3.1085 \(\int (e x)^m (A+B x) (a+b x+c x^2) \, dx\)

Optimal. Leaf size=83 \[ \frac{(e x)^{m+2} (a B+A b)}{e^2 (m+2)}+\frac{a A (e x)^{m+1}}{e (m+1)}+\frac{(e x)^{m+3} (A c+b B)}{e^3 (m+3)}+\frac{B c (e x)^{m+4}}{e^4 (m+4)} \]

[Out]

(a*A*(e*x)^(1 + m))/(e*(1 + m)) + ((A*b + a*B)*(e*x)^(2 + m))/(e^2*(2 + m)) + ((b*B + A*c)*(e*x)^(3 + m))/(e^3
*(3 + m)) + (B*c*(e*x)^(4 + m))/(e^4*(4 + m))

________________________________________________________________________________________

Rubi [A]  time = 0.0424643, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {765} \[ \frac{(e x)^{m+2} (a B+A b)}{e^2 (m+2)}+\frac{a A (e x)^{m+1}}{e (m+1)}+\frac{(e x)^{m+3} (A c+b B)}{e^3 (m+3)}+\frac{B c (e x)^{m+4}}{e^4 (m+4)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*A*(e*x)^(1 + m))/(e*(1 + m)) + ((A*b + a*B)*(e*x)^(2 + m))/(e^2*(2 + m)) + ((b*B + A*c)*(e*x)^(3 + m))/(e^3
*(3 + m)) + (B*c*(e*x)^(4 + m))/(e^4*(4 + m))

Rule 765

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand
Integrand[(e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, e, f, g, m}, x] && IntegerQ[p] && (
GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int (e x)^m (A+B x) \left (a+b x+c x^2\right ) \, dx &=\int \left (a A (e x)^m+\frac{(A b+a B) (e x)^{1+m}}{e}+\frac{(b B+A c) (e x)^{2+m}}{e^2}+\frac{B c (e x)^{3+m}}{e^3}\right ) \, dx\\ &=\frac{a A (e x)^{1+m}}{e (1+m)}+\frac{(A b+a B) (e x)^{2+m}}{e^2 (2+m)}+\frac{(b B+A c) (e x)^{3+m}}{e^3 (3+m)}+\frac{B c (e x)^{4+m}}{e^4 (4+m)}\\ \end{align*}

Mathematica [A]  time = 0.113281, size = 91, normalized size = 1.1 \[ \frac{x (e x)^m \left (a \left (m^2+7 m+12\right ) (A (m+2)+B (m+1) x)+(m+1) x (A (m+4) (b (m+3)+c (m+2) x)+B (m+2) x (b (m+4)+c (m+3) x))\right )}{(m+1) (m+2) (m+3) (m+4)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(e*x)^m*(a*(12 + 7*m + m^2)*(A*(2 + m) + B*(1 + m)*x) + (1 + m)*x*(A*(4 + m)*(b*(3 + m) + c*(2 + m)*x) + B*
(2 + m)*x*(b*(4 + m) + c*(3 + m)*x))))/((1 + m)*(2 + m)*(3 + m)*(4 + m))

________________________________________________________________________________________

Maple [B]  time = 0.004, size = 205, normalized size = 2.5 \begin{align*}{\frac{ \left ( Bc{m}^{3}{x}^{3}+Ac{m}^{3}{x}^{2}+Bb{m}^{3}{x}^{2}+6\,Bc{m}^{2}{x}^{3}+Ab{m}^{3}x+7\,Ac{m}^{2}{x}^{2}+Ba{m}^{3}x+7\,Bb{m}^{2}{x}^{2}+11\,Bcm{x}^{3}+Aa{m}^{3}+8\,Ab{m}^{2}x+14\,Acm{x}^{2}+8\,Ba{m}^{2}x+14\,Bbm{x}^{2}+6\,Bc{x}^{3}+9\,Aa{m}^{2}+19\,Abmx+8\,Ac{x}^{2}+19\,Bamx+8\,Bb{x}^{2}+26\,Aam+12\,Abx+12\,aBx+24\,aA \right ) x \left ( ex \right ) ^{m}}{ \left ( 4+m \right ) \left ( 3+m \right ) \left ( 2+m \right ) \left ( 1+m \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m*(B*x+A)*(c*x^2+b*x+a),x)

