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

Optimal. Leaf size=75 \[ \frac{d (c d-b e) (d+e x)^{m+1}}{e^3 (m+1)}-\frac{(2 c d-b e) (d+e x)^{m+2}}{e^3 (m+2)}+\frac{c (d+e x)^{m+3}}{e^3 (m+3)} \]

[Out]

(d*(c*d - b*e)*(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 = 0.105912, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{d (c d-b e) (d+e x)^{m+1}}{e^3 (m+1)}-\frac{(2 c d-b e) (d+e x)^{m+2}}{e^3 (m+2)}+\frac{c (d+e x)^{m+3}}{e^3 (m+3)} \]

Antiderivative was successfully verified.

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

[Out]

(d*(c*d - b*e)*(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 = 18.3548, size = 63, normalized size = 0.84 \[ \frac{c \left (d + e x\right )^{m + 3}}{e^{3} \left (m + 3\right )} - \frac{d \left (d + e x\right )^{m + 1} \left (b e - c d\right )}{e^{3} \left (m + 1\right )} + \frac{\left (d + e x\right )^{m + 2} \left (b e - 2 c d\right )}{e^{3} \left (m + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*(d + e*x)**(m + 3)/(e**3*(m + 3)) - d*(d + e*x)**(m + 1)*(b*e - c*d)/(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 = 0.0738313, size = 76, normalized size = 1.01 \[ \frac{(d+e x)^{m+1} \left (b e (m+3) (e (m+1) x-d)+c \left (2 d^2-2 d e (m+1) x+e^2 \left (m^2+3 m+2\right ) x^2\right )\right )}{e^3 (m+1) (m+2) (m+3)} \]

Antiderivative was successfully verified.

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

[Out]

((d + e*x)^(1 + m)*(b*e*(3 + m)*(-d + 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 = 116, normalized size = 1.6 \[ -{\frac{ \left ( ex+d \right ) ^{1+m} \left ( -c{e}^{2}{m}^{2}{x}^{2}-b{e}^{2}{m}^{2}x-3\,c{e}^{2}m{x}^{2}-4\,b{e}^{2}mx+2\,cdemx-2\,c{e}^{2}{x}^{2}+bdem-3\,b{e}^{2}x+2\,cdex+3\,bde-2\,c{d}^{2} \right ) }{{e}^{3} \left ({m}^{3}+6\,{m}^{2}+11\,m+6 \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^m*(c*x^2+b*x),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^2-4*b*e^2*m*x+2*c*d*e*m*x
-2*c*e^2*x^2+b*d*e*m-3*b*e^2*x+2*c*d*e*x+3*b*d*e-2*c*d^2)/e^3/(m^3+6*m^2+11*m+6)

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Maxima [A]  time = 0.732006, size = 153, normalized size = 2.04 \[ \frac{{\left (e^{2}{\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )}{\left (e x + d\right )}^{m} b}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac{{\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} +{\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )}{\left (e x + d\right )}^{m} c}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.225483, size = 216, normalized size = 2.88 \[ -\frac{{\left (b d^{2} e m - 2 \, c d^{3} + 3 \, b d^{2} e -{\left (c e^{3} m^{2} + 3 \, c e^{3} m + 2 \, c e^{3}\right )} x^{3} -{\left (3 \, b e^{3} +{\left (c d e^{2} + b e^{3}\right )} m^{2} +{\left (c d e^{2} + 4 \, b e^{3}\right )} m\right )} x^{2} -{\left (b d e^{2} m^{2} -{\left (2 \, c d^{2} e - 3 \, b d e^{2}\right )} m\right )} x\right )}{\left (e x + d\right )}^{m}}{e^{3} m^{3} + 6 \, e^{3} m^{2} + 11 \, e^{3} m + 6 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(b*d^2*e*m - 2*c*d^3 + 3*b*d^2*e - (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*e^2*m^2 - (2*c*
d^2*e - 3*b*d*e^2)*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 = 4.60914, size = 1093, normalized size = 14.57 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((d**m*(b*x**2/2 + c*x**3/3), Eq(e, 0)), (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)), (b*d*e*log(d/e + x
)/(d*e**3 + e**4*x) + b*e**2*x*log(d/e + x)/(d*e**3 + e**4*x) - b*e**2*x/(d*e**3
+ e**4*x) - 2*c*d**2*log(d/e + x)/(d*e**3 + e**4*x) - 2*c*d*e*x*log(d/e + x)/(d
*e**3 + e**4*x) + 2*c*d*e*x/(d*e**3 + e**4*x) + c*e**2*x**2/(d*e**3 + e**4*x), E
q(m, -2)), (-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)), (-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 = 0.208758, size = 393, normalized size = 5.24 \[ \frac{c m^{2} x^{3} e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + c d m^{2} x^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )} + b m^{2} x^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + 3 \, c m x^{3} e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + b d m^{2} x e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )} + c d m x^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )} - 2 \, c d^{2} m x e^{\left (m{\rm ln}\left (x e + d\right ) + 1\right )} + 4 \, b m x^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + 2 \, c x^{3} e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + 3 \, b d m x e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )} - b d^{2} m e^{\left (m{\rm ln}\left (x e + d\right ) + 1\right )} + 2 \, c d^{3} e^{\left (m{\rm ln}\left (x e + d\right )\right )} + 3 \, b x^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} - 3 \, b d^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 1\right )}}{m^{3} e^{3} + 6 \, m^{2} e^{3} + 11 \, m e^{3} + 6 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)*(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)
+ 4*b*m*x^2*e^(m*ln(x*e + d) + 3) + 2*c*x^3*e^(m*ln(x*e + d) + 3) + 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))
+ 3*b*x^2*e^(m*ln(x*e + d) + 3) - 3*b*d^2*e^(m*ln(x*e + d) + 1))/(m^3*e^3 + 6*m
^2*e^3 + 11*m*e^3 + 6*e^3)