3.2170 \(\int (a+b \sqrt{x})^{15} x^4 \, dx\)

Optimal. Leaf size=202 \[ \frac{72 a^2 \left (a+b \sqrt{x}\right )^{23}}{23 b^{10}}-\frac{84 a^3 \left (a+b \sqrt{x}\right )^{22}}{11 b^{10}}+\frac{12 a^4 \left (a+b \sqrt{x}\right )^{21}}{b^{10}}-\frac{63 a^5 \left (a+b \sqrt{x}\right )^{20}}{5 b^{10}}+\frac{168 a^6 \left (a+b \sqrt{x}\right )^{19}}{19 b^{10}}-\frac{4 a^7 \left (a+b \sqrt{x}\right )^{18}}{b^{10}}+\frac{18 a^8 \left (a+b \sqrt{x}\right )^{17}}{17 b^{10}}-\frac{a^9 \left (a+b \sqrt{x}\right )^{16}}{8 b^{10}}+\frac{2 \left (a+b \sqrt{x}\right )^{25}}{25 b^{10}}-\frac{3 a \left (a+b \sqrt{x}\right )^{24}}{4 b^{10}} \]

[Out]

-(a^9*(a + b*Sqrt[x])^16)/(8*b^10) + (18*a^8*(a + b*Sqrt[x])^17)/(17*b^10) - (4*a^7*(a + b*Sqrt[x])^18)/b^10 +
(168*a^6*(a + b*Sqrt[x])^19)/(19*b^10) - (63*a^5*(a + b*Sqrt[x])^20)/(5*b^10) + (12*a^4*(a + b*Sqrt[x])^21)/b
^10 - (84*a^3*(a + b*Sqrt[x])^22)/(11*b^10) + (72*a^2*(a + b*Sqrt[x])^23)/(23*b^10) - (3*a*(a + b*Sqrt[x])^24)
/(4*b^10) + (2*(a + b*Sqrt[x])^25)/(25*b^10)

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Rubi [A]  time = 0.11175, antiderivative size = 202, 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} \[ \frac{72 a^2 \left (a+b \sqrt{x}\right )^{23}}{23 b^{10}}-\frac{84 a^3 \left (a+b \sqrt{x}\right )^{22}}{11 b^{10}}+\frac{12 a^4 \left (a+b \sqrt{x}\right )^{21}}{b^{10}}-\frac{63 a^5 \left (a+b \sqrt{x}\right )^{20}}{5 b^{10}}+\frac{168 a^6 \left (a+b \sqrt{x}\right )^{19}}{19 b^{10}}-\frac{4 a^7 \left (a+b \sqrt{x}\right )^{18}}{b^{10}}+\frac{18 a^8 \left (a+b \sqrt{x}\right )^{17}}{17 b^{10}}-\frac{a^9 \left (a+b \sqrt{x}\right )^{16}}{8 b^{10}}+\frac{2 \left (a+b \sqrt{x}\right )^{25}}{25 b^{10}}-\frac{3 a \left (a+b \sqrt{x}\right )^{24}}{4 b^{10}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^15*x^4,x]

[Out]

-(a^9*(a + b*Sqrt[x])^16)/(8*b^10) + (18*a^8*(a + b*Sqrt[x])^17)/(17*b^10) - (4*a^7*(a + b*Sqrt[x])^18)/b^10 +
(168*a^6*(a + b*Sqrt[x])^19)/(19*b^10) - (63*a^5*(a + b*Sqrt[x])^20)/(5*b^10) + (12*a^4*(a + b*Sqrt[x])^21)/b
^10 - (84*a^3*(a + b*Sqrt[x])^22)/(11*b^10) + (72*a^2*(a + b*Sqrt[x])^23)/(23*b^10) - (3*a*(a + b*Sqrt[x])^24)
/(4*b^10) + (2*(a + b*Sqrt[x])^25)/(25*b^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 )^{15} x^4 \, dx &=2 \operatorname{Subst}\left (\int x^9 (a+b x)^{15} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-\frac{a^9 (a+b x)^{15}}{b^9}+\frac{9 a^8 (a+b x)^{16}}{b^9}-\frac{36 a^7 (a+b x)^{17}}{b^9}+\frac{84 a^6 (a+b x)^{18}}{b^9}-\frac{126 a^5 (a+b x)^{19}}{b^9}+\frac{126 a^4 (a+b x)^{20}}{b^9}-\frac{84 a^3 (a+b x)^{21}}{b^9}+\frac{36 a^2 (a+b x)^{22}}{b^9}-\frac{9 a (a+b x)^{23}}{b^9}+\frac{(a+b x)^{24}}{b^9}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{a^9 \left (a+b \sqrt{x}\right )^{16}}{8 b^{10}}+\frac{18 a^8 \left (a+b \sqrt{x}\right )^{17}}{17 b^{10}}-\frac{4 a^7 \left (a+b \sqrt{x}\right )^{18}}{b^{10}}+\frac{168 a^6 \left (a+b \sqrt{x}\right )^{19}}{19 b^{10}}-\frac{63 a^5 \left (a+b \sqrt{x}\right )^{20}}{5 b^{10}}+\frac{12 a^4 \left (a+b \sqrt{x}\right )^{21}}{b^{10}}-\frac{84 a^3 \left (a+b \sqrt{x}\right )^{22}}{11 b^{10}}+\frac{72 a^2 \left (a+b \sqrt{x}\right )^{23}}{23 b^{10}}-\frac{3 a \left (a+b \sqrt{x}\right )^{24}}{4 b^{10}}+\frac{2 \left (a+b \sqrt{x}\right )^{25}}{25 b^{10}}\\ \end{align*}