[Out]

x*(B*c*m^3*x^3+A*c*m^3*x^2+B*b*m^3*x^2+6*B*c*m^2*x^3+A*b*m^3*x+7*A*c*m^2*x^2+B*a*m^3*x+7*B*b*m^2*x^2+11*B*c*m*
x^3+A*a*m^3+8*A*b*m^2*x+14*A*c*m*x^2+8*B*a*m^2*x+14*B*b*m*x^2+6*B*c*x^3+9*A*a*m^2+19*A*b*m*x+8*A*c*x^2+19*B*a*
m*x+8*B*b*x^2+26*A*a*m+12*A*b*x+12*B*a*x+24*A*a)*(e*x)^m/(4+m)/(3+m)/(2+m)/(1+m)

________________________________________________________________________________________

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*(B*x+A)*(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 1.6691, size = 402, normalized size = 4.84 \begin{align*} \frac{{\left ({\left (B c m^{3} + 6 \, B c m^{2} + 11 \, B c m + 6 \, B c\right )} x^{4} +{\left ({\left (B b + A c\right )} m^{3} + 7 \,{\left (B b + A c\right )} m^{2} + 8 \, B b + 8 \, A c + 14 \,{\left (B b + A c\right )} m\right )} x^{3} +{\left ({\left (B a + A b\right )} m^{3} + 8 \,{\left (B a + A b\right )} m^{2} + 12 \, B a + 12 \, A b + 19 \,{\left (B a + A b\right )} m\right )} x^{2} +{\left (A a m^{3} + 9 \, A a m^{2} + 26 \, A a m + 24 \, A a\right )} x\right )} \left (e x\right )^{m}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((B*c*m^3 + 6*B*c*m^2 + 11*B*c*m + 6*B*c)*x^4 + ((B*b + A*c)*m^3 + 7*(B*b + A*c)*m^2 + 8*B*b + 8*A*c + 14*(B*b
+ A*c)*m)*x^3 + ((B*a + A*b)*m^3 + 8*(B*a + A*b)*m^2 + 12*B*a + 12*A*b + 19*(B*a + A*b)*m)*x^2 + (A*a*m^3 + 9
*A*a*m^2 + 26*A*a*m + 24*A*a)*x)*(e*x)^m/(m^4 + 10*m^3 + 35*m^2 + 50*m + 24)

________________________________________________________________________________________

Sympy [A]  time = 1.37997, size = 1022, normalized size = 12.31 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**m*(B*x+A)*(c*x**2+b*x+a),x)

[Out]

Piecewise(((-A*a/(3*x**3) - A*b/(2*x**2) - A*c/x - B*a/(2*x**2) - B*b/x + B*c*log(x))/e**4, Eq(m, -4)), ((-A*a
/(2*x**2) - A*b/x + A*c*log(x) - B*a/x + B*b*log(x) + B*c*x)/e**3, Eq(m, -3)), ((-A*a/x + A*b*log(x) + A*c*x +
B*a*log(x) + B*b*x + B*c*x**2/2)/e**2, Eq(m, -2)), ((A*a*log(x) + A*b*x + A*c*x**2/2 + B*a*x + B*b*x**2/2 + B
*c*x**3/3)/e, Eq(m, -1)), (A*a*e**m*m**3*x*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 9*A*a*e**m*m**2*x*x**
m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 26*A*a*e**m*m*x*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 24*A*
a*e**m*x*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + A*b*e**m*m**3*x**2*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*
m + 24) + 8*A*b*e**m*m**2*x**2*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 19*A*b*e**m*m*x**2*x**m/(m**4 + 1
0*m**3 + 35*m**2 + 50*m + 24) + 12*A*b*e**m*x**2*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + A*c*e**m*m**3*x
**3*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 7*A*c*e**m*m**2*x**3*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m +
24) + 14*A*c*e**m*m*x**3*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 8*A*c*e**m*x**3*x**m/(m**4 + 10*m**3 +
35*m**2 + 50*m + 24) + B*a*e**m*m**3*x**2*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 8*B*a*e**m*m**2*x**2*
x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 19*B*a*e**m*m*x**2*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) +
12*B*a*e**m*x**2*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + B*b*e**m*m**3*x**3*x**m/(m**4 + 10*m**3 + 35*m
**2 + 50*m + 24) + 7*B*b*e**m*m**2*x**3*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 14*B*b*e**m*m*x**3*x**m/
(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 8*B*b*e**m*x**3*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + B*c*e**
m*m**3*x**4*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 6*B*c*e**m*m**2*x**4*x**m/(m**4 + 10*m**3 + 35*m**2
+ 50*m + 24) + 11*B*c*e**m*m*x**4*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 6*B*c*e**m*x**4*x**m/(m**4 + 1
0*m**3 + 35*m**2 + 50*m + 24), True))

