3.2075 \(\int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx\)

Optimal. Leaf size=52 \[ \frac{c d (d+e x)^{m+3}}{e^2 (m+3)}-\frac{\left (c d^2-a e^2\right ) (d+e x)^{m+2}}{e^2 (m+2)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0796572, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061 \[ \frac{c d (d+e x)^{m+3}}{e^2 (m+3)}-\frac{\left (c d^2-a e^2\right ) (d+e x)^{m+2}}{e^2 (m+2)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 20.9217, size = 42, normalized size = 0.81 \[ \frac{c d \left (d + e x\right )^{m + 3}}{e^{2} \left (m + 3\right )} + \frac{\left (d + e x\right )^{m + 2} \left (a e^{2} - c d^{2}\right )}{e^{2} \left (m + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0692338, size = 45, normalized size = 0.87 \[ \frac{(d+e x)^{m+2} \left (a e^2 (m+3)+c d (e (m+2) x-d)\right )}{e^2 (m+2) (m+3)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.004, size = 55, normalized size = 1.1 \[{\frac{ \left ( ex+d \right ) ^{2+m} \left ( cdemx+a{e}^{2}m+2\,cdex+3\,a{e}^{2}-c{d}^{2} \right ) }{{e}^{2} \left ({m}^{2}+5\,m+6 \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.221286, size = 184, normalized size = 3.54 \[ \frac{{\left (a d^{2} e^{2} m - c d^{4} + 3 \, a d^{2} e^{2} +{\left (c d e^{3} m + 2 \, c d e^{3}\right )} x^{3} +{\left (3 \, c d^{2} e^{2} + 3 \, a e^{4} +{\left (2 \, c d^{2} e^{2} + a e^{4}\right )} m\right )} x^{2} +{\left (6 \, a d e^{3} +{\left (c d^{3} e + 2 \, a d e^{3}\right )} m\right )} x\right )}{\left (e x + d\right )}^{m}}{e^{2} m^{2} + 5 \, e^{2} m + 6 \, e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 3.82997, size = 639, normalized size = 12.29 \[ \begin{cases} \frac{c d^{2} d^{m} x^{2}}{2} & \text{for}\: e = 0 \\- \frac{a d e^{2}}{2 d^{2} e^{2} + 2 d e^{3} x} + \frac{a e^{3} x}{2 d^{2} e^{2} + 2 d e^{3} x} + \frac{2 c d^{3} \log{\left (\frac{d}{e} + x \right )}}{2 d^{2} e^{2} + 2 d e^{3} x} + \frac{c d^{3}}{2 d^{2} e^{2} + 2 d e^{3} x} + \frac{2 c d^{2} e x \log{\left (\frac{d}{e} + x \right )}}{2 d^{2} e^{2} + 2 d e^{3} x} - \frac{c d^{2} e x}{2 d^{2} e^{2} + 2 d e^{3} x} & \text{for}\: m = -3 \\a \log{\left (\frac{d}{e} + x \right )} - \frac{c d^{2} \log{\left (\frac{d}{e} + x \right )}}{e^{2}} + \frac{c d x}{e} & \text{for}\: m = -2 \\\frac{a d^{2} e^{2} m \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac{3 a d^{2} e^{2} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac{2 a d e^{3} m x \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac{6 a d e^{3} x \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac{a e^{4} m x^{2} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac{3 a e^{4} x^{2} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} - \frac{c d^{4} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac{c d^{3} e m x \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac{2 c d^{2} e^{2} m x^{2} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac{3 c d^{2} e^{2} x^{2} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac{c d e^{3} m x^{3} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac{2 c d e^{3} x^{3} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((c*d**2*d**m*x**2/2, Eq(e, 0)), (-a*d*e**2/(2*d**2*e**2 + 2*d*e**3*x)
+ a*e**3*x/(2*d**2*e**2 + 2*d*e**3*x) + 2*c*d**3*log(d/e + x)/(2*d**2*e**2 + 2*d
*e**3*x) + c*d**3/(2*d**2*e**2 + 2*d*e**3*x) + 2*c*d**2*e*x*log(d/e + x)/(2*d**2
*e**2 + 2*d*e**3*x) - c*d**2*e*x/(2*d**2*e**2 + 2*d*e**3*x), Eq(m, -3)), (a*log(
d/e + x) - c*d**2*log(d/e + x)/e**2 + c*d*x/e, Eq(m, -2)), (a*d**2*e**2*m*(d + e
*x)**m/(e**2*m**2 + 5*e**2*m + 6*e**2) + 3*a*d**2*e**2*(d + e*x)**m/(e**2*m**2 +
5*e**2*m + 6*e**2) + 2*a*d*e**3*m*x*(d + e*x)**m/(e**2*m**2 + 5*e**2*m + 6*e**2
) + 6*a*d*e**3*x*(d + e*x)**m/(e**2*m**2 + 5*e**2*m + 6*e**2) + a*e**4*m*x**2*(d
+ e*x)**m/(e**2*m**2 + 5*e**2*m + 6*e**2) + 3*a*e**4*x**2*(d + e*x)**m/(e**2*m*
*2 + 5*e**2*m + 6*e**2) - c*d**4*(d + e*x)**m/(e**2*m**2 + 5*e**2*m + 6*e**2) +
c*d**3*e*m*x*(d + e*x)**m/(e**2*m**2 + 5*e**2*m + 6*e**2) + 2*c*d**2*e**2*m*x**2
*(d + e*x)**m/(e**2*m**2 + 5*e**2*m + 6*e**2) + 3*c*d**2*e**2*x**2*(d + e*x)**m/
(e**2*m**2 + 5*e**2*m + 6*e**2) + c*d*e**3*m*x**3*(d + e*x)**m/(e**2*m**2 + 5*e*
*2*m + 6*e**2) + 2*c*d*e**3*x**3*(d + e*x)**m/(e**2*m**2 + 5*e**2*m + 6*e**2), T
rue))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.208348, size = 328, normalized size = 6.31 \[ \frac{c d m x^{3} e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + 2 \, c d^{2} m x^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )} + c d^{3} m x e^{\left (m{\rm ln}\left (x e + d\right ) + 1\right )} + 2 \, c d x^{3} e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + 3 \, c d^{2} x^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )} - c d^{4} e^{\left (m{\rm ln}\left (x e + d\right )\right )} + a m x^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 4\right )} + 2 \, a d m x e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + a d^{2} m e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )} + 3 \, a x^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 4\right )} + 6 \, a d x e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + 3 \, a d^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )}}{m^{2} e^{2} + 5 \, m e^{2} + 6 \, e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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