### 3.1089 $$\int (d+e x)^m (c d^2+2 c d e x+c e^2 x^2)^2 \, dx$$

Optimal. Leaf size=21 $\frac{c^2 (d+e x)^{m+5}}{e (m+5)}$

[Out]

(c^2*(d + e*x)^(5 + m))/(e*(5 + m))

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Rubi [A]  time = 0.0093606, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 30, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.1, Rules used = {27, 12, 32} $\frac{c^2 (d+e x)^{m+5}}{e (m+5)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(c^2*(d + e*x)^(5 + m))/(e*(5 + 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 )^2 \, dx &=\int c^2 (d+e x)^{4+m} \, dx\\ &=c^2 \int (d+e x)^{4+m} \, dx\\ &=\frac{c^2 (d+e x)^{5+m}}{e (5+m)}\\ \end{align*}

Mathematica [A]  time = 0.0168074, size = 22, normalized size = 1.05 $\frac{c^2 (d+e x)^{m+5}}{e m+5 e}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(c^2*(d + e*x)^(5 + m))/(5*e + e*m)

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Maple [A]  time = 0.04, size = 40, normalized size = 1.9 \begin{align*}{\frac{ \left ( ex+d \right ) ^{1+m}{c}^{2} \left ({e}^{2}{x}^{2}+2\,dex+{d}^{2} \right ) ^{2}}{e \left ( 5+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)^2,x)

[Out]

(e*x+d)^(1+m)*c^2*(e^2*x^2+2*d*e*x+d^2)^2/e/(5+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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.44608, size = 169, normalized size = 8.05 \begin{align*} \frac{{\left (c^{2} e^{5} x^{5} + 5 \, c^{2} d e^{4} x^{4} + 10 \, c^{2} d^{2} e^{3} x^{3} + 10 \, c^{2} d^{3} e^{2} x^{2} + 5 \, c^{2} d^{4} e x + c^{2} d^{5}\right )}{\left (e x + d\right )}^{m}}{e m + 5 \, 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)^2,x, algorithm="fricas")

[Out]

(c^2*e^5*x^5 + 5*c^2*d*e^4*x^4 + 10*c^2*d^2*e^3*x^3 + 10*c^2*d^3*e^2*x^2 + 5*c^2*d^4*e*x + c^2*d^5)*(e*x + d)^
m/(e*m + 5*e)

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Sympy [A]  time = 1.45693, size = 185, normalized size = 8.81 \begin{align*} \begin{cases} \frac{c^{2} x}{d} & \text{for}\: e = 0 \wedge m = -5 \\c^{2} d^{4} d^{m} x & \text{for}\: e = 0 \\\frac{c^{2} \log{\left (\frac{d}{e} + x \right )}}{e} & \text{for}\: m = -5 \\\frac{c^{2} d^{5} \left (d + e x\right )^{m}}{e m + 5 e} + \frac{5 c^{2} d^{4} e x \left (d + e x\right )^{m}}{e m + 5 e} + \frac{10 c^{2} d^{3} e^{2} x^{2} \left (d + e x\right )^{m}}{e m + 5 e} + \frac{10 c^{2} d^{2} e^{3} x^{3} \left (d + e x\right )^{m}}{e m + 5 e} + \frac{5 c^{2} d e^{4} x^{4} \left (d + e x\right )^{m}}{e m + 5 e} + \frac{c^{2} e^{5} x^{5} \left (d + e x\right )^{m}}{e m + 5 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)**2,x)

[Out]

Piecewise((c**2*x/d, Eq(e, 0) & Eq(m, -5)), (c**2*d**4*d**m*x, Eq(e, 0)), (c**2*log(d/e + x)/e, Eq(m, -5)), (c
**2*d**5*(d + e*x)**m/(e*m + 5*e) + 5*c**2*d**4*e*x*(d + e*x)**m/(e*m + 5*e) + 10*c**2*d**3*e**2*x**2*(d + e*x
)**m/(e*m + 5*e) + 10*c**2*d**2*e**3*x**3*(d + e*x)**m/(e*m + 5*e) + 5*c**2*d*e**4*x**4*(d + e*x)**m/(e*m + 5*
e) + c**2*e**5*x**5*(d + e*x)**m/(e*m + 5*e), True))

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Giac [B]  time = 1.33514, size = 169, normalized size = 8.05 \begin{align*} \frac{{\left (x e + d\right )}^{m} c^{2} x^{5} e^{5} + 5 \,{\left (x e + d\right )}^{m} c^{2} d x^{4} e^{4} + 10 \,{\left (x e + d\right )}^{m} c^{2} d^{2} x^{3} e^{3} + 10 \,{\left (x e + d\right )}^{m} c^{2} d^{3} x^{2} e^{2} + 5 \,{\left (x e + d\right )}^{m} c^{2} d^{4} x e +{\left (x e + d\right )}^{m} c^{2} d^{5}}{m e + 5 \, 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)^2,x, algorithm="giac")

[Out]

((x*e + d)^m*c^2*x^5*e^5 + 5*(x*e + d)^m*c^2*d*x^4*e^4 + 10*(x*e + d)^m*c^2*d^2*x^3*e^3 + 10*(x*e + d)^m*c^2*d
^3*x^2*e^2 + 5*(x*e + d)^m*c^2*d^4*x*e + (x*e + d)^m*c^2*d^5)/(m*e + 5*e)