3.2153 \(\int (a+b \sqrt{x})^{10} x^4 \, dx\)

Optimal. Leaf size=140 \[ \frac{15}{2} a^8 b^2 x^6+\frac{240}{13} a^7 b^3 x^{13/2}+30 a^6 b^4 x^7+\frac{168}{5} a^5 b^5 x^{15/2}+\frac{105}{4} a^4 b^6 x^8+\frac{240}{17} a^3 b^7 x^{17/2}+5 a^2 b^8 x^9+\frac{20}{11} a^9 b x^{11/2}+\frac{a^{10} x^5}{5}+\frac{20}{19} a b^9 x^{19/2}+\frac{b^{10} x^{10}}{10} \]

[Out]

(a^10*x^5)/5 + (20*a^9*b*x^(11/2))/11 + (15*a^8*b^2*x^6)/2 + (240*a^7*b^3*x^(13/2))/13 + 30*a^6*b^4*x^7 + (168
*a^5*b^5*x^(15/2))/5 + (105*a^4*b^6*x^8)/4 + (240*a^3*b^7*x^(17/2))/17 + 5*a^2*b^8*x^9 + (20*a*b^9*x^(19/2))/1
9 + (b^10*x^10)/10

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Rubi [A]  time = 0.0967475, antiderivative size = 140, 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} \[ \frac{15}{2} a^8 b^2 x^6+\frac{240}{13} a^7 b^3 x^{13/2}+30 a^6 b^4 x^7+\frac{168}{5} a^5 b^5 x^{15/2}+\frac{105}{4} a^4 b^6 x^8+\frac{240}{17} a^3 b^7 x^{17/2}+5 a^2 b^8 x^9+\frac{20}{11} a^9 b x^{11/2}+\frac{a^{10} x^5}{5}+\frac{20}{19} a b^9 x^{19/2}+\frac{b^{10} x^{10}}{10} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^10*x^4,x]

[Out]

(a^10*x^5)/5 + (20*a^9*b*x^(11/2))/11 + (15*a^8*b^2*x^6)/2 + (240*a^7*b^3*x^(13/2))/13 + 30*a^6*b^4*x^7 + (168
*a^5*b^5*x^(15/2))/5 + (105*a^4*b^6*x^8)/4 + (240*a^3*b^7*x^(17/2))/17 + 5*a^2*b^8*x^9 + (20*a*b^9*x^(19/2))/1
9 + (b^10*x^10)/10

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^4 \, dx &=2 \operatorname{Subst}\left (\int x^9 (a+b x)^{10} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (a^{10} x^9+10 a^9 b x^{10}+45 a^8 b^2 x^{11}+120 a^7 b^3 x^{12}+210 a^6 b^4 x^{13}+252 a^5 b^5 x^{14}+210 a^4 b^6 x^{15}+120 a^3 b^7 x^{16}+45 a^2 b^8 x^{17}+10 a b^9 x^{18}+b^{10} x^{19}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{a^{10} x^5}{5}+\frac{20}{11} a^9 b x^{11/2}+\frac{15}{2} a^8 b^2 x^6+\frac{240}{13} a^7 b^3 x^{13/2}+30 a^6 b^4 x^7+\frac{168}{5} a^5 b^5 x^{15/2}+\frac{105}{4} a^4 b^6 x^8+\frac{240}{17} a^3 b^7 x^{17/2}+5 a^2 b^8 x^9+\frac{20}{19} a b^9 x^{19/2}+\frac{b^{10} x^{10}}{10}\\ \end{align*}

Mathematica [A]  time = 0.0655292, size = 140, normalized size = 1. \[ \frac{15}{2} a^8 b^2 x^6+\frac{240}{13} a^7 b^3 x^{13/2}+30 a^6 b^4 x^7+\frac{168}{5} a^5 b^5 x^{15/2}+\frac{105}{4} a^4 b^6 x^8+\frac{240}{17} a^3 b^7 x^{17/2}+5 a^2 b^8 x^9+\frac{20}{11} a^9 b x^{11/2}+\frac{a^{10} x^5}{5}+\frac{20}{19} a b^9 x^{19/2}+\frac{b^{10} x^{10}}{10} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^10*x^4,x]

[Out]

(a^10*x^5)/5 + (20*a^9*b*x^(11/2))/11 + (15*a^8*b^2*x^6)/2 + (240*a^7*b^3*x^(13/2))/13 + 30*a^6*b^4*x^7 + (168
*a^5*b^5*x^(15/2))/5 + (105*a^4*b^6*x^8)/4 + (240*a^3*b^7*x^(17/2))/17 + 5*a^2*b^8*x^9 + (20*a*b^9*x^(19/2))/1
9 + (b^10*x^10)/10

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Maple [A]  time = 0.003, size = 113, normalized size = 0.8 \begin{align*}{\frac{{a}^{10}{x}^{5}}{5}}+{\frac{20\,{a}^{9}b}{11}{x}^{{\frac{11}{2}}}}+{\frac{15\,{a}^{8}{b}^{2}{x}^{6}}{2}}+{\frac{240\,{a}^{7}{b}^{3}}{13}{x}^{{\frac{13}{2}}}}+30\,{a}^{6}{b}^{4}{x}^{7}+{\frac{168\,{a}^{5}{b}^{5}}{5}{x}^{{\frac{15}{2}}}}+{\frac{105\,{a}^{4}{b}^{6}{x}^{8}}{4}}+{\frac{240\,{a}^{3}{b}^{7}}{17}{x}^{{\frac{17}{2}}}}+5\,{a}^{2}{b}^{8}{x}^{9}+{\frac{20\,a{b}^{9}}{19}{x}^{{\frac{19}{2}}}}+{\frac{{b}^{10}{x}^{10}}{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a+b*x^(1/2))^10,x)

[Out]

1/5*a^10*x^5+20/11*a^9*b*x^(11/2)+15/2*a^8*b^2*x^6+240/13*a^7*b^3*x^(13/2)+30*a^6*b^4*x^7+168/5*a^5*b^5*x^(15/
2)+105/4*a^4*b^6*x^8+240/17*a^3*b^7*x^(17/2)+5*a^2*b^8*x^9+20/19*a*b^9*x^(19/2)+1/10*b^10*x^10

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Maxima [A]  time = 0.989901, size = 224, normalized size = 1.6 \begin{align*} \frac{{\left (b \sqrt{x} + a\right )}^{20}}{10 \, b^{10}} - \frac{18 \,{\left (b \sqrt{x} + a\right )}^{19} a}{19 \, b^{10}} + \frac{4 \,{\left (b \sqrt{x} + a\right )}^{18} a^{2}}{b^{10}} - \frac{168 \,{\left (b \sqrt{x} + a\right )}^{17} a^{3}}{17 \, b^{10}} + \frac{63 \,{\left (b \sqrt{x} + a\right )}^{16} a^{4}}{4 \, b^{10}} - \frac{84 \,{\left (b \sqrt{x} + a\right )}^{15} a^{5}}{5 \, b^{10}} + \frac{12 \,{\left (b \sqrt{x} + a\right )}^{14} a^{6}}{b^{10}} - \frac{72 \,{\left (b \sqrt{x} + a\right )}^{13} a^{7}}{13 \, b^{10}} + \frac{3 \,{\left (b \sqrt{x} + a\right )}^{12} a^{8}}{2 \, b^{10}} - \frac{2 \,{\left (b \sqrt{x} + a\right )}^{11} a^{9}}{11 \, b^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*x^(1/2))^10,x, algorithm="maxima")

[Out]

1/10*(b*sqrt(x) + a)^20/b^10 - 18/19*(b*sqrt(x) + a)^19*a/b^10 + 4*(b*sqrt(x) + a)^18*a^2/b^10 - 168/17*(b*sqr
t(x) + a)^17*a^3/b^10 + 63/4*(b*sqrt(x) + a)^16*a^4/b^10 - 84/5*(b*sqrt(x) + a)^15*a^5/b^10 + 12*(b*sqrt(x) +
a)^14*a^6/b^10 - 72/13*(b*sqrt(x) + a)^13*a^7/b^10 + 3/2*(b*sqrt(x) + a)^12*a^8/b^10 - 2/11*(b*sqrt(x) + a)^11
*a^9/b^10

