3.2143 \(\int \left (a+b \sqrt{x}\right )^5 \, dx\)

Optimal. Leaf size=38 \[ \frac{2 \left (a+b \sqrt{x}\right )^7}{7 b^2}-\frac{a \left (a+b \sqrt{x}\right )^6}{3 b^2} \]

[Out]

-(a*(a + b*Sqrt[x])^6)/(3*b^2) + (2*(a + b*Sqrt[x])^7)/(7*b^2)

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Rubi [A]  time = 0.0450174, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{2 \left (a+b \sqrt{x}\right )^7}{7 b^2}-\frac{a \left (a+b \sqrt{x}\right )^6}{3 b^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^5,x]

[Out]

-(a*(a + b*Sqrt[x])^6)/(3*b^2) + (2*(a + b*Sqrt[x])^7)/(7*b^2)

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Rubi in Sympy [A]  time = 8.07235, size = 32, normalized size = 0.84 \[ - \frac{a \left (a + b \sqrt{x}\right )^{6}}{3 b^{2}} + \frac{2 \left (a + b \sqrt{x}\right )^{7}}{7 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**(1/2))**5,x)

[Out]

-a*(a + b*sqrt(x))**6/(3*b**2) + 2*(a + b*sqrt(x))**7/(7*b**2)

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Mathematica [A]  time = 0.0127571, size = 66, normalized size = 1.74 \[ a^5 x+\frac{10}{3} a^4 b x^{3/2}+5 a^3 b^2 x^2+4 a^2 b^3 x^{5/2}+\frac{5}{3} a b^4 x^3+\frac{2}{7} b^5 x^{7/2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*Sqrt[x])^5,x]

[Out]

a^5*x + (10*a^4*b*x^(3/2))/3 + 5*a^3*b^2*x^2 + 4*a^2*b^3*x^(5/2) + (5*a*b^4*x^3)
/3 + (2*b^5*x^(7/2))/7

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Maple [A]  time = 0.003, size = 55, normalized size = 1.5 \[{\frac{2\,{b}^{5}}{7}{x}^{{\frac{7}{2}}}}+{\frac{5\,a{b}^{4}{x}^{3}}{3}}+4\,{a}^{2}{b}^{3}{x}^{5/2}+5\,{a}^{3}{b}^{2}{x}^{2}+{\frac{10\,{a}^{4}b}{3}{x}^{{\frac{3}{2}}}}+x{a}^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/2))^5,x)

[Out]

2/7*x^(7/2)*b^5+5/3*a*b^4*x^3+4*a^2*b^3*x^(5/2)+5*a^3*b^2*x^2+10/3*a^4*b*x^(3/2)
+x*a^5

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Maxima [A]  time = 1.43983, size = 73, normalized size = 1.92 \[ \frac{2}{7} \, b^{5} x^{\frac{7}{2}} + \frac{5}{3} \, a b^{4} x^{3} + 4 \, a^{2} b^{3} x^{\frac{5}{2}} + 5 \, a^{3} b^{2} x^{2} + \frac{10}{3} \, a^{4} b x^{\frac{3}{2}} + a^{5} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^5,x, algorithm="maxima")

[Out]

2/7*b^5*x^(7/2) + 5/3*a*b^4*x^3 + 4*a^2*b^3*x^(5/2) + 5*a^3*b^2*x^2 + 10/3*a^4*b
*x^(3/2) + a^5*x

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Fricas [A]  time = 0.229263, size = 78, normalized size = 2.05 \[ \frac{5}{3} \, a b^{4} x^{3} + 5 \, a^{3} b^{2} x^{2} + a^{5} x + \frac{2}{21} \,{\left (3 \, b^{5} x^{3} + 42 \, a^{2} b^{3} x^{2} + 35 \, a^{4} b x\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^5,x, algorithm="fricas")

[Out]

5/3*a*b^4*x^3 + 5*a^3*b^2*x^2 + a^5*x + 2/21*(3*b^5*x^3 + 42*a^2*b^3*x^2 + 35*a^
4*b*x)*sqrt(x)

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Sympy [A]  time = 1.43243, size = 66, normalized size = 1.74 \[ a^{5} x + \frac{10 a^{4} b x^{\frac{3}{2}}}{3} + 5 a^{3} b^{2} x^{2} + 4 a^{2} b^{3} x^{\frac{5}{2}} + \frac{5 a b^{4} x^{3}}{3} + \frac{2 b^{5} x^{\frac{7}{2}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**(1/2))**5,x)

[Out]

a**5*x + 10*a**4*b*x**(3/2)/3 + 5*a**3*b**2*x**2 + 4*a**2*b**3*x**(5/2) + 5*a*b*
*4*x**3/3 + 2*b**5*x**(7/2)/7

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GIAC/XCAS [A]  time = 0.213633, size = 73, normalized size = 1.92 \[ \frac{2}{7} \, b^{5} x^{\frac{7}{2}} + \frac{5}{3} \, a b^{4} x^{3} + 4 \, a^{2} b^{3} x^{\frac{5}{2}} + 5 \, a^{3} b^{2} x^{2} + \frac{10}{3} \, a^{4} b x^{\frac{3}{2}} + a^{5} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^5,x, algorithm="giac")

[Out]

2/7*b^5*x^(7/2) + 5/3*a*b^4*x^3 + 4*a^2*b^3*x^(5/2) + 5*a^3*b^2*x^2 + 10/3*a^4*b
*x^(3/2) + a^5*x