3.2169 \(\int (a+b \sqrt{x})^{15} x^5 \, dx\)

Optimal. Leaf size=242 \[ \frac{22 a^2 \left (a+b \sqrt{x}\right )^{25}}{5 b^{12}}-\frac{55 a^3 \left (a+b \sqrt{x}\right )^{24}}{4 b^{12}}+\frac{660 a^4 \left (a+b \sqrt{x}\right )^{23}}{23 b^{12}}-\frac{42 a^5 \left (a+b \sqrt{x}\right )^{22}}{b^{12}}+\frac{44 a^6 \left (a+b \sqrt{x}\right )^{21}}{b^{12}}-\frac{33 a^7 \left (a+b \sqrt{x}\right )^{20}}{b^{12}}+\frac{330 a^8 \left (a+b \sqrt{x}\right )^{19}}{19 b^{12}}-\frac{55 a^9 \left (a+b \sqrt{x}\right )^{18}}{9 b^{12}}+\frac{22 a^{10} \left (a+b \sqrt{x}\right )^{17}}{17 b^{12}}-\frac{a^{11} \left (a+b \sqrt{x}\right )^{16}}{8 b^{12}}+\frac{2 \left (a+b \sqrt{x}\right )^{27}}{27 b^{12}}-\frac{11 a \left (a+b \sqrt{x}\right )^{26}}{13 b^{12}} \]

[Out]

-(a^11*(a + b*Sqrt[x])^16)/(8*b^12) + (22*a^10*(a + b*Sqrt[x])^17)/(17*b^12) - (55*a^9*(a + b*Sqrt[x])^18)/(9*
b^12) + (330*a^8*(a + b*Sqrt[x])^19)/(19*b^12) - (33*a^7*(a + b*Sqrt[x])^20)/b^12 + (44*a^6*(a + b*Sqrt[x])^21
)/b^12 - (42*a^5*(a + b*Sqrt[x])^22)/b^12 + (660*a^4*(a + b*Sqrt[x])^23)/(23*b^12) - (55*a^3*(a + b*Sqrt[x])^2
4)/(4*b^12) + (22*a^2*(a + b*Sqrt[x])^25)/(5*b^12) - (11*a*(a + b*Sqrt[x])^26)/(13*b^12) + (2*(a + b*Sqrt[x])^
27)/(27*b^12)

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Rubi [A]  time = 0.161866, antiderivative size = 242, 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} \[ \frac{22 a^2 \left (a+b \sqrt{x}\right )^{25}}{5 b^{12}}-\frac{55 a^3 \left (a+b \sqrt{x}\right )^{24}}{4 b^{12}}+\frac{660 a^4 \left (a+b \sqrt{x}\right )^{23}}{23 b^{12}}-\frac{42 a^5 \left (a+b \sqrt{x}\right )^{22}}{b^{12}}+\frac{44 a^6 \left (a+b \sqrt{x}\right )^{21}}{b^{12}}-\frac{33 a^7 \left (a+b \sqrt{x}\right )^{20}}{b^{12}}+\frac{330 a^8 \left (a+b \sqrt{x}\right )^{19}}{19 b^{12}}-\frac{55 a^9 \left (a+b \sqrt{x}\right )^{18}}{9 b^{12}}+\frac{22 a^{10} \left (a+b \sqrt{x}\right )^{17}}{17 b^{12}}-\frac{a^{11} \left (a+b \sqrt{x}\right )^{16}}{8 b^{12}}+\frac{2 \left (a+b \sqrt{x}\right )^{27}}{27 b^{12}}-\frac{11 a \left (a+b \sqrt{x}\right )^{26}}{13 b^{12}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^15*x^5,x]

[Out]

