3.610 \(\int \frac{-1+\sqrt{a+b x}}{1+\sqrt{a+b x}} \, dx\)

Optimal. Leaf size=33 \[ -\frac{4 \sqrt{a+b x}}{b}+\frac{4 \log \left (\sqrt{a+b x}+1\right )}{b}+x \]

[Out]

x - (4*Sqrt[a + b*x])/b + (4*Log[1 + Sqrt[a + b*x]])/b

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Rubi [A]  time = 0.020707, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {431, 376, 77} \[ -\frac{4 \sqrt{a+b x}}{b}+\frac{4 \log \left (\sqrt{a+b x}+1\right )}{b}+x \]

Antiderivative was successfully verified.

[In]

Int[(-1 + Sqrt[a + b*x])/(1 + Sqrt[a + b*x]),x]

[Out]

x - (4*Sqrt[a + b*x])/b + (4*Log[1 + Sqrt[a + b*x]])/b

Rule 431

Int[((a_.) + (b_.)*(u_)^(n_))^(p_.)*((c_.) + (d_.)*(u_)^(n_))^(q_.), x_Symbol] :> Dist[1/Coefficient[u, x, 1],
Subst[Int[(a + b*x^n)^p*(c + d*x^n)^q, x], x, u], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && LinearQ[u, x] && N
eQ[u, x]

Rule 376

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{g = Denominator[n]}, Dis
t[g, Subst[Int[x^(g - 1)*(a + b*x^(g*n))^p*(c + d*x^(g*n))^q, x], x, x^(1/g)], x]] /; FreeQ[{a, b, c, d, p, q}
, x] && NeQ[b*c - a*d, 0] && FractionQ[n]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{-1+\sqrt{a+b x}}{1+\sqrt{a+b x}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{-1+\sqrt{x}}{1+\sqrt{x}} \, dx,x,a+b x\right )}{b}\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{(-1+x) x}{1+x} \, dx,x,\sqrt{a+b x}\right )}{b}\\ &=\frac{2 \operatorname{Subst}\left (\int \left (-2+x+\frac{2}{1+x}\right ) \, dx,x,\sqrt{a+b x}\right )}{b}\\ &=x-\frac{4 \sqrt{a+b x}}{b}+\frac{4 \log \left (1+\sqrt{a+b x}\right )}{b}\\ \end{align*}

Mathematica [A]  time = 0.0149851, size = 33, normalized size = 1. \[ -\frac{4 \sqrt{a+b x}}{b}+\frac{4 \log \left (\sqrt{a+b x}+1\right )}{b}+x \]

Antiderivative was successfully verified.

[In]

Integrate[(-1 + Sqrt[a + b*x])/(1 + Sqrt[a + b*x]),x]

[Out]

x - (4*Sqrt[a + b*x])/b + (4*Log[1 + Sqrt[a + b*x]])/b

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Maple [A]  time = 0.001, size = 35, normalized size = 1.1 \begin{align*} -4\,{\frac{\sqrt{bx+a}}{b}}+x+{\frac{a}{b}}+4\,{\frac{\ln \left ( 1+\sqrt{bx+a} \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*(b*x+a)^(1/2)/b+x+a/b+4*ln(1+(b*x+a)^(1/2))/b

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Maxima [A]  time = 1.10034, size = 41, normalized size = 1.24 \begin{align*} \frac{b x + a - 4 \, \sqrt{b x + a} + 4 \, \log \left (\sqrt{b x + a} + 1\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x + a - 4*sqrt(b*x + a) + 4*log(sqrt(b*x + a) + 1))/b

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Fricas [A]  time = 1.68102, size = 73, normalized size = 2.21 \begin{align*} \frac{b x - 4 \, \sqrt{b x + a} + 4 \, \log \left (\sqrt{b x + a} + 1\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x - 4*sqrt(b*x + a) + 4*log(sqrt(b*x + a) + 1))/b

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Sympy [A]  time = 0.416818, size = 42, normalized size = 1.27 \begin{align*} \begin{cases} x - \frac{4 \sqrt{a + b x}}{b} + \frac{4 \log{\left (\sqrt{a + b x} + 1 \right )}}{b} & \text{for}\: b \neq 0 \\\frac{x \left (\sqrt{a} - 1\right )}{\sqrt{a} + 1} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x - 4*sqrt(a + b*x)/b + 4*log(sqrt(a + b*x) + 1)/b, Ne(b, 0)), (x*(sqrt(a) - 1)/(sqrt(a) + 1), True
))

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Giac [A]  time = 1.14317, size = 51, normalized size = 1.55 \begin{align*} \frac{4 \, \log \left (\sqrt{b x + a} + 1\right )}{b} + \frac{{\left (b x + a\right )} b - 4 \, \sqrt{b x + a} b}{b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*log(sqrt(b*x + a) + 1)/b + ((b*x + a)*b - 4*sqrt(b*x + a)*b)/b^2