3.2265 \(\int (a+b \sqrt{x})^p x^2 \, dx\)

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 = 0.0812685, 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, Rules used = {266, 43} \[ -\frac{2 a^5 \left (a+b \sqrt{x}\right )^{p+1}}{b^6 (p+1)}+\frac{10 a^4 \left (a+b \sqrt{x}\right )^{p+2}}{b^6 (p+2)}-\frac{20 a^3 \left (a+b \sqrt{x}\right )^{p+3}}{b^6 (p+3)}+\frac{20 a^2 \left (a+b \sqrt{x}\right )^{p+4}}{b^6 (p+4)}-\frac{10 a \left (a+b \sqrt{x}\right )^{p+5}}{b^6 (p+5)}+\frac{2 \left (a+b \sqrt{x}\right )^{p+6}}{b^6 (p+6)} \]

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))

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \left (a+b \sqrt{x}\right )^p x^2 \, dx &=2 \operatorname{Subst}\left (\int x^5 (a+b x)^p \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-\frac{a^5 (a+b x)^p}{b^5}+\frac{5 a^4 (a+b x)^{1+p}}{b^5}-\frac{10 a^3 (a+b x)^{2+p}}{b^5}+\frac{10 a^2 (a+b x)^{3+p}}{b^5}-\frac{5 a (a+b x)^{4+p}}{b^5}+\frac{(a+b x)^{5+p}}{b^5}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{2 a^5 \left (a+b \sqrt{x}\right )^{1+p}}{b^6 (1+p)}+\frac{10 a^4 \left (a+b \sqrt{x}\right )^{2+p}}{b^6 (2+p)}-\frac{20 a^3 \left (a+b \sqrt{x}\right )^{3+p}}{b^6 (3+p)}+\frac{20 a^2 \left (a+b \sqrt{x}\right )^{4+p}}{b^6 (4+p)}-\frac{10 a \left (a+b \sqrt{x}\right )^{5+p}}{b^6 (5+p)}+\frac{2 \left (a+b \sqrt{x}\right )^{6+p}}{b^6 (6+p)}\\ \end{align*}

Mathematica [A]  time = 0.0920903, size = 126, normalized size = 0.83 \[ \frac{2 \left (\frac{5 a^4 \left (a+b \sqrt{x}\right )}{p+2}-\frac{10 a^3 \left (a+b \sqrt{x}\right )^2}{p+3}+\frac{10 a^2 \left (a+b \sqrt{x}\right )^3}{p+4}-\frac{a^5}{p+1}-\frac{5 a \left (a+b \sqrt{x}\right )^4}{p+5}+\frac{\left (a+b \sqrt{x}\right )^5}{p+6}\right ) \left (a+b \sqrt{x}\right )^{p+1}}{b^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^p*x^2,x]

[Out]

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

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Maple [F]  time = 0.018, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( a+b\sqrt{x} \right ) ^{p}\, dx \end{align*}

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 = 1.00312, size = 250, normalized size = 1.64 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*x^(1/2))^p,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 [B]  time = 1.5453, size = 610, normalized size = 4.01 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*x^(1/2))^p,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. \begin{align*} \text{Timed out} \end{align*}

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 [B]  time = 1.64324, size = 1245, normalized size = 8.19 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*x^(1/2))^p,x, algorithm="giac")

[Out]

2*((b*sqrt(x) + a)^6*(b*sqrt(x) + a)^p*p^5 - 5*(b*sqrt(x) + a)^5*(b*sqrt(x) + a)^p*a*p^5 + 10*(b*sqrt(x) + a)^
4*(b*sqrt(x) + a)^p*a^2*p^5 - 10*(b*sqrt(x) + a)^3*(b*sqrt(x) + a)^p*a^3*p^5 + 5*(b*sqrt(x) + a)^2*(b*sqrt(x)
+ a)^p*a^4*p^5 - (b*sqrt(x) + a)*(b*sqrt(x) + a)^p*a^5*p^5 + 15*(b*sqrt(x) + a)^6*(b*sqrt(x) + a)^p*p^4 - 80*(
b*sqrt(x) + a)^5*(b*sqrt(x) + a)^p*a*p^4 + 170*(b*sqrt(x) + a)^4*(b*sqrt(x) + a)^p*a^2*p^4 - 180*(b*sqrt(x) +
a)^3*(b*sqrt(x) + a)^p*a^3*p^4 + 95*(b*sqrt(x) + a)^2*(b*sqrt(x) + a)^p*a^4*p^4 - 20*(b*sqrt(x) + a)*(b*sqrt(x
) + a)^p*a^5*p^4 + 85*(b*sqrt(x) + a)^6*(b*sqrt(x) + a)^p*p^3 - 475*(b*sqrt(x) + a)^5*(b*sqrt(x) + a)^p*a*p^3
+ 1070*(b*sqrt(x) + a)^4*(b*sqrt(x) + a)^p*a^2*p^3 - 1210*(b*sqrt(x) + a)^3*(b*sqrt(x) + a)^p*a^3*p^3 + 685*(b
*sqrt(x) + a)^2*(b*sqrt(x) + a)^p*a^4*p^3 - 155*(b*sqrt(x) + a)*(b*sqrt(x) + a)^p*a^5*p^3 + 225*(b*sqrt(x) + a
)^6*(b*sqrt(x) + a)^p*p^2 - 1300*(b*sqrt(x) + a)^5*(b*sqrt(x) + a)^p*a*p^2 + 3070*(b*sqrt(x) + a)^4*(b*sqrt(x)
+ a)^p*a^2*p^2 - 3720*(b*sqrt(x) + a)^3*(b*sqrt(x) + a)^p*a^3*p^2 + 2305*(b*sqrt(x) + a)^2*(b*sqrt(x) + a)^p*
a^4*p^2 - 580*(b*sqrt(x) + a)*(b*sqrt(x) + a)^p*a^5*p^2 + 274*(b*sqrt(x) + a)^6*(b*sqrt(x) + a)^p*p - 1620*(b*
sqrt(x) + a)^5*(b*sqrt(x) + a)^p*a*p + 3960*(b*sqrt(x) + a)^4*(b*sqrt(x) + a)^p*a^2*p - 5080*(b*sqrt(x) + a)^3
*(b*sqrt(x) + a)^p*a^3*p + 3510*(b*sqrt(x) + a)^2*(b*sqrt(x) + a)^p*a^4*p - 1044*(b*sqrt(x) + a)*(b*sqrt(x) +
a)^p*a^5*p + 120*(b*sqrt(x) + a)^6*(b*sqrt(x) + a)^p - 720*(b*sqrt(x) + a)^5*(b*sqrt(x) + a)^p*a + 1800*(b*sqr
t(x) + a)^4*(b*sqrt(x) + a)^p*a^2 - 2400*(b*sqrt(x) + a)^3*(b*sqrt(x) + a)^p*a^3 + 1800*(b*sqrt(x) + a)^2*(b*s
qrt(x) + a)^p*a^4 - 720*(b*sqrt(x) + a)*(b*sqrt(x) + a)^p*a^5)/((b^4*p^6 + 21*b^4*p^5 + 175*b^4*p^4 + 735*b^4*
p^3 + 1624*b^4*p^2 + 1764*b^4*p + 720*b^4)*b^2)