3.1.4 \(\int (e x)^m (A+B x^n) (c+d x^n) \, dx\) [4]

Optimal. Leaf size=66 \[ \frac {(B c+A d) x^{1+n} (e x)^m}{1+m+n}+\frac {B d x^{1+2 n} (e x)^m}{1+m+2 n}+\frac {A c (e x)^{1+m}}{e (1+m)} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {459, 20, 30} \begin {gather*} \frac {x^{n+1} (e x)^m (A d+B c)}{m+n+1}+\frac {A c (e x)^{m+1}}{e (m+1)}+\frac {B d x^{2 n+1} (e x)^m}{m+2 n+1} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]
time = 0.06, size = 49, normalized size = 0.74 \begin {gather*} x (e x)^m \left (\frac {A c}{1+m}+\frac {(B c+A d) x^n}{1+m+n}+\frac {B d x^{2 n}}{1+m+2 n}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.06, size = 262, normalized size = 3.97

method	result	size
risch	\(\frac {x \left (B d \,m^{2} x^{2 n}+B d m n \,x^{2 n}+A d \,m^{2} x^{n}+2 A d m n \,x^{n}+B c \,m^{2} x^{n}+2 B c m n \,x^{n}+2 B \,x^{2 n} d m +B \,x^{2 n} d n +A c \,m^{2}+3 A c m n +2 A c \,n^{2}+2 A \,x^{n} d m +2 A \,x^{n} d n +2 B \,x^{n} c m +2 B \,x^{n} c n +d \,x^{2 n} B +2 A c m +3 A c n +d \,x^{n} A +c B \,x^{n}+A c \right ) {\mathrm e}^{\frac {m \left (-i \pi \mathrm {csgn}\left (i e x \right )^{3}+i \pi \mathrm {csgn}\left (i e x \right )^{2} \mathrm {csgn}\left (i e \right )+i \pi \mathrm {csgn}\left (i e x \right )^{2} \mathrm {csgn}\left (i x \right )-i \pi \,\mathrm {csgn}\left (i e x \right ) \mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i x \right )+2 \ln \left (e \right )+2 \ln \left (x \right )\right )}{2}}}{\left (1+m \right ) \left (1+m +n \right ) \left (1+m +2 n \right )}\)	\(262\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*(B*d*m^2*(x^n)^2+B*d*m*n*(x^n)^2+A*d*m^2*x^n+2*A*d*m*n*x^n+B*c*m^2*x^n+2*B*c*m*n*x^n+2*B*(x^n)^2*d*m+B*(x^n)
^2*d*n+A*c*m^2+3*A*c*m*n+2*A*c*n^2+2*A*x^n*d*m+2*A*x^n*d*n+2*B*x^n*c*m+2*B*x^n*c*n+d*(x^n)^2*B+2*A*c*m+3*A*c*n
+d*x^n*A+c*B*x^n+A*c)/(1+m)/(1+m+n)/(1+m+2*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 = 0.31, size = 85, normalized size = 1.29 \begin {gather*} \frac {\left (x e\right )^{m + 1} A c e^{\left (-1\right )}}{m + 1} + \frac {B d x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right ) + m\right )}}{m + 2 \, n + 1} + \frac {B c x e^{\left (m \log \left (x\right ) + n \log \left (x\right ) + m\right )}}{m + n + 1} + \frac {A 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),x, algorithm="maxima")

[Out]

(x*e)^(m + 1)*A*c*e^(-1)/(m + 1) + B*d*x*e^(m*log(x) + 2*n*log(x) + m)/(m + 2*n + 1) + B*c*x*e^(m*log(x) + n*l
og(x) + m)/(m + n + 1) + A*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. 176 vs. \(2 (68) = 136\).
time = 1.21, size = 176, normalized size = 2.67 \begin {gather*} \frac {{\left (B d m^{2} + 2 \, B d m + B d + {\left (B d m + B d\right )} n\right )} x x^{2 \, n} e^{\left (m \log \left (x\right ) + m\right )} + {\left ({\left (B c + A d\right )} m^{2} + B c + A d + 2 \, {\left (B c + A d\right )} m + 2 \, {\left (B c + A d + {\left (B c + A d\right )} m\right )} n\right )} x x^{n} e^{\left (m \log \left (x\right ) + m\right )} + {\left (A c m^{2} + 2 \, A c n^{2} + 2 \, A c m + A c + 3 \, {\left (A c m + A c\right )} n\right )} x e^{\left (m \log \left (x\right ) + m\right )}}{m^{3} + 2 \, {\left (m + 1\right )} n^{2} + 3 \, m^{2} + 3 \, {\left (m^{2} + 2 \, m + 1\right )} n + 3 \, 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),x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1499 vs. \(2 (58) = 116\).
time = 13.54, size = 1499, normalized size = 22.71 \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),x)

