3.2155 \(\int (a+b \sqrt{x})^{10} x^2 \, dx\)

Optimal. Leaf size=122 \[ \frac{10 a^2 \left (a+b \sqrt{x}\right )^{14}}{7 b^6}-\frac{20 a^3 \left (a+b \sqrt{x}\right )^{13}}{13 b^6}+\frac{5 a^4 \left (a+b \sqrt{x}\right )^{12}}{6 b^6}-\frac{2 a^5 \left (a+b \sqrt{x}\right )^{11}}{11 b^6}+\frac{\left (a+b \sqrt{x}\right )^{16}}{8 b^6}-\frac{2 a \left (a+b \sqrt{x}\right )^{15}}{3 b^6} \]

[Out]

(-2*a^5*(a + b*Sqrt[x])^11)/(11*b^6) + (5*a^4*(a + b*Sqrt[x])^12)/(6*b^6) - (20*a^3*(a + b*Sqrt[x])^13)/(13*b^
6) + (10*a^2*(a + b*Sqrt[x])^14)/(7*b^6) - (2*a*(a + b*Sqrt[x])^15)/(3*b^6) + (a + b*Sqrt[x])^16/(8*b^6)

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Rubi [A]  time = 0.0642877, antiderivative size = 122, 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} \[ \frac{10 a^2 \left (a+b \sqrt{x}\right )^{14}}{7 b^6}-\frac{20 a^3 \left (a+b \sqrt{x}\right )^{13}}{13 b^6}+\frac{5 a^4 \left (a+b \sqrt{x}\right )^{12}}{6 b^6}-\frac{2 a^5 \left (a+b \sqrt{x}\right )^{11}}{11 b^6}+\frac{\left (a+b \sqrt{x}\right )^{16}}{8 b^6}-\frac{2 a \left (a+b \sqrt{x}\right )^{15}}{3 b^6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^10*x^2,x]

[Out]

(-2*a^5*(a + b*Sqrt[x])^11)/(11*b^6) + (5*a^4*(a + b*Sqrt[x])^12)/(6*b^6) - (20*a^3*(a + b*Sqrt[x])^13)/(13*b^
6) + (10*a^2*(a + b*Sqrt[x])^14)/(7*b^6) - (2*a*(a + b*Sqrt[x])^15)/(3*b^6) + (a + b*Sqrt[x])^16/(8*b^6)

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^2 \, dx &=2 \operatorname{Subst}\left (\int x^5 (a+b x)^{10} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-\frac{a^5 (a+b x)^{10}}{b^5}+\frac{5 a^4 (a+b x)^{11}}{b^5}-\frac{10 a^3 (a+b x)^{12}}{b^5}+\frac{10 a^2 (a+b x)^{13}}{b^5}-\frac{5 a (a+b x)^{14}}{b^5}+\frac{(a+b x)^{15}}{b^5}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{2 a^5 \left (a+b \sqrt{x}\right )^{11}}{11 b^6}+\frac{5 a^4 \left (a+b \sqrt{x}\right )^{12}}{6 b^6}-\frac{20 a^3 \left (a+b \sqrt{x}\right )^{13}}{13 b^6}+\frac{10 a^2 \left (a+b \sqrt{x}\right )^{14}}{7 b^6}-\frac{2 a \left (a+b \sqrt{x}\right )^{15}}{3 b^6}+\frac{\left (a+b \sqrt{x}\right )^{16}}{8 b^6}\\ \end{align*}

Mathematica [A]  time = 0.0472047, size = 74, normalized size = 0.61 \[ -\frac{\left (a+b \sqrt{x}\right )^{11} \left (-286 a^2 b^3 x^{3/2}+66 a^3 b^2 x-11 a^4 b \sqrt{x}+a^5+1001 a b^4 x^2-3003 b^5 x^{5/2}\right )}{24024 b^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^10*x^2,x]

[Out]

-((a + b*Sqrt[x])^11*(a^5 - 11*a^4*b*Sqrt[x] + 66*a^3*b^2*x - 286*a^2*b^3*x^(3/2) + 1001*a*b^4*x^2 - 3003*b^5*
x^(5/2)))/(24024*b^6)

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Maple [A]  time = 0.003, size = 113, normalized size = 0.9 \begin{align*}{\frac{{x}^{8}{b}^{10}}{8}}+{\frac{4\,a{b}^{9}}{3}{x}^{{\frac{15}{2}}}}+{\frac{45\,{x}^{7}{a}^{2}{b}^{8}}{7}}+{\frac{240\,{a}^{3}{b}^{7}}{13}{x}^{{\frac{13}{2}}}}+35\,{a}^{4}{b}^{6}{x}^{6}+{\frac{504\,{a}^{5}{b}^{5}}{11}{x}^{{\frac{11}{2}}}}+42\,{x}^{5}{a}^{6}{b}^{4}+{\frac{80\,{a}^{7}{b}^{3}}{3}{x}^{{\frac{9}{2}}}}+{\frac{45\,{x}^{4}{a}^{8}{b}^{2}}{4}}+{\frac{20\,{a}^{9}b}{7}{x}^{{\frac{7}{2}}}}+{\frac{{x}^{3}{a}^{10}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*x^8*b^10+4/3*x^(15/2)*a*b^9+45/7*x^7*a^2*b^8+240/13*x^(13/2)*a^3*b^7+35*a^4*b^6*x^6+504/11*x^(11/2)*a^5*b^
5+42*x^5*a^6*b^4+80/3*x^(9/2)*a^7*b^3+45/4*x^4*a^8*b^2+20/7*x^(7/2)*a^9*b+1/3*x^3*a^10