-(a^11*(a + b*Sqrt[x])^16)/(8*b^12) + (22*a^10*(a + b*Sqrt[x])^17)/(17*b^12) - (55*a^9*(a + b*Sqrt[x])^18)/(9*
b^12) + (330*a^8*(a + b*Sqrt[x])^19)/(19*b^12) - (33*a^7*(a + b*Sqrt[x])^20)/b^12 + (44*a^6*(a + b*Sqrt[x])^21
)/b^12 - (42*a^5*(a + b*Sqrt[x])^22)/b^12 + (660*a^4*(a + b*Sqrt[x])^23)/(23*b^12) - (55*a^3*(a + b*Sqrt[x])^2
4)/(4*b^12) + (22*a^2*(a + b*Sqrt[x])^25)/(5*b^12) - (11*a*(a + b*Sqrt[x])^26)/(13*b^12) + (2*(a + b*Sqrt[x])^
27)/(27*b^12)

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 )^{15} x^5 \, dx &=2 \operatorname{Subst}\left (\int x^{11} (a+b x)^{15} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-\frac{a^{11} (a+b x)^{15}}{b^{11}}+\frac{11 a^{10} (a+b x)^{16}}{b^{11}}-\frac{55 a^9 (a+b x)^{17}}{b^{11}}+\frac{165 a^8 (a+b x)^{18}}{b^{11}}-\frac{330 a^7 (a+b x)^{19}}{b^{11}}+\frac{462 a^6 (a+b x)^{20}}{b^{11}}-\frac{462 a^5 (a+b x)^{21}}{b^{11}}+\frac{330 a^4 (a+b x)^{22}}{b^{11}}-\frac{165 a^3 (a+b x)^{23}}{b^{11}}+\frac{55 a^2 (a+b x)^{24}}{b^{11}}-\frac{11 a (a+b x)^{25}}{b^{11}}+\frac{(a+b x)^{26}}{b^{11}}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{a^{11} \left (a+b \sqrt{x}\right )^{16}}{8 b^{12}}+\frac{22 a^{10} \left (a+b \sqrt{x}\right )^{17}}{17 b^{12}}-\frac{55 a^9 \left (a+b \sqrt{x}\right )^{18}}{9 b^{12}}+\frac{330 a^8 \left (a+b \sqrt{x}\right )^{19}}{19 b^{12}}-\frac{33 a^7 \left (a+b \sqrt{x}\right )^{20}}{b^{12}}+\frac{44 a^6 \left (a+b \sqrt{x}\right )^{21}}{b^{12}}-\frac{42 a^5 \left (a+b \sqrt{x}\right )^{22}}{b^{12}}+\frac{660 a^4 \left (a+b \sqrt{x}\right )^{23}}{23 b^{12}}-\frac{55 a^3 \left (a+b \sqrt{x}\right )^{24}}{4 b^{12}}+\frac{22 a^2 \left (a+b \sqrt{x}\right )^{25}}{5 b^{12}}-\frac{11 a \left (a+b \sqrt{x}\right )^{26}}{13 b^{12}}+\frac{2 \left (a+b \sqrt{x}\right )^{27}}{27 b^{12}}\\ \end{align*}

Mathematica [A]  time = 0.112604, size = 211, normalized size = 0.87 \[ 15 a^{13} b^2 x^7+\frac{182}{3} a^{12} b^3 x^{15/2}+\frac{1365}{8} a^{11} b^4 x^8+\frac{6006}{17} a^{10} b^5 x^{17/2}+\frac{5005}{9} a^9 b^6 x^9+\frac{12870}{19} a^8 b^7 x^{19/2}+\frac{1287}{2} a^7 b^8 x^{10}+\frac{1430}{3} a^6 b^9 x^{21/2}+273 a^5 b^{10} x^{11}+\frac{2730}{23} a^4 b^{11} x^{23/2}+\frac{455}{12} a^3 b^{12} x^{12}+\frac{42}{5} a^2 b^{13} x^{25/2}+\frac{30}{13} a^{14} b x^{13/2}+\frac{a^{15} x^6}{6}+\frac{15}{13} a b^{14} x^{13}+\frac{2}{27} b^{15} x^{27/2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^15*x^5,x]

