3.403 \(\int \frac{1}{\sqrt{a+b x}+\sqrt{c+b x}} \, dx\)

Optimal. Leaf size=47 \[ \frac{2 (a+b x)^{3/2}}{3 b (a-c)}-\frac{2 (b x+c)^{3/2}}{3 b (a-c)} \]

[Out]

(2*(a + b*x)^(3/2))/(3*b*(a - c)) - (2*(c + b*x)^(3/2))/(3*b*(a - c))

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Rubi [A]  time = 0.0466709, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {6689} \[ \frac{2 (a+b x)^{3/2}}{3 b (a-c)}-\frac{2 (b x+c)^{3/2}}{3 b (a-c)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a + b*x)^(3/2))/(3*b*(a - c)) - (2*(c + b*x)^(3/2))/(3*b*(a - c))

Rule 6689

Int[(u_.)*((e_.)*Sqrt[(a_.) + (b_.)*(x_)^(n_.)] + (f_.)*Sqrt[(c_.) + (d_.)*(x_)^(n_.)])^(m_), x_Symbol] :> Dis
t[(a*e^2 - c*f^2)^m, Int[ExpandIntegrand[u/(e*Sqrt[a + b*x^n] - f*Sqrt[c + d*x^n])^m, x], x], x] /; FreeQ[{a,
b, c, d, e, f, n}, x] && ILtQ[m, 0] && EqQ[b*e^2 - d*f^2, 0]

Rubi steps

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

Mathematica [A]  time = 0.0480739, size = 35, normalized size = 0.74 \[ \frac{2 \left ((a+b x)^{3/2}-(b x+c)^{3/2}\right )}{3 b (a-c)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((a + b*x)^(3/2) - (c + b*x)^(3/2)))/(3*b*(a - c))

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Maple [A]  time = 0.003, size = 40, normalized size = 0.9 \begin{align*}{\frac{2}{3\,b \left ( a-c \right ) } \left ( bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{2}{3\,b \left ( a-c \right ) } \left ( bx+c \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(b*x+a)^(3/2)/b/(a-c)-2/3*(b*x+c)^(3/2)/b/(a-c)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b x + a} + \sqrt{b x + c}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.966095, size = 72, normalized size = 1.53 \begin{align*} \frac{2 \,{\left ({\left (b x + a\right )}^{\frac{3}{2}} -{\left (b x + c\right )}^{\frac{3}{2}}\right )}}{3 \,{\left (a b - b c\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*((b*x + a)^(3/2) - (b*x + c)^(3/2))/(a*b - b*c)

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Sympy [A]  time = 0.647716, size = 136, normalized size = 2.89 \begin{align*} \begin{cases} \frac{2 a}{3 b \sqrt{a + b x} + 3 b \sqrt{b x + c}} + \frac{4 b x}{3 b \sqrt{a + b x} + 3 b \sqrt{b x + c}} + \frac{2 c}{3 b \sqrt{a + b x} + 3 b \sqrt{b x + c}} + \frac{2 \sqrt{a + b x} \sqrt{b x + c}}{3 b \sqrt{a + b x} + 3 b \sqrt{b x + c}} & \text{for}\: b \neq 0 \\\frac{x}{\sqrt{a} + \sqrt{c}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((2*a/(3*b*sqrt(a + b*x) + 3*b*sqrt(b*x + c)) + 4*b*x/(3*b*sqrt(a + b*x) + 3*b*sqrt(b*x + c)) + 2*c/(
3*b*sqrt(a + b*x) + 3*b*sqrt(b*x + c)) + 2*sqrt(a + b*x)*sqrt(b*x + c)/(3*b*sqrt(a + b*x) + 3*b*sqrt(b*x + c))
, Ne(b, 0)), (x/(sqrt(a) + sqrt(c)), True))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError