Optimal. Leaf size=37 \[ \frac{x^4}{4 a (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]
[Out]
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Rubi [A] time = 0.0603818, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{x^4}{4 a (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
[In] Int[x^3/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 6.45057, size = 34, normalized size = 0.92 \[ \frac{x^{4} \left (2 a + 2 b x\right )}{8 a \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)
[Out]
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Mathematica [A] time = 0.0248611, size = 55, normalized size = 1.49 \[ \frac{-a^3-4 a^2 b x-6 a b^2 x^2-4 b^3 x^3}{4 b^4 (a+b x)^3 \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^3/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
[Out]
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Maple [A] time = 0.009, size = 48, normalized size = 1.3 \[ -{\frac{ \left ( bx+a \right ) \left ( 4\,{b}^{3}{x}^{3}+6\,a{b}^{2}{x}^{2}+4\,{a}^{2}bx+{a}^{3} \right ) }{4\,{b}^{4}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)
[Out]
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Maxima [A] time = 0.798552, size = 181, normalized size = 4.89 \[ -\frac{x^{2}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{3}{2}} b^{2}} - \frac{2 \, a^{2}}{3 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{3}{2}} b^{4}} - \frac{a^{3} b}{4 \,{\left (b^{2}\right )}^{\frac{9}{2}}{\left (x + \frac{a}{b}\right )}^{4}} + \frac{2 \, a^{2}}{3 \,{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x + \frac{a}{b}\right )}^{3}} - \frac{a}{2 \,{\left (b^{2}\right )}^{\frac{5}{2}} b{\left (x + \frac{a}{b}\right )}^{2}} + \frac{a^{3}}{2 \,{\left (b^{2}\right )}^{\frac{5}{2}} b^{3}{\left (x + \frac{a}{b}\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.224518, size = 103, normalized size = 2.78 \[ -\frac{4 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 4 \, a^{2} b x + a^{3}}{4 \,{\left (b^{8} x^{4} + 4 \, a b^{7} x^{3} + 6 \, a^{2} b^{6} x^{2} + 4 \, a^{3} b^{5} x + a^{4} b^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{\left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.56971, size = 4, normalized size = 0.11 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="giac")
[Out]