3.1079 \(\int (d+e x)^m \left (c d^2+2 c d e x+c e^2 x^2\right ) \, dx\)

Optimal. Leaf size=19 \[ \frac{c (d+e x)^{m+3}}{e (m+3)} \]

[Out]

(c*(d + e*x)^(3 + m))/(e*(3 + m))

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Rubi [A]  time = 0.0234116, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107 \[ \frac{c (d+e x)^{m+3}}{e (m+3)} \]

Antiderivative was successfully verified.

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

[Out]

(c*(d + e*x)^(3 + m))/(e*(3 + m))

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Rubi in Sympy [A]  time = 16.8165, size = 14, normalized size = 0.74 \[ \frac{c \left (d + e x\right )^{m + 3}}{e \left (m + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*(d + e*x)**(m + 3)/(e*(m + 3))

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Mathematica [A]  time = 0.022809, size = 20, normalized size = 1.05 \[ \frac{c (d+e x)^{m+3}}{e m+3 e} \]

Antiderivative was successfully verified.

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

[Out]

(c*(d + e*x)^(3 + m))/(3*e + e*m)

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Maple [A]  time = 0.006, size = 36, normalized size = 1.9 \[{\frac{ \left ( ex+d \right ) ^{1+m}c \left ({e}^{2}{x}^{2}+2\,dex+{d}^{2} \right ) }{e \left ( 3+m \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23119, size = 66, normalized size = 3.47 \[ \frac{{\left (c e^{3} x^{3} + 3 \, c d e^{2} x^{2} + 3 \, c d^{2} e x + c d^{3}\right )}{\left (e x + d\right )}^{m}}{e m + 3 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.83499, size = 116, normalized size = 6.11 \[ \begin{cases} \frac{c x}{d} & \text{for}\: e = 0 \wedge m = -3 \\c d^{2} d^{m} x & \text{for}\: e = 0 \\\frac{c \log{\left (\frac{d}{e} + x \right )}}{e} & \text{for}\: m = -3 \\\frac{c d^{3} \left (d + e x\right )^{m}}{e m + 3 e} + \frac{3 c d^{2} e x \left (d + e x\right )^{m}}{e m + 3 e} + \frac{3 c d e^{2} x^{2} \left (d + e x\right )^{m}}{e m + 3 e} + \frac{c e^{3} x^{3} \left (d + e x\right )^{m}}{e m + 3 e} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.218172, size = 112, normalized size = 5.89 \[ \frac{c x^{3} e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + 3 \, c d x^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )} + 3 \, c d^{2} x e^{\left (m{\rm ln}\left (x e + d\right ) + 1\right )} + c d^{3} e^{\left (m{\rm ln}\left (x e + d\right )\right )}}{m e + 3 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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