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

Optimal. Leaf size=137 \[ \frac {c^2 x^{n+1} (e x)^m (3 A d+B c)}{m+n+1}+\frac {d^2 x^{3 n+1} (e x)^m (A d+3 B c)}{m+3 n+1}+\frac {3 c d x^{2 n+1} (e x)^m (A d+B c)}{m+2 n+1}+\frac {A c^3 (e x)^{m+1}}{e (m+1)}+\frac {B d^3 x^{4 n+1} (e x)^m}{m+4 n+1} \]

[Out]

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

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Rubi [A]  time = 0.11, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {448, 20, 30} \[ \frac {c^2 x^{n+1} (e x)^m (3 A d+B c)}{m+n+1}+\frac {d^2 x^{3 n+1} (e x)^m (A d+3 B c)}{m+3 n+1}+\frac {3 c d x^{2 n+1} (e x)^m (A d+B c)}{m+2 n+1}+\frac {A c^3 (e x)^{m+1}}{e (m+1)}+\frac {B d^3 x^{4 n+1} (e x)^m}{m+4 n+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^2*(B*c + 3*A*d)*x^(1 + n)*(e*x)^m)/(1 + m + n) + (3*c*d*(B*c + A*d)*x^(1 + 2*n)*(e*x)^m)/(1 + m + 2*n) + (d
^2*(3*B*c + A*d)*x^(1 + 3*n)*(e*x)^m)/(1 + m + 3*n) + (B*d^3*x^(1 + 4*n)*(e*x)^m)/(1 + m + 4*n) + (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 448

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]

