### 3.1021 $$\int \frac{(d+e x)^5}{(c d^2+2 c d e x+c e^2 x^2)^3} \, dx$$

Optimal. Leaf size=13 $\frac{\log (d+e x)}{c^3 e}$

[Out]

Log[d + e*x]/(c^3*e)

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Rubi [A]  time = 0.0046667, 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} $\frac{\log (d+e x)}{c^3 e}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^5/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^3,x]

[Out]

Log[d + e*x]/(c^3*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{(d+e x)^5}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx &=\int \frac{1}{c^3 (d+e x)} \, dx\\ &=\frac{\int \frac{1}{d+e x} \, dx}{c^3}\\ &=\frac{\log (d+e x)}{c^3 e}\\ \end{align*}

Mathematica [A]  time = 0.0014406, size = 13, normalized size = 1. $\frac{\log (d+e x)}{c^3 e}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^5/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^3,x]

[Out]

Log[d + e*x]/(c^3*e)

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Maple [A]  time = 0.039, size = 14, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( ex+d \right ) }{{c}^{3}e}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^5/(c*e^2*x^2+2*c*d*e*x+c*d^2)^3,x)

[Out]

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

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Maxima [A]  time = 0.976387, size = 18, normalized size = 1.38 \begin{align*} \frac{\log \left (e x + d\right )}{c^{3} e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(c*e^2*x^2+2*c*d*e*x+c*d^2)^3,x, algorithm="maxima")

[Out]

log(e*x + d)/(c^3*e)

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Fricas [A]  time = 2.06933, size = 30, normalized size = 2.31 \begin{align*} \frac{\log \left (e x + d\right )}{c^{3} e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(c*e^2*x^2+2*c*d*e*x+c*d^2)^3,x, algorithm="fricas")

[Out]

log(e*x + d)/(c^3*e)

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Sympy [A]  time = 0.162682, size = 17, normalized size = 1.31 \begin{align*} \frac{\log{\left (c^{3} d + c^{3} e x \right )}}{c^{3} e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**5/(c*e**2*x**2+2*c*d*e*x+c*d**2)**3,x)

[Out]

log(c**3*d + c**3*e*x)/(c**3*e)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(c*e^2*x^2+2*c*d*e*x+c*d^2)^3,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError