3.266 \(\int \frac{x^2}{\sqrt{a+b x}+\sqrt{a+c x}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.181601, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ \frac{2 (a+b x)^{5/2}}{5 b^2 (b-c)}-\frac{2 a (a+b x)^{3/2}}{3 b^2 (b-c)}-\frac{2 (a+c x)^{5/2}}{5 c^2 (b-c)}+\frac{2 a (a+c x)^{3/2}}{3 c^2 (b-c)} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(Sqrt[a + b*x] + Sqrt[a + c*x]),x]

[Out]

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

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Rubi in Sympy [A]  time = 18.5712, size = 76, normalized size = 0.8 \[ \frac{2 a \left (a + c x\right )^{\frac{3}{2}}}{3 c^{2} \left (b - c\right )} - \frac{2 a \left (a + b x\right )^{\frac{3}{2}}}{3 b^{2} \left (b - c\right )} - \frac{2 \left (a + c x\right )^{\frac{5}{2}}}{5 c^{2} \left (b - c\right )} + \frac{2 \left (a + b x\right )^{\frac{5}{2}}}{5 b^{2} \left (b - c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.106714, size = 113, normalized size = 1.19 \[ \frac{a^2 \left (4 b^2 \sqrt{a+c x}-4 c^2 \sqrt{a+b x}\right )+6 b^2 c^2 x^2 \left (\sqrt{a+b x}-\sqrt{a+c x}\right )+2 a b c x \left (c \sqrt{a+b x}-b \sqrt{a+c x}\right )}{15 b^2 c^2 (b-c)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(Sqrt[a + b*x] + Sqrt[a + c*x]),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{b x + a} + \sqrt{c x + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(sqrt(b*x + a) + sqrt(c*x + a)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.270817, size = 124, normalized size = 1.31 \[ \frac{2 \,{\left ({\left (3 \, b^{2} c^{2} x^{2} + a b c^{2} x - 2 \, a^{2} c^{2}\right )} \sqrt{b x + a} -{\left (3 \, b^{2} c^{2} x^{2} + a b^{2} c x - 2 \, a^{2} b^{2}\right )} \sqrt{c x + a}\right )}}{15 \,{\left (b^{3} c^{2} - b^{2} c^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(sqrt(b*x + a) + sqrt(c*x + a)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{a + b x} + \sqrt{a + c x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out