3.4 \(\int (e x)^m \left (A+B x^n\right ) \left (c+d x^n\right ) \, dx\)

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

[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))

_______________________________________________________________________________________

Rubi [A]  time = 0.107305, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{x^{n+1} (e x)^m (A d+B c)}{m+n+1}+\frac{A c (e x)^{m+1}}{e (m+1)}+\frac{B d x^{2 n+1} (e x)^m}{m+2 n+1} \]

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))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 17.5508, size = 75, normalized size = 1.14 \[ \frac{A c \left (e x\right )^{m + 1}}{e \left (m + 1\right )} + \frac{B d x^{2 n} \left (e x\right )^{- 2 n} \left (e x\right )^{m + 2 n + 1}}{e \left (m + 2 n + 1\right )} + \frac{x^{- m} x^{m + n + 1} \left (e x\right )^{m} \left (A d + B c\right )}{m + n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**m*(A+B*x**n)*(c+d*x**n),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0603808, size = 49, normalized size = 0.74 \[ x (e x)^m \left (\frac{x^n (A d+B c)}{m+n+1}+\frac{A c}{m+1}+\frac{B d x^{2 n}}{m+2 n+1}\right ) \]

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))

_______________________________________________________________________________________

Maple [C]  time = 0.067, size = 262, normalized size = 4. \[{\frac{x \left ( Bd{m}^{2} \left ({x}^{n} \right ) ^{2}+Bdmn \left ({x}^{n} \right ) ^{2}+Ad{m}^{2}{x}^{n}+2\,Admn{x}^{n}+Bc{m}^{2}{x}^{n}+2\,Bcmn{x}^{n}+2\,B \left ({x}^{n} \right ) ^{2}dm+B \left ({x}^{n} \right ) ^{2}dn+Ac{m}^{2}+3\,Acmn+2\,Ac{n}^{2}+2\,A{x}^{n}dm+2\,A{x}^{n}dn+2\,B{x}^{n}cm+2\,B{x}^{n}cn+B \left ({x}^{n} \right ) ^{2}d+2\,Acm+3\,Acn+A{x}^{n}d+B{x}^{n}c+Ac \right ) }{ \left ( 1+m \right ) \left ( 1+m+n \right ) \left ( 1+m+2\,n \right ) }{{\rm e}^{{\frac{m \left ( -i\pi \, \left ({\it csgn} \left ( iex \right ) \right ) ^{3}+i\pi \, \left ({\it csgn} \left ( iex \right ) \right ) ^{2}{\it csgn} \left ( ie \right ) +i\pi \, \left ({\it csgn} \left ( iex \right ) \right ) ^{2}{\it csgn} \left ( ix \right ) -i\pi \,{\it csgn} \left ( iex \right ){\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ) +2\,\ln \left ( e \right ) +2\,\ln \left ( x \right ) \right ) }{2}}}}} \]

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]

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

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.234471, size = 250, normalized size = 3.79 \[ \frac{{\left (B d m^{2} + 2 \, B d m + B d +{\left (B d m + B d\right )} n\right )} x x^{2 \, n} e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )} +{\left ({\left (B c + A d\right )} m^{2} + B c + A d + 2 \,{\left (B c + A d\right )} m + 2 \,{\left (B c + A d +{\left (B c + A d\right )} m\right )} n\right )} x x^{n} e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )} +{\left (A c m^{2} + 2 \, A c n^{2} + 2 \, A c m + A c + 3 \,{\left (A c m + A c\right )} n\right )} x e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )}}{m^{3} + 2 \,{\left (m + 1\right )} n^{2} + 3 \, m^{2} + 3 \,{\left (m^{2} + 2 \, m + 1\right )} n + 3 \, m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^n + A)*(d*x^n + c)*(e*x)^m,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 [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

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]

Exception raised: TypeError

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.218853, size = 512, normalized size = 7.76 \[ \frac{B d m^{2} x e^{\left (m{\rm ln}\left (x\right ) + 2 \, n{\rm ln}\left (x\right ) + m\right )} + B d m n x e^{\left (m{\rm ln}\left (x\right ) + 2 \, n{\rm ln}\left (x\right ) + m\right )} + B c m^{2} x e^{\left (m{\rm ln}\left (x\right ) + n{\rm ln}\left (x\right ) + m\right )} + A d m^{2} x e^{\left (m{\rm ln}\left (x\right ) + n{\rm ln}\left (x\right ) + m\right )} + 2 \, B c m n x e^{\left (m{\rm ln}\left (x\right ) + n{\rm ln}\left (x\right ) + m\right )} + 2 \, A d m n x e^{\left (m{\rm ln}\left (x\right ) + n{\rm ln}\left (x\right ) + m\right )} + A c m^{2} x e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 3 \, A c m n x e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 2 \, A c n^{2} x e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 2 \, B d m x e^{\left (m{\rm ln}\left (x\right ) + 2 \, n{\rm ln}\left (x\right ) + m\right )} + B d n x e^{\left (m{\rm ln}\left (x\right ) + 2 \, n{\rm ln}\left (x\right ) + m\right )} + 2 \, B c m x e^{\left (m{\rm ln}\left (x\right ) + n{\rm ln}\left (x\right ) + m\right )} + 2 \, A d m x e^{\left (m{\rm ln}\left (x\right ) + n{\rm ln}\left (x\right ) + m\right )} + 2 \, B c n x e^{\left (m{\rm ln}\left (x\right ) + n{\rm ln}\left (x\right ) + m\right )} + 2 \, A d n x e^{\left (m{\rm ln}\left (x\right ) + n{\rm ln}\left (x\right ) + m\right )} + 2 \, A c m x e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 3 \, A c n x e^{\left (m{\rm ln}\left (x\right ) + m\right )} + B d x e^{\left (m{\rm ln}\left (x\right ) + 2 \, n{\rm ln}\left (x\right ) + m\right )} + B c x e^{\left (m{\rm ln}\left (x\right ) + n{\rm ln}\left (x\right ) + m\right )} + A d x e^{\left (m{\rm ln}\left (x\right ) + n{\rm ln}\left (x\right ) + m\right )} + A c x e^{\left (m{\rm ln}\left (x\right ) + m\right )}}{m^{3} + 3 \, m^{2} n + 2 \, m n^{2} + 3 \, m^{2} + 6 \, m n + 2 \, n^{2} + 3 \, m + 3 \, n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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