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

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))

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Rubi [A]  time = 0.112224, 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 \[ \frac{(e x)^{m+2} (a B+A b)}{e^2 (m+2)}+\frac{a A (e x)^{m+1}}{e (m+1)}+\frac{(e x)^{m+3} (A c+b B)}{e^3 (m+3)}+\frac{B c (e x)^{m+4}}{e^4 (m+4)} \]

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))

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Rubi in Sympy [A]  time = 19.0096, size = 71, normalized size = 0.86 \[ \frac{A a \left (e x\right )^{m + 1}}{e \left (m + 1\right )} + \frac{B c \left (e x\right )^{m + 4}}{e^{4} \left (m + 4\right )} + \frac{\left (e x\right )^{m + 2} \left (A b + B a\right )}{e^{2} \left (m + 2\right )} + \frac{\left (e x\right )^{m + 3} \left (A c + B b\right )}{e^{3} \left (m + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0963501, size = 59, normalized size = 0.71 \[ (e x)^m \left (\frac{x^2 (a B+A b)}{m+2}+\frac{a A x}{m+1}+\frac{x^3 (A c+b B)}{m+3}+\frac{B c x^4}{m+4}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.006, size = 205, normalized size = 2.5 \[{\frac{ \left ( Bc{m}^{3}{x}^{3}+Ac{m}^{3}{x}^{2}+Bb{m}^{3}{x}^{2}+6\,Bc{m}^{2}{x}^{3}+Ab{m}^{3}x+7\,Ac{m}^{2}{x}^{2}+Ba{m}^{3}x+7\,Bb{m}^{2}{x}^{2}+11\,Bcm{x}^{3}+Aa{m}^{3}+8\,Ab{m}^{2}x+14\,Acm{x}^{2}+8\,Ba{m}^{2}x+14\,Bbm{x}^{2}+6\,Bc{x}^{3}+9\,Aa{m}^{2}+19\,Abmx+8\,Ac{x}^{2}+19\,Bamx+8\,Bb{x}^{2}+26\,Aam+12\,Abx+12\,aBx+24\,aA \right ) x \left ( ex \right ) ^{m}}{ \left ( 4+m \right ) \left ( 3+m \right ) \left ( 2+m \right ) \left ( 1+m \right ) }} \]

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+1
4*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)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.339195, size = 231, normalized size = 2.78 \[ \frac{{\left ({\left (B c m^{3} + 6 \, B c m^{2} + 11 \, B c m + 6 \, B c\right )} x^{4} +{\left ({\left (B b + A c\right )} m^{3} + 7 \,{\left (B b + A c\right )} m^{2} + 8 \, B b + 8 \, A c + 14 \,{\left (B b + A c\right )} m\right )} x^{3} +{\left ({\left (B a + A b\right )} m^{3} + 8 \,{\left (B a + A b\right )} m^{2} + 12 \, B a + 12 \, A b + 19 \,{\left (B a + A b\right )} m\right )} x^{2} +{\left (A a m^{3} + 9 \, A a m^{2} + 26 \, A a m + 24 \, A a\right )} x\right )} \left (e x\right )^{m}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Sympy [A]  time = 3.04762, size = 1022, normalized size = 12.31 \[ \text{result too large to display} \]

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*lo
g(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 + 10*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 + 3
5*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 + 10*m**3 + 35*m**2 + 50*m + 24), True))

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GIAC/XCAS [A]  time = 0.286807, size = 521, normalized size = 6.28 \[ \frac{B c m^{3} x^{4} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + B b m^{3} x^{3} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + A c m^{3} x^{3} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 6 \, B c m^{2} x^{4} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + B a m^{3} x^{2} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + A b m^{3} x^{2} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 7 \, B b m^{2} x^{3} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 7 \, A c m^{2} x^{3} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 11 \, B c m x^{4} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + A a m^{3} x e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 8 \, B a m^{2} x^{2} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 8 \, A b m^{2} x^{2} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 14 \, B b m x^{3} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 14 \, A c m x^{3} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 6 \, B c x^{4} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 9 \, A a m^{2} x e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 19 \, B a m x^{2} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 19 \, A b m x^{2} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 8 \, B b x^{3} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 8 \, A c x^{3} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 26 \, A a m x e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 12 \, B a x^{2} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 12 \, A b x^{2} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 24 \, A a x e^{\left (m{\rm ln}\left (x\right ) + m\right )}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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