### 3.997 $$\int \frac{(c d^2+2 c d e x+c e^2 x^2)^2}{(d+e x)^8} \, dx$$

Optimal. Leaf size=17 $-\frac{c^2}{3 e (d+e x)^3}$

[Out]

-c^2/(3*e*(d + e*x)^3)

________________________________________________________________________________________

Rubi [A]  time = 0.0047521, antiderivative size = 17, 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} $-\frac{c^2}{3 e (d+e x)^3}$

Antiderivative was successfully veriﬁed.

[In]

Int[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2/(d + e*x)^8,x]

[Out]

-c^2/(3*e*(d + e*x)^3)

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 \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^2}{(d+e x)^8} \, dx &=\int \frac{c^2}{(d+e x)^4} \, dx\\ &=c^2 \int \frac{1}{(d+e x)^4} \, dx\\ &=-\frac{c^2}{3 e (d+e x)^3}\\ \end{align*}

Mathematica [A]  time = 0.0019206, size = 17, normalized size = 1. $-\frac{c^2}{3 e (d+e x)^3}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2/(d + e*x)^8,x]

[Out]

-c^2/(3*e*(d + e*x)^3)

________________________________________________________________________________________

Maple [A]  time = 0.039, size = 16, normalized size = 0.9 \begin{align*} -{\frac{{c}^{2}}{3\,e \left ( ex+d \right ) ^{3}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c*e^2*x^2+2*c*d*e*x+c*d^2)^2/(e*x+d)^8,x)

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.19668, size = 51, normalized size = 3. \begin{align*} -\frac{c^{2}}{3 \,{\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} \end{align*}

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)^2/(e*x+d)^8,x, algorithm="maxima")

[Out]

-1/3*c^2/(e^4*x^3 + 3*d*e^3*x^2 + 3*d^2*e^2*x + d^3*e)

________________________________________________________________________________________

Fricas [B]  time = 1.93112, size = 76, normalized size = 4.47 \begin{align*} -\frac{c^{2}}{3 \,{\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} \end{align*}

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)^2/(e*x+d)^8,x, algorithm="fricas")

[Out]

-1/3*c^2/(e^4*x^3 + 3*d*e^3*x^2 + 3*d^2*e^2*x + d^3*e)

________________________________________________________________________________________

Sympy [B]  time = 0.66286, size = 39, normalized size = 2.29 \begin{align*} - \frac{c^{2}}{3 d^{3} e + 9 d^{2} e^{2} x + 9 d e^{3} x^{2} + 3 e^{4} x^{3}} \end{align*}

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)**2/(e*x+d)**8,x)

[Out]

-c**2/(3*d**3*e + 9*d**2*e**2*x + 9*d*e**3*x**2 + 3*e**4*x**3)

________________________________________________________________________________________

Giac [B]  time = 1.26211, size = 89, normalized size = 5.24 \begin{align*} -\frac{{\left (c^{2} x^{4} e^{8} + 4 \, c^{2} d x^{3} e^{7} + 6 \, c^{2} d^{2} x^{2} e^{6} + 4 \, c^{2} d^{3} x e^{5} + c^{2} d^{4} e^{4}\right )} e^{\left (-5\right )}}{3 \,{\left (x e + d\right )}^{7}} \end{align*}

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)^2/(e*x+d)^8,x, algorithm="giac")

[Out]

-1/3*(c^2*x^4*e^8 + 4*c^2*d*x^3*e^7 + 6*c^2*d^2*x^2*e^6 + 4*c^2*d^3*x*e^5 + c^2*d^4*e^4)*e^(-5)/(x*e + d)^7