[Out]

(a^15*x^6)/6 + (30*a^14*b*x^(13/2))/13 + 15*a^13*b^2*x^7 + (182*a^12*b^3*x^(15/2))/3 + (1365*a^11*b^4*x^8)/8 +
(6006*a^10*b^5*x^(17/2))/17 + (5005*a^9*b^6*x^9)/9 + (12870*a^8*b^7*x^(19/2))/19 + (1287*a^7*b^8*x^10)/2 + (1
430*a^6*b^9*x^(21/2))/3 + 273*a^5*b^10*x^11 + (2730*a^4*b^11*x^(23/2))/23 + (455*a^3*b^12*x^12)/12 + (42*a^2*b
^13*x^(25/2))/5 + (15*a*b^14*x^13)/13 + (2*b^15*x^(27/2))/27

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Maple [A]  time = 0.004, size = 168, normalized size = 0.7 \begin{align*}{\frac{2\,{b}^{15}}{27}{x}^{{\frac{27}{2}}}}+{\frac{15\,{x}^{13}a{b}^{14}}{13}}+{\frac{42\,{a}^{2}{b}^{13}}{5}{x}^{{\frac{25}{2}}}}+{\frac{455\,{x}^{12}{a}^{3}{b}^{12}}{12}}+{\frac{2730\,{a}^{4}{b}^{11}}{23}{x}^{{\frac{23}{2}}}}+273\,{x}^{11}{a}^{5}{b}^{10}+{\frac{1430\,{a}^{6}{b}^{9}}{3}{x}^{{\frac{21}{2}}}}+{\frac{1287\,{x}^{10}{a}^{7}{b}^{8}}{2}}+{\frac{12870\,{a}^{8}{b}^{7}}{19}{x}^{{\frac{19}{2}}}}+{\frac{5005\,{x}^{9}{a}^{9}{b}^{6}}{9}}+{\frac{6006\,{a}^{10}{b}^{5}}{17}{x}^{{\frac{17}{2}}}}+{\frac{1365\,{x}^{8}{a}^{11}{b}^{4}}{8}}+{\frac{182\,{a}^{12}{b}^{3}}{3}{x}^{{\frac{15}{2}}}}+15\,{x}^{7}{a}^{13}{b}^{2}+{\frac{30\,{a}^{14}b}{13}{x}^{{\frac{13}{2}}}}+{\frac{{a}^{15}{x}^{6}}{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a+b*x^(1/2))^15,x)

[Out]

2/27*b^15*x^(27/2)+15/13*x^13*a*b^14+42/5*a^2*b^13*x^(25/2)+455/12*x^12*a^3*b^12+2730/23*a^4*b^11*x^(23/2)+273
*x^11*a^5*b^10+1430/3*a^6*b^9*x^(21/2)+1287/2*x^10*a^7*b^8+12870/19*a^8*b^7*x^(19/2)+5005/9*x^9*a^9*b^6+6006/1
7*a^10*b^5*x^(17/2)+1365/8*x^8*a^11*b^4+182/3*a^12*b^3*x^(15/2)+15*x^7*a^13*b^2+30/13*a^14*b*x^(13/2)+1/6*a^15
*x^6

