)^4+44*B*b^3*c*m^3*n*(x^n)^4+123*B*b^3*c*m^2*n^2*(x^n)^4+122*B*b^3*c*m*n^3*(x^n)^4+1
1*B*b^3*c*m^4*n*(x^n)^4+15*B*a^2*b*c*m^4*(x^n)^2+180*B*a^2*b*c*n^4*(x^n)^2+30*B*a^2*b*d*m^3*(x^n)^3+234*B*a^2*
b*d*n^3*(x^n)^3+78*A*b^3*c*m^2*n^3*(x^n)^3+40*A*b^3*c*m*n^4*(x^n)^3+44*A*b^3*d*m^3*n*(x^n)^4+123*A*b^3*d*m^2*n
^2*(x^n)^4+42*A*a^2*b*c*x^n*n+168*A*a^2*b*c*m^3*n*x^n+639*A*a^2*b*c*m^2*n^2*x^n+b^3*B*d*(x^n)^5+A*b^3*d*(x^n)^
4+3*B*a^2*b*d*m^5*(x^n)^3+3*B*a*b^2*c*m^5*(x^n)^3+15*B*a*b^2*d*m^4*(x^n)^4+642*A*a^2*b*d*m*n^3*(x^n)^2+156*A*a
*b^2*c*m^3*n*(x^n)^2+531*A*a*b^2*c*m^2*n^2*(x^n)^2+642*A*a*b^2*c*m*n^3*(x^n)^2+60*A*a^3*c*m^3*n+255*A*a^3*c*m^
2*n^2+450*A*a^3*c*m*n^3+90*A*a^3*c*m^2*n+255*A*a^3*c*m*n^2+156*A*b^3*c*m*n^3*(x^n)^3+66*A*b^3*d*m^2*n*(x^n)^4+
123*A*b^3*d*m*n^2*(x^n)^4+15*B*a^2*b*d*m^4*(x^n)^3+120*B*a^2*b*d*n^4*(x^n)^3+15*B*a*b^2*c*m^4*(x^n)^3+120*B*a*
b^2*c*n^4*(x^n)^3+24*B*b^3*d*n^4*(x^n)^5+A*b^3*c*m^5*(x^n)^3+3*B*a*b^2*d*m^5*(x^n)^4+15*A*a^3*c*m^4*n+85*A*a^3
*c*m^3*n^2+225*A*a^3*c*m^2*n^3+274*A*a^3*c*m*n^4+144*B*a*b^2*c*m*n*(x^n)^3+252*A*a^2*b*c*m^2*n*x^n+120*A*a*b^2
*d*n^4*(x^n)^3+48*A*b^3*c*m^3*n*(x^n)^3+147*A*b^3*c*m^2*n^2*(x^n)^3+105*B*b^3*d*m*n^2*(x^n)^5+3*A*a^2*b*d*m^5*
(x^n)^2+3*A*a*b^2*c*m^5*(x^n)^2+15*A*a*b^2*d*m^4*(x^n)^3+30*B*a^2*b*c*m^2*(x^n)^2+177*B*a^2*b*c*n^2*(x^n)^2+15
*B*a^2*b*d*(x^n)^3*m+36*B*a^2*b*d*(x^n)^3*n+60*B*b^3*d*m^2*n*(x^n)^5+36*A*a*b^2*d*(x^n)^3*n+84*B*a^3*c*m^2*n*x
^n+213*B*a^3*c*m*n^2*x^n+52*B*a^3*d*m*n*(x^n)^2+14*A*a^3*d*m^4*n*x^n+71*A*a^3*d*m^3*n^2*x^n+154*A*a^3*d*m^2*n^
3*x^n+177*A*a*b^2*c*n^2*(x^n)^2+15*A*a*b^2*d*(x^n)^3*m+15*A*a^2*b*c*x^n*m+183*B*a*b^2*d*n^3*(x^n)^4+66*B*b^3*c
*m^2*n*(x^n)^4+123*B*b^3*c*m*n^2*(x^n)^4+40*B*b^3*d*m*n*(x^n)^5+30*A*b^3*d*m*n^4*(x^n)^4+39*B*a^2*b*c*(x^n)^2*
n+11*A*b^3*d*m^4*n*(x^n)^4+41*A*b^3*d*m^3*n^2*(x^n)^4+61*A*b^3*d*m^2*n^3*(x^n)^4+180*A*a^2*b*d*n^4*(x^n)^2+15*
A*a*b^2*c*m^4*(x^n)^2+180*A*a*b^2*c*n^4*(x^n)^2+35*B*b^3*d*m^3*n^2*(x^n)^5+50*B*b^3*d*m^2*n^3*(x^n)^5+24*B*b^3
*d*m*n^4*(x^n)^5+120*A*a^3*d*m*n^4*x^n+3*A*a^2*...

