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

Optimal. Leaf size=41 \[ \frac{c (d+e x)^{m+3}}{e^2 (m+3)}-\frac{c d (d+e x)^{m+2}}{e^2 (m+2)} \]

[Out]

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

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Rubi [A]  time = 0.0625484, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{c (d+e x)^{m+3}}{e^2 (m+3)}-\frac{c d (d+e x)^{m+2}}{e^2 (m+2)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 13.2458, size = 34, normalized size = 0.83 \[ - \frac{c d \left (d + e x\right )^{m + 2}}{e^{2} \left (m + 2\right )} + \frac{c \left (d + e x\right )^{m + 3}}{e^{2} \left (m + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0281332, size = 34, normalized size = 0.83 \[ \frac{c (d+e x)^{m+2} (e (m+2) x-d)}{e^2 (m+2) (m+3)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.003, size = 37, normalized size = 0.9 \[ -{\frac{c \left ( ex+d \right ) ^{2+m} \left ( -mex-2\,ex+d \right ) }{{e}^{2} \left ({m}^{2}+5\,m+6 \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.751921, size = 154, normalized size = 3.76 \[ \frac{{\left (e^{2}{\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )}{\left (e x + d\right )}^{m} c d}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac{{\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} +{\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )}{\left (e x + d\right )}^{m} c}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*e*x^2 + c*d*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*c*d/((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^2)

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Fricas [A]  time = 0.233581, size = 109, normalized size = 2.66 \[ \frac{{\left (c d^{2} e m x - c d^{3} +{\left (c e^{3} m + 2 \, c e^{3}\right )} x^{3} +{\left (2 \, c d e^{2} m + 3 \, c d e^{2}\right )} x^{2}\right )}{\left (e x + d\right )}^{m}}{e^{2} m^{2} + 5 \, e^{2} m + 6 \, e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.92419, size = 332, normalized size = 8.1 \[ \begin{cases} \frac{c d d^{m} x^{2}}{2} & \text{for}\: e = 0 \\\frac{2 c d \log{\left (\frac{d}{e} + x \right )}}{2 d e^{2} + 2 e^{3} x} + \frac{c d}{2 d e^{2} + 2 e^{3} x} + \frac{2 c e x \log{\left (\frac{d}{e} + x \right )}}{2 d e^{2} + 2 e^{3} x} - \frac{c e x}{2 d e^{2} + 2 e^{3} x} & \text{for}\: m = -3 \\- \frac{c d \log{\left (\frac{d}{e} + x \right )}}{e^{2}} + \frac{c x}{e} & \text{for}\: m = -2 \\- \frac{c d^{3} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac{c d^{2} e m x \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac{2 c d e^{2} m x^{2} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac{3 c d e^{2} x^{2} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac{c e^{3} m x^{3} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac{2 c e^{3} x^{3} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.209863, size = 176, normalized size = 4.29 \[ \frac{c m x^{3} e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + 2 \, c d m x^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )} + c d^{2} m x e^{\left (m{\rm ln}\left (x e + d\right ) + 1\right )} + 2 \, c x^{3} e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + 3 \, c d x^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )} - c d^{3} e^{\left (m{\rm ln}\left (x e + d\right )\right )}}{m^{2} e^{2} + 5 \, m e^{2} + 6 \, e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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