3.2264 \(\int \left (a+b \sqrt{x}\right )^p x \, dx\)

Optimal. Leaf size=100 \[ -\frac{2 a^3 \left (a+b \sqrt{x}\right )^{p+1}}{b^4 (p+1)}+\frac{6 a^2 \left (a+b \sqrt{x}\right )^{p+2}}{b^4 (p+2)}-\frac{6 a \left (a+b \sqrt{x}\right )^{p+3}}{b^4 (p+3)}+\frac{2 \left (a+b \sqrt{x}\right )^{p+4}}{b^4 (p+4)} \]

[Out]

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

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Rubi [A]  time = 0.120948, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{2 a^3 \left (a+b \sqrt{x}\right )^{p+1}}{b^4 (p+1)}+\frac{6 a^2 \left (a+b \sqrt{x}\right )^{p+2}}{b^4 (p+2)}-\frac{6 a \left (a+b \sqrt{x}\right )^{p+3}}{b^4 (p+3)}+\frac{2 \left (a+b \sqrt{x}\right )^{p+4}}{b^4 (p+4)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^p*x,x]

[Out]

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

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Rubi in Sympy [A]  time = 23.2331, size = 88, normalized size = 0.88 \[ - \frac{2 a^{3} \left (a + b \sqrt{x}\right )^{p + 1}}{b^{4} \left (p + 1\right )} + \frac{6 a^{2} \left (a + b \sqrt{x}\right )^{p + 2}}{b^{4} \left (p + 2\right )} - \frac{6 a \left (a + b \sqrt{x}\right )^{p + 3}}{b^{4} \left (p + 3\right )} + \frac{2 \left (a + b \sqrt{x}\right )^{p + 4}}{b^{4} \left (p + 4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0746156, size = 95, normalized size = 0.95 \[ \frac{2 \left (a+b \sqrt{x}\right )^{p+1} \left (-6 a^3+6 a^2 b (p+1) \sqrt{x}-3 a b^2 \left (p^2+3 p+2\right ) x+b^3 \left (p^3+6 p^2+11 p+6\right ) x^{3/2}\right )}{b^4 (p+1) (p+2) (p+3) (p+4)} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a + b*Sqrt[x])^(1 + p)*(-6*a^3 + 6*a^2*b*(1 + p)*Sqrt[x] - 3*a*b^2*(2 + 3*p
+ p^2)*x + b^3*(6 + 11*p + 6*p^2 + p^3)*x^(3/2)))/(b^4*(1 + p)*(2 + p)*(3 + p)*(
4 + p))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(a+b*x^(1/2))^p,x)

[Out]

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

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Maxima [A]  time = 1.43222, size = 140, normalized size = 1.4 \[ \frac{2 \,{\left ({\left (p^{3} + 6 \, p^{2} + 11 \, p + 6\right )} b^{4} x^{2} +{\left (p^{3} + 3 \, p^{2} + 2 \, p\right )} a b^{3} x^{\frac{3}{2}} - 3 \,{\left (p^{2} + p\right )} a^{2} b^{2} x + 6 \, a^{3} b p \sqrt{x} - 6 \, a^{4}\right )}{\left (b \sqrt{x} + a\right )}^{p}}{{\left (p^{4} + 10 \, p^{3} + 35 \, p^{2} + 50 \, p + 24\right )} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*((p^3 + 6*p^2 + 11*p + 6)*b^4*x^2 + (p^3 + 3*p^2 + 2*p)*a*b^3*x^(3/2) - 3*(p^2
+ p)*a^2*b^2*x + 6*a^3*b*p*sqrt(x) - 6*a^4)*(b*sqrt(x) + a)^p/((p^4 + 10*p^3 +
35*p^2 + 50*p + 24)*b^4)

