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

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 = 0.0816449, 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 \[ \frac{2 (a+b x)^{3/2}}{3 b (a-c)}-\frac{2 (b x+c)^{3/2}}{3 b (a-c)} \]

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))

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Rubi in Sympy [A]  time = 5.57357, size = 32, normalized size = 0.68 \[ \frac{2 \left (a + b x\right )^{\frac{3}{2}}}{3 b \left (a - c\right )} - \frac{2 \left (b x + c\right )^{\frac{3}{2}}}{3 b \left (a - c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/((b*x+a)**(1/2)+(b*x+c)**(1/2)),x)

[Out]

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

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Mathematica [A]  time = 0.0766401, size = 63, normalized size = 1.34 \[ \frac{2 a \sqrt{a+b x}+2 b x \sqrt{a+b x}-2 c \sqrt{b x+c}-2 b x \sqrt{b x+c}}{3 a b-3 b c} \]

Antiderivative was successfully verified.

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

[Out]

(2*a*Sqrt[a + b*x] + 2*b*x*Sqrt[a + b*x] - 2*c*Sqrt[c + b*x] - 2*b*x*Sqrt[c + b*
x])/(3*a*b - 3*b*c)

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

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. \[ \int \frac{1}{\sqrt{b x + a} + \sqrt{b x + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + a) + sqrt(b*x + c)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.274852, size = 39, normalized size = 0.83 \[ \frac{2 \,{\left ({\left (b x + a\right )}^{\frac{3}{2}} -{\left (b x + c\right )}^{\frac{3}{2}}\right )}}{3 \,{\left (a b - b c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + a) + sqrt(b*x + c)),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 = 2.4676, size = 136, normalized size = 2.89 \[ \begin{cases} \frac{2 a}{3 b \sqrt{a + b x} + 3 b \sqrt{b x + c}} + \frac{4 b x}{3 b \sqrt{a + b x} + 3 b \sqrt{b x + c}} + \frac{2 c}{3 b \sqrt{a + b x} + 3 b \sqrt{b x + c}} + \frac{2 \sqrt{a + b x} \sqrt{b x + c}}{3 b \sqrt{a + b x} + 3 b \sqrt{b x + c}} & \text{for}\: b \neq 0 \\\frac{x}{\sqrt{a} + \sqrt{c}} & \text{otherwise} \end{cases} \]

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/(sqr
t(a) + sqrt(c)), True))

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + a) + sqrt(b*x + c)),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError