3.2265 \(\int \left (a+b \sqrt{x}\right )^p \, dx\)

Optimal. Leaf size=48 \[ \frac{2 \left (a+b \sqrt{x}\right )^{p+2}}{b^2 (p+2)}-\frac{2 a \left (a+b \sqrt{x}\right )^{p+1}}{b^2 (p+1)} \]

[Out]

(-2*a*(a + b*Sqrt[x])^(1 + p))/(b^2*(1 + p)) + (2*(a + b*Sqrt[x])^(2 + p))/(b^2*
(2 + p))

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Rubi [A]  time = 0.0566575, antiderivative size = 48, 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 \[ \frac{2 \left (a+b \sqrt{x}\right )^{p+2}}{b^2 (p+2)}-\frac{2 a \left (a+b \sqrt{x}\right )^{p+1}}{b^2 (p+1)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^p,x]

[Out]

(-2*a*(a + b*Sqrt[x])^(1 + p))/(b^2*(1 + p)) + (2*(a + b*Sqrt[x])^(2 + p))/(b^2*
(2 + p))

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Rubi in Sympy [A]  time = 9.21465, size = 41, normalized size = 0.85 \[ - \frac{2 a \left (a + b \sqrt{x}\right )^{p + 1}}{b^{2} \left (p + 1\right )} + \frac{2 \left (a + b \sqrt{x}\right )^{p + 2}}{b^{2} \left (p + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a*(a + b*sqrt(x))**(p + 1)/(b**2*(p + 1)) + 2*(a + b*sqrt(x))**(p + 2)/(b**2*
(p + 2))

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Mathematica [A]  time = 0.0286932, size = 42, normalized size = 0.88 \[ \frac{2 \left (a+b \sqrt{x}\right )^{p+1} \left (b (p+1) \sqrt{x}-a\right )}{b^2 (p+1) (p+2)} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*Sqrt[x])^p,x]

[Out]

(2*(a + b*Sqrt[x])^(1 + p)*(-a + b*(1 + p)*Sqrt[x]))/(b^2*(1 + p)*(2 + p))

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Maple [F]  time = 0.021, size = 0, normalized size = 0. \[ \int \left ( a+b\sqrt{x} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/2))^p,x)

[Out]

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

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Maxima [A]  time = 1.44858, size = 61, normalized size = 1.27 \[ \frac{2 \,{\left (b^{2}{\left (p + 1\right )} x + a b p \sqrt{x} - a^{2}\right )}{\left (b \sqrt{x} + a\right )}^{p}}{{\left (p^{2} + 3 \, p + 2\right )} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(b^2*(p + 1)*x + a*b*p*sqrt(x) - a^2)*(b*sqrt(x) + a)^p/((p^2 + 3*p + 2)*b^2)

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Fricas [A]  time = 0.2708, size = 76, normalized size = 1.58 \[ \frac{2 \,{\left (a b p \sqrt{x} - a^{2} +{\left (b^{2} p + b^{2}\right )} x\right )}{\left (b \sqrt{x} + a\right )}^{p}}{b^{2} p^{2} + 3 \, b^{2} p + 2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(a*b*p*sqrt(x) - a^2 + (b^2*p + b^2)*x)*(b*sqrt(x) + a)^p/(b^2*p^2 + 3*b^2*p +
2*b^2)

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Sympy [A]  time = 5.80834, size = 823, normalized size = 17.15 \[ - \frac{2 a^{3} a^{p} x^{2} \left (1 + \frac{b \sqrt{x}}{a}\right )^{p}}{a b^{2} p^{2} x^{2} + 3 a b^{2} p x^{2} + 2 a b^{2} x^{2} + b^{3} p^{2} x^{\frac{5}{2}} + 3 b^{3} p x^{\frac{5}{2}} + 2 b^{3} x^{\frac{5}{2}}} + \frac{2 a^{3} a^{p} x^{2}}{a b^{2} p^{2} x^{2} + 3 a b^{2} p x^{2} + 2 a b^{2} x^{2} + b^{3} p^{2} x^{\frac{5}{2}} + 3 b^{3} p x^{\frac{5}{2}} + 2 b^{3} x^{\frac{5}{2}}} + \frac{2 a^{2} a^{p} b p x^{\frac{5}{2}} \left (1 + \frac{b \sqrt{x}}{a}\right )^{p}}{a b^{2} p^{2} x^{2} + 3 a b^{2} p x^{2} + 2 a b^{2} x^{2} + b^{3} p^{2} x^{\frac{5}{2}} + 3 b^{3} p x^{\frac{5}{2}} + 2 b^{3} x^{\frac{5}{2}}} - \frac{2 a^{2} a^{p} b x^{\frac{5}{2}} \left (1 + \frac{b \sqrt{x}}{a}\right )^{p}}{a b^{2} p^{2} x^{2} + 3 a b^{2} p x^{2} + 2 a b^{2} x^{2} + b^{3} p^{2} x^{\frac{5}{2}} + 3 b^{3} p x^{\frac{5}{2}} + 2 b^{3} x^{\frac{5}{2}}} + \frac{2 a^{2} a^{p} b x^{\frac{5}{2}}}{a b^{2} p^{2} x^{2} + 3 a b^{2} p x^{2} + 2 a b^{2} x^{2} + b^{3} p^{2} x^{\frac{5}{2}} + 3 b^{3} p x^{\frac{5}{2}} + 2 b^{3} x^{\frac{5}{2}}} + \frac{4 a a^{p} b^{2} p x^{3} \left (1 + \frac{b \sqrt{x}}{a}\right )^{p}}{a b^{2} p^{2} x^{2} + 3 a b^{2} p x^{2} + 2 a b^{2} x^{2} + b^{3} p^{2} x^{\frac{5}{2}} + 3 b^{3} p x^{\frac{5}{2}} + 2 b^{3} x^{\frac{5}{2}}} + \frac{2 a a^{p} b^{2} x^{3} \left (1 + \frac{b \sqrt{x}}{a}\right )^{p}}{a b^{2} p^{2} x^{2} + 3 a b^{2} p x^{2} + 2 a b^{2} x^{2} + b^{3} p^{2} x^{\frac{5}{2}} + 3 b^{3} p x^{\frac{5}{2}} + 2 b^{3} x^{\frac{5}{2}}} + \frac{2 a^{p} b^{3} p x^{\frac{7}{2}} \left (1 + \frac{b \sqrt{x}}{a}\right )^{p}}{a b^{2} p^{2} x^{2} + 3 a b^{2} p x^{2} + 2 a b^{2} x^{2} + b^{3} p^{2} x^{\frac{5}{2}} + 3 b^{3} p x^{\frac{5}{2}} + 2 b^{3} x^{\frac{5}{2}}} + \frac{2 a^{p} b^{3} x^{\frac{7}{2}} \left (1 + \frac{b \sqrt{x}}{a}\right )^{p}}{a b^{2} p^{2} x^{2} + 3 a b^{2} p x^{2} + 2 a b^{2} x^{2} + b^{3} p^{2} x^{\frac{5}{2}} + 3 b^{3} p x^{\frac{5}{2}} + 2 b^{3} x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**(1/2))**p,x)

