3.1.11 \(\int (e x)^m (A+B x^n) (c+d x^n)^2 \, dx\) [11]

Optimal. Leaf size=102 \[ \frac {c (B c+2 A d) x^{1+n} (e x)^m}{1+m+n}+\frac {d (2 B c+A d) x^{1+2 n} (e x)^m}{1+m+2 n}+\frac {B d^2 x^{1+3 n} (e x)^m}{1+m+3 n}+\frac {A c^2 (e x)^{1+m}}{e (1+m)} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {459, 20, 30} \begin {gather*} \frac {c x^{n+1} (e x)^m (2 A d+B c)}{m+n+1}+\frac {d x^{2 n+1} (e x)^m (A d+2 B c)}{m+2 n+1}+\frac {A c^2 (e x)^{m+1}}{e (m+1)}+\frac {B d^2 x^{3 n+1} (e x)^m}{m+3 n+1} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 20

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[b^IntPart[n]*((b*v)^FracPart[n]/(a^IntPart[n]
*(a*v)^FracPart[n])), Int[u*(a*v)^(m + n), x], x] /; FreeQ[{a, b, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n]
&&  !IntegerQ[m + n]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 459

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

Rubi steps

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

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Mathematica [A]
time = 0.11, size = 78, normalized size = 0.76 \begin {gather*} x (e x)^m \left (\frac {A c^2}{1+m}+\frac {c (B c+2 A d) x^n}{1+m+n}+\frac {d (2 B c+A d) x^{2 n}}{1+m+2 n}+\frac {B d^2 x^{3 n}}{1+m+3 n}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.36, size = 732, normalized size = 7.18

