3.2541 \(\int (d+e x)^m \left (a+b x+c x^2\right ) \, dx\)

Optimal. Leaf size=82 \[ \frac{(d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )}{e^3 (m+1)}-\frac{(2 c d-b e) (d+e x)^{m+2}}{e^3 (m+2)}+\frac{c (d+e x)^{m+3}}{e^3 (m+3)} \]

[Out]

((c*d^2 - b*d*e + a*e^2)*(d + e*x)^(1 + m))/(e^3*(1 + m)) - ((2*c*d - b*e)*(d +
e*x)^(2 + m))/(e^3*(2 + m)) + (c*(d + e*x)^(3 + m))/(e^3*(3 + m))

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Rubi [A]  time = 0.107984, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ \frac{(d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )}{e^3 (m+1)}-\frac{(2 c d-b e) (d+e x)^{m+2}}{e^3 (m+2)}+\frac{c (d+e x)^{m+3}}{e^3 (m+3)} \]

Antiderivative was successfully verified.

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

[Out]

((c*d^2 - b*d*e + a*e^2)*(d + e*x)^(1 + m))/(e^3*(1 + m)) - ((2*c*d - b*e)*(d +
e*x)^(2 + m))/(e^3*(2 + m)) + (c*(d + e*x)^(3 + m))/(e^3*(3 + m))

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Rubi in Sympy [A]  time = 19.479, size = 70, normalized size = 0.85 \[ \frac{c \left (d + e x\right )^{m + 3}}{e^{3} \left (m + 3\right )} + \frac{\left (d + e x\right )^{m + 1} \left (a e^{2} - b d e + c d^{2}\right )}{e^{3} \left (m + 1\right )} + \frac{\left (d + e x\right )^{m + 2} \left (b e - 2 c d\right )}{e^{3} \left (m + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*(d + e*x)**(m + 3)/(e**3*(m + 3)) + (d + e*x)**(m + 1)*(a*e**2 - b*d*e + c*d**
2)/(e**3*(m + 1)) + (d + e*x)**(m + 2)*(b*e - 2*c*d)/(e**3*(m + 2))

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Mathematica [A]  time = 0.113427, size = 83, normalized size = 1.01 \[ \frac{(d+e x)^{m+1} \left (e (m+3) (a e (m+2)-b d+b e (m+1) x)+c \left (2 d^2-2 d e (m+1) x+e^2 \left (m^2+3 m+2\right ) x^2\right )\right )}{e^3 (m+1) (m+2) (m+3)} \]

Antiderivative was successfully verified.

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

[Out]

((d + e*x)^(1 + m)*(e*(3 + m)*(-(b*d) + a*e*(2 + m) + b*e*(1 + m)*x) + c*(2*d^2
- 2*d*e*(1 + m)*x + e^2*(2 + 3*m + m^2)*x^2)))/(e^3*(1 + m)*(2 + m)*(3 + m))

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Maple [A]  time = 0.007, size = 135, normalized size = 1.7 \[{\frac{ \left ( ex+d \right ) ^{1+m} \left ( c{e}^{2}{m}^{2}{x}^{2}+b{e}^{2}{m}^{2}x+3\,c{e}^{2}m{x}^{2}+a{e}^{2}{m}^{2}+4\,b{e}^{2}mx-2\,cdemx+2\,c{e}^{2}{x}^{2}+5\,a{e}^{2}m-bdem+3\,b{e}^{2}x-2\,cdex+6\,a{e}^{2}-3\,bde+2\,c{d}^{2} \right ) }{{e}^{3} \left ({m}^{3}+6\,{m}^{2}+11\,m+6 \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^m*(c*x^2+b*x+a),x)

[Out]

(e*x+d)^(1+m)*(c*e^2*m^2*x^2+b*e^2*m^2*x+3*c*e^2*m*x^2+a*e^2*m^2+4*b*e^2*m*x-2*c
*d*e*m*x+2*c*e^2*x^2+5*a*e^2*m-b*d*e*m+3*b*e^2*x-2*c*d*e*x+6*a*e^2-3*b*d*e+2*c*d
^2)/e^3/(m^3+6*m^2+11*m+6)

