3.2263 \(\int \left (a+b \sqrt{x}\right )^p x^2 \, dx\)

Optimal. Leaf size=152 \[ -\frac{2 a^5 \left (a+b \sqrt{x}\right )^{p+1}}{b^6 (p+1)}+\frac{10 a^4 \left (a+b \sqrt{x}\right )^{p+2}}{b^6 (p+2)}-\frac{20 a^3 \left (a+b \sqrt{x}\right )^{p+3}}{b^6 (p+3)}+\frac{20 a^2 \left (a+b \sqrt{x}\right )^{p+4}}{b^6 (p+4)}-\frac{10 a \left (a+b \sqrt{x}\right )^{p+5}}{b^6 (p+5)}+\frac{2 \left (a+b \sqrt{x}\right )^{p+6}}{b^6 (p+6)} \]

[Out]

(-2*a^5*(a + b*Sqrt[x])^(1 + p))/(b^6*(1 + p)) + (10*a^4*(a + b*Sqrt[x])^(2 + p)
)/(b^6*(2 + p)) - (20*a^3*(a + b*Sqrt[x])^(3 + p))/(b^6*(3 + p)) + (20*a^2*(a +
b*Sqrt[x])^(4 + p))/(b^6*(4 + p)) - (10*a*(a + b*Sqrt[x])^(5 + p))/(b^6*(5 + p))
+ (2*(a + b*Sqrt[x])^(6 + p))/(b^6*(6 + p))

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Rubi [A]  time = 0.184023, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{2 a^5 \left (a+b \sqrt{x}\right )^{p+1}}{b^6 (p+1)}+\frac{10 a^4 \left (a+b \sqrt{x}\right )^{p+2}}{b^6 (p+2)}-\frac{20 a^3 \left (a+b \sqrt{x}\right )^{p+3}}{b^6 (p+3)}+\frac{20 a^2 \left (a+b \sqrt{x}\right )^{p+4}}{b^6 (p+4)}-\frac{10 a \left (a+b \sqrt{x}\right )^{p+5}}{b^6 (p+5)}+\frac{2 \left (a+b \sqrt{x}\right )^{p+6}}{b^6 (p+6)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^p*x^2,x]

[Out]

(-2*a^5*(a + b*Sqrt[x])^(1 + p))/(b^6*(1 + p)) + (10*a^4*(a + b*Sqrt[x])^(2 + p)
)/(b^6*(2 + p)) - (20*a^3*(a + b*Sqrt[x])^(3 + p))/(b^6*(3 + p)) + (20*a^2*(a +
b*Sqrt[x])^(4 + p))/(b^6*(4 + p)) - (10*a*(a + b*Sqrt[x])^(5 + p))/(b^6*(5 + p))
+ (2*(a + b*Sqrt[x])^(6 + p))/(b^6*(6 + p))

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Rubi in Sympy [A]  time = 36.3819, size = 136, normalized size = 0.89 \[ - \frac{2 a^{5} \left (a + b \sqrt{x}\right )^{p + 1}}{b^{6} \left (p + 1\right )} + \frac{10 a^{4} \left (a + b \sqrt{x}\right )^{p + 2}}{b^{6} \left (p + 2\right )} - \frac{20 a^{3} \left (a + b \sqrt{x}\right )^{p + 3}}{b^{6} \left (p + 3\right )} + \frac{20 a^{2} \left (a + b \sqrt{x}\right )^{p + 4}}{b^{6} \left (p + 4\right )} - \frac{10 a \left (a + b \sqrt{x}\right )^{p + 5}}{b^{6} \left (p + 5\right )} + \frac{2 \left (a + b \sqrt{x}\right )^{p + 6}}{b^{6} \left (p + 6\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**5*(a + b*sqrt(x))**(p + 1)/(b**6*(p + 1)) + 10*a**4*(a + b*sqrt(x))**(p +
2)/(b**6*(p + 2)) - 20*a**3*(a + b*sqrt(x))**(p + 3)/(b**6*(p + 3)) + 20*a**2*(a
+ b*sqrt(x))**(p + 4)/(b**6*(p + 4)) - 10*a*(a + b*sqrt(x))**(p + 5)/(b**6*(p +
5)) + 2*(a + b*sqrt(x))**(p + 6)/(b**6*(p + 6))

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Mathematica [A]  time = 0.198549, size = 170, normalized size = 1.12 \[ \frac{2 \left (a+b \sqrt{x}\right )^{p+1} \left (-120 a^5+120 a^4 b (p+1) \sqrt{x}-60 a^3 b^2 \left (p^2+3 p+2\right ) x+20 a^2 b^3 \left (p^3+6 p^2+11 p+6\right ) x^{3/2}-5 a b^4 \left (p^4+10 p^3+35 p^2+50 p+24\right ) x^2+b^5 \left (p^5+15 p^4+85 p^3+225 p^2+274 p+120\right ) x^{5/2}\right )}{b^6 (p+1) (p+2) (p+3) (p+4) (p+5) (p+6)} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*Sqrt[x])^p*x^2,x]

[Out]