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Fricas [A]  time = 0.276088, size = 200, normalized size = 2. \[ -\frac{2 \,{\left (6 \, a^{4} -{\left (b^{4} p^{3} + 6 \, b^{4} p^{2} + 11 \, b^{4} p + 6 \, b^{4}\right )} x^{2} + 3 \,{\left (a^{2} b^{2} p^{2} + a^{2} b^{2} p\right )} x -{\left (6 \, a^{3} b p +{\left (a b^{3} p^{3} + 3 \, a b^{3} p^{2} + 2 \, a b^{3} p\right )} x\right )} \sqrt{x}\right )}{\left (b \sqrt{x} + a\right )}^{p}}{b^{4} p^{4} + 10 \, b^{4} p^{3} + 35 \, b^{4} p^{2} + 50 \, b^{4} p + 24 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(6*a^4 - (b^4*p^3 + 6*b^4*p^2 + 11*b^4*p + 6*b^4)*x^2 + 3*(a^2*b^2*p^2 + a^2*
b^2*p)*x - (6*a^3*b*p + (a*b^3*p^3 + 3*a*b^3*p^2 + 2*a*b^3*p)*x)*sqrt(x))*(b*sqr
t(x) + a)^p/(b^4*p^4 + 10*b^4*p^3 + 35*b^4*p^2 + 50*b^4*p + 24*b^4)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.242723, size = 597, normalized size = 5.97 \[ \frac{2 \,{\left ({\left (b \sqrt{x} + a\right )}^{4} p^{3} e^{\left (p{\rm ln}\left (b \sqrt{x} + a\right )\right )} - 3 \,{\left (b \sqrt{x} + a\right )}^{3} a p^{3} e^{\left (p{\rm ln}\left (b \sqrt{x} + a\right )\right )} + 3 \,{\left (b \sqrt{x} + a\right )}^{2} a^{2} p^{3} e^{\left (p{\rm ln}\left (b \sqrt{x} + a\right )\right )} -{\left (b \sqrt{x} + a\right )} a^{3} p^{3} e^{\left (p{\rm ln}\left (b \sqrt{x} + a\right )\right )} + 6 \,{\left (b \sqrt{x} + a\right )}^{4} p^{2} e^{\left (p{\rm ln}\left (b \sqrt{x} + a\right )\right )} - 21 \,{\left (b \sqrt{x} + a\right )}^{3} a p^{2} e^{\left (p{\rm ln}\left (b \sqrt{x} + a\right )\right )} + 24 \,{\left (b \sqrt{x} + a\right )}^{2} a^{2} p^{2} e^{\left (p{\rm ln}\left (b \sqrt{x} + a\right )\right )} - 9 \,{\left (b \sqrt{x} + a\right )} a^{3} p^{2} e^{\left (p{\rm ln}\left (b \sqrt{x} + a\right )\right )} + 11 \,{\left (b \sqrt{x} + a\right )}^{4} p e^{\left (p{\rm ln}\left (b \sqrt{x} + a\right )\right )} - 42 \,{\left (b \sqrt{x} + a\right )}^{3} a p e^{\left (p{\rm ln}\left (b \sqrt{x} + a\right )\right )} + 57 \,{\left (b \sqrt{x} + a\right )}^{2} a^{2} p e^{\left (p{\rm ln}\left (b \sqrt{x} + a\right )\right )} - 26 \,{\left (b \sqrt{x} + a\right )} a^{3} p e^{\left (p{\rm ln}\left (b \sqrt{x} + a\right )\right )} + 6 \,{\left (b \sqrt{x} + a\right )}^{4} e^{\left (p{\rm ln}\left (b \sqrt{x} + a\right )\right )} - 24 \,{\left (b \sqrt{x} + a\right )}^{3} a e^{\left (p{\rm ln}\left (b \sqrt{x} + a\right )\right )} + 36 \,{\left (b \sqrt{x} + a\right )}^{2} a^{2} e^{\left (p{\rm ln}\left (b \sqrt{x} + a\right )\right )} - 24 \,{\left (b \sqrt{x} + a\right )} a^{3} e^{\left (p{\rm ln}\left (b \sqrt{x} + a\right )\right )}\right )}}{{\left (b^{2} p^{4} + 10 \, b^{2} p^{3} + 35 \, b^{2} p^{2} + 50 \, b^{2} p + 24 \, b^{2}\right )} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*((b*sqrt(x) + a)^4*p^3*e^(p*ln(b*sqrt(x) + a)) - 3*(b*sqrt(x) + a)^3*a*p^3*e^(
p*ln(b*sqrt(x) + a)) + 3*(b*sqrt(x) + a)^2*a^2*p^3*e^(p*ln(b*sqrt(x) + a)) - (b*
sqrt(x) + a)*a^3*p^3*e^(p*ln(b*sqrt(x) + a)) + 6*(b*sqrt(x) + a)^4*p^2*e^(p*ln(b
*sqrt(x) + a)) - 21*(b*sqrt(x) + a)^3*a*p^2*e^(p*ln(b*sqrt(x) + a)) + 24*(b*sqrt
(x) + a)^2*a^2*p^2*e^(p*ln(b*sqrt(x) + a)) - 9*(b*sqrt(x) + a)*a^3*p^2*e^(p*ln(b
*sqrt(x) + a)) + 11*(b*sqrt(x) + a)^4*p*e^(p*ln(b*sqrt(x) + a)) - 42*(b*sqrt(x)
+ a)^3*a*p*e^(p*ln(b*sqrt(x) + a)) + 57*(b*sqrt(x) + a)^2*a^2*p*e^(p*ln(b*sqrt(x
) + a)) - 26*(b*sqrt(x) + a)*a^3*p*e^(p*ln(b*sqrt(x) + a)) + 6*(b*sqrt(x) + a)^4
*e^(p*ln(b*sqrt(x) + a)) - 24*(b*sqrt(x) + a)^3*a*e^(p*ln(b*sqrt(x) + a)) + 36*(
b*sqrt(x) + a)^2*a^2*e^(p*ln(b*sqrt(x) + a)) - 24*(b*sqrt(x) + a)*a^3*e^(p*ln(b*
sqrt(x) + a)))/((b^2*p^4 + 10*b^2*p^3 + 35*b^2*p^2 + 50*b^2*p + 24*b^2)*b^2)