### 3.1599 $$\int \frac{d+e x}{(a^2+2 a b x+b^2 x^2)^{3/2}} \, dx$$

Optimal. Leaf size=69 $-\frac{b d-a e}{2 b^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{e}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}$

[Out]

-(e/(b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])) - (b*d - a*e)/(2*b^2*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.0218046, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.077, Rules used = {640, 607} $-\frac{b d-a e}{2 b^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{e}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(e/(b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])) - (b*d - a*e)/(2*b^2*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

Rule 640

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

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

Mathematica [A]  time = 0.014965, size = 39, normalized size = 0.57 $\frac{-a e-b (d+2 e x)}{2 b^2 (a+b x) \sqrt{(a+b x)^2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-(a*e) - b*(d + 2*e*x))/(2*b^2*(a + b*x)*Sqrt[(a + b*x)^2])

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Maple [A]  time = 0.154, size = 32, normalized size = 0.5 \begin{align*} -{\frac{ \left ( bx+a \right ) \left ( 2\,bxe+ae+bd \right ) }{2\,{b}^{2}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)/(b^2*x^2+2*a*b*x+a^2)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.01296, size = 85, normalized size = 1.23 \begin{align*} -\frac{e}{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} - \frac{d}{2 \,{\left (b^{2}\right )}^{\frac{3}{2}}{\left (x + \frac{a}{b}\right )}^{2}} + \frac{a e}{2 \,{\left (b^{2}\right )}^{\frac{3}{2}} b{\left (x + \frac{a}{b}\right )}^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^2) - 1/2*d/((b^2)^(3/2)*(x + a/b)^2) + 1/2*a*e/((b^2)^(3/2)*b*(x + a/b)^2)

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Fricas [A]  time = 1.53212, size = 81, normalized size = 1.17 \begin{align*} -\frac{2 \, b e x + b d + a e}{2 \,{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*b*e*x + b*d + a*e)/(b^4*x^2 + 2*a*b^3*x + a^2*b^2)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Integral((d + e*x)/((a + b*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((e*x+d)/(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="giac")

[Out]

sage0*x