### 3.1075 $$\int \frac{1}{(c d^2+2 c d e x+c e^2 x^2)^{3/2}} \, dx$$

Optimal. Leaf size=41 $-\frac{1}{2 c e (d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}$

[Out]

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

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Rubi [A]  time = 0.0051423, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.042, Rules used = {607} $-\frac{1}{2 c e (d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 607

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(2*(a + b*x + c*x^2)^(p + 1))/((2*p + 1)*(b + 2
*c*x)), x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && LtQ[p, -1]

Rubi steps

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

Mathematica [A]  time = 0.0026425, size = 25, normalized size = 0.61 $-\frac{d+e x}{2 e \left (c (d+e x)^2\right )^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.042, size = 33, normalized size = 0.8 \begin{align*} -{\frac{ex+d}{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(1/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.25243, size = 24, normalized size = 0.59 \begin{align*} -\frac{1}{2 \, \left (c e^{2}\right )^{\frac{3}{2}}{\left (x + \frac{d}{e}\right )}^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2),x, algorithm="maxima")

[Out]

-1/2/((c*e^2)^(3/2)*(x + d/e)^2)

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Fricas [A]  time = 2.30546, size = 140, normalized size = 3.41 \begin{align*} -\frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{2 \,{\left (c^{2} e^{4} x^{3} + 3 \, c^{2} d e^{3} x^{2} + 3 \, c^{2} d^{2} e^{2} x + c^{2} d^{3} e\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(3/2),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2),x, algorithm="giac")

[Out]

sage0*x