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Maxima [A]  time = 0.962464, size = 132, normalized size = 1.08 \begin{align*} \frac{{\left (b \sqrt{x} + a\right )}^{16}}{8 \, b^{6}} - \frac{2 \,{\left (b \sqrt{x} + a\right )}^{15} a}{3 \, b^{6}} + \frac{10 \,{\left (b \sqrt{x} + a\right )}^{14} a^{2}}{7 \, b^{6}} - \frac{20 \,{\left (b \sqrt{x} + a\right )}^{13} a^{3}}{13 \, b^{6}} + \frac{5 \,{\left (b \sqrt{x} + a\right )}^{12} a^{4}}{6 \, b^{6}} - \frac{2 \,{\left (b \sqrt{x} + a\right )}^{11} a^{5}}{11 \, b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(b*sqrt(x) + a)^16/b^6 - 2/3*(b*sqrt(x) + a)^15*a/b^6 + 10/7*(b*sqrt(x) + a)^14*a^2/b^6 - 20/13*(b*sqrt(x)
+ a)^13*a^3/b^6 + 5/6*(b*sqrt(x) + a)^12*a^4/b^6 - 2/11*(b*sqrt(x) + a)^11*a^5/b^6

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Fricas [A]  time = 1.46003, size = 286, normalized size = 2.34 \begin{align*} \frac{1}{8} \, b^{10} x^{8} + \frac{45}{7} \, a^{2} b^{8} x^{7} + 35 \, a^{4} b^{6} x^{6} + 42 \, a^{6} b^{4} x^{5} + \frac{45}{4} \, a^{8} b^{2} x^{4} + \frac{1}{3} \, a^{10} x^{3} + \frac{4}{3003} \,{\left (1001 \, a b^{9} x^{7} + 13860 \, a^{3} b^{7} x^{6} + 34398 \, a^{5} b^{5} x^{5} + 20020 \, a^{7} b^{3} x^{4} + 2145 \, a^{9} b x^{3}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*b^10*x^8 + 45/7*a^2*b^8*x^7 + 35*a^4*b^6*x^6 + 42*a^6*b^4*x^5 + 45/4*a^8*b^2*x^4 + 1/3*a^10*x^3 + 4/3003*(
1001*a*b^9*x^7 + 13860*a^3*b^7*x^6 + 34398*a^5*b^5*x^5 + 20020*a^7*b^3*x^4 + 2145*a^9*b*x^3)*sqrt(x)

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Sympy [A]  time = 3.39806, size = 139, normalized size = 1.14 \begin{align*} \frac{a^{10} x^{3}}{3} + \frac{20 a^{9} b x^{\frac{7}{2}}}{7} + \frac{45 a^{8} b^{2} x^{4}}{4} + \frac{80 a^{7} b^{3} x^{\frac{9}{2}}}{3} + 42 a^{6} b^{4} x^{5} + \frac{504 a^{5} b^{5} x^{\frac{11}{2}}}{11} + 35 a^{4} b^{6} x^{6} + \frac{240 a^{3} b^{7} x^{\frac{13}{2}}}{13} + \frac{45 a^{2} b^{8} x^{7}}{7} + \frac{4 a b^{9} x^{\frac{15}{2}}}{3} + \frac{b^{10} x^{8}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**10*x**3/3 + 20*a**9*b*x**(7/2)/7 + 45*a**8*b**2*x**4/4 + 80*a**7*b**3*x**(9/2)/3 + 42*a**6*b**4*x**5 + 504*
a**5*b**5*x**(11/2)/11 + 35*a**4*b**6*x**6 + 240*a**3*b**7*x**(13/2)/13 + 45*a**2*b**8*x**7/7 + 4*a*b**9*x**(1
5/2)/3 + b**10*x**8/8

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Giac [A]  time = 1.09286, size = 151, normalized size = 1.24 \begin{align*} \frac{1}{8} \, b^{10} x^{8} + \frac{4}{3} \, a b^{9} x^{\frac{15}{2}} + \frac{45}{7} \, a^{2} b^{8} x^{7} + \frac{240}{13} \, a^{3} b^{7} x^{\frac{13}{2}} + 35 \, a^{4} b^{6} x^{6} + \frac{504}{11} \, a^{5} b^{5} x^{\frac{11}{2}} + 42 \, a^{6} b^{4} x^{5} + \frac{80}{3} \, a^{7} b^{3} x^{\frac{9}{2}} + \frac{45}{4} \, a^{8} b^{2} x^{4} + \frac{20}{7} \, a^{9} b x^{\frac{7}{2}} + \frac{1}{3} \, a^{10} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*b^10*x^8 + 4/3*a*b^9*x^(15/2) + 45/7*a^2*b^8*x^7 + 240/13*a^3*b^7*x^(13/2) + 35*a^4*b^6*x^6 + 504/11*a^5*b
^5*x^(11/2) + 42*a^6*b^4*x^5 + 80/3*a^7*b^3*x^(9/2) + 45/4*a^8*b^2*x^4 + 20/7*a^9*b*x^(7/2) + 1/3*a^10*x^3