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Maxima [A]  time = 0.970632, size = 270, normalized size = 1.12 \begin{align*} \frac{2 \,{\left (b \sqrt{x} + a\right )}^{27}}{27 \, b^{12}} - \frac{11 \,{\left (b \sqrt{x} + a\right )}^{26} a}{13 \, b^{12}} + \frac{22 \,{\left (b \sqrt{x} + a\right )}^{25} a^{2}}{5 \, b^{12}} - \frac{55 \,{\left (b \sqrt{x} + a\right )}^{24} a^{3}}{4 \, b^{12}} + \frac{660 \,{\left (b \sqrt{x} + a\right )}^{23} a^{4}}{23 \, b^{12}} - \frac{42 \,{\left (b \sqrt{x} + a\right )}^{22} a^{5}}{b^{12}} + \frac{44 \,{\left (b \sqrt{x} + a\right )}^{21} a^{6}}{b^{12}} - \frac{33 \,{\left (b \sqrt{x} + a\right )}^{20} a^{7}}{b^{12}} + \frac{330 \,{\left (b \sqrt{x} + a\right )}^{19} a^{8}}{19 \, b^{12}} - \frac{55 \,{\left (b \sqrt{x} + a\right )}^{18} a^{9}}{9 \, b^{12}} + \frac{22 \,{\left (b \sqrt{x} + a\right )}^{17} a^{10}}{17 \, b^{12}} - \frac{{\left (b \sqrt{x} + a\right )}^{16} a^{11}}{8 \, b^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*x^(1/2))^15,x, algorithm="maxima")

[Out]

2/27*(b*sqrt(x) + a)^27/b^12 - 11/13*(b*sqrt(x) + a)^26*a/b^12 + 22/5*(b*sqrt(x) + a)^25*a^2/b^12 - 55/4*(b*sq
rt(x) + a)^24*a^3/b^12 + 660/23*(b*sqrt(x) + a)^23*a^4/b^12 - 42*(b*sqrt(x) + a)^22*a^5/b^12 + 44*(b*sqrt(x) +
a)^21*a^6/b^12 - 33*(b*sqrt(x) + a)^20*a^7/b^12 + 330/19*(b*sqrt(x) + a)^19*a^8/b^12 - 55/9*(b*sqrt(x) + a)^1
8*a^9/b^12 + 22/17*(b*sqrt(x) + a)^17*a^10/b^12 - 1/8*(b*sqrt(x) + a)^16*a^11/b^12

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Fricas [A]  time = 1.31041, size = 506, normalized size = 2.09 \begin{align*} \frac{15}{13} \, a b^{14} x^{13} + \frac{455}{12} \, a^{3} b^{12} x^{12} + 273 \, a^{5} b^{10} x^{11} + \frac{1287}{2} \, a^{7} b^{8} x^{10} + \frac{5005}{9} \, a^{9} b^{6} x^{9} + \frac{1365}{8} \, a^{11} b^{4} x^{8} + 15 \, a^{13} b^{2} x^{7} + \frac{1}{6} \, a^{15} x^{6} + \frac{2}{13037895} \,{\left (482885 \, b^{15} x^{13} + 54759159 \, a^{2} b^{13} x^{12} + 773770725 \, a^{4} b^{11} x^{11} + 3107364975 \, a^{6} b^{9} x^{10} + 4415729175 \, a^{8} b^{7} x^{9} + 2303105805 \, a^{10} b^{5} x^{8} + 395482815 \, a^{12} b^{3} x^{7} + 15043725 \, a^{14} b x^{6}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*x^(1/2))^15,x, algorithm="fricas")

[Out]

15/13*a*b^14*x^13 + 455/12*a^3*b^12*x^12 + 273*a^5*b^10*x^11 + 1287/2*a^7*b^8*x^10 + 5005/9*a^9*b^6*x^9 + 1365
/8*a^11*b^4*x^8 + 15*a^13*b^2*x^7 + 1/6*a^15*x^6 + 2/13037895*(482885*b^15*x^13 + 54759159*a^2*b^13*x^12 + 773
770725*a^4*b^11*x^11 + 3107364975*a^6*b^9*x^10 + 4415729175*a^8*b^7*x^9 + 2303105805*a^10*b^5*x^8 + 395482815*
a^12*b^3*x^7 + 15043725*a^14*b*x^6)*sqrt(x)