Rubi steps

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

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Mathematica [A]  time = 0.18, size = 106, normalized size = 0.77 \[ x (e x)^m \left (\frac {c^2 x^n (3 A d+B c)}{m+n+1}+\frac {d^2 x^{3 n} (A d+3 B c)}{m+3 n+1}+\frac {3 c d x^{2 n} (A d+B c)}{m+2 n+1}+\frac {A c^3}{m+1}+\frac {B d^3 x^{4 n}}{m+4 n+1}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.72, size = 1104, normalized size = 8.06 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.75, size = 2278, normalized size = 16.63 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(B*d^3*m^4*x*x^m*x^(4*n)*e^m + 6*B*d^3*m^3*n*x*x^m*x^(4*n)*e^m + 11*B*d^3*m^2*n^2*x*x^m*x^(4*n)*e^m + 6*B*d^3*
m*n^3*x*x^m*x^(4*n)*e^m + 3*B*c*d^2*m^4*x*x^m*x^(3*n)*e^m + A*d^3*m^4*x*x^m*x^(3*n)*e^m + 21*B*c*d^2*m^3*n*x*x
^m*x^(3*n)*e^m + 7*A*d^3*m^3*n*x*x^m*x^(3*n)*e^m + 42*B*c*d^2*m^2*n^2*x*x^m*x^(3*n)*e^m + 14*A*d^3*m^2*n^2*x*x
^m*x^(3*n)*e^m + 24*B*c*d^2*m*n^3*x*x^m*x^(3*n)*e^m + 8*A*d^3*m*n^3*x*x^m*x^(3*n)*e^m + 3*B*c^2*d*m^4*x*x^m*x^
(2*n)*e^m + 3*A*c*d^2*m^4*x*x^m*x^(2*n)*e^m + 24*B*c^2*d*m^3*n*x*x^m*x^(2*n)*e^m + 24*A*c*d^2*m^3*n*x*x^m*x^(2
*n)*e^m + 57*B*c^2*d*m^2*n^2*x*x^m*x^(2*n)*e^m + 57*A*c*d^2*m^2*n^2*x*x^m*x^(2*n)*e^m + 36*B*c^2*d*m*n^3*x*x^m
*x^(2*n)*e^m + 36*A*c*d^2*m*n^3*x*x^m*x^(2*n)*e^m + B*c^3*m^4*x*x^m*x^n*e^m + 3*A*c^2*d*m^4*x*x^m*x^n*e^m + 9*
B*c^3*m^3*n*x*x^m*x^n*e^m + 27*A*c^2*d*m^3*n*x*x^m*x^n*e^m + 26*B*c^3*m^2*n^2*x*x^m*x^n*e^m + 78*A*c^2*d*m^2*n
^2*x*x^m*x^n*e^m + 24*B*c^3*m*n^3*x*x^m*x^n*e^m + 72*A*c^2*d*m*n^3*x*x^m*x^n*e^m + A*c^3*m^4*x*x^m*e^m + 10*A*
c^3*m^3*n*x*x^m*e^m + 35*A*c^3*m^2*n^2*x*x^m*e^m + 50*A*c^3*m*n^3*x*x^m*e^m + 24*A*c^3*n^4*x*x^m*e^m + 4*B*d^3
*m^3*x*x^m*x^(4*n)*e^m + 18*B*d^3*m^2*n*x*x^m*x^(4*n)*e^m + 22*B*d^3*m*n^2*x*x^m*x^(4*n)*e^m + 6*B*d^3*n^3*x*x
^m*x^(4*n)*e^m + 12*B*c*d^2*m^3*x*x^m*x^(3*n)*e^m + 4*A*d^3*m^3*x*x^m*x^(3*n)*e^m + 63*B*c*d^2*m^2*n*x*x^m*x^(
3*n)*e^m + 21*A*d^3*m^2*n*x*x^m*x^(3*n)*e^m + 84*B*c*d^2*m*n^2*x*x^m*x^(3*n)*e^m + 28*A*d^3*m*n^2*x*x^m*x^(3*n
)*e^m + 24*B*c*d^2*n^3*x*x^m*x^(3*n)*e^m + 8*A*d^3*n^3*x*x^m*x^(3*n)*e^m + 12*B*c^2*d*m^3*x*x^m*x^(2*n)*e^m +