________________________________________________________________________________________

Giac [B]  time = 1.24267, size = 456, normalized size = 5.49 \begin{align*} \frac{B c m^{3} x^{4} x^{m} e^{m} + B b m^{3} x^{3} x^{m} e^{m} + A c m^{3} x^{3} x^{m} e^{m} + 6 \, B c m^{2} x^{4} x^{m} e^{m} + B a m^{3} x^{2} x^{m} e^{m} + A b m^{3} x^{2} x^{m} e^{m} + 7 \, B b m^{2} x^{3} x^{m} e^{m} + 7 \, A c m^{2} x^{3} x^{m} e^{m} + 11 \, B c m x^{4} x^{m} e^{m} + A a m^{3} x x^{m} e^{m} + 8 \, B a m^{2} x^{2} x^{m} e^{m} + 8 \, A b m^{2} x^{2} x^{m} e^{m} + 14 \, B b m x^{3} x^{m} e^{m} + 14 \, A c m x^{3} x^{m} e^{m} + 6 \, B c x^{4} x^{m} e^{m} + 9 \, A a m^{2} x x^{m} e^{m} + 19 \, B a m x^{2} x^{m} e^{m} + 19 \, A b m x^{2} x^{m} e^{m} + 8 \, B b x^{3} x^{m} e^{m} + 8 \, A c x^{3} x^{m} e^{m} + 26 \, A a m x x^{m} e^{m} + 12 \, B a x^{2} x^{m} e^{m} + 12 \, A b x^{2} x^{m} e^{m} + 24 \, A a x x^{m} e^{m}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(B*c*m^3*x^4*x^m*e^m + B*b*m^3*x^3*x^m*e^m + A*c*m^3*x^3*x^m*e^m + 6*B*c*m^2*x^4*x^m*e^m + B*a*m^3*x^2*x^m*e^m
+ A*b*m^3*x^2*x^m*e^m + 7*B*b*m^2*x^3*x^m*e^m + 7*A*c*m^2*x^3*x^m*e^m + 11*B*c*m*x^4*x^m*e^m + A*a*m^3*x*x^m*
e^m + 8*B*a*m^2*x^2*x^m*e^m + 8*A*b*m^2*x^2*x^m*e^m + 14*B*b*m*x^3*x^m*e^m + 14*A*c*m*x^3*x^m*e^m + 6*B*c*x^4*
x^m*e^m + 9*A*a*m^2*x*x^m*e^m + 19*B*a*m*x^2*x^m*e^m + 19*A*b*m*x^2*x^m*e^m + 8*B*b*x^3*x^m*e^m + 8*A*c*x^3*x^
m*e^m + 26*A*a*m*x*x^m*e^m + 12*B*a*x^2*x^m*e^m + 12*A*b*x^2*x^m*e^m + 24*A*a*x*x^m*e^m)/(m^4 + 10*m^3 + 35*m^
2 + 50*m + 24)