\(\int (e x)^m (A+B x^n) (c+d x^n) \, dx\) [4]
Optimal result
Integrand size = 20, antiderivative size = 66 \[
\int (e x)^m \left (A+B x^n\right ) \left (c+d x^n\right ) \, dx=\frac {(B c+A d) x^{1+n} (e x)^m}{1+m+n}+\frac {B d x^{1+2 n} (e x)^m}{1+m+2 n}+\frac {A c (e x)^{1+m}}{e (1+m)}
\]
[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)
Rubi [A] (verified)
Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number
of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {459, 20, 30}
\[
\int (e x)^m \left (A+B x^n\right ) \left (c+d x^n\right ) \, dx=\frac {x^{n+1} (e x)^m (A d+B c)}{m+n+1}+\frac {A c (e x)^{m+1}}{e (m+1)}+\frac {B d x^{2 n+1} (e x)^m}{m+2 n+1}
\]
[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*}
\text {integral}& = \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*}
Mathematica [A] (verified)
Time = 0.10 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.74
\[
\int (e x)^m \left (A+B x^n\right ) \left (c+d x^n\right ) \, dx=x (e x)^m \left (\frac {A c}{1+m}+\frac {(B c+A d) x^n}{1+m+n}+\frac {B d x^{2 n}}{1+m+2 n}\right )
\]
[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))
Maple [C] (warning: unable to verify)
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.17 (sec) , antiderivative size = 229, normalized size of antiderivative = 3.47
| | |
| method | result | size |
| | |
| risch |
\(\frac {x \left (B d \,m^{2} x^{2 n}+B d m n \,x^{2 n}+A d \,m^{2} x^{n}+2 A d m n \,x^{n}+B c \,m^{2} x^{n}+2 B c m n \,x^{n}+2 B \,x^{2 n} d m +B \,x^{2 n} d n +A c \,m^{2}+3 A c m n +2 A c \,n^{2}+2 A \,x^{n} d m +2 A \,x^{n} d n +2 B \,x^{n} c m +2 B \,x^{n} c n +d \,x^{2 n} B +2 A c m +3 A c n +d \,x^{n} A +c B \,x^{n}+A c \right ) e^{m} x^{m} {\mathrm e}^{\frac {i \operatorname {csgn}\left (i e x \right ) \pi m \left (\operatorname {csgn}\left (i e x \right )-\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (i e x \right )+\operatorname {csgn}\left (i e \right )\right )}{2}}}{\left (1+m \right ) \left (1+m +n \right ) \left (1+m +2 n \right )}\) |
\(229\) |
| parallelrisch |
\(\frac {2 A x \,x^{n} \left (e x \right )^{m} d n +3 A x \left (e x \right )^{m} c m n +A x \,x^{n} \left (e x \right )^{m} d \,m^{2}+2 B x \,x^{2 n} \left (e x \right )^{m} d m +B x \,x^{2 n} \left (e x \right )^{m} d n +B x \,x^{n} \left (e x \right )^{m} c \,m^{2}+2 B x \,x^{n} \left (e x \right )^{m} c n +B x \,x^{2 n} \left (e x \right )^{m} d \,m^{2}+B x \,x^{2 n} \left (e x \right )^{m} d m n +2 A x \,x^{n} \left (e x \right )^{m} d m n +2 B x \,x^{n} \left (e x \right )^{m} c m n +2 B x \,x^{n} \left (e x \right )^{m} c m +2 A x \,x^{n} \left (e x \right )^{m} d m +3 A x \left (e x \right )^{m} c n +A x \left (e x \right )^{m} c \,m^{2}+2 A x \left (e x \right )^{m} c \,n^{2}+B x \,x^{2 n} \left (e x \right )^{m} d +A x \,x^{n} \left (e x \right )^{m} d +2 A x \left (e x \right )^{m} c m +A x \left (e x \right )^{m} c +B x \,x^{n} \left (e x \right )^{m} c}{\left (1+m \right ) \left (1+m +n \right ) \left (1+m +2 n \right )}\) |
\(308\) |
| | |
|
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[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)*e^m*x^m*exp(1/2*I*csgn(I*e*x)*Pi*m*(csgn(I*e*x)-csgn(I*x))*(-csg
n(I*e*x)+csgn(I*e)))
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 185 vs. \(2 (66) = 132\).