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 434 vs. \(2 (215) = 430\).
time = 0.34, size = 434, normalized size = 2.07 \begin {gather*} \frac {\left (x e\right )^{m + 1} A a^{3} c e^{\left (-1\right )}}{m + 1} + \frac {B b^{3} d x e^{\left (m \log \left (x\right ) + 5 \, n \log \left (x\right ) + m\right )}}{m + 5 \, n + 1} + \frac {B b^{3} c x e^{\left (m \log \left (x\right ) + 4 \, n \log \left (x\right ) + m\right )}}{m + 4 \, n + 1} + \frac {3 \, B a b^{2} d x e^{\left (m \log \left (x\right ) + 4 \, n \log \left (x\right ) + m\right )}}{m + 4 \, n + 1} + \frac {A b^{3} d x e^{\left (m \log \left (x\right ) + 4 \, n \log \left (x\right ) + m\right )}}{m + 4 \, n + 1} + \frac {3 \, B a b^{2} c x e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right ) + m\right )}}{m + 3 \, n + 1} + \frac {A b^{3} c x e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right ) + m\right )}}{m + 3 \, n + 1} + \frac {3 \, B a^{2} b d x e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right ) + m\right )}}{m + 3 \, n + 1} + \frac {3 \, A 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 {3 \, B a^{2} b c x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right ) + m\right )}}{m + 2 \, n + 1} + \frac {3 \, A 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^{3} d x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right ) + m\right )}}{m + 2 \, n + 1} + \frac {3 \, A a^{2} 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^{3} c x e^{\left (m \log \left (x\right ) + n \log \left (x\right ) + m\right )}}{m + n + 1} + \frac {3 \, A a^{2} b c x e^{\left (m \log \left (x\right ) + n \log \left (x\right ) + m\right )}}{m + n + 1} + \frac {A a^{3} 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)^3*(A+B*x^n)*(c+d*x^n),x, algorithm="maxima")

[Out]

(x*e)^(m + 1)*A*a^3*c*e^(-1)/(m + 1) + B*b^3*d*x*e^(m*log(x) + 5*n*log(x) + m)/(m + 5*n + 1) + B*b^3*c*x*e^(m*
log(x) + 4*n*log(x) + m)/(m + 4*n + 1) + 3*B*a*b^2*d*x*e^(m*log(x) + 4*n*log(x) + m)/(m + 4*n + 1) + A*b^3*d*x
*e^(m*log(x) + 4*n*log(x) + m)/(m + 4*n + 1) + 3*B*a*b^2*c*x*e^(m*log(x) + 3*n*log(x) + m)/(m + 3*n + 1) + A*b
^3*c*x*e^(m*log(x) + 3*n*log(x) + m)/(m + 3*n + 1) + 3*B*a^2*b*d*x*e^(m*log(x) + 3*n*log(x) + m)/(m + 3*n + 1)
+ 3*A*a*b^2*d*x*e^(m*log(x) + 3*n*log(x) + m)/(m + 3*n + 1) + 3*B*a^2*b*c*x*e^(m*log(x) + 2*n*log(x) + m)/(m
+ 2*n + 1) + 3*A*a*b^2*c*x*e^(m*log(x) + 2*n*log(x) + m)/(m + 2*n + 1) + B*a^3*d*x*e^(m*log(x) + 2*n*log(x) +
m)/(m + 2*n + 1) + 3*A*a^2*b*d*x*e^(m*log(x) + 2*n*log(x) + m)/(m + 2*n + 1) + B*a^3*c*x*e^(m*log(x) + n*log(x