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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)*(e*x + d)^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.246143, size = 271, normalized size = 3.3 \[ \frac{{\left (a d e^{2} m^{2} + 2 \, c d^{3} - 3 \, b d^{2} e + 6 \, a d e^{2} +{\left (c e^{3} m^{2} + 3 \, c e^{3} m + 2 \, c e^{3}\right )} x^{3} +{\left (3 \, b e^{3} +{\left (c d e^{2} + b e^{3}\right )} m^{2} +{\left (c d e^{2} + 4 \, b e^{3}\right )} m\right )} x^{2} -{\left (b d^{2} e - 5 \, a d e^{2}\right )} m +{\left (6 \, a e^{3} +{\left (b d e^{2} + a e^{3}\right )} m^{2} -{\left (2 \, c d^{2} e - 3 \, b d e^{2} - 5 \, a e^{3}\right )} m\right )} x\right )}{\left (e x + d\right )}^{m}}{e^{3} m^{3} + 6 \, e^{3} m^{2} + 11 \, e^{3} m + 6 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a*d*e^2*m^2 + 2*c*d^3 - 3*b*d^2*e + 6*a*d*e^2 + (c*e^3*m^2 + 3*c*e^3*m + 2*c*e^
3)*x^3 + (3*b*e^3 + (c*d*e^2 + b*e^3)*m^2 + (c*d*e^2 + 4*b*e^3)*m)*x^2 - (b*d^2*
e - 5*a*d*e^2)*m + (6*a*e^3 + (b*d*e^2 + a*e^3)*m^2 - (2*c*d^2*e - 3*b*d*e^2 - 5
*a*e^3)*m)*x)*(e*x + d)^m/(e^3*m^3 + 6*e^3*m^2 + 11*e^3*m + 6*e^3)

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Sympy [A]  time = 5.49834, size = 1459, normalized size = 17.79 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((d**m*(a*x + b*x**2/2 + c*x**3/3), Eq(e, 0)), (-a*d*e**2/(2*d**3*e**3
+ 4*d**2*e**4*x + 2*d*e**5*x**2) + b*e**3*x**2/(2*d**3*e**3 + 4*d**2*e**4*x + 2*
d*e**5*x**2) + 2*c*d**3*log(d/e + x)/(2*d**3*e**3 + 4*d**2*e**4*x + 2*d*e**5*x**
2) + c*d**3/(2*d**3*e**3 + 4*d**2*e**4*x + 2*d*e**5*x**2) + 4*c*d**2*e*x*log(d/e
+ x)/(2*d**3*e**3 + 4*d**2*e**4*x + 2*d*e**5*x**2) + 2*c*d*e**2*x**2*log(d/e +
x)/(2*d**3*e**3 + 4*d**2*e**4*x + 2*d*e**5*x**2) - 2*c*d*e**2*x**2/(2*d**3*e**3
+ 4*d**2*e**4*x + 2*d*e**5*x**2), Eq(m, -3)), (a*e**3*x/(d**2*e**3 + d*e**4*x) +
b*d**2*e*log(d/e + x)/(d**2*e**3 + d*e**4*x) + b*d*e**2*x*log(d/e + x)/(d**2*e*
*3 + d*e**4*x) - b*d*e**2*x/(d**2*e**3 + d*e**4*x) - 2*c*d**3*log(d/e + x)/(d**2
*e**3 + d*e**4*x) - 2*c*d**2*e*x*log(d/e + x)/(d**2*e**3 + d*e**4*x) + 2*c*d**2*
e*x/(d**2*e**3 + d*e**4*x) + c*d*e**2*x**2/(d**2*e**3 + d*e**4*x), Eq(m, -2)), (
a*log(d/e + x)/e - b*d*log(d/e + x)/e**2 + b*x/e + c*d**2*log(d/e + x)/e**3 - c*
d*x/e**2 + c*x**2/(2*e), Eq(m, -1)), (a*d*e**2*m**2*(d + e*x)**m/(e**3*m**3 + 6*
e**3*m**2 + 11*e**3*m + 6*e**3) + 5*a*d*e**2*m*(d + e*x)**m/(e**3*m**3 + 6*e**3*
m**2 + 11*e**3*m + 6*e**3) + 6*a*d*e**2*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 +
11*e**3*m + 6*e**3) + a*e**3*m**2*x*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e
**3*m + 6*e**3) + 5*a*e**3*m*x*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m
+ 6*e**3) + 6*a*e**3*x*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e*
*3) - b*d**2*e*m*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) - 3
*b*d**2*e*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + b*d*e**2
*m**2*x*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + 3*b*d*e**2
*m*x*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + b*e**3*m**2*x
**2*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + 4*b*e**3*m*x**
2*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + 3*b*e**3*x**2*(d
+ e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + 2*c*d**3*(d + e*x)**
m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) - 2*c*d**2*e*m*x*(d + e*x)**m/(
e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + c*d*e**2*m**2*x**2*(d + e*x)**m/
(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + c*d*e**2*m*x**2*(d + e*x)**m/(e
**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + c*e**3*m**2*x**3*(d + e*x)**m/(e*
*3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + 3*c*e**3*m*x**3*(d + e*x)**m/(e**3
*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + 2*c*e**3*x**3*(d + e*x)**m/(e**3*m**
3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3), True))

