3.1722 \(\int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right ) \, dx\)

Optimal. Leaf size=78 \[ \frac{(b d-a e)^2 (d+e x)^{m+1}}{e^3 (m+1)}-\frac{2 b (b d-a e) (d+e x)^{m+2}}{e^3 (m+2)}+\frac{b^2 (d+e x)^{m+3}}{e^3 (m+3)} \]

[Out]

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

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Rubi [A]  time = 0.0958531, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{(b d-a e)^2 (d+e x)^{m+1}}{e^3 (m+1)}-\frac{2 b (b d-a e) (d+e x)^{m+2}}{e^3 (m+2)}+\frac{b^2 (d+e x)^{m+3}}{e^3 (m+3)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 41.8517, size = 66, normalized size = 0.85 \[ \frac{b^{2} \left (d + e x\right )^{m + 3}}{e^{3} \left (m + 3\right )} + \frac{2 b \left (d + e x\right )^{m + 2} \left (a e - b d\right )}{e^{3} \left (m + 2\right )} + \frac{\left (d + e x\right )^{m + 1} \left (a e - b d\right )^{2}}{e^{3} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.109159, size = 95, normalized size = 1.22 \[ \frac{(d+e x)^{m+1} \left (a^2 e^2 \left (m^2+5 m+6\right )+2 a b e (m+3) (e (m+1) x-d)+b^2 \left (2 d^2-2 d e (m+1) x+e^2 \left (m^2+3 m+2\right ) x^2\right )\right )}{e^3 (m+1) (m+2) (m+3)} \]

Antiderivative was successfully verified.

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

[Out]

