### 3.1065 $$\int \frac{d+e x}{\sqrt{c d^2+2 c d e x+c e^2 x^2}} \, dx$$

Optimal. Leaf size=31 $\frac{\sqrt{c d^2+2 c d e x+c e^2 x^2}}{c e}$

[Out]

Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]/(c*e)

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Rubi [A]  time = 0.0087882, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 30, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.033, Rules used = {629} $\frac{\sqrt{c d^2+2 c d e x+c e^2 x^2}}{c e}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]/(c*e)

Rule 629

Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d*(a + b*x + c*x^2)^(p +
1))/(b*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \frac{d+e x}{\sqrt{c d^2+2 c d e x+c e^2 x^2}} \, dx &=\frac{\sqrt{c d^2+2 c d e x+c e^2 x^2}}{c e}\\ \end{align*}

Mathematica [A]  time = 0.0025526, size = 20, normalized size = 0.65 $\frac{x (d+e x)}{\sqrt{c (d+e x)^2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(x*(d + e*x))/Sqrt[c*(d + e*x)^2]

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Maple [A]  time = 0.04, size = 30, normalized size = 1. \begin{align*}{ \left ( ex+d \right ) x{\frac{1}{\sqrt{c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2}}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.20373, size = 39, normalized size = 1.26 \begin{align*} \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{c e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.27197, size = 72, normalized size = 2.32 \begin{align*} \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} x}{c e x + c d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.827076, size = 39, normalized size = 1.26 \begin{align*} \begin{cases} \frac{\sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{c e} & \text{for}\: e \neq 0 \\\frac{d x}{\sqrt{c d^{2}}} & \text{otherwise} \end{cases} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(1/2),x)

[Out]

Piecewise((sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/(c*e), Ne(e, 0)), (d*x/sqrt(c*d**2), True))

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Giac [A]  time = 1.34488, size = 38, normalized size = 1.23 \begin{align*} \frac{\sqrt{c x^{2} e^{2} + 2 \, c d x e + c d^{2}} e^{\left (-1\right )}}{c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2),x, algorithm="giac")

[Out]

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