3.2021 \(\int \frac{a+b x}{(d+e x)^2 \sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=38 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{(d+e x) (b d-a e)} \]

[Out]

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

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Rubi [A]  time = 0.0228127, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.03, Rules used = {767} \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{(d+e x) (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 767

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_) + (g_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Sim
p[(f*g*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(b*(p + 1)*(e*f - d*g)), x] /; FreeQ[{a, b, c, d, e, f, g,
m, p}, x] && EqQ[b^2 - 4*a*c, 0] && EqQ[m + 2*p + 3, 0] && EqQ[2*c*f - b*g, 0]

Rubi steps

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

Mathematica [A]  time = 0.0099172, size = 28, normalized size = 0.74 \[ -\frac{a+b x}{e \sqrt{(a+b x)^2} (d+e x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 27, normalized size = 0.7 \begin{align*} -{\frac{bx+a}{e \left ( ex+d \right ) }{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.53316, size = 24, normalized size = 0.63 \begin{align*} -\frac{1}{e^{2} x + d e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(e^2*x + d*e)

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Sympy [A]  time = 0.318102, size = 10, normalized size = 0.26 \begin{align*} - \frac{1}{d e + e^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(d*e + e**2*x)

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Giac [A]  time = 1.16842, size = 24, normalized size = 0.63 \begin{align*} -\frac{e^{\left (-1\right )} \mathrm{sgn}\left (b x + a\right )}{x e + d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e^(-1)*sgn(b*x + a)/(x*e + d)