\(\int \sqrt {\frac {c}{(a+b x)^2}} \, dx\) [2827]

Optimal result

Integrand size = 13, antiderivative size = 28 \[ \int \sqrt {\frac {c}{(a+b x)^2}} \, dx=\frac {\sqrt {\frac {c}{(a+b x)^2}} (a+b x) \log (a+b x)}{b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {253, 15, 29} \[ \int \sqrt {\frac {c}{(a+b x)^2}} \, dx=\frac {(a+b x) \sqrt {\frac {c}{(a+b x)^2}} \log (a+b x)}{b} \]

[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 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]

Rule 253

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {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 ) \text {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] (verified)

Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \sqrt {\frac {c}{(a+b x)^2}} \, dx=\frac {\sqrt {\frac {c}{(a+b x)^2}} (a+b x) \log (a+b x)}{b} \]

[In]

Integrate[Sqrt[c/(a + b*x)^2],x]

[Out]

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

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96

method	result	size
default	\(\frac {\left (b x +a \right ) \ln \left (b x +a \right ) \sqrt {\frac {c}{\left (b x +a \right )^{2}}}}{b}\)	\(27\)
risch	\(\frac {\left (b x +a \right ) \ln \left (b x +a \right ) \sqrt {\frac {c}{\left (b x +a \right )^{2}}}}{b}\)	\(27\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32 \[ \int \sqrt {\frac {c}{(a+b x)^2}} \, dx=\frac {{\left (b x + a\right )} \sqrt {\frac {c}{b^{2} x^{2} + 2 \, a b x + a^{2}}} \log \left (b x + a\right )}{b} \]

[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

Sympy [A] (verification not implemented)

Time = 0.51 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \sqrt {\frac {c}{(a+b x)^2}} \, dx=\begin {cases} \sqrt {\frac {c}{\left (a + b x\right )^{2}}} \left (\frac {a}{b} + x\right ) \log {\left (\frac {a}{b} + x \right )} & \text {for}\: b \neq 0 \\x \sqrt {\frac {c}{a^{2}}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((sqrt(c/(a + b*x)**2)*(a/b + x)*log(a/b + x), Ne(b, 0)), (x*sqrt(c/a**2), True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.46 \[ \int \sqrt {\frac {c}{(a+b x)^2}} \, dx=\frac {\sqrt {c} \log \left (b x + a\right )}{b} \]

[In]

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

[Out]

sqrt(c)*log(b*x + a)/b

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int \sqrt {\frac {c}{(a+b x)^2}} \, dx=\frac {\sqrt {c} \log \left ({\left | b x + a \right |}\right ) \mathrm {sgn}\left (b x + a\right )}{b} \]

[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

Mupad [F(-1)]

Timed out. \[ \int \sqrt {\frac {c}{(a+b x)^2}} \, dx=\int \sqrt {\frac {c}{{\left (a+b\,x\right )}^2}} \,d x \]

[In]

int((c/(a + b*x)^2)^(1/2),x)

[Out]

int((c/(a + b*x)^2)^(1/2), x)