) + m)/(m + n + 1) + 3*A*a^2*b*c*x*e^(m*log(x) + n*log(x) + m)/(m + n + 1) + A*a^3*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. 3055 vs. \(2 (215) = 430\).
time = 3.75, size = 3055, normalized size = 14.55 \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)^3*(A+B*x^n)*(c+d*x^n),x, algorithm="fricas")

[Out]

((B*b^3*d*m^5 + 5*B*b^3*d*m^4 + 10*B*b^3*d*m^3 + 10*B*b^3*d*m^2 + 5*B*b^3*d*m + B*b^3*d + 24*(B*b^3*d*m + B*b^
3*d)*n^4 + 50*(B*b^3*d*m^2 + 2*B*b^3*d*m + B*b^3*d)*n^3 + 35*(B*b^3*d*m^3 + 3*B*b^3*d*m^2 + 3*B*b^3*d*m + B*b^
3*d)*n^2 + 10*(B*b^3*d*m^4 + 4*B*b^3*d*m^3 + 6*B*b^3*d*m^2 + 4*B*b^3*d*m + B*b^3*d)*n)*x*x^(5*n)*e^(m*log(x) +
m) + ((B*b^3*c + (3*B*a*b^2 + A*b^3)*d)*m^5 + B*b^3*c + 5*(B*b^3*c + (3*B*a*b^2 + A*b^3)*d)*m^4 + 30*(B*b^3*c
+ (3*B*a*b^2 + A*b^3)*d + (B*b^3*c + (3*B*a*b^2 + A*b^3)*d)*m)*n^4 + 10*(B*b^3*c + (3*B*a*b^2 + A*b^3)*d)*m^3
+ 61*(B*b^3*c + (B*b^3*c + (3*B*a*b^2 + A*b^3)*d)*m^2 + (3*B*a*b^2 + A*b^3)*d + 2*(B*b^3*c + (3*B*a*b^2 + A*b
^3)*d)*m)*n^3 + 10*(B*b^3*c + (3*B*a*b^2 + A*b^3)*d)*m^2 + 41*(B*b^3*c + (B*b^3*c + (3*B*a*b^2 + A*b^3)*d)*m^3
+ 3*(B*b^3*c + (3*B*a*b^2 + A*b^3)*d)*m^2 + (3*B*a*b^2 + A*b^3)*d + 3*(B*b^3*c + (3*B*a*b^2 + A*b^3)*d)*m)*n^
2 + (3*B*a*b^2 + A*b^3)*d + 5*(B*b^3*c + (3*B*a*b^2 + A*b^3)*d)*m + 11*(B*b^3*c + (B*b^3*c + (3*B*a*b^2 + A*b^
3)*d)*m^4 + 4*(B*b^3*c + (3*B*a*b^2 + A*b^3)*d)*m^3 + 6*(B*b^3*c + (3*B*a*b^2 + A*b^3)*d)*m^2 + (3*B*a*b^2 + A
*b^3)*d + 4*(B*b^3*c + (3*B*a*b^2 + A*b^3)*d)*m)*n)*x*x^(4*n)*e^(m*log(x) + m) + (((3*B*a*b^2 + A*b^3)*c + 3*(
B*a^2*b + A*a*b^2)*d)*m^5 + 5*((3*B*a*b^2 + A*b^3)*c + 3*(B*a^2*b + A*a*b^2)*d)*m^4 + 40*((3*B*a*b^2 + A*b^3)*
c + 3*(B*a^2*b + A*a*b^2)*d + ((3*B*a*b^2 + A*b^3)*c + 3*(B*a^2*b + A*a*b^2)*d)*m)*n^4 + 10*((3*B*a*b^2 + A*b^
3)*c + 3*(B*a^2*b + A*a*b^2)*d)*m^3 + 78*(((3*B*a*b^2 + A*b^3)*c + 3*(B*a^2*b + A*a*b^2)*d)*m^2 + (3*B*a*b^2 +
A*b^3)*c + 3*(B*a^2*b + A*a*b^2)*d + 2*((3*B*a*b^2 + A*b^3)*c + 3*(B*a^2*b + A*a*b^2)*d)*m)*n^3 + 10*((3*B*a*
b^2 + A*b^3)*c + 3*(B*a^2*b + A*a*b^2)*d)*m^2 + 49*(((3*B*a*b^2 + A*b^3)*c + 3*(B*a^2*b + A*a*b^2)*d)*m^3 + 3*
((3*B*a*b^2 + A*b^3)*c + 3*(B*a^2*b + A*a*b^2)*d)*m^2 + (3*B*a*b^2 + A*b^3)*c + 3*(B*a^2*b + A*a*b^2)*d + 3*((
3*B*a*b^2 + A*b^3)*c + 3*(B*a^2*b + A*a*b^2)*d)*m)*n^2 + (3*B*a*b^2 + A*b^3)*c + 3*(B*a^2*b + A*a*b^2)*d + 5*(
(3*B*a*b^2 + A*b^3)*c + 3*(B*a^2*b + A*a*b^2)*d)*m + 12*(((3*B*a*b^2 + A*b^3)*c + 3*(B*a^2*b + A*a*b^2)*d)*m^4
+ 4*((3*B*a*b^2 + A*b^3)*c + 3*(B*a^2*b + A*a*b^2)*d)*m^3 + 6*((3*B*a*b^2 + A*b^3)*c + 3*(B*a^2*b + A*a*b^2)*
d)*m^2 + (3*B*a*b^2 + A*b^3)*c + 3*(B*a^2*b + A*a*b^2)*d + 4*((3*B*a*b^2 + A*b^3)*c + 3*(B*a^2*b + A*a*b^2)*d)