((d + e*x)^(1 + m)*(a^2*e^2*(6 + 5*m + m^2) + 2*a*b*e*(3 + m)*(-d + e*(1 + m)*x)
+ b^2*(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 [B]  time = 0.013, size = 159, normalized size = 2. \[{\frac{ \left ( ex+d \right ) ^{1+m} \left ({b}^{2}{e}^{2}{m}^{2}{x}^{2}+2\,ab{e}^{2}{m}^{2}x+3\,{b}^{2}{e}^{2}m{x}^{2}+{a}^{2}{e}^{2}{m}^{2}+8\,ab{e}^{2}mx-2\,{b}^{2}demx+2\,{x}^{2}{b}^{2}{e}^{2}+5\,{a}^{2}{e}^{2}m-2\,abdem+6\,xab{e}^{2}-2\,x{b}^{2}de+6\,{a}^{2}{e}^{2}-6\,abde+2\,{b}^{2}{d}^{2} \right ) }{{e}^{3} \left ({m}^{3}+6\,{m}^{2}+11\,m+6 \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(e*x+d)^(1+m)*(b^2*e^2*m^2*x^2+2*a*b*e^2*m^2*x+3*b^2*e^2*m*x^2+a^2*e^2*m^2+8*a*b
*e^2*m*x-2*b^2*d*e*m*x+2*b^2*e^2*x^2+5*a^2*e^2*m-2*a*b*d*e*m+6*a*b*e^2*x-2*b^2*d
*e*x+6*a^2*e^2-6*a*b*d*e+2*b^2*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((b^2*x^2 + 2*a*b*x + a^2)*(e*x + d)^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221741, size = 320, normalized size = 4.1 \[ \frac{{\left (a^{2} d e^{2} m^{2} + 2 \, b^{2} d^{3} - 6 \, a b d^{2} e + 6 \, a^{2} d e^{2} +{\left (b^{2} e^{3} m^{2} + 3 \, b^{2} e^{3} m + 2 \, b^{2} e^{3}\right )} x^{3} +{\left (6 \, a b e^{3} +{\left (b^{2} d e^{2} + 2 \, a b e^{3}\right )} m^{2} +{\left (b^{2} d e^{2} + 8 \, a b e^{3}\right )} m\right )} x^{2} -{\left (2 \, a b d^{2} e - 5 \, a^{2} d e^{2}\right )} m +{\left (6 \, a^{2} e^{3} +{\left (2 \, a b d e^{2} + a^{2} e^{3}\right )} m^{2} -{\left (2 \, b^{2} d^{2} e - 6 \, a b d e^{2} - 5 \, a^{2} e^{3}\right )} m\right )} x\right )}{\left (e x + d\right )}^{m}}{e^{3} m^{3} + 6 \, e^{3} m^{2} + 11 \, e^{3} m + 6 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a^2*d*e^2*m^2 + 2*b^2*d^3 - 6*a*b*d^2*e + 6*a^2*d*e^2 + (b^2*e^3*m^2 + 3*b^2*e^
3*m + 2*b^2*e^3)*x^3 + (6*a*b*e^3 + (b^2*d*e^2 + 2*a*b*e^3)*m^2 + (b^2*d*e^2 + 8
*a*b*e^3)*m)*x^2 - (2*a*b*d^2*e - 5*a^2*d*e^2)*m + (6*a^2*e^3 + (2*a*b*d*e^2 + a
^2*e^3)*m^2 - (2*b^2*d^2*e - 6*a*b*d*e^2 - 5*a^2*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 = 5.79729, size = 1547, normalized size = 19.83 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((d**m*(a**2*x + a*b*x**2 + b**2*x**3/3), Eq(e, 0)), (-a**2*d*e**2/(2*d
**3*e**3 + 4*d**2*e**4*x + 2*d*e**5*x**2) + 2*a*b*e**3*x**2/(2*d**3*e**3 + 4*d**
2*e**4*x + 2*d*e**5*x**2) + 2*b**2*d**3*log(d/e + x)/(2*d**3*e**3 + 4*d**2*e**4*
x + 2*d*e**5*x**2) + b**2*d**3/(2*d**3*e**3 + 4*d**2*e**4*x + 2*d*e**5*x**2) + 4
*b**2*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*b*
*2*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*b*
*2*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**2*
e**3*x/(d**2*e**3 + d*e**4*x) + 2*a*b*d**2*e*log(d/e + x)/(d**2*e**3 + d*e**4*x)
+ 2*a*b*d*e**2*x*log(d/e + x)/(d**2*e**3 + d*e**4*x) - 2*a*b*d*e**2*x/(d**2*e**
3 + d*e**4*x) - 2*b**2*d**3*log(d/e + x)/(d**2*e**3 + d*e**4*x) - 2*b**2*d**2*e*
x*log(d/e + x)/(d**2*e**3 + d*e**4*x) + 2*b**2*d**2*e*x/(d**2*e**3 + d*e**4*x) +
b**2*d*e**2*x**2/(d**2*e**3 + d*e**4*x), Eq(m, -2)), (a**2*log(d/e + x)/e - 2*a
*b*d*log(d/e + x)/e**2 + 2*a*b*x/e + b**2*d**2*log(d/e + x)/e**3 - b**2*d*x/e**2
+ b**2*x**2/(2*e), Eq(m, -1)), (a**2*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**2*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**2*d*e**2*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**
2 + 11*e**3*m + 6*e**3) + a**2*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**2*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**2*e**3*x*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*
e**3*m + 6*e**3) - 2*a*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) - 6*a*b*d**2*e*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m +
6*e**3) + 2*a*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) + 6*a*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) + 2*a*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) + 8*a*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) + 6*a*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*b**2*d**3*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e
**3) - 2*b**2*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) + b**2*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) + b**2*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) + b**2*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*b**2*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*b**2*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 = 0.215667, size = 578, normalized size = 7.41 \[ \frac{b^{2} m^{2} x^{3} e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + b^{2} d m^{2} x^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )} + 2 \, a b m^{2} x^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + 3 \, b^{2} m x^{3} e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + 2 \, a b d m^{2} x e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )} + b^{2} d m x^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )} - 2 \, b^{2} d^{2} m x e^{\left (m{\rm ln}\left (x e + d\right ) + 1\right )} + a^{2} m^{2} x e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + 8 \, a b m x^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + 2 \, b^{2} x^{3} e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + a^{2} d m^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )} + 6 \, a b d m x e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )} - 2 \, a b d^{2} m e^{\left (m{\rm ln}\left (x e + d\right ) + 1\right )} + 2 \, b^{2} d^{3} e^{\left (m{\rm ln}\left (x e + d\right )\right )} + 5 \, a^{2} m x e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + 6 \, a b x^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + 5 \, a^{2} d m e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )} - 6 \, a b d^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 1\right )} + 6 \, a^{2} x e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + 6 \, a^{2} d e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )}}{m^{3} e^{3} + 6 \, m^{2} e^{3} + 11 \, m e^{3} + 6 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b^2*m^2*x^3*e^(m*ln(x*e + d) + 3) + b^2*d*m^2*x^2*e^(m*ln(x*e + d) + 2) + 2*a*b
*m^2*x^2*e^(m*ln(x*e + d) + 3) + 3*b^2*m*x^3*e^(m*ln(x*e + d) + 3) + 2*a*b*d*m^2
*x*e^(m*ln(x*e + d) + 2) + b^2*d*m*x^2*e^(m*ln(x*e + d) + 2) - 2*b^2*d^2*m*x*e^(
m*ln(x*e + d) + 1) + a^2*m^2*x*e^(m*ln(x*e + d) + 3) + 8*a*b*m*x^2*e^(m*ln(x*e +
d) + 3) + 2*b^2*x^3*e^(m*ln(x*e + d) + 3) + a^2*d*m^2*e^(m*ln(x*e + d) + 2) + 6
*a*b*d*m*x*e^(m*ln(x*e + d) + 2) - 2*a*b*d^2*m*e^(m*ln(x*e + d) + 1) + 2*b^2*d^3
*e^(m*ln(x*e + d)) + 5*a^2*m*x*e^(m*ln(x*e + d) + 3) + 6*a*b*x^2*e^(m*ln(x*e + d
) + 3) + 5*a^2*d*m*e^(m*ln(x*e + d) + 2) - 6*a*b*d^2*e^(m*ln(x*e + d) + 1) + 6*a
^2*x*e^(m*ln(x*e + d) + 3) + 6*a^2*d*e^(m*ln(x*e + d) + 2))/(m^3*e^3 + 6*m^2*e^3
+ 11*m*e^3 + 6*e^3)