[Out]

-2*a**3*a**p*x**2*(1 + b*sqrt(x)/a)**p/(a*b**2*p**2*x**2 + 3*a*b**2*p*x**2 + 2*a
*b**2*x**2 + b**3*p**2*x**(5/2) + 3*b**3*p*x**(5/2) + 2*b**3*x**(5/2)) + 2*a**3*
a**p*x**2/(a*b**2*p**2*x**2 + 3*a*b**2*p*x**2 + 2*a*b**2*x**2 + b**3*p**2*x**(5/
2) + 3*b**3*p*x**(5/2) + 2*b**3*x**(5/2)) + 2*a**2*a**p*b*p*x**(5/2)*(1 + b*sqrt
(x)/a)**p/(a*b**2*p**2*x**2 + 3*a*b**2*p*x**2 + 2*a*b**2*x**2 + b**3*p**2*x**(5/
2) + 3*b**3*p*x**(5/2) + 2*b**3*x**(5/2)) - 2*a**2*a**p*b*x**(5/2)*(1 + b*sqrt(x
)/a)**p/(a*b**2*p**2*x**2 + 3*a*b**2*p*x**2 + 2*a*b**2*x**2 + b**3*p**2*x**(5/2)
+ 3*b**3*p*x**(5/2) + 2*b**3*x**(5/2)) + 2*a**2*a**p*b*x**(5/2)/(a*b**2*p**2*x*
*2 + 3*a*b**2*p*x**2 + 2*a*b**2*x**2 + b**3*p**2*x**(5/2) + 3*b**3*p*x**(5/2) +
2*b**3*x**(5/2)) + 4*a*a**p*b**2*p*x**3*(1 + b*sqrt(x)/a)**p/(a*b**2*p**2*x**2 +
3*a*b**2*p*x**2 + 2*a*b**2*x**2 + b**3*p**2*x**(5/2) + 3*b**3*p*x**(5/2) + 2*b*
*3*x**(5/2)) + 2*a*a**p*b**2*x**3*(1 + b*sqrt(x)/a)**p/(a*b**2*p**2*x**2 + 3*a*b
**2*p*x**2 + 2*a*b**2*x**2 + b**3*p**2*x**(5/2) + 3*b**3*p*x**(5/2) + 2*b**3*x**
(5/2)) + 2*a**p*b**3*p*x**(7/2)*(1 + b*sqrt(x)/a)**p/(a*b**2*p**2*x**2 + 3*a*b**
2*p*x**2 + 2*a*b**2*x**2 + b**3*p**2*x**(5/2) + 3*b**3*p*x**(5/2) + 2*b**3*x**(5
/2)) + 2*a**p*b**3*x**(7/2)*(1 + b*sqrt(x)/a)**p/(a*b**2*p**2*x**2 + 3*a*b**2*p*
x**2 + 2*a*b**2*x**2 + b**3*p**2*x**(5/2) + 3*b**3*p*x**(5/2) + 2*b**3*x**(5/2))

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GIAC/XCAS [A]  time = 0.269291, size = 138, normalized size = 2.88 \[ \frac{2 \,{\left ({\left (b \sqrt{x} + a\right )}^{2} p e^{\left (p{\rm ln}\left (b \sqrt{x} + a\right )\right )} -{\left (b \sqrt{x} + a\right )} a p e^{\left (p{\rm ln}\left (b \sqrt{x} + a\right )\right )} +{\left (b \sqrt{x} + a\right )}^{2} e^{\left (p{\rm ln}\left (b \sqrt{x} + a\right )\right )} - 2 \,{\left (b \sqrt{x} + a\right )} a e^{\left (p{\rm ln}\left (b \sqrt{x} + a\right )\right )}\right )}}{{\left (p^{2} + 3 \, p + 2\right )} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*((b*sqrt(x) + a)^2*p*e^(p*ln(b*sqrt(x) + a)) - (b*sqrt(x) + a)*a*p*e^(p*ln(b*s
qrt(x) + a)) + (b*sqrt(x) + a)^2*e^(p*ln(b*sqrt(x) + a)) - 2*(b*sqrt(x) + a)*a*e
^(p*ln(b*sqrt(x) + a)))/((p^2 + 3*p + 2)*b^2)