method	result	size
risch	\(\frac {x \left (2 A c d \,m^{3} x^{n}+5 B \,c^{2} m^{2} n \,x^{n}+6 B \,c^{2} m \,n^{2} x^{n}+16 B c d m n \,x^{2 n}+6 B c d m \,n^{2} x^{2 n}+8 B c d \,m^{2} n \,x^{2 n}+3 A \,d^{2} n^{2} x^{2 n}+3 A \,d^{2} x^{2 n} m +4 A \,d^{2} x^{2 n} n +3 m B \,d^{2} x^{3 n}+3 B \,d^{2} x^{3 n} n +B \,d^{2} m^{3} x^{3 n}+2 B \,d^{2} n^{2} x^{3 n}+3 A \,d^{2} m^{2} x^{2 n}+A \,d^{2} m^{3} x^{2 n}+3 B \,d^{2} m^{2} x^{3 n}+2 B c d \,x^{2 n}+x^{2 n} A \,d^{2}+B \,c^{2} m^{3} x^{n}+6 A \,c^{2} m^{2} n +11 A \,c^{2} m \,n^{2}+12 A \,c^{2} m n +A \,c^{2}+2 B c d \,m^{3} x^{2 n}+3 B \,c^{2} m^{2} x^{n}+12 A c d m \,n^{2} x^{n}+x^{3 n} B \,d^{2}+x^{n} B \,c^{2}+3 B \,c^{2} x^{n} m +A \,c^{2} m^{3}+3 A \,c^{2} m +6 A \,c^{2} n +6 A \,c^{2} n^{3}+3 A \,c^{2} m^{2}+11 A \,c^{2} n^{2}+20 A c d m n \,x^{n}+10 A c d \,m^{2} n \,x^{n}+6 B \,c^{2} n^{2} x^{n}+2 A c d \,x^{n}+5 B \,c^{2} x^{n} n +6 A c d \,x^{n} m +6 A c d \,m^{2} x^{n}+12 A c d \,n^{2} x^{n}+10 B \,c^{2} m n \,x^{n}+10 A c d \,x^{n} n +6 B \,d^{2} m n \,x^{3 n}+8 A \,d^{2} m n \,x^{2 n}+6 B c d \,m^{2} x^{2 n}+6 B c d \,n^{2} x^{2 n}+6 B c d \,x^{2 n} m +8 B c d \,x^{2 n} n +3 B \,d^{2} m^{2} n \,x^{3 n}+2 B \,d^{2} m \,n^{2} x^{3 n}+4 A \,d^{2} m^{2} n \,x^{2 n}+3 A \,d^{2} m \,n^{2} x^{2 n}\right ) {\mathrm e}^{\frac {m \left (-i \pi \mathrm {csgn}\left (i e x \right )^{3}+i \pi \mathrm {csgn}\left (i e x \right )^{2} \mathrm {csgn}\left (i e \right )+i \pi \mathrm {csgn}\left (i e x \right )^{2} \mathrm {csgn}\left (i x \right )-i \pi \,\mathrm {csgn}\left (i e x \right ) \mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i x \right )+2 \ln \left (e \right )+2 \ln \left (x \right )\right )}{2}}}{\left (1+m \right ) \left (1+m +n \right ) \left (1+m +2 n \right ) \left (1+m +3 n \right )}\)	\(732\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*(2*B*c*d*m^3*(x^n)^2+6*B*d^2*m*n*(x^n)^3+2*A*c*d*m^3*x^n+8*A*d^2*m*n*(x^n)^2+5*B*c^2*m^2*n*x^n+6*B*c^2*m*n^2
*x^n+3*A*d^2*n^2*(x^n)^2+B*c^2*m^3*x^n+3*A*d^2*(x^n)^2*m+6*A*c^2*m^2*n+11*A*c^2*m*n^2+12*A*c^2*m*n+A*c^2+(x^n)
^3*B*d^2+(x^n)^2*A*d^2+4*A*d^2*(x^n)^2*n+3*B*c^2*m^2*x^n+3*m*B*d^2*(x^n)^3+3*B*d^2*(x^n)^3*n+12*A*c*d*m*n^2*x^
n+16*B*c*d*m*n*(x^n)^2+6*B*c*d*m*n^2*(x^n)^2+x^n*B*c^2+3*B*c^2*x^n*m+A*c^2*m^3+8*B*c*d*m^2*n*(x^n)^2+B*d^2*m^3
*(x^n)^3+3*A*c^2*m+6*A*c^2*n+6*A*c^2*n^3+3*A*c^2*m^2+11*A*c^2*n^2+20*A*c*d*m*n*x^n+10*A*c*d*m^2*n*x^n+2*B*d^2*
n^2*(x^n)^3+3*A*d^2*m^2*(x^n)^2+A*d^2*m^3*(x^n)^2+3*B*d^2*m^2*(x^n)^3+6*B*c^2*n^2*x^n+2*B*c*d*(x^n)^2+2*A*c*d*
x^n+5*B*c^2*x^n*n+6*A*c*d*x^n*m+6*B*c*d*m^2*(x^n)^2+6*B*c*d*n^2*(x^n)^2+6*A*c*d*m^2*x^n+12*A*c*d*n^2*x^n+10*B*
c^2*m*n*x^n+10*A*c*d*x^n*n+6*B*c*d*(x^n)^2*m+8*B*c*d*(x^n)^2*n+3*B*d^2*m^2*n*(x^n)^3+2*B*d^2*m*n^2*(x^n)^3+4*A
*d^2*m^2*n*(x^n)^2+3*A*d^2*m*n^2*(x^n)^2)/(1+m)/(1+m+n)/(1+m+2*n)/(1+m+3*n)*exp(1/2*m*(-I*Pi*csgn(I*e*x)^3+I*P
i*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 = 0.32, size = 145, normalized size = 1.42 \begin {gather*} \frac {\left (x e\right )^{m + 1} A c^{2} e^{\left (-1\right )}}{m + 1} + \frac {B d^{2} x e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right ) + m\right )}}{m + 3 \, n + 1} + \frac {2 \, B c d x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right ) + m\right )}}{m + 2 \, n + 1} + \frac {A d^{2} x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right ) + m\right )}}{m + 2 \, n + 1} + \frac {B c^{2} x e^{\left (m \log \left (x\right ) + n \log \left (x\right ) + m\right )}}{m + n + 1} + \frac {2 \, A c 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)*(c+d*x^n)^2,x, algorithm="maxima")

[Out]