(2*(a + b*Sqrt[x])^(1 + p)*(-120*a^5 + 120*a^4*b*(1 + p)*Sqrt[x] - 60*a^3*b^2*(2
+ 3*p + p^2)*x + 20*a^2*b^3*(6 + 11*p + 6*p^2 + p^3)*x^(3/2) - 5*a*b^4*(24 + 50
*p + 35*p^2 + 10*p^3 + p^4)*x^2 + b^5*(120 + 274*p + 225*p^2 + 85*p^3 + 15*p^4 +
p^5)*x^(5/2)))/(b^6*(1 + p)*(2 + p)*(3 + p)*(4 + p)*(5 + p)*(6 + p))

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Maple [F]  time = 0.022, size = 0, normalized size = 0. \[ \int{x}^{2} \left ( a+b\sqrt{x} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(a+b*x^(1/2))^p,x)

[Out]

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

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Maxima [A]  time = 1.43506, size = 250, normalized size = 1.64 \[ \frac{2 \,{\left ({\left (p^{5} + 15 \, p^{4} + 85 \, p^{3} + 225 \, p^{2} + 274 \, p + 120\right )} b^{6} x^{3} +{\left (p^{5} + 10 \, p^{4} + 35 \, p^{3} + 50 \, p^{2} + 24 \, p\right )} a b^{5} x^{\frac{5}{2}} - 5 \,{\left (p^{4} + 6 \, p^{3} + 11 \, p^{2} + 6 \, p\right )} a^{2} b^{4} x^{2} + 20 \,{\left (p^{3} + 3 \, p^{2} + 2 \, p\right )} a^{3} b^{3} x^{\frac{3}{2}} - 60 \,{\left (p^{2} + p\right )} a^{4} b^{2} x + 120 \, a^{5} b p \sqrt{x} - 120 \, a^{6}\right )}{\left (b \sqrt{x} + a\right )}^{p}}{{\left (p^{6} + 21 \, p^{5} + 175 \, p^{4} + 735 \, p^{3} + 1624 \, p^{2} + 1764 \, p + 720\right )} b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^p*x^2,x, algorithm="maxima")

[Out]

2*((p^5 + 15*p^4 + 85*p^3 + 225*p^2 + 274*p + 120)*b^6*x^3 + (p^5 + 10*p^4 + 35*
p^3 + 50*p^2 + 24*p)*a*b^5*x^(5/2) - 5*(p^4 + 6*p^3 + 11*p^2 + 6*p)*a^2*b^4*x^2
+ 20*(p^3 + 3*p^2 + 2*p)*a^3*b^3*x^(3/2) - 60*(p^2 + p)*a^4*b^2*x + 120*a^5*b*p*
sqrt(x) - 120*a^6)*(b*sqrt(x) + a)^p/((p^6 + 21*p^5 + 175*p^4 + 735*p^3 + 1624*p
^2 + 1764*p + 720)*b^6)

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Fricas [A]  time = 0.29176, size = 379, normalized size = 2.49 \[ -\frac{2 \,{\left (120 \, a^{6} -{\left (b^{6} p^{5} + 15 \, b^{6} p^{4} + 85 \, b^{6} p^{3} + 225 \, b^{6} p^{2} + 274 \, b^{6} p + 120 \, b^{6}\right )} x^{3} + 5 \,{\left (a^{2} b^{4} p^{4} + 6 \, a^{2} b^{4} p^{3} + 11 \, a^{2} b^{4} p^{2} + 6 \, a^{2} b^{4} p\right )} x^{2} + 60 \,{\left (a^{4} b^{2} p^{2} + a^{4} b^{2} p\right )} x -{\left (120 \, a^{5} b p +{\left (a b^{5} p^{5} + 10 \, a b^{5} p^{4} + 35 \, a b^{5} p^{3} + 50 \, a b^{5} p^{2} + 24 \, a b^{5} p\right )} x^{2} + 20 \,{\left (a^{3} b^{3} p^{3} + 3 \, a^{3} b^{3} p^{2} + 2 \, a^{3} b^{3} p\right )} x\right )} \sqrt{x}\right )}{\left (b \sqrt{x} + a\right )}^{p}}{b^{6} p^{6} + 21 \, b^{6} p^{5} + 175 \, b^{6} p^{4} + 735 \, b^{6} p^{3} + 1624 \, b^{6} p^{2} + 1764 \, b^{6} p + 720 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^p*x^2,x, algorithm="fricas")

[Out]

-2*(120*a^6 - (b^6*p^5 + 15*b^6*p^4 + 85*b^6*p^3 + 225*b^6*p^2 + 274*b^6*p + 120
*b^6)*x^3 + 5*(a^2*b^4*p^4 + 6*a^2*b^4*p^3 + 11*a^2*b^4*p^2 + 6*a^2*b^4*p)*x^2 +
60*(a^4*b^2*p^2 + a^4*b^2*p)*x - (120*a^5*b*p + (a*b^5*p^5 + 10*a*b^5*p^4 + 35*
a*b^5*p^3 + 50*a*b^5*p^2 + 24*a*b^5*p)*x^2 + 20*(a^3*b^3*p^3 + 3*a^3*b^3*p^2 + 2
*a^3*b^3*p)*x)*sqrt(x))*(b*sqrt(x) + a)^p/(b^6*p^6 + 21*b^6*p^5 + 175*b^6*p^4 +
735*b^6*p^3 + 1624*b^6*p^2 + 1764*b^6*p + 720*b^6)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.266615, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^p*x^2,x, algorithm="giac")

[Out]

Done