3.1090 \(\int (d+e x)^m (c d^2+2 c d e x+c e^2 x^2) \, dx\)

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 = 0.0077997, 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, Rules used = {27, 12, 32} \[ \frac{c (d+e x)^{m+3}}{e (m+3)} \]

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))

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int (d+e x)^m \left (c d^2+2 c d e x+c e^2 x^2\right ) \, dx &=\int c (d+e x)^{2+m} \, dx\\ &=c \int (d+e x)^{2+m} \, dx\\ &=\frac{c (d+e x)^{3+m}}{e (3+m)}\\ \end{align*}

Mathematica [A]  time = 0.0140483, 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 = 0.043, size = 36, normalized size = 1.9 \begin{align*}{\frac{ \left ( ex+d \right ) ^{1+m}c \left ({e}^{2}{x}^{2}+2\,dex+{d}^{2} \right ) }{e \left ( 3+m \right ) }} \end{align*}

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(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.48514, size = 101, normalized size = 5.32 \begin{align*} \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} \end{align*}

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, 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 = 0.70723, size = 116, normalized size = 6.11 \begin{align*} \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} \end{align*}

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 [B]  time = 1.27151, size = 101, normalized size = 5.32 \begin{align*} \frac{{\left (x e + d\right )}^{m} c x^{3} e^{3} + 3 \,{\left (x e + d\right )}^{m} c d x^{2} e^{2} + 3 \,{\left (x e + d\right )}^{m} c d^{2} x e +{\left (x e + d\right )}^{m} c d^{3}}{m e + 3 \, e} \end{align*}

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, algorithm="giac")

[Out]

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