(x*e)^(m + 1)*A*c^2*e^(-1)/(m + 1) + B*d^2*x*e^(m*log(x) + 3*n*log(x) + m)/(m + 3*n + 1) + 2*B*c*d*x*e^(m*log(
x) + 2*n*log(x) + m)/(m + 2*n + 1) + A*d^2*x*e^(m*log(x) + 2*n*log(x) + m)/(m + 2*n + 1) + B*c^2*x*e^(m*log(x)
+ n*log(x) + m)/(m + n + 1) + 2*A*c*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. 515 vs. \(2 (105) = 210\).
time = 3.53, size = 515, normalized size = 5.05 \begin {gather*} \frac {{\left (B d^{2} m^{3} + 3 \, B d^{2} m^{2} + 3 \, B d^{2} m + B d^{2} + 2 \, {\left (B d^{2} m + B d^{2}\right )} n^{2} + 3 \, {\left (B d^{2} m^{2} + 2 \, B d^{2} m + B d^{2}\right )} n\right )} x x^{3 \, n} e^{\left (m \log \left (x\right ) + m\right )} + {\left ({\left (2 \, B c d + A d^{2}\right )} m^{3} + 2 \, B c d + A d^{2} + 3 \, {\left (2 \, B c d + A d^{2}\right )} m^{2} + 3 \, {\left (2 \, B c d + A d^{2} + {\left (2 \, B c d + A d^{2}\right )} m\right )} n^{2} + 3 \, {\left (2 \, B c d + A d^{2}\right )} m + 4 \, {\left (2 \, B c d + A d^{2} + {\left (2 \, B c d + A d^{2}\right )} m^{2} + 2 \, {\left (2 \, B c d + A d^{2}\right )} m\right )} n\right )} x x^{2 \, n} e^{\left (m \log \left (x\right ) + m\right )} + {\left ({\left (B c^{2} + 2 \, A c d\right )} m^{3} + B c^{2} + 2 \, A c d + 3 \, {\left (B c^{2} + 2 \, A c d\right )} m^{2} + 6 \, {\left (B c^{2} + 2 \, A c d + {\left (B c^{2} + 2 \, A c d\right )} m\right )} n^{2} + 3 \, {\left (B c^{2} + 2 \, A c d\right )} m + 5 \, {\left (B c^{2} + 2 \, A c d + {\left (B c^{2} + 2 \, A c d\right )} m^{2} + 2 \, {\left (B c^{2} + 2 \, A c d\right )} m\right )} n\right )} x x^{n} e^{\left (m \log \left (x\right ) + m\right )} + {\left (A c^{2} m^{3} + 6 \, A c^{2} n^{3} + 3 \, A c^{2} m^{2} + 3 \, A c^{2} m + A c^{2} + 11 \, {\left (A c^{2} m + A c^{2}\right )} n^{2} + 6 \, {\left (A c^{2} m^{2} + 2 \, A c^{2} m + A c^{2}\right )} n\right )} x e^{\left (m \log \left (x\right ) + m\right )}}{m^{4} + 6 \, {\left (m + 1\right )} n^{3} + 4 \, m^{3} + 11 \, {\left (m^{2} + 2 \, m + 1\right )} n^{2} + 6 \, m^{2} + 6 \, {\left (m^{3} + 3 \, m^{2} + 3 \, m + 1\right )} n + 4 \, m + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((B*d^2*m^3 + 3*B*d^2*m^2 + 3*B*d^2*m + B*d^2 + 2*(B*d^2*m + B*d^2)*n^2 + 3*(B*d^2*m^2 + 2*B*d^2*m + B*d^2)*n)
*x*x^(3*n)*e^(m*log(x) + m) + ((2*B*c*d + A*d^2)*m^3 + 2*B*c*d + A*d^2 + 3*(2*B*c*d + A*d^2)*m^2 + 3*(2*B*c*d
+ A*d^2 + (2*B*c*d + A*d^2)*m)*n^2 + 3*(2*B*c*d + A*d^2)*m + 4*(2*B*c*d + A*d^2 + (2*B*c*d + A*d^2)*m^2 + 2*(2
*B*c*d + A*d^2)*m)*n)*x*x^(2*n)*e^(m*log(x) + m) + ((B*c^2 + 2*A*c*d)*m^3 + B*c^2 + 2*A*c*d + 3*(B*c^2 + 2*A*c
*d)*m^2 + 6*(B*c^2 + 2*A*c*d + (B*c^2 + 2*A*c*d)*m)*n^2 + 3*(B*c^2 + 2*A*c*d)*m + 5*(B*c^2 + 2*A*c*d + (B*c^2
+ 2*A*c*d)*m^2 + 2*(B*c^2 + 2*A*c*d)*m)*n)*x*x^n*e^(m*log(x) + m) + (A*c^2*m^3 + 6*A*c^2*n^3 + 3*A*c^2*m^2 + 3
*A*c^2*m + A*c^2 + 11*(A*c^2*m + A*c^2)*n^2 + 6*(A*c^2*m^2 + 2*A*c^2*m + A*c^2)*n)*x*e^(m*log(x) + m))/(m^4 +
6*(m + 1)*n^3 + 4*m^3 + 11*(m^2 + 2*m + 1)*n^2 + 6*m^2 + 6*(m^3 + 3*m^2 + 3*m + 1)*n + 4*m + 1)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 5794 vs. \(2 (94) = 188\).
time = 35.26, size = 5794, normalized size = 56.80 \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)*(c+d*x**n)**2,x)

