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

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 = 0.0399767, 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, Rules used = {448, 20, 30} \[ \frac{x^{n+1} (e x)^m (A d+B c)}{m+n+1}+\frac{A c (e x)^{m+1}}{e (m+1)}+\frac{B d x^{2 n+1} (e x)^m}{m+2 n+1} \]

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))

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]

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]

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*}

Mathematica [A]  time = 0.059067, 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 = 0.09, size = 262, normalized size = 4. \begin{align*}{\frac{ \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+d \left ({x}^{n} \right ) ^{2}B+2\,Acm+3\,Acn+d{x}^{n}A+cB{x}^{n}+Ac \right ) x}{ \left ( 1+m \right ) \left ( m+n+1 \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}}}}} \end{align*}

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+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)/(m+n+1)/(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(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 1.08085, size = 463, normalized size = 7.02 \begin{align*} \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} \end{align*}

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(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. \begin{align*} \text{Exception raised: TypeError} \end{align*}

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 [B]  time = 1.25673, size = 441, normalized size = 6.68 \begin{align*} \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{align*}

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)