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GIAC/XCAS [A]  time = 0.223376, size = 531, normalized size = 6.48 \[ \frac{c m^{2} x^{3} e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + c d m^{2} x^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )} + b m^{2} x^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + 3 \, c m x^{3} e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + b d m^{2} x e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )} + c d m x^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )} - 2 \, c d^{2} m x e^{\left (m{\rm ln}\left (x e + d\right ) + 1\right )} + a m^{2} x e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + 4 \, b m x^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + 2 \, c x^{3} e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + a d m^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )} + 3 \, b d m x e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )} - b d^{2} m e^{\left (m{\rm ln}\left (x e + d\right ) + 1\right )} + 2 \, c d^{3} e^{\left (m{\rm ln}\left (x e + d\right )\right )} + 5 \, a m x e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + 3 \, b x^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + 5 \, a d m e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )} - 3 \, b d^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 1\right )} + 6 \, a x e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + 6 \, a d e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )}}{m^{3} e^{3} + 6 \, m^{2} e^{3} + 11 \, m e^{3} + 6 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c*m^2*x^3*e^(m*ln(x*e + d) + 3) + c*d*m^2*x^2*e^(m*ln(x*e + d) + 2) + b*m^2*x^2
*e^(m*ln(x*e + d) + 3) + 3*c*m*x^3*e^(m*ln(x*e + d) + 3) + b*d*m^2*x*e^(m*ln(x*e
+ d) + 2) + c*d*m*x^2*e^(m*ln(x*e + d) + 2) - 2*c*d^2*m*x*e^(m*ln(x*e + d) + 1)
+ a*m^2*x*e^(m*ln(x*e + d) + 3) + 4*b*m*x^2*e^(m*ln(x*e + d) + 3) + 2*c*x^3*e^(
m*ln(x*e + d) + 3) + a*d*m^2*e^(m*ln(x*e + d) + 2) + 3*b*d*m*x*e^(m*ln(x*e + d)
+ 2) - b*d^2*m*e^(m*ln(x*e + d) + 1) + 2*c*d^3*e^(m*ln(x*e + d)) + 5*a*m*x*e^(m*
ln(x*e + d) + 3) + 3*b*x^2*e^(m*ln(x*e + d) + 3) + 5*a*d*m*e^(m*ln(x*e + d) + 2)
- 3*b*d^2*e^(m*ln(x*e + d) + 1) + 6*a*x*e^(m*ln(x*e + d) + 3) + 6*a*d*e^(m*ln(x
*e + d) + 2))/(m^3*e^3 + 6*m^2*e^3 + 11*m*e^3 + 6*e^3)