[Out]

Piecewise(((A + B)*(c + d)**2*log(x)/e, Eq(m, -1) & Eq(n, 0)), ((A*c**2*log(x) + 2*A*c*d*x**n/n + A*d**2*x**(2
*n)/(2*n) + B*c**2*x**n/n + B*c*d*x**(2*n)/n + B*d**2*x**(3*n)/(3*n))/e, Eq(m, -1)), (A*c**2*Piecewise((-1/(3*
n*(e*x)**(3*n)), Ne(n, 0)), (log(x), True))/e + 2*A*c*d*Piecewise((-x**n/(2*n*(e*x)**(3*n)), Ne(n, 0)), (log(x
), True))/e + A*d**2*Piecewise((-x**(2*n)/(n*(e*x)**(3*n)), Ne(n, 0)), (log(x), True))/e + B*c**2*Piecewise((-
x**n/(2*n*(e*x)**(3*n)), Ne(n, 0)), (log(x), True))/e + 2*B*c*d*Piecewise((-x**(2*n)/(n*(e*x)**(3*n)), Ne(n, 0
)), (log(x), True))/e + B*d**2*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, Eq(m, -3*n - 1)), (A*c**2*Piecewise((-1/(2*n*(e*x)**(2*n)), Ne(n, 0)), (log
(x), True))/e + 2*A*c*d*Piecewise((-x**n/(n*(e*x)**(2*n)), Ne(n, 0)), (log(x), True))/e + A*d**2*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 + B*c**2*P
iecewise((-x**n/(n*(e*x)**(2*n)), Ne(n, 0)), (log(x), True))/e + 2*B*c*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 + B*d**2*Piecewise((x**(3*n)/(n*(e
*x)**(2*n)), Ne(n, 0)), (log(x), True))/e, Eq(m, -2*n - 1)), (A*c**2*Piecewise((-1/(n*(e*x)**n), Ne(n, 0)), (l
og(x), True))/e + 2*A*c*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 + A*d**2*Piecewise((x**(2*n)/(n*(e*x)**n), Ne(n, 0)), (log(x), True))/e + B*c**2*Piecewise((0, (A
bs(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*B*c*d*Piecewise((x**(2*n)
/(n*(e*x)**n), Ne(n, 0)), (log(x), True))/e + B*d**2*Piecewise((x**(3*n)/(2*n*(e*x)**n), Ne(n, 0)), (log(x), T
rue))/e, Eq(m, -n - 1)), (A*c**2*m**3*x*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2
+ 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + 6*A*c**2*m**2*n*x*(e*x)**m/(m**4 + 6*m*
*3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6
*n + 1) + 3*A*c**2*m**2*x*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 +
22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + 11*A*c**2*m*n**2*x*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3
+ 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + 12*
A*c**2*m*n*x*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 1
8*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + 3*A*c**2*m*x*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 1
8*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + 6*A*c**2*n**3*x*(e*x)*
*m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**
3 + 11*n**2 + 6*n + 1) + 11*A*c**2*n**2*x*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m*
*2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + 6*A*c**2*n*x*(e*x)**m/(m**4 + 6*m**3*
n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n
+ 1) + A*c**2*x*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2
+ 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + 2*A*c*d*m**3*x*x**n*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**
2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + 10*A*c*d*m**
2*n*x*x**n*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*
m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + 6*A*c*d*m**2*x*x**n*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**
2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + 12*A*c*d*m*n**2*x
*x**n*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n +
4*m + 6*n**3 + 11*n**2 + 6*n + 1) + 20*A*c*d*m*n*x*x**n*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 1
8*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + 6*A*c*d*m*x*x**n*(e*x)
**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n*
*3 + 11*n**2 + 6*n + 1) + 12*A*c*d*n**2*x*x**n*(e*x)**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n +
6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m +...

