Optimal. Leaf size=47 \[ \frac{a^2 x^{m+1}}{m+1}+\frac{4 a b x^{m+\frac{3}{2}}}{2 m+3}+\frac{b^2 x^{m+2}}{m+2} \]
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Rubi [A] time = 0.0511278, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{a^2 x^{m+1}}{m+1}+\frac{4 a b x^{m+\frac{3}{2}}}{2 m+3}+\frac{b^2 x^{m+2}}{m+2} \]
Antiderivative was successfully verified.
[In] Int[(a + b*Sqrt[x])^2*x^m,x]
[Out]
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Rubi in Sympy [A] time = 8.32769, size = 39, normalized size = 0.83 \[ \frac{a^{2} x^{m + 1}}{m + 1} + \frac{4 a b x^{m + \frac{3}{2}}}{2 m + 3} + \frac{b^{2} x^{m + 2}}{m + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**m*(a+b*x**(1/2))**2,x)
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Mathematica [A] time = 0.0663917, size = 49, normalized size = 1.04 \[ x^m \left (\frac{2 a^2 x}{2 m+2}+\frac{4 a b x^{3/2}}{2 m+3}+\frac{2 b^2 x^2}{2 m+4}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*Sqrt[x])^2*x^m,x]
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Maple [F] time = 0.014, size = 0, normalized size = 0. \[ \int{x}^{m} \left ( a+b\sqrt{x} \right ) ^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^m*(a+b*x^(1/2))^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*sqrt(x) + a)^2*x^m,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.252343, size = 120, normalized size = 2.55 \[ \frac{{\left ({\left (2 \, b^{2} m^{2} + 5 \, b^{2} m + 3 \, b^{2}\right )} x^{2} + 4 \,{\left (a b m^{2} + 3 \, a b m + 2 \, a b\right )} x^{\frac{3}{2}} +{\left (2 \, a^{2} m^{2} + 7 \, a^{2} m + 6 \, a^{2}\right )} x\right )} x^{m}}{2 \, m^{3} + 9 \, m^{2} + 13 \, m + 6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*sqrt(x) + a)^2*x^m,x, algorithm="fricas")
[Out]
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**m*(a+b*x**(1/2))**2,x)
[Out]
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GIAC/XCAS [A] time = 0.258311, size = 82, normalized size = 1.74 \[ \frac{b^{2} x^{2} e^{\left (2 \, m{\rm ln}\left (\sqrt{x}\right )\right )}}{m + 2} + \frac{4 \, a b x^{\frac{3}{2}} e^{\left (2 \, m{\rm ln}\left (\sqrt{x}\right )\right )}}{2 \, m + 3} + \frac{a^{2} x e^{\left (2 \, m{\rm ln}\left (\sqrt{x}\right )\right )}}{m + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*sqrt(x) + a)^2*x^m,x, algorithm="giac")
[Out]