3.2256 \(\int \left (a+b \sqrt{x}\right )^3 x^m \, dx\)

Optimal. Leaf size=70 \[ \frac{a^3 x^{m+1}}{m+1}+\frac{6 a^2 b x^{m+\frac{3}{2}}}{2 m+3}+\frac{3 a b^2 x^{m+2}}{m+2}+\frac{2 b^3 x^{m+\frac{5}{2}}}{2 m+5} \]

[Out]

(a^3*x^(1 + m))/(1 + m) + (6*a^2*b*x^(3/2 + m))/(3 + 2*m) + (3*a*b^2*x^(2 + m))/
(2 + m) + (2*b^3*x^(5/2 + m))/(5 + 2*m)

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Rubi [A]  time = 0.0720506, antiderivative size = 70, 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 \[ \frac{a^3 x^{m+1}}{m+1}+\frac{6 a^2 b x^{m+\frac{3}{2}}}{2 m+3}+\frac{3 a b^2 x^{m+2}}{m+2}+\frac{2 b^3 x^{m+\frac{5}{2}}}{2 m+5} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^3*x^m,x]

[Out]

(a^3*x^(1 + m))/(1 + m) + (6*a^2*b*x^(3/2 + m))/(3 + 2*m) + (3*a*b^2*x^(2 + m))/
(2 + m) + (2*b^3*x^(5/2 + m))/(5 + 2*m)

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Rubi in Sympy [A]  time = 12.0738, size = 61, normalized size = 0.87 \[ \frac{a^{3} x^{m + 1}}{m + 1} + \frac{6 a^{2} b x^{m + \frac{3}{2}}}{2 m + 3} + \frac{3 a b^{2} x^{m + 2}}{m + 2} + \frac{2 b^{3} x^{m + \frac{5}{2}}}{2 m + 5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m*(a+b*x**(1/2))**3,x)

[Out]

a**3*x**(m + 1)/(m + 1) + 6*a**2*b*x**(m + 3/2)/(2*m + 3) + 3*a*b**2*x**(m + 2)/
(m + 2) + 2*b**3*x**(m + 5/2)/(2*m + 5)

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Mathematica [A]  time = 0.0560137, size = 69, normalized size = 0.99 \[ x^m \left (\frac{2 a^3 x}{2 m+2}+\frac{6 a^2 b x^{3/2}}{2 m+3}+\frac{6 a b^2 x^2}{2 m+4}+\frac{2 b^3 x^{5/2}}{2 m+5}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*Sqrt[x])^3*x^m,x]

[Out]

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

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Maple [F]  time = 0.015, size = 0, normalized size = 0. \[ \int{x}^{m} \left ( a+b\sqrt{x} \right ) ^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(a+b*x^(1/2))^3,x)

[Out]

int(x^m*(a+b*x^(1/2))^3,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)^3*x^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.259144, size = 225, normalized size = 3.21 \[ \frac{{\left (3 \,{\left (4 \, a b^{2} m^{3} + 20 \, a b^{2} m^{2} + 31 \, a b^{2} m + 15 \, a b^{2}\right )} x^{2} +{\left (4 \, a^{3} m^{3} + 24 \, a^{3} m^{2} + 47 \, a^{3} m + 30 \, a^{3}\right )} x + 2 \,{\left ({\left (2 \, b^{3} m^{3} + 9 \, b^{3} m^{2} + 13 \, b^{3} m + 6 \, b^{3}\right )} x^{2} + 3 \,{\left (2 \, a^{2} b m^{3} + 11 \, a^{2} b m^{2} + 19 \, a^{2} b m + 10 \, a^{2} b\right )} x\right )} \sqrt{x}\right )} x^{m}}{4 \, m^{4} + 28 \, m^{3} + 71 \, m^{2} + 77 \, m + 30} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^3*x^m,x, algorithm="fricas")

[Out]

(3*(4*a*b^2*m^3 + 20*a*b^2*m^2 + 31*a*b^2*m + 15*a*b^2)*x^2 + (4*a^3*m^3 + 24*a^
3*m^2 + 47*a^3*m + 30*a^3)*x + 2*((2*b^3*m^3 + 9*b^3*m^2 + 13*b^3*m + 6*b^3)*x^2
+ 3*(2*a^2*b*m^3 + 11*a^2*b*m^2 + 19*a^2*b*m + 10*a^2*b)*x)*sqrt(x))*x^m/(4*m^4
+ 28*m^3 + 71*m^2 + 77*m + 30)

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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))**3,x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [A]  time = 0.276776, size = 119, normalized size = 1.7 \[ \frac{2 \, b^{3} x^{\frac{5}{2}} e^{\left (2 \, m{\rm ln}\left (\sqrt{x}\right )\right )}}{2 \, m + 5} + \frac{3 \, a b^{2} x^{2} e^{\left (2 \, m{\rm ln}\left (\sqrt{x}\right )\right )}}{m + 2} + \frac{6 \, a^{2} b x^{\frac{3}{2}} e^{\left (2 \, m{\rm ln}\left (\sqrt{x}\right )\right )}}{2 \, m + 3} + \frac{a^{3} x e^{\left (2 \, m{\rm ln}\left (\sqrt{x}\right )\right )}}{m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^3*x^m,x, algorithm="giac")

[Out]

2*b^3*x^(5/2)*e^(2*m*ln(sqrt(x)))/(2*m + 5) + 3*a*b^2*x^2*e^(2*m*ln(sqrt(x)))/(m
+ 2) + 6*a^2*b*x^(3/2)*e^(2*m*ln(sqrt(x)))/(2*m + 3) + a^3*x*e^(2*m*ln(sqrt(x))
)/(m + 1)