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1023 vs. \(2 (105) = 210\).
time = 1.45, size = 1023, normalized size = 10.03 \begin {gather*} \frac {B d^{2} m^{3} x x^{m} x^{3 \, n} e^{m} + 3 \, B d^{2} m^{2} n x x^{m} x^{3 \, n} e^{m} + 2 \, B d^{2} m n^{2} x x^{m} x^{3 \, n} e^{m} + 2 \, B c d m^{3} x x^{m} x^{2 \, n} e^{m} + A d^{2} m^{3} x x^{m} x^{2 \, n} e^{m} + 8 \, B c d m^{2} n x x^{m} x^{2 \, n} e^{m} + 4 \, A d^{2} m^{2} n x x^{m} x^{2 \, n} e^{m} + 6 \, B c d m n^{2} x x^{m} x^{2 \, n} e^{m} + 3 \, A d^{2} m n^{2} x x^{m} x^{2 \, n} e^{m} + B c^{2} m^{3} x x^{m} x^{n} e^{m} + 2 \, A c d m^{3} x x^{m} x^{n} e^{m} + 5 \, B c^{2} m^{2} n x x^{m} x^{n} e^{m} + 10 \, A c d m^{2} n x x^{m} x^{n} e^{m} + 6 \, B c^{2} m n^{2} x x^{m} x^{n} e^{m} + 12 \, A c d m n^{2} x x^{m} x^{n} e^{m} + A c^{2} m^{3} x x^{m} e^{m} + 6 \, A c^{2} m^{2} n x x^{m} e^{m} + 11 \, A c^{2} m n^{2} x x^{m} e^{m} + 6 \, A c^{2} n^{3} x x^{m} e^{m} + 3 \, B d^{2} m^{2} x x^{m} x^{3 \, n} e^{m} + 6 \, B d^{2} m n x x^{m} x^{3 \, n} e^{m} + 2 \, B d^{2} n^{2} x x^{m} x^{3 \, n} e^{m} + 6 \, B c d m^{2} x x^{m} x^{2 \, n} e^{m} + 3 \, A d^{2} m^{2} x x^{m} x^{2 \, n} e^{m} + 16 \, B c d m n x x^{m} x^{2 \, n} e^{m} + 8 \, A d^{2} m n x x^{m} x^{2 \, n} e^{m} + 6 \, B c d n^{2} x x^{m} x^{2 \, n} e^{m} + 3 \, A d^{2} n^{2} x x^{m} x^{2 \, n} e^{m} + 3 \, B c^{2} m^{2} x x^{m} x^{n} e^{m} + 6 \, A c d m^{2} x x^{m} x^{n} e^{m} + 10 \, B c^{2} m n x x^{m} x^{n} e^{m} + 20 \, A c d m n x x^{m} x^{n} e^{m} + 6 \, B c^{2} n^{2} x x^{m} x^{n} e^{m} + 12 \, A c d n^{2} x x^{m} x^{n} e^{m} + 3 \, A c^{2} m^{2} x x^{m} e^{m} + 12 \, A c^{2} m n x x^{m} e^{m} + 11 \, A c^{2} n^{2} x x^{m} e^{m} + 3 \, B d^{2} m x x^{m} x^{3 \, n} e^{m} + 3 \, B d^{2} n x x^{m} x^{3 \, n} e^{m} + 6 \, B c d m x x^{m} x^{2 \, n} e^{m} + 3 \, A d^{2} m x x^{m} x^{2 \, n} e^{m} + 8 \, B c d n x x^{m} x^{2 \, n} e^{m} + 4 \, A d^{2} n x x^{m} x^{2 \, n} e^{m} + 3 \, B c^{2} m x x^{m} x^{n} e^{m} + 6 \, A c d m x x^{m} x^{n} e^{m} + 5 \, B c^{2} n x x^{m} x^{n} e^{m} + 10 \, A c d n x x^{m} x^{n} e^{m} + 3 \, A c^{2} m x x^{m} e^{m} + 6 \, A c^{2} n x