Time = 0.27 (sec) , antiderivative size = 185, normalized size of antiderivative = 2.80
\[
\int (e x)^m \left (A+B x^n\right ) \left (c+d x^n\right ) \, dx=\frac {{\left (B d m^{2} + 2 \, B d m + B d + {\left (B d m + B d\right )} n\right )} x x^{2 \, n} e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )} + {\left ({\left (B c + A d\right )} m^{2} + B c + A d + 2 \, {\left (B c + A d\right )} m + 2 \, {\left (B c + A d + {\left (B c + A d\right )} m\right )} n\right )} x x^{n} e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )} + {\left (A c m^{2} + 2 \, A c n^{2} + 2 \, A c m + A c + 3 \, {\left (A c m + A c\right )} n\right )} x e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )}}{m^{3} + 2 \, {\left (m + 1\right )} n^{2} + 3 \, m^{2} + 3 \, {\left (m^{2} + 2 \, m + 1\right )} n + 3 \, m + 1}
\]
[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(e) + m*log(x)) + ((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(e) + m*log(x)) + (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(e) + m*log(x)))/(m^3 + 2*(m + 1)*n^2 + 3*m^2 + 3*(m^2 + 2*m + 1)*n
+ 3*m + 1)
Sympy [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 1498 vs. \(2 (58) = 116\).
Time = 1.66 (sec) , antiderivative size = 1498, normalized size of antiderivative = 22.70
\[
\int (e x)^m \left (A+B x^n\right ) \left (c+d x^n\right ) \, dx=\text {Too large to display}
\]
[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((0**(-2*n - 1)*x, Eq(e, 0)), (Piecewise((-1/(2*n*(e*x)**(2*n)), Ne(n,
0)), (log(e*x), True))/e, True)) + A*d*Piecewise((-x*x**n*(e*x)**(-2*n - 1)/n, Ne(n, 0)), (x*x**n*(e*x)**(-2*
n - 1)*log(x), True)) + B*c*Piecewise((-x*x**n*(e*x)**(-2*n - 1)/n, Ne(n, 0)), (x*x**n*(e*x)**(-2*n - 1)*log(x
), True)) + B*d*x*x**(2*n)*(e*x)**(-2*n - 1)*log(x), Eq(m, -2*n - 1)), (A*c*Piecewise((0**(-n - 1)*x, Eq(e, 0)
), (Piecewise((-1/(n*(e*x)**n), Ne(n, 0)), (log(e*x), True))/e, True)) + A*d*x*x**n*(e*x)**(-n - 1)*log(x) + B
*c*x*x**n*(e*x)**(-n - 1)*log(x) + B*d*Piecewise((x*x**(2*n)*(e*x)**(-n - 1)/n, Ne(n, 0)), (x*x**(2*n)*(e*x)**
(-n - 1)*log(x), True)), 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), True))
Maxima [A] (verification not implemented)
none
Time = 0.20 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.38
\[
\int (e x)^m \left (A+B x^n\right ) \left (c+d x^n\right ) \, dx=\frac {B d e^{m} x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right )\right )}}{m + 2 \, n + 1} + \frac {B c e^{m} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{m + n + 1} + \frac {A d e^{m} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{m + n + 1} + \frac {\left (e x\right )^{m + 1} A c}{e {\left (m + 1\right )}}
\]
[In]
integrate((e*x)^m*(A+B*x^n)*(c+d*x^n),x, algorithm="maxima")
[Out]
B*d*e^m*x*e^(m*log(x) + 2*n*log(x))/(m + 2*n + 1) + B*c*e^m*x*e^(m*log(x) + n*log(x))/(m + n + 1) + A*d*e^m*x*
e^(m*log(x) + n*log(x))/(m + n + 1) + (e*x)^(m + 1)*A*c/(e*(m + 1))
Giac [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 763 vs. \(2 (66) = 132\).
