Optimal. Leaf size=51 \[ \frac{2 a^2 \sqrt{a+b x}}{b^3}+\frac{2 (a+b x)^{5/2}}{5 b^3}-\frac{4 a (a+b x)^{3/2}}{3 b^3} \]
[Out]
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Rubi [A] time = 0.0389682, antiderivative size = 51, 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 \[ \frac{2 a^2 \sqrt{a+b x}}{b^3}+\frac{2 (a+b x)^{5/2}}{5 b^3}-\frac{4 a (a+b x)^{3/2}}{3 b^3} \]
Antiderivative was successfully verified.
[In] Int[x^2/Sqrt[a + b*x],x]
[Out]
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Rubi in Sympy [A] time = 7.93041, size = 48, normalized size = 0.94 \[ \frac{2 a^{2} \sqrt{a + b x}}{b^{3}} - \frac{4 a \left (a + b x\right )^{\frac{3}{2}}}{3 b^{3}} + \frac{2 \left (a + b x\right )^{\frac{5}{2}}}{5 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(b*x+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0165262, size = 35, normalized size = 0.69 \[ \frac{2 \sqrt{a+b x} \left (8 a^2-4 a b x+3 b^2 x^2\right )}{15 b^3} \]
Antiderivative was successfully verified.
[In] Integrate[x^2/Sqrt[a + b*x],x]
[Out]
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Maple [A] time = 0.007, size = 32, normalized size = 0.6 \[{\frac{6\,{b}^{2}{x}^{2}-8\,abx+16\,{a}^{2}}{15\,{b}^{3}}\sqrt{bx+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(b*x+a)^(1/2),x)
[Out]
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Maxima [A] time = 1.33882, size = 55, normalized size = 1.08 \[ \frac{2 \,{\left (b x + a\right )}^{\frac{5}{2}}}{5 \, b^{3}} - \frac{4 \,{\left (b x + a\right )}^{\frac{3}{2}} a}{3 \, b^{3}} + \frac{2 \, \sqrt{b x + a} a^{2}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/sqrt(b*x + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.231636, size = 42, normalized size = 0.82 \[ \frac{2 \,{\left (3 \, b^{2} x^{2} - 4 \, a b x + 8 \, a^{2}\right )} \sqrt{b x + a}}{15 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/sqrt(b*x + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.73924, size = 600, normalized size = 11.76 \[ \frac{16 a^{\frac{21}{2}} \sqrt{1 + \frac{b x}{a}}}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} - \frac{16 a^{\frac{21}{2}}}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} + \frac{40 a^{\frac{19}{2}} b x \sqrt{1 + \frac{b x}{a}}}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} - \frac{48 a^{\frac{19}{2}} b x}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} + \frac{30 a^{\frac{17}{2}} b^{2} x^{2} \sqrt{1 + \frac{b x}{a}}}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} - \frac{48 a^{\frac{17}{2}} b^{2} x^{2}}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} + \frac{10 a^{\frac{15}{2}} b^{3} x^{3} \sqrt{1 + \frac{b x}{a}}}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} - \frac{16 a^{\frac{15}{2}} b^{3} x^{3}}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} + \frac{10 a^{\frac{13}{2}} b^{4} x^{4} \sqrt{1 + \frac{b x}{a}}}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} + \frac{6 a^{\frac{11}{2}} b^{5} x^{5} \sqrt{1 + \frac{b x}{a}}}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(b*x+a)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.20821, size = 62, normalized size = 1.22 \[ \frac{2 \,{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} b^{8} - 10 \,{\left (b x + a\right )}^{\frac{3}{2}} a b^{8} + 15 \, \sqrt{b x + a} a^{2} b^{8}\right )}}{15 \, b^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/sqrt(b*x + a),x, algorithm="giac")
[Out]