3.2156 \(\int (a+b \sqrt{x})^{10} x \, dx\)

Optimal. Leaf size=80 \[ \frac{a^2 \left (a+b \sqrt{x}\right )^{12}}{2 b^4}-\frac{2 a^3 \left (a+b \sqrt{x}\right )^{11}}{11 b^4}+\frac{\left (a+b \sqrt{x}\right )^{14}}{7 b^4}-\frac{6 a \left (a+b \sqrt{x}\right )^{13}}{13 b^4} \]

[Out]

(-2*a^3*(a + b*Sqrt[x])^11)/(11*b^4) + (a^2*(a + b*Sqrt[x])^12)/(2*b^4) - (6*a*(a + b*Sqrt[x])^13)/(13*b^4) +
(a + b*Sqrt[x])^14/(7*b^4)

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Rubi [A]  time = 0.045463, 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} \[ \frac{a^2 \left (a+b \sqrt{x}\right )^{12}}{2 b^4}-\frac{2 a^3 \left (a+b \sqrt{x}\right )^{11}}{11 b^4}+\frac{\left (a+b \sqrt{x}\right )^{14}}{7 b^4}-\frac{6 a \left (a+b \sqrt{x}\right )^{13}}{13 b^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^10*x,x]

[Out]

(-2*a^3*(a + b*Sqrt[x])^11)/(11*b^4) + (a^2*(a + b*Sqrt[x])^12)/(2*b^4) - (6*a*(a + b*Sqrt[x])^13)/(13*b^4) +
(a + b*Sqrt[x])^14/(7*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 )^{10} x \, dx &=2 \operatorname{Subst}\left (\int x^3 (a+b x)^{10} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-\frac{a^3 (a+b x)^{10}}{b^3}+\frac{3 a^2 (a+b x)^{11}}{b^3}-\frac{3 a (a+b x)^{12}}{b^3}+\frac{(a+b x)^{13}}{b^3}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{2 a^3 \left (a+b \sqrt{x}\right )^{11}}{11 b^4}+\frac{a^2 \left (a+b \sqrt{x}\right )^{12}}{2 b^4}-\frac{6 a \left (a+b \sqrt{x}\right )^{13}}{13 b^4}+\frac{\left (a+b \sqrt{x}\right )^{14}}{7 b^4}\\ \end{align*}

Mathematica [A]  time = 0.03694, size = 50, normalized size = 0.62 \[ -\frac{\left (a+b \sqrt{x}\right )^{11} \left (-11 a^2 b \sqrt{x}+a^3+66 a b^2 x-286 b^3 x^{3/2}\right )}{2002 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^10*x,x]

[Out]

-((a + b*Sqrt[x])^11*(a^3 - 11*a^2*b*Sqrt[x] + 66*a*b^2*x - 286*b^3*x^(3/2)))/(2002*b^4)

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Maple [A]  time = 0.003, size = 113, normalized size = 1.4 \begin{align*}{\frac{{x}^{7}{b}^{10}}{7}}+{\frac{20\,a{b}^{9}}{13}{x}^{{\frac{13}{2}}}}+{\frac{15\,{x}^{6}{a}^{2}{b}^{8}}{2}}+{\frac{240\,{a}^{3}{b}^{7}}{11}{x}^{{\frac{11}{2}}}}+42\,{x}^{5}{a}^{4}{b}^{6}+56\,{x}^{9/2}{a}^{5}{b}^{5}+{\frac{105\,{x}^{4}{a}^{6}{b}^{4}}{2}}+{\frac{240\,{a}^{7}{b}^{3}}{7}{x}^{{\frac{7}{2}}}}+15\,{x}^{3}{a}^{8}{b}^{2}+4\,{x}^{5/2}{a}^{9}b+{\frac{{x}^{2}{a}^{10}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a+b*x^(1/2))^10,x)

[Out]

1/7*x^7*b^10+20/13*x^(13/2)*a*b^9+15/2*x^6*a^2*b^8+240/11*x^(11/2)*a^3*b^7+42*x^5*a^4*b^6+56*x^(9/2)*a^5*b^5+1
05/2*x^4*a^6*b^4+240/7*x^(7/2)*a^7*b^3+15*x^3*a^8*b^2+4*x^(5/2)*a^9*b+1/2*x^2*a^10