Time = 0.28 (sec) , antiderivative size = 763, normalized size of antiderivative = 11.56
\[
\int (e x)^m \left (A+B x^n\right ) \left (c+d x^n\right ) \, dx=\text {Too large to display}
\]
[In]
integrate((e*x)^m*(A+B*x^n)*(c+d*x^n),x, algorithm="giac")
[Out]
(B*d*m^2*x*x^(2*n)*e^(m*log(e) + m*log(x)) + B*d*m*n*x*x^(2*n)*e^(m*log(e) + m*log(x)) + B*c*m^2*x*x^n*e^(m*lo
g(e) + m*log(x)) + A*d*m^2*x*x^n*e^(m*log(e) + m*log(x)) + B*d*m^2*x*x^n*e^(m*log(e) + m*log(x)) + 2*B*c*m*n*x
*x^n*e^(m*log(e) + m*log(x)) + 2*A*d*m*n*x*x^n*e^(m*log(e) + m*log(x)) + B*d*m*n*x*x^n*e^(m*log(e) + m*log(x))
+ A*c*m^2*x*e^(m*log(e) + m*log(x)) + B*c*m^2*x*e^(m*log(e) + m*log(x)) + A*d*m^2*x*e^(m*log(e) + m*log(x)) +
B*d*m^2*x*e^(m*log(e) + m*log(x)) + 3*A*c*m*n*x*e^(m*log(e) + m*log(x)) + 2*B*c*m*n*x*e^(m*log(e) + m*log(x))
+ 2*A*d*m*n*x*e^(m*log(e) + m*log(x)) + B*d*m*n*x*e^(m*log(e) + m*log(x)) + 2*A*c*n^2*x*e^(m*log(e) + m*log(x
)) + 2*B*d*m*x*x^(2*n)*e^(m*log(e) + m*log(x)) + B*d*n*x*x^(2*n)*e^(m*log(e) + m*log(x)) + 2*B*c*m*x*x^n*e^(m*
log(e) + m*log(x)) + 2*A*d*m*x*x^n*e^(m*log(e) + m*log(x)) + 2*B*d*m*x*x^n*e^(m*log(e) + m*log(x)) + 2*B*c*n*x
*x^n*e^(m*log(e) + m*log(x)) + 2*A*d*n*x*x^n*e^(m*log(e) + m*log(x)) + B*d*n*x*x^n*e^(m*log(e) + m*log(x)) + 2
*A*c*m*x*e^(m*log(e) + m*log(x)) + 2*B*c*m*x*e^(m*log(e) + m*log(x)) + 2*A*d*m*x*e^(m*log(e) + m*log(x)) + 2*B
*d*m*x*e^(m*log(e) + m*log(x)) + 3*A*c*n*x*e^(m*log(e) + m*log(x)) + 2*B*c*n*x*e^(m*log(e) + m*log(x)) + 2*A*d
*n*x*e^(m*log(e) + m*log(x)) + B*d*n*x*e^(m*log(e) + m*log(x)) + B*d*x*x^(2*n)*e^(m*log(e) + m*log(x)) + B*c*x
*x^n*e^(m*log(e) + m*log(x)) + A*d*x*x^n*e^(m*log(e) + m*log(x)) + B*d*x*x^n*e^(m*log(e) + m*log(x)) + A*c*x*e
^(m*log(e) + m*log(x)) + B*c*x*e^(m*log(e) + m*log(x)) + A*d*x*e^(m*log(e) + m*log(x)) + B*d*x*e^(m*log(e) + m
*log(x)))/(m^3 + 3*m^2*n + 2*m*n^2 + 3*m^2 + 6*m*n + 2*n^2 + 3*m + 3*n + 1)
Mupad [B] (verification not implemented)
Time = 9.06 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.38
\[
\int (e x)^m \left (A+B x^n\right ) \left (c+d x^n\right ) \, dx={\left (e\,x\right )}^m\,\left (\frac {A\,c\,x}{m+1}+\frac {x\,x^n\,\left (A\,d+B\,c\right )\,\left (m+2\,n+1\right )}{m^2+3\,m\,n+2\,m+2\,n^2+3\,n+1}+\frac {B\,d\,x\,x^{2\,n}\,\left (m+n+1\right )}{m^2+3\,m\,n+2\,m+2\,n^2+3\,n+1}\right )
\]
[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))