Optimal. Leaf size=61 \[ \frac{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 b^2}-\frac{a (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{4 b^2} \]
[Out]
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Rubi [A] time = 0.0651242, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 b^2}-\frac{a (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{4 b^2} \]
Antiderivative was successfully verified.
[In] Int[x*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 7.21996, size = 60, normalized size = 0.98 \[ - \frac{a \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{8 b^{2}} + \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{5 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0236704, size = 55, normalized size = 0.9 \[ \frac{x^2 \sqrt{(a+b x)^2} \left (10 a^3+20 a^2 b x+15 a b^2 x^2+4 b^3 x^3\right )}{20 (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[x*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.005, size = 52, normalized size = 0.9 \[{\frac{{x}^{2} \left ( 4\,{b}^{3}{x}^{3}+15\,a{b}^{2}{x}^{2}+20\,{a}^{2}bx+10\,{a}^{3} \right ) }{20\, \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(b^2*x^2+2*a*b*x+a^2)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*x,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.237009, size = 46, normalized size = 0.75 \[ \frac{1}{5} \, b^{3} x^{5} + \frac{3}{4} \, a b^{2} x^{4} + a^{2} b x^{3} + \frac{1}{2} \, a^{3} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*x,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int x \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.209737, size = 97, normalized size = 1.59 \[ \frac{1}{5} \, b^{3} x^{5}{\rm sign}\left (b x + a\right ) + \frac{3}{4} \, a b^{2} x^{4}{\rm sign}\left (b x + a\right ) + a^{2} b x^{3}{\rm sign}\left (b x + a\right ) + \frac{1}{2} \, a^{3} x^{2}{\rm sign}\left (b x + a\right ) - \frac{a^{5}{\rm sign}\left (b x + a\right )}{20 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*x,x, algorithm="giac")
[Out]