3.2827 \(\int \sqrt{\frac{c}{(a+b x)^2}} \, dx\)

Optimal. Leaf size=28 \[ \frac{(a+b x) \sqrt{\frac{c}{(a+b x)^2}} \log (a+b x)}{b} \]

[Out]

(Sqrt[c/(a + b*x)^2]*(a + b*x)*Log[a + b*x])/b

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Rubi [A]  time = 0.0096413, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {247, 15, 29} \[ \frac{(a+b x) \sqrt{\frac{c}{(a+b x)^2}} \log (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c/(a + b*x)^2],x]

[Out]

(Sqrt[c/(a + b*x)^2]*(a + b*x)*Log[a + b*x])/b

Rule 247

Int[((a_.) + (b_.)*(v_)^(n_))^(p_), x_Symbol] :> Dist[1/Coefficient[v, x, 1], Subst[Int[(a + b*x^n)^p, x], x,
v], x] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && NeQ[v, x]

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps

\begin{align*} \int \sqrt{\frac{c}{(a+b x)^2}} \, dx &=\frac{\operatorname{Subst}\left (\int \sqrt{\frac{c}{x^2}} \, dx,x,a+b x\right )}{b}\\ &=\frac{\left (\sqrt{\frac{c}{(a+b x)^2}} (a+b x)\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,a+b x\right )}{b}\\ &=\frac{\sqrt{\frac{c}{(a+b x)^2}} (a+b x) \log (a+b x)}{b}\\ \end{align*}

Mathematica [A]  time = 0.0070481, size = 28, normalized size = 1. \[ \frac{(a+b x) \sqrt{\frac{c}{(a+b x)^2}} \log (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c/(a + b*x)^2],x]

[Out]

(Sqrt[c/(a + b*x)^2]*(a + b*x)*Log[a + b*x])/b

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Maple [A]  time = 0.004, size = 27, normalized size = 1. \begin{align*}{\frac{ \left ( bx+a \right ) \ln \left ( bx+a \right ) }{b}\sqrt{{\frac{c}{ \left ( bx+a \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x+a)*ln(b*x+a)*(c/(b*x+a)^2)^(1/2)/b

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Maxima [A]  time = 1.22164, size = 18, normalized size = 0.64 \begin{align*} \frac{\sqrt{c} \log \left (b x + a\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(c)*log(b*x + a)/b

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Fricas [A]  time = 1.31028, size = 81, normalized size = 2.89 \begin{align*} \frac{{\left (b x + a\right )} \sqrt{\frac{c}{b^{2} x^{2} + 2 \, a b x + a^{2}}} \log \left (b x + a\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x + a)*sqrt(c/(b^2*x^2 + 2*a*b*x + a^2))*log(b*x + a)/b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(c/(a + b*x)**2), x)

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Giac [A]  time = 1.10624, size = 27, normalized size = 0.96 \begin{align*} \frac{\sqrt{c} \log \left ({\left | b x + a \right |}\right ) \mathrm{sgn}\left (b x + a\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(c)*log(abs(b*x + a))*sgn(b*x + a)/b