*m)*n)*x*x^(3*n)*e^(m*log(x) + m) + ((3*(B*a^2*b + A*a*b^2)*c + (B*a^3 + 3*A*a^2*b)*d)*m^5 + 5*(3*(B*a^2*b + A
*a*b^2)*c + (B*a^3 + 3*A*a^2*b)*d)*m^4 + 60*(3*(B*a^2*b + A*a*b^2)*c + (B*a^3 + 3*A*a^2*b)*d + (3*(B*a^2*b + A
*a*b^2)*c + (B*a^3 + 3*A*a^2*b)*d)*m)*n^4 + 10*(3*(B*a^2*b + A*a*b^2)*c + (B*a^3 + 3*A*a^2*b)*d)*m^3 + 107*((3
*(B*a^2*b + A*a*b^2)*c + (B*a^3 + 3*A*a^2*b)*d)*m^2 + 3*(B*a^2*b + A*a*b^2)*c + (B*a^3 + 3*A*a^2*b)*d + 2*(3*(
B*a^2*b + A*a*b^2)*c + (B*a^3 + 3*A*a^2*b)*d)*m)*n^3 + 10*(3*(B*a^2*b + A*a*b^2)*c + (B*a^3 + 3*A*a^2*b)*d)*m^
2 + 59*((3*(B*a^2*b + A*a*b^2)*c + (B*a^3 + 3*A*a^2*b)*d)*m^3 + 3*(3*(B*a^2*b + A*a*b^2)*c + (B*a^3 + 3*A*a^2*
b)*d)*m^2 + 3*(B*a^2*b + A*a*b^2)*c + (B*a^3 + 3*A*a^2*b)*d + 3*(3*(B*a^2*b + A*a*b^2)*c + (B*a^3 + 3*A*a^2*b)
*d)*m)*n^2 + 3*(B*a^2*b + A*a*b^2)*c + (B*a^3 + 3*A*a^2*b)*d + 5*(3*(B*a^2*b + A*a*b^2)*c + (B*a^3 + 3*A*a^2*b
)*d)*m + 13*((3*(B*a^2*b + A*a*b^2)*c + (B*a^3 + 3*A*a^2*b)*d)*m^4 + 4*(3*(B*a^2*b + A*a*b^2)*c + (B*a^3 + 3*A
*a^2*b)*d)*m^3 + 6*(3*(B*a^2*b + A*a*b^2)*c + (B*a^3 + 3*A*a^2*b)*d)*m^2 + 3*(B*a^2*b + A*a*b^2)*c + (B*a^3 +
3*A*a^2*b)*d + 4*(3*(B*a^2*b + A*a*b^2)*c + (B*a^3 + 3*A*a^2*b)*d)*m)*n)*x*x^(2*n)*e^(m*log(x) + m) + ((A*a^3*
d + (B*a^3 + 3*A*a^2*b)*c)*m^5 + A*a^3*d + 5*(A*a^3*d + (B*a^3 + 3*A*a^2*b)*c)*m^4 + 120*(A*a^3*d + (B*a^3 + 3
*A*a^2*b)*c + (A*a^3*d + (B*a^3 + 3*A*a^2*b)*c)*m)*n^4 + 10*(A*a^3*d + (B*a^3 + 3*A*a^2*b)*c)*m^3 + 154*(A*a^3
*d + (A*a^3*d + (B*a^3 + 3*A*a^2*b)*c)*m^2 + (B*a^3 + 3*A*a^2*b)*c + 2*(A*a^3*d + (B*a^3 + 3*A*a^2*b)*c)*m)*n^
3 + 10*(A*a^3*d + (B*a^3 + 3*A*a^2*b)*c)*m^2 + 71*(A*a^3*d + (A*a^3*d + (B*a^3 + 3*A*a^2*b)*c)*m^3 + 3*(A*a^3*
d + (B*a^3 + 3*A*a^2*b)*c)*m^2 + (B*a^3 + 3*A*a^2*b)*c + 3*(A*a^3*d + (B*a^3 + 3*A*a^2*b)*c)*m)*n^2 + (B*a^3 +
3*A*a^2*b)*c + 5*(A*a^3*d + (B*a^3 + 3*A*a^2*b)*c)*m + 14*(A*a^3*d + (A*a^3*d + (B*a^3 + 3*A*a^2*b)*c)*m^4 +
4*(A*a^3*d + (B*a^3 + 3*A*a^2*b)*c)*m^3 + 6*(A*a^3*d + (B*a^3 + 3*A*a^2*b)*c)*m^2 + (B*a^3 + 3*A*a^2*b)*c + 4*
(A*a^3*d + (B*a^3 + 3*A*a^2*b)*c)*m)*n)*x*x^n*e^(m*log(x) + m) + (A*a^3*c*m^5 + 120*A*a^3*c*n^5 + 5*A*a^3*c*m^
4 + 10*A*a^3*c*m^3 + 10*A*a^3*c*m^2 + 5*A*a^3*c*m + A*a^3*c + 274*(A*a^3*c*m + A*a^3*c)*n^4 + 225*(A*a^3*c*m^2
+ 2*A*a^3*c*m + A*a^3*c)*n^3 + 85*(A*a^3*c*m^3 + 3*A*a^3*c*m^2 + 3*A*a^3*c*m + A*a^3*c)*n^2 + 15*(A*a^3*c*m^4
+ 4*A*a^3*c*m^3 + 6*A*a^3*c*m^2 + 4*A*a^3*c*m + A*a^3*c)*n)*x*e^(m*log(x) + m))/(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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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 63250 vs. \(2 (206) = 412\).