12*A*c*d^2*m^3*x*x^m*x^(2*n)*e^m + 72*B*c^2*d*m^2*n*x*x^m*x^(2*n)*e^m + 72*A*c*d^2*m^2*n*x*x^m*x^(2*n)*e^m + 1
14*B*c^2*d*m*n^2*x*x^m*x^(2*n)*e^m + 114*A*c*d^2*m*n^2*x*x^m*x^(2*n)*e^m + 36*B*c^2*d*n^3*x*x^m*x^(2*n)*e^m +
36*A*c*d^2*n^3*x*x^m*x^(2*n)*e^m + 4*B*c^3*m^3*x*x^m*x^n*e^m + 12*A*c^2*d*m^3*x*x^m*x^n*e^m + 27*B*c^3*m^2*n*x
*x^m*x^n*e^m + 81*A*c^2*d*m^2*n*x*x^m*x^n*e^m + 52*B*c^3*m*n^2*x*x^m*x^n*e^m + 156*A*c^2*d*m*n^2*x*x^m*x^n*e^m
+ 24*B*c^3*n^3*x*x^m*x^n*e^m + 72*A*c^2*d*n^3*x*x^m*x^n*e^m + 4*A*c^3*m^3*x*x^m*e^m + 30*A*c^3*m^2*n*x*x^m*e^
m + 70*A*c^3*m*n^2*x*x^m*e^m + 50*A*c^3*n^3*x*x^m*e^m + 6*B*d^3*m^2*x*x^m*x^(4*n)*e^m + 18*B*d^3*m*n*x*x^m*x^(
4*n)*e^m + 11*B*d^3*n^2*x*x^m*x^(4*n)*e^m + 18*B*c*d^2*m^2*x*x^m*x^(3*n)*e^m + 6*A*d^3*m^2*x*x^m*x^(3*n)*e^m +
63*B*c*d^2*m*n*x*x^m*x^(3*n)*e^m + 21*A*d^3*m*n*x*x^m*x^(3*n)*e^m + 42*B*c*d^2*n^2*x*x^m*x^(3*n)*e^m + 14*A*d
^3*n^2*x*x^m*x^(3*n)*e^m + 18*B*c^2*d*m^2*x*x^m*x^(2*n)*e^m + 18*A*c*d^2*m^2*x*x^m*x^(2*n)*e^m + 72*B*c^2*d*m*
n*x*x^m*x^(2*n)*e^m + 72*A*c*d^2*m*n*x*x^m*x^(2*n)*e^m + 57*B*c^2*d*n^2*x*x^m*x^(2*n)*e^m + 57*A*c*d^2*n^2*x*x
^m*x^(2*n)*e^m + 6*B*c^3*m^2*x*x^m*x^n*e^m + 18*A*c^2*d*m^2*x*x^m*x^n*e^m + 27*B*c^3*m*n*x*x^m*x^n*e^m + 81*A*
c^2*d*m*n*x*x^m*x^n*e^m + 26*B*c^3*n^2*x*x^m*x^n*e^m + 78*A*c^2*d*n^2*x*x^m*x^n*e^m + 6*A*c^3*m^2*x*x^m*e^m +
30*A*c^3*m*n*x*x^m*e^m + 35*A*c^3*n^2*x*x^m*e^m + 4*B*d^3*m*x*x^m*x^(4*n)*e^m + 6*B*d^3*n*x*x^m*x^(4*n)*e^m +
12*B*c*d^2*m*x*x^m*x^(3*n)*e^m + 4*A*d^3*m*x*x^m*x^(3*n)*e^m + 21*B*c*d^2*n*x*x^m*x^(3*n)*e^m + 7*A*d^3*n*x*x^
m*x^(3*n)*e^m + 12*B*c^2*d*m*x*x^m*x^(2*n)*e^m + 12*A*c*d^2*m*x*x^m*x^(2*n)*e^m + 24*B*c^2*d*n*x*x^m*x^(2*n)*e
^m + 24*A*c*d^2*n*x*x^m*x^(2*n)*e^m + 4*B*c^3*m*x*x^m*x^n*e^m + 12*A*c^2*d*m*x*x^m*x^n*e^m + 9*B*c^3*n*x*x^m*x
^n*e^m + 27*A*c^2*d*n*x*x^m*x^n*e^m + 4*A*c^3*m*x*x^m*e^m + 10*A*c^3*n*x*x^m*e^m + B*d^3*x*x^m*x^(4*n)*e^m + 3
*B*c*d^2*x*x^m*x^(3*n)*e^m + A*d^3*x*x^m*x^(3*n)*e^m + 3*B*c^2*d*x*x^m*x^(2*n)*e^m + 3*A*c*d^2*x*x^m*x^(2*n)*e
^m + B*c^3*x*x^m*x^n*e^m + 3*A*c^2*d*x*x^m*x^n*e^m + A*c^3*x*x^m*e^m)/(m^5 + 10*m^4*n + 35*m^3*n^2 + 50*m^2*n^
3 + 24*m*n^4 + 5*m^4 + 40*m^3*n + 105*m^2*n^2 + 100*m*n^3 + 24*n^4 + 10*m^3 + 60*m^2*n + 105*m*n^2 + 50*n^3 +
10*m^2 + 40*m*n + 35*n^2 + 5*m + 10*n + 1)

