Optimal. Leaf size=49 \[ -\frac{2 a^2}{b^3 \sqrt{a+b x}}-\frac{4 a \sqrt{a+b x}}{b^3}+\frac{2 (a+b x)^{3/2}}{3 b^3} \]
[Out]
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Rubi [A] time = 0.0382472, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{2 a^2}{b^3 \sqrt{a+b x}}-\frac{4 a \sqrt{a+b x}}{b^3}+\frac{2 (a+b x)^{3/2}}{3 b^3} \]
Antiderivative was successfully verified.
[In] Int[x^2/(a + b*x)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 7.88459, size = 46, normalized size = 0.94 \[ - \frac{2 a^{2}}{b^{3} \sqrt{a + b x}} - \frac{4 a \sqrt{a + b x}}{b^{3}} + \frac{2 \left (a + b x\right )^{\frac{3}{2}}}{3 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(b*x+a)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0207877, size = 34, normalized size = 0.69 \[ \frac{2 \left (-8 a^2-4 a b x+b^2 x^2\right )}{3 b^3 \sqrt{a+b x}} \]
Antiderivative was successfully verified.
[In] Integrate[x^2/(a + b*x)^(3/2),x]
[Out]
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Maple [A] time = 0.009, size = 32, normalized size = 0.7 \[ -{\frac{-2\,{b}^{2}{x}^{2}+8\,abx+16\,{a}^{2}}{3\,{b}^{3}}{\frac{1}{\sqrt{bx+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(b*x+a)^(3/2),x)
[Out]
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Maxima [A] time = 1.34362, size = 55, normalized size = 1.12 \[ \frac{2 \,{\left (b x + a\right )}^{\frac{3}{2}}}{3 \, b^{3}} - \frac{4 \, \sqrt{b x + a} a}{b^{3}} - \frac{2 \, a^{2}}{\sqrt{b x + a} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(b*x + a)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.224363, size = 41, normalized size = 0.84 \[ \frac{2 \,{\left (b^{2} x^{2} - 4 \, a b x - 8 \, a^{2}\right )}}{3 \, \sqrt{b x + a} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(b*x + a)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.53629, size = 534, normalized size = 10.9 \[ - \frac{16 a^{\frac{19}{2}} \sqrt{1 + \frac{b x}{a}}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} + \frac{16 a^{\frac{19}{2}}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} - \frac{40 a^{\frac{17}{2}} b x \sqrt{1 + \frac{b x}{a}}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} + \frac{48 a^{\frac{17}{2}} b x}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} - \frac{30 a^{\frac{15}{2}} b^{2} x^{2} \sqrt{1 + \frac{b x}{a}}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} + \frac{48 a^{\frac{15}{2}} b^{2} x^{2}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} - \frac{4 a^{\frac{13}{2}} b^{3} x^{3} \sqrt{1 + \frac{b x}{a}}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} + \frac{16 a^{\frac{13}{2}} b^{3} x^{3}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} + \frac{2 a^{\frac{11}{2}} b^{4} x^{4} \sqrt{1 + \frac{b x}{a}}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(b*x+a)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.204529, size = 62, normalized size = 1.27 \[ -\frac{2 \, a^{2}}{\sqrt{b x + a} b^{3}} + \frac{2 \,{\left ({\left (b x + a\right )}^{\frac{3}{2}} b^{6} - 6 \, \sqrt{b x + a} a b^{6}\right )}}{3 \, b^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(b*x + a)^(3/2),x, algorithm="giac")
[Out]