time = 175.91, size = 63250, normalized size = 301.19 \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)**3*(A+B*x**n)*(c+d*x**n),x)

[Out]

Piecewise(((A + B)*(a + b)**3*(c + d)*log(x)/e, Eq(m, -1) & Eq(n, 0)), ((A*a**3*c*log(x) + A*a**3*d*x**n/n + 3
*A*a**2*b*c*x**n/n + 3*A*a**2*b*d*x**(2*n)/(2*n) + 3*A*a*b**2*c*x**(2*n)/(2*n) + A*a*b**2*d*x**(3*n)/n + A*b**
3*c*x**(3*n)/(3*n) + A*b**3*d*x**(4*n)/(4*n) + B*a**3*c*x**n/n + B*a**3*d*x**(2*n)/(2*n) + 3*B*a**2*b*c*x**(2*
n)/(2*n) + B*a**2*b*d*x**(3*n)/n + B*a*b**2*c*x**(3*n)/n + 3*B*a*b**2*d*x**(4*n)/(4*n) + B*b**3*c*x**(4*n)/(4*
n) + B*b**3*d*x**(5*n)/(5*n))/e, Eq(m, -1)), (A*a**3*c*Piecewise((-1/(5*n*(e*x)**(5*n)), Ne(n, 0)), (log(x), T
rue))/e + A*a**3*d*Piecewise((-x**n/(4*n*(e*x)**(5*n)), Ne(n, 0)), (log(x), True))/e + 3*A*a**2*b*c*Piecewise(
(-x**n/(4*n*(e*x)**(5*n)), Ne(n, 0)), (log(x), True))/e + 3*A*a**2*b*d*Piecewise((-x**(2*n)/(3*n*(e*x)**(5*n))
, Ne(n, 0)), (log(x), True))/e + 3*A*a*b**2*c*Piecewise((-x**(2*n)/(3*n*(e*x)**(5*n)), Ne(n, 0)), (log(x), Tru
e))/e + 3*A*a*b**2*d*Piecewise((-x**(3*n)/(2*n*(e*x)**(5*n)), Ne(n, 0)), (log(x), True))/e + A*b**3*c*Piecewis
e((-x**(3*n)/(2*n*(e*x)**(5*n)), Ne(n, 0)), (log(x), True))/e + A*b**3*d*Piecewise((-x**(4*n)/(n*(e*x)**(5*n))
, Ne(n, 0)), (log(x), True))/e + B*a**3*c*Piecewise((-x**n/(4*n*(e*x)**(5*n)), Ne(n, 0)), (log(x), True))/e +
B*a**3*d*Piecewise((-x**(2*n)/(3*n*(e*x)**(5*n)), Ne(n, 0)), (log(x), True))/e + 3*B*a**2*b*c*Piecewise((-x**(
2*n)/(3*n*(e*x)**(5*n)), Ne(n, 0)), (log(x), True))/e + 3*B*a**2*b*d*Piecewise((-x**(3*n)/(2*n*(e*x)**(5*n)),
Ne(n, 0)), (log(x), True))/e + 3*B*a*b**2*c*Piecewise((-x**(3*n)/(2*n*(e*x)**(5*n)), Ne(n, 0)), (log(x), True)
)/e + 3*B*a*b**2*d*Piecewise((-x**(4*n)/(n*(e*x)**(5*n)), Ne(n, 0)), (log(x), True))/e + B*b**3*c*Piecewise((-
x**(4*n)/(n*(e*x)**(5*n)), Ne(n, 0)), (log(x), True))/e + B*b**3*d*Piecewise((0, (Abs(x) < 1) & (1/Abs(x) < 1)
), (log(x)/e**(5*n), Abs(x) < 1), (-log(1/x)/e**(5*n), 1/Abs(x) < 1), (-meijerg(((), (1, 1)), ((0, 0), ()), x)
/e**(5*n) + meijerg(((1, 1), ()), ((), (0, 0)), x)/e**(5*n), True))/e, Eq(m, -5*n - 1)), (A*a**3*c*Piecewise((
-1/(4*n*(e*x)**(4*n)), Ne(n, 0)), (log(x), True))/e + A*a**3*d*Piecewise((-x**n/(3*n*(e*x)**(4*n)), Ne(n, 0)),
(log(x), True))/e + 3*A*a**2*b*c*Piecewise((-x**n/(3*n*(e*x)**(4*n)), Ne(n, 0)), (log(x), True))/e + 3*A*a**2
*b*d*Piecewise((-x**(2*n)/(2*n*(e*x)**(4*n)), Ne(n, 0)), (log(x), True))/e + 3*A*a*b**2*c*Piecewise((-x**(2*n)
/(2*n*(e*x)**(4*n)), Ne(n, 0)), (log(x), True))/e + 3*A*a*b**2*d*Piecewise((-x**(3*n)/(n*(e*x)**(4*n)), Ne(n,
0)), (log(x), True))/e + A*b**3*c*Piecewise((-x**(3*n)/(n*(e*x)**(4*n)), Ne(n, 0)), (log(x), True))/e + A*b**3
