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

Optimal. Leaf size=95 \[ \frac{2 (a+b x)^{5/2}}{5 b^2 (a-c)}-\frac{2 a (a+b x)^{3/2}}{3 b^2 (a-c)}-\frac{2 (b x+c)^{5/2}}{5 b^2 (a-c)}+\frac{2 c (b x+c)^{3/2}}{3 b^2 (a-c)} \]

[Out]

(-2*a*(a + b*x)^(3/2))/(3*b^2*(a - c)) + (2*(a + b*x)^(5/2))/(5*b^2*(a - c)) + (
2*c*(c + b*x)^(3/2))/(3*b^2*(a - c)) - (2*(c + b*x)^(5/2))/(5*b^2*(a - c))

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Rubi [A]  time = 0.150951, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{2 (a+b x)^{5/2}}{5 b^2 (a-c)}-\frac{2 a (a+b x)^{3/2}}{3 b^2 (a-c)}-\frac{2 (b x+c)^{5/2}}{5 b^2 (a-c)}+\frac{2 c (b x+c)^{3/2}}{3 b^2 (a-c)} \]

Antiderivative was successfully verified.

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

[Out]

(-2*a*(a + b*x)^(3/2))/(3*b^2*(a - c)) + (2*(a + b*x)^(5/2))/(5*b^2*(a - c)) + (
2*c*(c + b*x)^(3/2))/(3*b^2*(a - c)) - (2*(c + b*x)^(5/2))/(5*b^2*(a - c))

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Rubi in Sympy [A]  time = 17.4956, size = 76, normalized size = 0.8 \[ - \frac{2 a \left (a + b x\right )^{\frac{3}{2}}}{3 b^{2} \left (a - c\right )} + \frac{2 c \left (b x + c\right )^{\frac{3}{2}}}{3 b^{2} \left (a - c\right )} + \frac{2 \left (a + b x\right )^{\frac{5}{2}}}{5 b^{2} \left (a - c\right )} - \frac{2 \left (b x + c\right )^{\frac{5}{2}}}{5 b^{2} \left (a - c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a*(a + b*x)**(3/2)/(3*b**2*(a - c)) + 2*c*(b*x + c)**(3/2)/(3*b**2*(a - c)) +
2*(a + b*x)**(5/2)/(5*b**2*(a - c)) - 2*(b*x + c)**(5/2)/(5*b**2*(a - c))

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Mathematica [A]  time = 0.0992523, size = 100, normalized size = 1.05 \[ \frac{-4 a^2 \sqrt{a+b x}+6 b^2 x^2 \left (\sqrt{a+b x}-\sqrt{b x+c}\right )+2 a b x \sqrt{a+b x}+4 c^2 \sqrt{b x+c}-2 b c x \sqrt{b x+c}}{15 b^2 (a-c)} \]

Antiderivative was successfully verified.

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

[Out]

(-4*a^2*Sqrt[a + b*x] + 2*a*b*x*Sqrt[a + b*x] + 4*c^2*Sqrt[c + b*x] - 2*b*c*x*Sq
rt[c + b*x] + 6*b^2*x^2*(Sqrt[a + b*x] - Sqrt[c + b*x]))/(15*b^2*(a - c))

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Maple [A]  time = 0.004, size = 66, normalized size = 0.7 \[ 2\,{\frac{1/5\, \left ( bx+a \right ) ^{5/2}-1/3\, \left ( bx+a \right ) ^{3/2}a}{ \left ( a-c \right ){b}^{2}}}-2\,{\frac{1/5\, \left ( bx+c \right ) ^{5/2}-1/3\, \left ( bx+c \right ) ^{3/2}c}{ \left ( a-c \right ){b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/(a-c)/b^2*(1/5*(b*x+a)^(5/2)-1/3*(b*x+a)^(3/2)*a)-2/(a-c)/b^2*(1/5*(b*x+c)^(5/
2)-1/3*(b*x+c)^(3/2)*c)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.273224, size = 95, normalized size = 1. \[ \frac{2 \,{\left ({\left (3 \, b^{2} x^{2} + a b x - 2 \, a^{2}\right )} \sqrt{b x + a} -{\left (3 \, b^{2} x^{2} + b c x - 2 \, c^{2}\right )} \sqrt{b x + c}\right )}}{15 \,{\left (a b^{2} - b^{2} c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*((3*b^2*x^2 + a*b*x - 2*a^2)*sqrt(b*x + a) - (3*b^2*x^2 + b*c*x - 2*c^2)*sq
rt(b*x + c))/(a*b^2 - b^2*c)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \mathit{undef} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

undef