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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]