*d*Piecewise((0, (Abs(x) < 1) & (1/Abs(x) < 1)), (log(x)/e**(4*n), Abs(x) < 1), (-log(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), Tru
e))/e + B*a**3*c*Piecewise((-x**n/(3*n*(e*x)**(4*n)), Ne(n, 0)), (log(x), True))/e + B*a**3*d*Piecewise((-x**(
2*n)/(2*n*(e*x)**(4*n)), Ne(n, 0)), (log(x), True))/e + 3*B*a**2*b*c*Piecewise((-x**(2*n)/(2*n*(e*x)**(4*n)),
Ne(n, 0)), (log(x), True))/e + 3*B*a**2*b*d*Piecewise((-x**(3*n)/(n*(e*x)**(4*n)), Ne(n, 0)), (log(x), True))/
e + 3*B*a*b**2*c*Piecewise((-x**(3*n)/(n*(e*x)**(4*n)), Ne(n, 0)), (log(x), True))/e + 3*B*a*b**2*d*Piecewise(
(0, (Abs(x) < 1) & (1/Abs(x) < 1)), (log(x)/e**(4*n), Abs(x) < 1), (-log(1/x)/e**(4*n), 1/Abs(x) < 1), (-meije
rg(((), (1, 1)), ((0, 0), ()), x)/e**(4*n) + meijerg(((1, 1), ()), ((), (0, 0)), x)/e**(4*n), True))/e + B*b**
3*c*Piecewise((0, (Abs(x) < 1) & (1/Abs(x) < 1)), (log(x)/e**(4*n), Abs(x) < 1), (-log(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), Tr
ue))/e + B*b**3*d*Piecewise((x**(5*n)/(n*(e*x)**(4*n)), Ne(n, 0)), (log(x), True))/e, Eq(m, -4*n - 1)), (A*a**
3*c*Piecewise((-1/(3*n*(e*x)**(3*n)), Ne(n, 0)), (log(x), True))/e + A*a**3*d*Piecewise((-x**n/(2*n*(e*x)**(3*
n)), Ne(n, 0)), (log(x), True))/e + 3*A*a**2*b*c*Piecewise((-x**n/(2*n*(e*x)**(3*n)), Ne(n, 0)), (log(x), True
))/e + 3*A*a**2*b*d*Piecewise((-x**(2*n)/(n*(e*x)**(3*n)), Ne(n, 0)), (log(x), True))/e + 3*A*a*b**2*c*Piecewi
se((-x**(2*n)/(n*(e*x)**(3*n)), Ne(n, 0)), (log(x), True))/e + 3*A*a*b**2*d*Piecewise((0, (Abs(x) < 1) & (1/Ab
s(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 + A*b**3*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) + meijerg(((1, 1), ()), ((), (0, 0)), x)/e**(3*n), True))/e + A*b**3*d*Piecew
ise((x**(4*n)/(n*(e*x)**(3*n)), Ne(n, 0)), (log(x), True))/e + B*a**3*c*Piecewise((-x**n/(2*n*(e*x)**(3*n)), N
e(n, 0)), (log(x), True))/e + B*a**3*d*Piecewise((-x**(2*n)/(n*(e*x)**(3*n)), Ne(n, 0)), (log(x), True))/e + 3
*B*a**2*b*c*Piecewise((-x**(2*n)/(n*(e*x)**(3*n)), Ne(n, 0)), (log(x), True))/e + 3*B*a**2*b*d*Piecewise((0, (
Abs(x) < 1) & (1/Abs(x) < 1)), (log(x)/e**(3*n)...

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 6927 vs. \(2 (215) = 430\).
time = 0.68, size = 6927, normalized size = 32.99 \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)^3*(A+B*x^n)*(c+d*x^n),x, algorithm="giac")

[Out]

(B*b^3*d*m^5*x*x^m*x^(5*n)*e^m + 10*B*b^3*d*m^4*n*x*x^m*x^(5*n)*e^m + 35*B*b^3*d*m^3*n^2*x*x^m*x^(5*n)*e^m + 5
0*B*b^3*d*m^2*n^3*x*x^m*x^(5*n)*e^m + 24*B*b^3*d*m*n^4*x*x^m*x^(5*n)*e^m + B*b^3*c*m^5*x*x^m*x^(4*n)*e^m + 3*B
*a*b^2*d*m^5*x*x^m*x^(4*n)*e^m + A*b^3*d*m^5*x*x^m*x^(4*n)*e^m + 11*B*b^3*c*m^4*n*x*x^m*x^(4*n)*e^m + 33*B*a*b
^2*d*m^4*n*x*x^m*x^(4*n)*e^m + 11*A*b^3*d*m^4*n*x*x^m*x^(4*n)*e^m + 41*B*b^3*c*m^3*n^2*x*x^m*x^(4*n)*e^m + 123
