3.2144 \(\int (a+b \sqrt{x})^5 x \, dx\)

Optimal. Leaf size=80 \[ \frac{6 a^2 \left (a+b \sqrt{x}\right )^7}{7 b^4}-\frac{a^3 \left (a+b \sqrt{x}\right )^6}{3 b^4}+\frac{2 \left (a+b \sqrt{x}\right )^9}{9 b^4}-\frac{3 a \left (a+b \sqrt{x}\right )^8}{4 b^4} \]

[Out]

-(a^3*(a + b*Sqrt[x])^6)/(3*b^4) + (6*a^2*(a + b*Sqrt[x])^7)/(7*b^4) - (3*a*(a + b*Sqrt[x])^8)/(4*b^4) + (2*(a
+ b*Sqrt[x])^9)/(9*b^4)

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Rubi [A]  time = 0.0370688, antiderivative size = 80, 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, Rules used = {266, 43} \[ \frac{6 a^2 \left (a+b \sqrt{x}\right )^7}{7 b^4}-\frac{a^3 \left (a+b \sqrt{x}\right )^6}{3 b^4}+\frac{2 \left (a+b \sqrt{x}\right )^9}{9 b^4}-\frac{3 a \left (a+b \sqrt{x}\right )^8}{4 b^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^5*x,x]

[Out]

-(a^3*(a + b*Sqrt[x])^6)/(3*b^4) + (6*a^2*(a + b*Sqrt[x])^7)/(7*b^4) - (3*a*(a + b*Sqrt[x])^8)/(4*b^4) + (2*(a
+ b*Sqrt[x])^9)/(9*b^4)

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 )^5 x \, dx &=2 \operatorname{Subst}\left (\int x^3 (a+b x)^5 \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-\frac{a^3 (a+b x)^5}{b^3}+\frac{3 a^2 (a+b x)^6}{b^3}-\frac{3 a (a+b x)^7}{b^3}+\frac{(a+b x)^8}{b^3}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{a^3 \left (a+b \sqrt{x}\right )^6}{3 b^4}+\frac{6 a^2 \left (a+b \sqrt{x}\right )^7}{7 b^4}-\frac{3 a \left (a+b \sqrt{x}\right )^8}{4 b^4}+\frac{2 \left (a+b \sqrt{x}\right )^9}{9 b^4}\\ \end{align*}

Mathematica [A]  time = 0.0228478, size = 73, normalized size = 0.91 \[ \frac{10}{3} a^3 b^2 x^3+\frac{20}{7} a^2 b^3 x^{7/2}+2 a^4 b x^{5/2}+\frac{a^5 x^2}{2}+\frac{5}{4} a b^4 x^4+\frac{2}{9} b^5 x^{9/2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^5*x,x]

[Out]

(a^5*x^2)/2 + 2*a^4*b*x^(5/2) + (10*a^3*b^2*x^3)/3 + (20*a^2*b^3*x^(7/2))/7 + (5*a*b^4*x^4)/4 + (2*b^5*x^(9/2)
)/9

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Maple [A]  time = 0.002, size = 58, normalized size = 0.7 \begin{align*}{\frac{2\,{b}^{5}}{9}{x}^{{\frac{9}{2}}}}+{\frac{5\,a{b}^{4}{x}^{4}}{4}}+{\frac{20\,{a}^{2}{b}^{3}}{7}{x}^{{\frac{7}{2}}}}+{\frac{10\,{a}^{3}{b}^{2}{x}^{3}}{3}}+2\,{x}^{5/2}{a}^{4}b+{\frac{{a}^{5}{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a+b*x^(1/2))^5,x)

[Out]

2/9*x^(9/2)*b^5+5/4*a*b^4*x^4+20/7*x^(7/2)*a^2*b^3+10/3*a^3*b^2*x^3+2*x^(5/2)*a^4*b+1/2*a^5*x^2

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Maxima [A]  time = 0.989521, size = 86, normalized size = 1.08 \begin{align*} \frac{2 \,{\left (b \sqrt{x} + a\right )}^{9}}{9 \, b^{4}} - \frac{3 \,{\left (b \sqrt{x} + a\right )}^{8} a}{4 \, b^{4}} + \frac{6 \,{\left (b \sqrt{x} + a\right )}^{7} a^{2}}{7 \, b^{4}} - \frac{{\left (b \sqrt{x} + a\right )}^{6} a^{3}}{3 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*x^(1/2))^5,x, algorithm="maxima")

[Out]

2/9*(b*sqrt(x) + a)^9/b^4 - 3/4*(b*sqrt(x) + a)^8*a/b^4 + 6/7*(b*sqrt(x) + a)^7*a^2/b^4 - 1/3*(b*sqrt(x) + a)^
6*a^3/b^4

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Fricas [A]  time = 1.46842, size = 144, normalized size = 1.8 \begin{align*} \frac{5}{4} \, a b^{4} x^{4} + \frac{10}{3} \, a^{3} b^{2} x^{3} + \frac{1}{2} \, a^{5} x^{2} + \frac{2}{63} \,{\left (7 \, b^{5} x^{4} + 90 \, a^{2} b^{3} x^{3} + 63 \, a^{4} b x^{2}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*x^(1/2))^5,x, algorithm="fricas")

[Out]

5/4*a*b^4*x^4 + 10/3*a^3*b^2*x^3 + 1/2*a^5*x^2 + 2/63*(7*b^5*x^4 + 90*a^2*b^3*x^3 + 63*a^4*b*x^2)*sqrt(x)

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Sympy [A]  time = 2.35238, size = 71, normalized size = 0.89 \begin{align*} \frac{a^{5} x^{2}}{2} + 2 a^{4} b x^{\frac{5}{2}} + \frac{10 a^{3} b^{2} x^{3}}{3} + \frac{20 a^{2} b^{3} x^{\frac{7}{2}}}{7} + \frac{5 a b^{4} x^{4}}{4} + \frac{2 b^{5} x^{\frac{9}{2}}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*x**(1/2))**5,x)

[Out]

a**5*x**2/2 + 2*a**4*b*x**(5/2) + 10*a**3*b**2*x**3/3 + 20*a**2*b**3*x**(7/2)/7 + 5*a*b**4*x**4/4 + 2*b**5*x**
(9/2)/9

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Giac [A]  time = 1.11483, size = 77, normalized size = 0.96 \begin{align*} \frac{2}{9} \, b^{5} x^{\frac{9}{2}} + \frac{5}{4} \, a b^{4} x^{4} + \frac{20}{7} \, a^{2} b^{3} x^{\frac{7}{2}} + \frac{10}{3} \, a^{3} b^{2} x^{3} + 2 \, a^{4} b x^{\frac{5}{2}} + \frac{1}{2} \, a^{5} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*x^(1/2))^5,x, algorithm="giac")

[Out]

2/9*b^5*x^(9/2) + 5/4*a*b^4*x^4 + 20/7*a^2*b^3*x^(7/2) + 10/3*a^3*b^2*x^3 + 2*a^4*b*x^(5/2) + 1/2*a^5*x^2