x^{m} e^{m} + B d^{2} x x^{m} x^{3 \, n} e^{m} + 2 \, B c d x x^{m} x^{2 \, n} e^{m} + A d^{2} x x^{m} x^{2 \, n} e^{m} + B c^{2} x x^{m} x^{n} e^{m} + 2 \, A c d x x^{m} x^{n} e^{m} + A c^{2} x x^{m} e^{m}}{m^{4} + 6 \, m^{3} n + 11 \, m^{2} n^{2} + 6 \, m n^{3} + 4 \, m^{3} + 18 \, m^{2} n + 22 \, m n^{2} + 6 \, n^{3} + 6 \, m^{2} + 18 \, m n + 11 \, n^{2} + 4 \, m + 6 \, n + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(B*d^2*m^3*x*x^m*x^(3*n)*e^m + 3*B*d^2*m^2*n*x*x^m*x^(3*n)*e^m + 2*B*d^2*m*n^2*x*x^m*x^(3*n)*e^m + 2*B*c*d*m^3
*x*x^m*x^(2*n)*e^m + A*d^2*m^3*x*x^m*x^(2*n)*e^m + 8*B*c*d*m^2*n*x*x^m*x^(2*n)*e^m + 4*A*d^2*m^2*n*x*x^m*x^(2*
n)*e^m + 6*B*c*d*m*n^2*x*x^m*x^(2*n)*e^m + 3*A*d^2*m*n^2*x*x^m*x^(2*n)*e^m + B*c^2*m^3*x*x^m*x^n*e^m + 2*A*c*d
*m^3*x*x^m*x^n*e^m + 5*B*c^2*m^2*n*x*x^m*x^n*e^m + 10*A*c*d*m^2*n*x*x^m*x^n*e^m + 6*B*c^2*m*n^2*x*x^m*x^n*e^m
+ 12*A*c*d*m*n^2*x*x^m*x^n*e^m + A*c^2*m^3*x*x^m*e^m + 6*A*c^2*m^2*n*x*x^m*e^m + 11*A*c^2*m*n^2*x*x^m*e^m + 6*
A*c^2*n^3*x*x^m*e^m + 3*B*d^2*m^2*x*x^m*x^(3*n)*e^m + 6*B*d^2*m*n*x*x^m*x^(3*n)*e^m + 2*B*d^2*n^2*x*x^m*x^(3*n
)*e^m + 6*B*c*d*m^2*x*x^m*x^(2*n)*e^m + 3*A*d^2*m^2*x*x^m*x^(2*n)*e^m + 16*B*c*d*m*n*x*x^m*x^(2*n)*e^m + 8*A*d
^2*m*n*x*x^m*x^(2*n)*e^m + 6*B*c*d*n^2*x*x^m*x^(2*n)*e^m + 3*A*d^2*n^2*x*x^m*x^(2*n)*e^m + 3*B*c^2*m^2*x*x^m*x
^n*e^m + 6*A*c*d*m^2*x*x^m*x^n*e^m + 10*B*c^2*m*n*x*x^m*x^n*e^m + 20*A*c*d*m*n*x*x^m*x^n*e^m + 6*B*c^2*n^2*x*x
^m*x^n*e^m + 12*A*c*d*n^2*x*x^m*x^n*e^m + 3*A*c^2*m^2*x*x^m*e^m + 12*A*c^2*m*n*x*x^m*e^m + 11*A*c^2*n^2*x*x^m*
e^m + 3*B*d^2*m*x*x^m*x^(3*n)*e^m + 3*B*d^2*n*x*x^m*x^(3*n)*e^m + 6*B*c*d*m*x*x^m*x^(2*n)*e^m + 3*A*d^2*m*x*x^
m*x^(2*n)*e^m + 8*B*c*d*n*x*x^m*x^(2*n)*e^m + 4*A*d^2*n*x*x^m*x^(2*n)*e^m + 3*B*c^2*m*x*x^m*x^n*e^m + 6*A*c*d*
m*x*x^m*x^n*e^m + 5*B*c^2*n*x*x^m*x^n*e^m + 10*A*c*d*n*x*x^m*x^n*e^m + 3*A*c^2*m*x*x^m*e^m + 6*A*c^2*n*x*x^m*e
^m + B*d^2*x*x^m*x^(3*n)*e^m + 2*B*c*d*x*x^m*x^(2*n)*e^m + A*d^2*x*x^m*x^(2*n)*e^m + B*c^2*x*x^m*x^n*e^m + 2*A
*c*d*x*x^m*x^n*e^m + A*c^2*x*x^m*e^m)/(m^4 + 6*m^3*n + 11*m^2*n^2 + 6*m*n^3 + 4*m^3 + 18*m^2*n + 22*m*n^2 + 6*
n^3 + 6*m^2 + 18*m*n + 11*n^2 + 4*m + 6*n + 1)