*B*a*b^2*d*m^3*n^2*x*x^m*x^(4*n)*e^m + 41*A*b^3*d*m^3*n^2*x*x^m*x^(4*n)*e^m + 61*B*b^3*c*m^2*n^3*x*x^m*x^(4*n)
*e^m + 183*B*a*b^2*d*m^2*n^3*x*x^m*x^(4*n)*e^m + 61*A*b^3*d*m^2*n^3*x*x^m*x^(4*n)*e^m + 30*B*b^3*c*m*n^4*x*x^m
*x^(4*n)*e^m + 90*B*a*b^2*d*m*n^4*x*x^m*x^(4*n)*e^m + 30*A*b^3*d*m*n^4*x*x^m*x^(4*n)*e^m + 3*B*a*b^2*c*m^5*x*x
^m*x^(3*n)*e^m + A*b^3*c*m^5*x*x^m*x^(3*n)*e^m + 3*B*a^2*b*d*m^5*x*x^m*x^(3*n)*e^m + 3*A*a*b^2*d*m^5*x*x^m*x^(
3*n)*e^m + 36*B*a*b^2*c*m^4*n*x*x^m*x^(3*n)*e^m + 12*A*b^3*c*m^4*n*x*x^m*x^(3*n)*e^m + 36*B*a^2*b*d*m^4*n*x*x^
m*x^(3*n)*e^m + 36*A*a*b^2*d*m^4*n*x*x^m*x^(3*n)*e^m + 147*B*a*b^2*c*m^3*n^2*x*x^m*x^(3*n)*e^m + 49*A*b^3*c*m^
3*n^2*x*x^m*x^(3*n)*e^m + 147*B*a^2*b*d*m^3*n^2*x*x^m*x^(3*n)*e^m + 147*A*a*b^2*d*m^3*n^2*x*x^m*x^(3*n)*e^m +
234*B*a*b^2*c*m^2*n^3*x*x^m*x^(3*n)*e^m + 78*A*b^3*c*m^2*n^3*x*x^m*x^(3*n)*e^m + 234*B*a^2*b*d*m^2*n^3*x*x^m*x
^(3*n)*e^m + 234*A*a*b^2*d*m^2*n^3*x*x^m*x^(3*n)*e^m + 120*B*a*b^2*c*m*n^4*x*x^m*x^(3*n)*e^m + 40*A*b^3*c*m*n^
4*x*x^m*x^(3*n)*e^m + 120*B*a^2*b*d*m*n^4*x*x^m*x^(3*n)*e^m + 120*A*a*b^2*d*m*n^4*x*x^m*x^(3*n)*e^m + 3*B*a^2*
b*c*m^5*x*x^m*x^(2*n)*e^m + 3*A*a*b^2*c*m^5*x*x^m*x^(2*n)*e^m + B*a^3*d*m^5*x*x^m*x^(2*n)*e^m + 3*A*a^2*b*d*m^
5*x*x^m*x^(2*n)*e^m + 39*B*a^2*b*c*m^4*n*x*x^m*x^(2*n)*e^m + 39*A*a*b^2*c*m^4*n*x*x^m*x^(2*n)*e^m + 13*B*a^3*d
*m^4*n*x*x^m*x^(2*n)*e^m + 39*A*a^2*b*d*m^4*n*x*x^m*x^(2*n)*e^m + 177*B*a^2*b*c*m^3*n^2*x*x^m*x^(2*n)*e^m + 17
7*A*a*b^2*c*m^3*n^2*x*x^m*x^(2*n)*e^m + 59*B*a^3*d*m^3*n^2*x*x^m*x^(2*n)*e^m + 177*A*a^2*b*d*m^3*n^2*x*x^m*x^(
2*n)*e^m + 321*B*a^2*b*c*m^2*n^3*x*x^m*x^(2*n)*e^m + 321*A*a*b^2*c*m^2*n^3*x*x^m*x^(2*n)*e^m + 107*B*a^3*d*m^2
*n^3*x*x^m*x^(2*n)*e^m + 321*A*a^2*b*d*m^2*n^3*x*x^m*x^(2*n)*e^m + 180*B*a^2*b*c*m*n^4*x*x^m*x^(2*n)*e^m + 180
*A*a*b^2*c*m*n^4*x*x^m*x^(2*n)*e^m + 60*B*a^3*d*m*n^4*x*x^m*x^(2*n)*e^m + 180*A*a^2*b*d*m*n^4*x*x^m*x^(2*n)*e^
m + B*a^3*c*m^5*x*x^m*x^n*e^m + 3*A*a^2*b*c*m^5*x*x^m*x^n*e^m + A*a^3*d*m^5*x*x^m*x^n*e^m + 14*B*a^3*c*m^4*n*x
*x^m*x^n*e^m + 42*A*a^2*b*c*m^4*n*x*x^m*x^n*e^m + 14*A*a^3*d*m^4*n*x*x^m*x^n*e^m + 71*B*a^3*c*m^3*n^2*x*x^m*x^
n*e^m + 213*A*a^2*b*c*m^3*n^2*x*x^m*x^n*e^m + 71*A*a^3*d*m^3*n^2*x*x^m*x^n*e^m + 154*B*a^3*c*m^2*n^3*x*x^m*x^n
*e^m + 462*A*a^2*b*c*m^2*n^3*x*x^m*x^n*e^m + 154*A*a^3*d*m^2*n^3*x*x^m*x^n*e^m + 120*B*a^3*c*m*n^4*x*x^m*x^n*e
^m + 360*A*a^2*b*c*m*n^4*x*x^m*x^n*e^m + 120*A*a^3*d*m*n^4*x*x^m*x^n*e^m + A*a^3*c*m^5*x*x^m*e^m + 15*A*a^3*c*
m^4*n*x*x^m*e^m + 85*A*a^3*c*m^3*n^2*x*x^m*e^m + 225*A*a^3*c*m^2*n^3*x*x^m*e^m + 274*A*a^3*c*m*n^4*x*x^m*e^m +
120*A*a^3*c*n^5*x*x^m*e^m + 5*B*b^3*d*m^4*x*x^m*x^(5*n)*e^m + 40*B*b^3*d*m^3*n*x*x^m*x^(5*n)*e^m + 105*B*b^3*
d*m^2*n^2*x*x^m*x^(5*n)*e^m + 100*B*b^3*d*m*n^3*x*x^m*x^(5*n)*e^m + 24*B*b^3*d*n^4*x*x^m*x^(5*n)*e^m + 5*B*b^3
