Optimal. Leaf size=51 \[ \frac{2 (a+2 b x)}{3 a^2 b \sqrt{a x+b x^2}}-\frac{2 x}{3 b \left (a x+b x^2\right )^{3/2}} \]
[Out]
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Rubi [A] time = 0.0535306, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{2 (a+2 b x)}{3 a^2 b \sqrt{a x+b x^2}}-\frac{2 x}{3 b \left (a x+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[x^2/(a*x + b*x^2)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 5.45877, size = 44, normalized size = 0.86 \[ - \frac{2 x}{3 b \left (a x + b x^{2}\right )^{\frac{3}{2}}} + \frac{2 a + 4 b x}{3 a^{2} b \sqrt{a x + b x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(b*x**2+a*x)**(5/2),x)
[Out]
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Mathematica [A] time = 0.0273374, size = 29, normalized size = 0.57 \[ \frac{2 x^2 (3 a+2 b x)}{3 a^2 (x (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^2/(a*x + b*x^2)^(5/2),x]
[Out]
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Maple [A] time = 0.006, size = 33, normalized size = 0.7 \[{\frac{2\,{x}^{3} \left ( bx+a \right ) \left ( 2\,bx+3\,a \right ) }{3\,{a}^{2}} \left ( b{x}^{2}+ax \right ) ^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(b*x^2+a*x)^(5/2),x)
[Out]
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Maxima [A] time = 0.708414, size = 73, normalized size = 1.43 \[ \frac{4 \, x}{3 \, \sqrt{b x^{2} + a x} a^{2}} - \frac{2 \, x}{3 \,{\left (b x^{2} + a x\right )}^{\frac{3}{2}} b} + \frac{2}{3 \, \sqrt{b x^{2} + a x} a b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(b*x^2 + a*x)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.219028, size = 49, normalized size = 0.96 \[ \frac{2 \,{\left (2 \, b x^{2} + 3 \, a x\right )}}{3 \,{\left (a^{2} b x + a^{3}\right )} \sqrt{b x^{2} + a x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(b*x^2 + a*x)^(5/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\left (x \left (a + b x\right )\right )^{\frac{5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(b*x**2+a*x)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.225079, size = 82, normalized size = 1.61 \[ \frac{2 \,{\left (3 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} b + 2 \, a \sqrt{b}\right )}}{3 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} \sqrt{b} + a\right )}^{3} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(b*x^2 + a*x)^(5/2),x, algorithm="giac")
[Out]