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

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

[Out]

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

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Rubi [A]  time = 0.0987983, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{2 (a+b x)^{3/2}}{3 b (b-c)}-\frac{2 (a+c x)^{3/2}}{3 c (b-c)} \]

Antiderivative was successfully verified.

[In]  Int[x/(Sqrt[a + b*x] + Sqrt[a + c*x]),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.23393, size = 32, normalized size = 0.68 \[ - \frac{2 \left (a + c x\right )^{\frac{3}{2}}}{3 c \left (b - c\right )} + \frac{2 \left (a + b x\right )^{\frac{3}{2}}}{3 b \left (b - c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0797935, size = 71, normalized size = 1.51 \[ \frac{2 b c x \sqrt{a+b x}-2 a b \sqrt{a+c x}-2 b c x \sqrt{a+c x}+2 a c \sqrt{a+b x}}{3 b^2 c-3 b c^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(Sqrt[a + b*x] + Sqrt[a + c*x]),x]

[Out]

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

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Maple [A]  time = 0.004, size = 40, normalized size = 0.9 \[{\frac{2}{3\,b \left ( b-c \right ) } \left ( bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{2}{ \left ( 3\,b-3\,c \right ) c} \left ( cx+a \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.275037, size = 68, normalized size = 1.45 \[ \frac{2 \,{\left ({\left (b c x + a c\right )} \sqrt{b x + a} -{\left (b c x + a b\right )} \sqrt{c x + a}\right )}}{3 \,{\left (b^{2} c - b c^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{a + b x} + \sqrt{a + c x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x/(sqrt(a + b*x) + sqrt(a + c*x)), x)

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out