[Out]

Piecewise(((A + B)*(c + d)*log(x)/e, Eq(m, -1) & Eq(n, 0)), ((A*c*log(x) + A*d*x**n/n + B*c*x**n/n + B*d*x**(2
*n)/(2*n))/e, Eq(m, -1)), (A*c*Piecewise((-1/(2*n*(e*x)**(2*n)), Ne(n, 0)), (log(x), True))/e + A*d*Piecewise(
(-x**n/(n*(e*x)**(2*n)), Ne(n, 0)), (log(x), True))/e + B*c*Piecewise((-x**n/(n*(e*x)**(2*n)), Ne(n, 0)), (log
(x), True))/e + 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, Eq(m, -2*n - 1)), (A*c*Piecewise((-1/(n*(e*x)**n), Ne(n, 0)), (log(x), True))/e + A*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), (-mei
jerg(((), (1, 1)), ((0, 0), ()), x)/e**n + meijerg(((1, 1), ()), ((), (0, 0)), x)/e**n, True))/e + B*c*Piecewi
se((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 + B*d*Piecewise((x**
(2*n)/(n*(e*x)**n), Ne(n, 0)), (log(x), True))/e, Eq(m, -n - 1)), (A*c*m**2*x*(e*x)**m/(m**3 + 3*m**2*n + 3*m*
*2 + 2*m*n**2 + 6*m*n + 3*m + 2*n**2 + 3*n + 1) + 3*A*c*m*n*x*(e*x)**m/(m**3 + 3*m**2*n + 3*m**2 + 2*m*n**2 +
6*m*n + 3*m + 2*n**2 + 3*n + 1) + 2*A*c*m*x*(e*x)**m/(m**3 + 3*m**2*n + 3*m**2 + 2*m*n**2 + 6*m*n + 3*m + 2*n*
*2 + 3*n + 1) + 2*A*c*n**2*x*(e*x)**m/(m**3 + 3*m**2*n + 3*m**2 + 2*m*n**2 + 6*m*n + 3*m + 2*n**2 + 3*n + 1) +
3*A*c*n*x*(e*x)**m/(m**3 + 3*m**2*n + 3*m**2 + 2*m*n**2 + 6*m*n + 3*m + 2*n**2 + 3*n + 1) + A*c*x*(e*x)**m/(m
**3 + 3*m**2*n + 3*m**2 + 2*m*n**2 + 6*m*n + 3*m + 2*n**2 + 3*n + 1) + A*d*m**2*x*x**n*(e*x)**m/(m**3 + 3*m**2
*n + 3*m**2 + 2*m*n**2 + 6*m*n + 3*m + 2*n**2 + 3*n + 1) + 2*A*d*m*n*x*x**n*(e*x)**m/(m**3 + 3*m**2*n + 3*m**2
+ 2*m*n**2 + 6*m*n + 3*m + 2*n**2 + 3*n + 1) + 2*A*d*m*x*x**n*(e*x)**m/(m**3 + 3*m**2*n + 3*m**2 + 2*m*n**2 +
6*m*n + 3*m + 2*n**2 + 3*n + 1) + 2*A*d*n*x*x**n*(e*x)**m/(m**3 + 3*m**2*n + 3*m**2 + 2*m*n**2 + 6*m*n + 3*m
+ 2*n**2 + 3*n + 1) + A*d*x*x**n*(e*x)**m/(m**3 + 3*m**2*n + 3*m**2 + 2*m*n**2 + 6*m*n + 3*m + 2*n**2 + 3*n +
1) + B*c*m**2*x*x**n*(e*x)**m/(m**3 + 3*m**2*n + 3*m**2 + 2*m*n**2 + 6*m*n + 3*m + 2*n**2 + 3*n + 1) + 2*B*c*m
*n*x*x**n*(e*x)**m/(m**3 + 3*m**2*n + 3*m**2 + 2*m*n**2 + 6*m*n + 3*m + 2*n**2 + 3*n + 1) + 2*B*c*m*x*x**n*(e*
x)**m/(m**3 + 3*m**2*n + 3*m**2 + 2*m*n**2 + 6*m*n + 3*m + 2*n**2 + 3*n + 1) + 2*B*c*n*x*x**n*(e*x)**m/(m**3 +
3*m**2*n + 3*m**2 + 2*m*n**2 + 6*m*n + 3*m + 2*n**2 + 3*n + 1) + B*c*x*x**n*(e*x)**m/(m**3 + 3*m**2*n + 3*m**
2 + 2*m*n**2 + 6*m*n + 3*m + 2*n**2 + 3*n + 1) + B*d*m**2*x*x**(2*n)*(e*x)**m/(m**3 + 3*m**2*n + 3*m**2 + 2*m*
n**2 + 6*m*n + 3*m + 2*n**2 + 3*n + 1) + B*d*m*n*x*x**(2*n)*(e*x)**m/(m**3 + 3*m**2*n + 3*m**2 + 2*m*n**2 + 6*
m*n + 3*m + 2*n**2 + 3*n + 1) + 2*B*d*m*x*x**(2*n)*(e*x)**m/(m**3 + 3*m**2*n + 3*m**2 + 2*m*n**2 + 6*m*n + 3*m
+ 2*n**2 + 3*n + 1) + B*d*n*x*x**(2*n)*(e*x)**m/(m**3 + 3*m**2*n + 3*m**2 + 2*m*n**2 + 6*m*n + 3*m + 2*n**2 +
3*n + 1) + B*d*x*x**(2*n)*(e*x)**m/(m**3 + 3*m**2*n + 3*m**2 + 2*m*n**2 + 6*m*n + 3*m + 2*n**2 + 3*n + 1), Tr
ue))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 327 vs. \(2 (68) = 136\).
time = 2.28, size = 327, normalized size = 4.95 \begin {gather*} \frac {B d m^{2} x x^{m} x^{2 \, n} e^{m} + B d m n x x^{m} x^{2 \, n} e^{m} + B c m^{2} x x^{m} x^{n} e^{m} + A d m^{2} x x^{m} x^{n} e^{m} + 2 \, B c m n x x^{m} x^{n} e^{m} + 2 \, A d m n x x^{m} x^{n} e^{m} + A c m^{2} x x^{m} e^{m} + 3 \, A c m n x x^{m} e^{m} + 2 \, A c n^{2} x x^{m} e^{m} + 2 \, B d m x x^{m} x^{2 \, n} e^{m} + B d n x x^{m} x^{2 \, n} e^{m} + 2 \, B c m x x^{m} x^{n} e^{m} + 2 \, A d m x x^{m} x^{n} e^{m} + 2 \, B c n x x^{m} x^{n} e^{m} + 2 \, A d n x x^{m} x^{n} e^{m} + 2 \, A c m x x^{m} e^{m} + 3 \, A c n x x^{m} e^{m} + B d x x^{m} x^{2 \, n} e^{m} + B c x x^{m} x^{n} e^{m} + A d x x^{m} x^{n} e^{m} + A c x x^{m} e^{m}}{m^{3} + 3 \, m^{2} n + 2 \, m n^{2} + 3 \, m^{2} + 6 \, m n + 2 \, n^{2} + 3 \, m + 3 \, 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),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 4.83, size = 91, normalized size = 1.38 \begin {gather*} {\left (e\,x\right )}^m\,\left (\frac {A\,c\,x}{m+1}+\frac {x\,x^n\,\left (A\,d+B\,c\right )\,\left (m+2\,n+1\right )}{m^2+3\,m\,n+2\,m+2\,n^2+3\,n+1}+\frac {B\,d\,x\,x^{2\,n}\,\left (m+n+1\right )}{m^2+3\,m\,n+2\,m+2\,n^2+3\,n+1}\right ) \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),x)

[Out]

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

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