### 3.994 $$\int \frac{(c d^2+2 c d e x+c e^2 x^2)^2}{(d+e x)^5} \, dx$$

Optimal. Leaf size=13 $\frac{c^2 \log (d+e x)}{e}$

[Out]

(c^2*Log[d + e*x])/e

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Rubi [A]  time = 0.0044983, antiderivative size = 13, 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, 31} $\frac{c^2 \log (d+e x)}{e}$

Antiderivative was successfully veriﬁed.

[In]

Int[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2/(d + e*x)^5,x]

[Out]

(c^2*Log[d + e*x])/e

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 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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)^5} \, dx &=\int \frac{c^2}{d+e x} \, dx\\ &=c^2 \int \frac{1}{d+e x} \, dx\\ &=\frac{c^2 \log (d+e x)}{e}\\ \end{align*}

Mathematica [A]  time = 0.0013471, size = 13, normalized size = 1. $\frac{c^2 \log (d+e x)}{e}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2/(d + e*x)^5,x]

[Out]

(c^2*Log[d + e*x])/e

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Maple [A]  time = 0.039, size = 14, normalized size = 1.1 \begin{align*}{\frac{{c}^{2}\ln \left ( ex+d \right ) }{e}} \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)^5,x)

[Out]

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

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Maxima [A]  time = 1.17349, size = 18, normalized size = 1.38 \begin{align*} \frac{c^{2} \log \left (e x + d\right )}{e} \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)^5,x, algorithm="maxima")

[Out]

c^2*log(e*x + d)/e

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Fricas [A]  time = 1.98157, size = 27, normalized size = 2.08 \begin{align*} \frac{c^{2} \log \left (e x + d\right )}{e} \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)^5,x, algorithm="fricas")

[Out]

c^2*log(e*x + d)/e

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Sympy [A]  time = 0.138259, size = 10, normalized size = 0.77 \begin{align*} \frac{c^{2} \log{\left (d + e x \right )}}{e} \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)**5,x)

[Out]

c**2*log(d + e*x)/e

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Giac [A]  time = 1.19822, size = 35, normalized size = 2.69 \begin{align*} -c^{2} e^{\left (-1\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\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)^5,x, algorithm="giac")

[Out]

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