3.2257 \(\int \left (a+b \sqrt{x}\right )^2 x^m \, dx\)

Optimal. Leaf size=47 \[ \frac{a^2 x^{m+1}}{m+1}+\frac{4 a b x^{m+\frac{3}{2}}}{2 m+3}+\frac{b^2 x^{m+2}}{m+2} \]

[Out]

(a^2*x^(1 + m))/(1 + m) + (4*a*b*x^(3/2 + m))/(3 + 2*m) + (b^2*x^(2 + m))/(2 + m
)

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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]

(a^2*x^(1 + m))/(1 + m) + (4*a*b*x^(3/2 + m))/(3 + 2*m) + (b^2*x^(2 + m))/(2 + m
)

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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)

[Out]

a**2*x**(m + 1)/(m + 1) + 4*a*b*x**(m + 3/2)/(2*m + 3) + b**2*x**(m + 2)/(m + 2)

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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]

[Out]

x^m*((2*a^2*x)/(2 + 2*m) + (4*a*b*x^(3/2))/(3 + 2*m) + (2*b^2*x^2)/(4 + 2*m))

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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]

int(x^m*(a+b*x^(1/2))^2,x)

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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]

Exception raised: ValueError

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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]

((2*b^2*m^2 + 5*b^2*m + 3*b^2)*x^2 + 4*(a*b*m^2 + 3*a*b*m + 2*a*b)*x^(3/2) + (2*
a^2*m^2 + 7*a^2*m + 6*a^2)*x)*x^m/(2*m^3 + 9*m^2 + 13*m + 6)

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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]

Exception raised: TypeError

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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]

b^2*x^2*e^(2*m*ln(sqrt(x)))/(m + 2) + 4*a*b*x^(3/2)*e^(2*m*ln(sqrt(x)))/(2*m + 3
) + a^2*x*e^(2*m*ln(sqrt(x)))/(m + 1)