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Maxima [A]  time = 0.964791, size = 86, normalized size = 1.08 \begin{align*} \frac{{\left (b \sqrt{x} + a\right )}^{14}}{7 \, b^{4}} - \frac{6 \,{\left (b \sqrt{x} + a\right )}^{13} a}{13 \, b^{4}} + \frac{{\left (b \sqrt{x} + a\right )}^{12} a^{2}}{2 \, b^{4}} - \frac{2 \,{\left (b \sqrt{x} + a\right )}^{11} a^{3}}{11 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*x^(1/2))^10,x, algorithm="maxima")

[Out]

1/7*(b*sqrt(x) + a)^14/b^4 - 6/13*(b*sqrt(x) + a)^13*a/b^4 + 1/2*(b*sqrt(x) + a)^12*a^2/b^4 - 2/11*(b*sqrt(x)
+ a)^11*a^3/b^4

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Fricas [A]  time = 1.40067, size = 284, normalized size = 3.55 \begin{align*} \frac{1}{7} \, b^{10} x^{7} + \frac{15}{2} \, a^{2} b^{8} x^{6} + 42 \, a^{4} b^{6} x^{5} + \frac{105}{2} \, a^{6} b^{4} x^{4} + 15 \, a^{8} b^{2} x^{3} + \frac{1}{2} \, a^{10} x^{2} + \frac{4}{1001} \,{\left (385 \, a b^{9} x^{6} + 5460 \, a^{3} b^{7} x^{5} + 14014 \, a^{5} b^{5} x^{4} + 8580 \, a^{7} b^{3} x^{3} + 1001 \, a^{9} 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))^10,x, algorithm="fricas")

[Out]

1/7*b^10*x^7 + 15/2*a^2*b^8*x^6 + 42*a^4*b^6*x^5 + 105/2*a^6*b^4*x^4 + 15*a^8*b^2*x^3 + 1/2*a^10*x^2 + 4/1001*
(385*a*b^9*x^6 + 5460*a^3*b^7*x^5 + 14014*a^5*b^5*x^4 + 8580*a^7*b^3*x^3 + 1001*a^9*b*x^2)*sqrt(x)

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Sympy [A]  time = 2.51321, size = 136, normalized size = 1.7 \begin{align*} \frac{a^{10} x^{2}}{2} + 4 a^{9} b x^{\frac{5}{2}} + 15 a^{8} b^{2} x^{3} + \frac{240 a^{7} b^{3} x^{\frac{7}{2}}}{7} + \frac{105 a^{6} b^{4} x^{4}}{2} + 56 a^{5} b^{5} x^{\frac{9}{2}} + 42 a^{4} b^{6} x^{5} + \frac{240 a^{3} b^{7} x^{\frac{11}{2}}}{11} + \frac{15 a^{2} b^{8} x^{6}}{2} + \frac{20 a b^{9} x^{\frac{13}{2}}}{13} + \frac{b^{10} x^{7}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*x**(1/2))**10,x)

[Out]

a**10*x**2/2 + 4*a**9*b*x**(5/2) + 15*a**8*b**2*x**3 + 240*a**7*b**3*x**(7/2)/7 + 105*a**6*b**4*x**4/2 + 56*a*
*5*b**5*x**(9/2) + 42*a**4*b**6*x**5 + 240*a**3*b**7*x**(11/2)/11 + 15*a**2*b**8*x**6/2 + 20*a*b**9*x**(13/2)/
13 + b**10*x**7/7

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Giac [A]  time = 1.11323, size = 151, normalized size = 1.89 \begin{align*} \frac{1}{7} \, b^{10} x^{7} + \frac{20}{13} \, a b^{9} x^{\frac{13}{2}} + \frac{15}{2} \, a^{2} b^{8} x^{6} + \frac{240}{11} \, a^{3} b^{7} x^{\frac{11}{2}} + 42 \, a^{4} b^{6} x^{5} + 56 \, a^{5} b^{5} x^{\frac{9}{2}} + \frac{105}{2} \, a^{6} b^{4} x^{4} + \frac{240}{7} \, a^{7} b^{3} x^{\frac{7}{2}} + 15 \, a^{8} b^{2} x^{3} + 4 \, a^{9} b x^{\frac{5}{2}} + \frac{1}{2} \, a^{10} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*x^(1/2))^10,x, algorithm="giac")

[Out]

1/7*b^10*x^7 + 20/13*a*b^9*x^(13/2) + 15/2*a^2*b^8*x^6 + 240/11*a^3*b^7*x^(11/2) + 42*a^4*b^6*x^5 + 56*a^5*b^5
*x^(9/2) + 105/2*a^6*b^4*x^4 + 240/7*a^7*b^3*x^(7/2) + 15*a^8*b^2*x^3 + 4*a^9*b*x^(5/2) + 1/2*a^10*x^2