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Sympy [A]  time = 7.83891, size = 214, normalized size = 0.88 \begin{align*} \frac{a^{15} x^{6}}{6} + \frac{30 a^{14} b x^{\frac{13}{2}}}{13} + 15 a^{13} b^{2} x^{7} + \frac{182 a^{12} b^{3} x^{\frac{15}{2}}}{3} + \frac{1365 a^{11} b^{4} x^{8}}{8} + \frac{6006 a^{10} b^{5} x^{\frac{17}{2}}}{17} + \frac{5005 a^{9} b^{6} x^{9}}{9} + \frac{12870 a^{8} b^{7} x^{\frac{19}{2}}}{19} + \frac{1287 a^{7} b^{8} x^{10}}{2} + \frac{1430 a^{6} b^{9} x^{\frac{21}{2}}}{3} + 273 a^{5} b^{10} x^{11} + \frac{2730 a^{4} b^{11} x^{\frac{23}{2}}}{23} + \frac{455 a^{3} b^{12} x^{12}}{12} + \frac{42 a^{2} b^{13} x^{\frac{25}{2}}}{5} + \frac{15 a b^{14} x^{13}}{13} + \frac{2 b^{15} x^{\frac{27}{2}}}{27} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(a+b*x**(1/2))**15,x)

[Out]

a**15*x**6/6 + 30*a**14*b*x**(13/2)/13 + 15*a**13*b**2*x**7 + 182*a**12*b**3*x**(15/2)/3 + 1365*a**11*b**4*x**
8/8 + 6006*a**10*b**5*x**(17/2)/17 + 5005*a**9*b**6*x**9/9 + 12870*a**8*b**7*x**(19/2)/19 + 1287*a**7*b**8*x**
10/2 + 1430*a**6*b**9*x**(21/2)/3 + 273*a**5*b**10*x**11 + 2730*a**4*b**11*x**(23/2)/23 + 455*a**3*b**12*x**12
/12 + 42*a**2*b**13*x**(25/2)/5 + 15*a*b**14*x**13/13 + 2*b**15*x**(27/2)/27

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Giac [A]  time = 1.17835, size = 225, normalized size = 0.93 \begin{align*} \frac{2}{27} \, b^{15} x^{\frac{27}{2}} + \frac{15}{13} \, a b^{14} x^{13} + \frac{42}{5} \, a^{2} b^{13} x^{\frac{25}{2}} + \frac{455}{12} \, a^{3} b^{12} x^{12} + \frac{2730}{23} \, a^{4} b^{11} x^{\frac{23}{2}} + 273 \, a^{5} b^{10} x^{11} + \frac{1430}{3} \, a^{6} b^{9} x^{\frac{21}{2}} + \frac{1287}{2} \, a^{7} b^{8} x^{10} + \frac{12870}{19} \, a^{8} b^{7} x^{\frac{19}{2}} + \frac{5005}{9} \, a^{9} b^{6} x^{9} + \frac{6006}{17} \, a^{10} b^{5} x^{\frac{17}{2}} + \frac{1365}{8} \, a^{11} b^{4} x^{8} + \frac{182}{3} \, a^{12} b^{3} x^{\frac{15}{2}} + 15 \, a^{13} b^{2} x^{7} + \frac{30}{13} \, a^{14} b x^{\frac{13}{2}} + \frac{1}{6} \, a^{15} x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*x^(1/2))^15,x, algorithm="giac")

[Out]

2/27*b^15*x^(27/2) + 15/13*a*b^14*x^13 + 42/5*a^2*b^13*x^(25/2) + 455/12*a^3*b^12*x^12 + 2730/23*a^4*b^11*x^(2
3/2) + 273*a^5*b^10*x^11 + 1430/3*a^6*b^9*x^(21/2) + 1287/2*a^7*b^8*x^10 + 12870/19*a^8*b^7*x^(19/2) + 5005/9*
a^9*b^6*x^9 + 6006/17*a^10*b^5*x^(17/2) + 1365/8*a^11*b^4*x^8 + 182/3*a^12*b^3*x^(15/2) + 15*a^13*b^2*x^7 + 30
/13*a^14*b*x^(13/2) + 1/6*a^15*x^6