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Mupad [B]
time = 5.11, size = 265, normalized size = 2.60 \begin {gather*} \frac {A\,c^2\,x\,{\left (e\,x\right )}^m}{m+1}+\frac {c\,x\,x^n\,{\left (e\,x\right )}^m\,\left (2\,A\,d+B\,c\right )\,\left (m^2+5\,m\,n+2\,m+6\,n^2+5\,n+1\right )}{m^3+6\,m^2\,n+3\,m^2+11\,m\,n^2+12\,m\,n+3\,m+6\,n^3+11\,n^2+6\,n+1}+\frac {d\,x\,x^{2\,n}\,{\left (e\,x\right )}^m\,\left (A\,d+2\,B\,c\right )\,\left (m^2+4\,m\,n+2\,m+3\,n^2+4\,n+1\right )}{m^3+6\,m^2\,n+3\,m^2+11\,m\,n^2+12\,m\,n+3\,m+6\,n^3+11\,n^2+6\,n+1}+\frac {B\,d^2\,x\,x^{3\,n}\,{\left (e\,x\right )}^m\,\left (m^2+3\,m\,n+2\,m+2\,n^2+3\,n+1\right )}{m^3+6\,m^2\,n+3\,m^2+11\,m\,n^2+12\,m\,n+3\,m+6\,n^3+11\,n^2+6\,n+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(A*c^2*x*(e*x)^m)/(m + 1) + (c*x*x^n*(e*x)^m*(2*A*d + B*c)*(2*m + 5*n + 5*m*n + m^2 + 6*n^2 + 1))/(3*m + 6*n +
12*m*n + 11*m*n^2 + 6*m^2*n + 3*m^2 + m^3 + 11*n^2 + 6*n^3 + 1) + (d*x*x^(2*n)*(e*x)^m*(A*d + 2*B*c)*(2*m + 4
*n + 4*m*n + m^2 + 3*n^2 + 1))/(3*m + 6*n + 12*m*n + 11*m*n^2 + 6*m^2*n + 3*m^2 + m^3 + 11*n^2 + 6*n^3 + 1) +
(B*d^2*x*x^(3*n)*(e*x)^m*(2*m + 3*n + 3*m*n + m^2 + 2*n^2 + 1))/(3*m + 6*n + 12*m*n + 11*m*n^2 + 6*m^2*n + 3*m
^2 + m^3 + 11*n^2 + 6*n^3 + 1)

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