*c*m^4*x*x^m*x^(4*n)*e^m + 15*B*a*b^2*d*m^4*x*x^m*x^(4*n)*e^m + 5*A*b^3*d*m^4*x*x^m*x^(4*n)*e^m + 44*B*b^3*c*m
^3*n*x*x^m*x^(4*n)*e^m + 132*B*a*b^2*d*m^3*n*x*x^m*x^(4*n)*e^m + 44*A*b^3*d*m^3*n*x*x^m*x^(4*n)*e^m + 123*B*b^
3*c*m^2*n^2*x*x^m*x^(4*n)*e^m + 369*B*a*b^2*d*m^2*n^2*x*x^m*x^(4*n)*e^m + 123*A*b^3*d*m^2*n^2*x*x^m*x^(4*n)*e^
m + 122*B*b^3*c*m*n^3*x*x^m*x^(4*n)*e^m + 366*B*a*b^2*d*m*n^3*x*x^m*x^(4*n)*e^m + 122*A*b^3*d*m*n^3*x*x^m*x^(4
*n)*e^m + 30*B*b^3*c*n^4*x*x^m*x^(4*n)*e^m + 90*B*a*b^2*d*n^4*x*x^m*x^(4*n)*e^m + 30*A*b^3*d*n^4*x*x^m*x^(4*n)
*e^m + 15*B*a*b^2*c*m^4*x*x^m*x^(3*n)*e^m + 5*A*b^3*c*m^4*x*x^m*x^(3*n)*e^m + 15*B*a^2*b*d*m^4*x*x^m*x^(3*n)*e
^m + 15*A*a*b^2*d*m^4*x*x^m*x^(3*n)*e^m + 144*B*a*b^2*c*m^3*n*x*x^m*x^(3*n)*e^m + 48*A*b^3*c*m^3*n*x*x^m*x^(3*
n)*e^m + 144*B*a^2*b*d*m^3*n*x*x^m*x^(3*n)*e^m + 144*A*a*b^2*d*m^3*n*x*x^m*x^(3*n)*e^m + 441*B*a*b^2*c*m^2*n^2
*x*x^m*x^(3*n)*e^m + 147*A*b^3*c*m^2*n^2*x*x^m*x^(3*n)*e^m + 441*B*a^2*b*d*m^2*n^2*x*x^m*x^(3*n)*e^m + 441*A*a
*b^2*d*m^2*n^2*x*x^m*x^(3*n)*e^m + 468*B*a*b^2*c*m*n^3*x*x^m*x^(3*n)*e^m + 156*A*b^3*c*m*n^3*x*x^m*x^(3*n)*e^m
+ 468*B*a^2*b*d*m*n^3*x*x^m*x^(3*n)*e^m + 468*A*a*b^2*d*m*n^3*x*x^m*x^(3*n)*e^m + 120*B*a*b^2*c*n^4*x*x^m*x^(
3*n)*e^m + 40*A*b^3*c*n^4*x*x^m*x^(3*n)*e^m + 120*B*a^2*b*d*n^4*x*x^m*x^(3*n)*e^m + 120*A*a*b^2*d*n^4*x*x^m*x^
(3*n)*e^m + 15*B*a^2*b*c*m^4*x*x^m*x^(2*n)*e^m + 15*A*a*b^2*c*m^4*x*x^m*x^(2*n)*e^m + 5*B*a^3*d*m^4*x*x^m*x^(2
*n)*e^m + 15*A*a^2*b*d*m^4*x*x^m*x^(2*n)*e^m + 156*B*a^2*b*c*m^3*n*x*x^m*x^(2*n)*e^m + 156*A*a*b^2*c*m^3*n*x*x
^m*x^(2*n)*e^m + 52*B*a^3*d*m^3*n*x*x^m*x^(2*n)*e^m + 156*A*a^2*b*d*m^3*n*x*x^m*x^(2*n)*e^m + 531*B*a^2*b*c*m^
2*n^2*x*x^m*x^(2*n)*e^m + 531*A*a*b^2*c*m^2*n^2*x*x^m*x^(2*n)*e^m + 177*B*a^3*d*m^2*n^2*x*x^m*x^(2*n)*e^m + 53
1*A*a^2*b*d*m^2*n^2*x*x^m*x^(2*n)*e^m + 642*B*a...

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Mupad [B]
time = 5.64, size = 1089, normalized size = 5.19 \begin {gather*} \frac {A\,a^3\,c\,x\,{\left (e\,x\right )}^m}{m+1}+\frac {b^2\,x\,x^{4\,n}\,{\left (e\,x\right )}^m\,\left (A\,b\,d+3\,B\,a\,d+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 {a\,x\,x^{2\,n}\,{\left (e\,x\right )}^m\,\left (3\,A\,b^2\,c+B\,a^2\,d+3\,A\,a\,b\,d+3\,B\,a\,b\,c\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 {b\,x\,x^{3\,n}\,{\left (e\,x\right )}^m\,\left (A\,b^2\,c+3\,B\,a^2\,d+3\,A\,a\,b\,d+3\,B\,a\,b\,c\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 {a^2\,x\,x^n\,{\left (e\,x\right )}^m\,\left (A\,a\,d+3\,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^3\,d\,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} \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)^3*(c + d*x^n),x)

[Out]

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