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maple [C]  time = 0.13, size = 1609, normalized size = 11.74 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*(57*B*c^2*d*m^2*n^2*(x^n)^2+36*B*c^2*d*m*n^3*(x^n)^2+24*B*c^2*d*m^3*n*(x^n)^2+11*B*d^3*n^2*(x^n)^4+B*d^3*m^4
*(x^n)^4+8*A*d^3*n^3*(x^n)^3+6*B*d^3*m^2*(x^n)^4+24*B*c^3*n^3*x^n+A*d^3*m^4*(x^n)^3+4*B*d^3*m^3*(x^n)^4+6*B*d^
3*n^3*(x^n)^4+4*A*d^3*m^3*(x^n)^3+4*B*c^3*m^3*x^n+6*B*d^3*(x^n)^4*n+4*A*d^3*(x^n)^3*m+7*A*d^3*(x^n)^3*n+B*c^3*
m^4*x^n+4*m*B*d^3*(x^n)^4+26*B*c^3*n^2*x^n+6*A*d^3*m^2*(x^n)^3+14*A*d^3*n^2*(x^n)^3+3*A*c^2*d*x^n+3*B*c*d^2*(x
^n)^3+3*A*c*d^2*(x^n)^2+4*B*c^3*x^n*m+9*B*c^3*x^n*n+6*B*c^3*m^2*x^n+3*B*c^2*d*(x^n)^2+10*A*c^3*m^3*n+35*A*c^3*
m^2*n^2+50*A*c^3*m*n^3+30*A*c^3*m^2*n+70*A*c^3*m*n^2+30*A*c^3*m*n+36*A*c*d^2*m*n^3*(x^n)^2+24*A*c*d^2*m^3*n*(x
^n)^2+57*A*c*d^2*m^2*n^2*(x^n)^2+72*B*c^2*d*m^2*n*(x^n)^2+114*B*c^2*d*m*n^2*(x^n)^2+63*B*c*d^2*m*n*(x^n)^3+21*
B*c*d^2*m^3*n*(x^n)^3+42*B*c*d^2*m^2*n^2*(x^n)^3+24*B*c*d^2*m*n^3*(x^n)^3+84*B*c*d^2*m*n^2*(x^n)^3+27*A*c^2*d*
m^3*n*x^n+78*A*c^2*d*m^2*n^2*x^n+72*A*c^2*d*m*n^3*x^n+72*A*c*d^2*m^2*n*(x^n)^2+114*A*c*d^2*m*n^2*(x^n)^2+81*A*
c^2*d*m^2*n*x^n+156*A*c^2*d*m*n^2*x^n+72*A*c*d^2*m*n*(x^n)^2+72*B*c^2*d*m*n*(x^n)^2+63*B*c*d^2*m^2*n*(x^n)^3+8
1*A*c^2*d*m*n*x^n+A*c^3+(x^n)^4*B*d^3+(x^n)^3*A*d^3+12*B*c^2*d*m^3*(x^n)^2+36*B*c^2*d*n^3*(x^n)^2+4*A*c^3*m+10
*A*c^3*n+24*A*c^3*n^4+4*A*c^3*m^3+50*A*c^3*n^3+6*A*c^3*m^2+35*A*c^3*n^2+x^n*B*c^3+A*c^3*m^4+72*A*c^2*d*n^3*x^n
+18*A*c*d^2*m^2*(x^n)^2+57*A*c*d^2*n^2*(x^n)^2+27*B*c^3*m^2*n*x^n+18*B*c*d^2*m^2*(x^n)^3+3*A*c*d^2*m^4*(x^n)^2
+21*A*d^3*m^2*n*(x^n)^3+28*A*d^3*m*n^2*(x^n)^3+3*B*c^2*d*m^4*(x^n)^2+26*B*c^3*m^2*n^2*x^n+7*A*d^3*m^3*n*(x^n)^
3+14*A*d^3*m^2*n^2*(x^n)^3+8*A*d^3*m*n^3*(x^n)^3+3*B*c*d^2*m^4*(x^n)^3+18*B*d^3*m^2*n*(x^n)^4+22*B*d^3*m*n^2*(
x^n)^4+78*A*c^2*d*n^2*x^n+12*A*c*d^2*(x^n)^2*m+24*A*c*d^2*(x^n)^2*n+27*B*c^3*m*n*x^n+12*B*c^2*d*(x^n)^2*m+24*B
*c^2*d*(x^n)^2*n+12*A*c^2*d*x^n*m+27*A*c^2*d*x^n*n+42*B*c*d^2*n^2*(x^n)^3+12*A*c^2*d*m^3*x^n+18*B*d^3*m*n*(x^n
)^4+3*A*c^2*d*m^4*x^n+12*A*c*d^2*m^3*(x^n)^2+36*A*c*d^2*n^3*(x^n)^2+21*A*d^3*m*n*(x^n)^3+9*B*c^3*m^3*n*x^n+24*
B*c^3*m*n^3*x^n+12*B*c*d^2*m^3*(x^n)^3+24*B*c*d^2*n^3*(x^n)^3+6*B*d^3*m^3*n*(x^n)^4+11*B*d^3*m^2*n^2*(x^n)^4+6
*B*d^3*m*n^3*(x^n)^4+52*B*c^3*m*n^2*x^n+18*B*c^2*d*m^2*(x^n)^2+57*B*c^2*d*n^2*(x^n)^2+12*B*c*d^2*(x^n)^3*m+21*
