### 3.1074 $$\int \frac{d+e x}{(c d^2+2 c d e x+c e^2 x^2)^{3/2}} \, dx$$

Optimal. Leaf size=32 $-\frac{1}{c e \sqrt{c d^2+2 c d e x+c e^2 x^2}}$

[Out]

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

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Rubi [A]  time = 0.0092454, antiderivative size = 32, 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} $-\frac{1}{c e \sqrt{c d^2+2 c d e x+c e^2 x^2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0064515, size = 21, normalized size = 0.66 $-\frac{1}{c e \sqrt{c (d+e x)^2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(1/(c*e*Sqrt[c*(d + e*x)^2]))

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Maple [A]  time = 0.04, size = 35, normalized size = 1.1 \begin{align*} -{\frac{ \left ( ex+d \right ) ^{2}}{e} \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{-{\frac{3}{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)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.19602, size = 41, normalized size = 1.28 \begin{align*} -\frac{1}{\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)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.44006, size = 108, normalized size = 3.38 \begin{align*} -\frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{c^{2} e^{3} x^{2} + 2 \, c^{2} d e^{2} x + c^{2} d^{2} 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)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.19376, size = 42, normalized size = 1.31 \begin{align*} \begin{cases} - \frac{1}{c e \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}} & \text{for}\: e \neq 0 \\\frac{d x}{\left (c d^{2}\right )^{\frac{3}{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)**(3/2),x)

[Out]

Piecewise((-1/(c*e*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)), Ne(e, 0)), (d*x/(c*d**2)**(3/2), True))

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Giac [A]  time = 1.36931, size = 55, normalized size = 1.72 \begin{align*} \frac{2 \, C_{0} d e^{\left (-1\right )} + 2 \, C_{0} x - \frac{e^{\left (-1\right )}}{c}}{\sqrt{c x^{2} e^{2} + 2 \, c d x e + c d^{2}}} \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)^(3/2),x, algorithm="giac")

[Out]

(2*C_0*d*e^(-1) + 2*C_0*x - e^(-1)/c)/sqrt(c*x^2*e^2 + 2*c*d*x*e + c*d^2)