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Fricas [A]  time = 1.53677, size = 302, normalized size = 2.16 \begin{align*} \frac{1}{10} \, b^{10} x^{10} + 5 \, a^{2} b^{8} x^{9} + \frac{105}{4} \, a^{4} b^{6} x^{8} + 30 \, a^{6} b^{4} x^{7} + \frac{15}{2} \, a^{8} b^{2} x^{6} + \frac{1}{5} \, a^{10} x^{5} + \frac{4}{230945} \,{\left (60775 \, a b^{9} x^{9} + 815100 \, a^{3} b^{7} x^{8} + 1939938 \, a^{5} b^{5} x^{7} + 1065900 \, a^{7} b^{3} x^{6} + 104975 \, a^{9} b x^{5}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*x^(1/2))^10,x, algorithm="fricas")

[Out]

1/10*b^10*x^10 + 5*a^2*b^8*x^9 + 105/4*a^4*b^6*x^8 + 30*a^6*b^4*x^7 + 15/2*a^8*b^2*x^6 + 1/5*a^10*x^5 + 4/2309
45*(60775*a*b^9*x^9 + 815100*a^3*b^7*x^8 + 1939938*a^5*b^5*x^7 + 1065900*a^7*b^3*x^6 + 104975*a^9*b*x^5)*sqrt(
x)

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Sympy [A]  time = 8.14843, size = 139, normalized size = 0.99 \begin{align*} \frac{a^{10} x^{5}}{5} + \frac{20 a^{9} b x^{\frac{11}{2}}}{11} + \frac{15 a^{8} b^{2} x^{6}}{2} + \frac{240 a^{7} b^{3} x^{\frac{13}{2}}}{13} + 30 a^{6} b^{4} x^{7} + \frac{168 a^{5} b^{5} x^{\frac{15}{2}}}{5} + \frac{105 a^{4} b^{6} x^{8}}{4} + \frac{240 a^{3} b^{7} x^{\frac{17}{2}}}{17} + 5 a^{2} b^{8} x^{9} + \frac{20 a b^{9} x^{\frac{19}{2}}}{19} + \frac{b^{10} x^{10}}{10} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(a+b*x**(1/2))**10,x)

[Out]

a**10*x**5/5 + 20*a**9*b*x**(11/2)/11 + 15*a**8*b**2*x**6/2 + 240*a**7*b**3*x**(13/2)/13 + 30*a**6*b**4*x**7 +
168*a**5*b**5*x**(15/2)/5 + 105*a**4*b**6*x**8/4 + 240*a**3*b**7*x**(17/2)/17 + 5*a**2*b**8*x**9 + 20*a*b**9*
x**(19/2)/19 + b**10*x**10/10

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Giac [A]  time = 1.12477, size = 151, normalized size = 1.08 \begin{align*} \frac{1}{10} \, b^{10} x^{10} + \frac{20}{19} \, a b^{9} x^{\frac{19}{2}} + 5 \, a^{2} b^{8} x^{9} + \frac{240}{17} \, a^{3} b^{7} x^{\frac{17}{2}} + \frac{105}{4} \, a^{4} b^{6} x^{8} + \frac{168}{5} \, a^{5} b^{5} x^{\frac{15}{2}} + 30 \, a^{6} b^{4} x^{7} + \frac{240}{13} \, a^{7} b^{3} x^{\frac{13}{2}} + \frac{15}{2} \, a^{8} b^{2} x^{6} + \frac{20}{11} \, a^{9} b x^{\frac{11}{2}} + \frac{1}{5} \, a^{10} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*x^(1/2))^10,x, algorithm="giac")

[Out]

1/10*b^10*x^10 + 20/19*a*b^9*x^(19/2) + 5*a^2*b^8*x^9 + 240/17*a^3*b^7*x^(17/2) + 105/4*a^4*b^6*x^8 + 168/5*a^
5*b^5*x^(15/2) + 30*a^6*b^4*x^7 + 240/13*a^7*b^3*x^(13/2) + 15/2*a^8*b^2*x^6 + 20/11*a^9*b*x^(11/2) + 1/5*a^10
*x^5