B*c*d^2*(x^n)^3*n+18*A*c^2*d*m^2*x^n)/(m+1)/(m+n+1)/(m+2*n+1)/(m+3*n+1)/(m+4*n+1)*exp(1/2*(-I*Pi*csgn(I*e)*csg
n(I*x)*csgn(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 [A]  time = 0.84, size = 219, normalized size = 1.60 \[ \frac {B d^{3} e^{m} x e^{\left (m \log \relax (x) + 4 \, n \log \relax (x)\right )}}{m + 4 \, n + 1} + \frac {3 \, B c d^{2} e^{m} x e^{\left (m \log \relax (x) + 3 \, n \log \relax (x)\right )}}{m + 3 \, n + 1} + \frac {A d^{3} e^{m} x e^{\left (m \log \relax (x) + 3 \, n \log \relax (x)\right )}}{m + 3 \, n + 1} + \frac {3 \, B c^{2} d e^{m} x e^{\left (m \log \relax (x) + 2 \, n \log \relax (x)\right )}}{m + 2 \, n + 1} + \frac {3 \, A c d^{2} e^{m} x e^{\left (m \log \relax (x) + 2 \, n \log \relax (x)\right )}}{m + 2 \, n + 1} + \frac {B c^{3} e^{m} x e^{\left (m \log \relax (x) + n \log \relax (x)\right )}}{m + n + 1} + \frac {3 \, A c^{2} d e^{m} x e^{\left (m \log \relax (x) + n \log \relax (x)\right )}}{m + n + 1} + \frac {\left (e x\right )^{m + 1} A c^{3}}{e {\left (m + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 5.31, size = 563, normalized size = 4.11 \[ \frac {A\,c^3\,x\,{\left (e\,x\right )}^m}{m+1}+\frac {d^2\,x\,x^{3\,n}\,{\left (e\,x\right )}^m\,\left (A\,d+3\,B\,c\right )\,\left (m^3+7\,m^2\,n+3\,m^2+14\,m\,n^2+14\,m\,n+3\,m+8\,n^3+14\,n^2+7\,n+1\right )}{m^4+10\,m^3\,n+4\,m^3+35\,m^2\,n^2+30\,m^2\,n+6\,m^2+50\,m\,n^3+70\,m\,n^2+30\,m\,n+4\,m+24\,n^4+50\,n^3+35\,n^2+10\,n+1}+\frac {c^2\,x\,x^n\,{\left (e\,x\right )}^m\,\left (3\,A\,d+B\,c\right )\,\left (m^3+9\,m^2\,n+3\,m^2+26\,m\,n^2+18\,m\,n+3\,m+24\,n^3+26\,n^2+9\,n+1\right )}{m^4+10\,m^3\,n+4\,m^3+35\,m^2\,n^2+30\,m^2\,n+6\,m^2+50\,m\,n^3+70\,m\,n^2+30\,m\,n+4\,m+24\,n^4+50\,n^3+35\,n^2+10\,n+1}+\frac {B\,d^3\,x\,x^{4\,n}\,{\left (e\,x\right )}^m\,\left (m^3+6\,m^2\,n+3\,m^2+11\,m\,n^2+12\,m\,n+3\,m+6\,n^3+11\,n^2+6\,n+1\right )}{m^4+10\,m^3\,n+4\,m^3+35\,m^2\,n^2+30\,m^2\,n+6\,m^2+50\,m\,n^3+70\,m\,n^2+30\,m\,n+4\,m+24\,n^4+50\,n^3+35\,n^2+10\,n+1}+\frac {3\,c\,d\,x\,x^{2\,n}\,{\left (e\,x\right )}^m\,\left (A\,d+B\,c\right )\,\left (m^3+8\,m^2\,n+3\,m^2+19\,m\,n^2+16\,m\,n+3\,m+12\,n^3+19\,n^2+8\,n+1\right )}{m^4+10\,m^3\,n+4\,m^3+35\,m^2\,n^2+30\,m^2\,n+6\,m^2+50\,m\,n^3+70\,m\,n^2+30\,m\,n+4\,m+24\,n^4+50\,n^3+35\,n^2+10\,n+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)*(c+d*x**n)**3,x)

[Out]

Timed out

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