Mathematica [A]  time = 0.0919145, size = 122, normalized size = 0.6 \[ -\frac{\left (a+b \sqrt{x}\right )^{16} \left (-816 a^6 b^3 x^{3/2}+3876 a^5 b^4 x^2-15504 a^4 b^5 x^{5/2}+54264 a^3 b^6 x^3-170544 a^2 b^7 x^{7/2}+136 a^7 b^2 x-16 a^8 b \sqrt{x}+a^9+490314 a b^8 x^4-1307504 b^9 x^{9/2}\right )}{16343800 b^{10}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^15*x^4,x]

[Out]

-((a + b*Sqrt[x])^16*(a^9 - 16*a^8*b*Sqrt[x] + 136*a^7*b^2*x - 816*a^6*b^3*x^(3/2) + 3876*a^5*b^4*x^2 - 15504*
a^4*b^5*x^(5/2) + 54264*a^3*b^6*x^3 - 170544*a^2*b^7*x^(7/2) + 490314*a*b^8*x^4 - 1307504*b^9*x^(9/2)))/(16343
800*b^10)

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Maple [A]  time = 0.003, size = 168, normalized size = 0.8 \begin{align*}{\frac{2\,{b}^{15}}{25}{x}^{{\frac{25}{2}}}}+{\frac{5\,{x}^{12}a{b}^{14}}{4}}+{\frac{210\,{a}^{2}{b}^{13}}{23}{x}^{{\frac{23}{2}}}}+{\frac{455\,{x}^{11}{a}^{3}{b}^{12}}{11}}+130\,{x}^{21/2}{a}^{4}{b}^{11}+{\frac{3003\,{x}^{10}{a}^{5}{b}^{10}}{10}}+{\frac{10010\,{a}^{6}{b}^{9}}{19}{x}^{{\frac{19}{2}}}}+715\,{x}^{9}{a}^{7}{b}^{8}+{\frac{12870\,{a}^{8}{b}^{7}}{17}{x}^{{\frac{17}{2}}}}+{\frac{5005\,{a}^{9}{b}^{6}{x}^{8}}{8}}+{\frac{2002\,{a}^{10}{b}^{5}}{5}{x}^{{\frac{15}{2}}}}+195\,{x}^{7}{a}^{11}{b}^{4}+70\,{x}^{13/2}{a}^{12}{b}^{3}+{\frac{35\,{x}^{6}{a}^{13}{b}^{2}}{2}}+{\frac{30\,{a}^{14}b}{11}{x}^{{\frac{11}{2}}}}+{\frac{{a}^{15}{x}^{5}}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a+b*x^(1/2))^15,x)

[Out]

2/25*x^(25/2)*b^15+5/4*x^12*a*b^14+210/23*x^(23/2)*a^2*b^13+455/11*x^11*a^3*b^12+130*x^(21/2)*a^4*b^11+3003/10
*x^10*a^5*b^10+10010/19*x^(19/2)*a^6*b^9+715*x^9*a^7*b^8+12870/17*x^(17/2)*a^8*b^7+5005/8*a^9*b^6*x^8+2002/5*x
^(15/2)*a^10*b^5+195*x^7*a^11*b^4+70*x^(13/2)*a^12*b^3+35/2*x^6*a^13*b^2+30/11*x^(11/2)*a^14*b+1/5*a^15*x^5

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Maxima [A]  time = 0.983489, size = 224, normalized size = 1.11 \begin{align*} \frac{2 \,{\left (b \sqrt{x} + a\right )}^{25}}{25 \, b^{10}} - \frac{3 \,{\left (b \sqrt{x} + a\right )}^{24} a}{4 \, b^{10}} + \frac{72 \,{\left (b \sqrt{x} + a\right )}^{23} a^{2}}{23 \, b^{10}} - \frac{84 \,{\left (b \sqrt{x} + a\right )}^{22} a^{3}}{11 \, b^{10}} + \frac{12 \,{\left (b \sqrt{x} + a\right )}^{21} a^{4}}{b^{10}} - \frac{63 \,{\left (b \sqrt{x} + a\right )}^{20} a^{5}}{5 \, b^{10}} + \frac{168 \,{\left (b \sqrt{x} + a\right )}^{19} a^{6}}{19 \, b^{10}} - \frac{4 \,{\left (b \sqrt{x} + a\right )}^{18} a^{7}}{b^{10}} + \frac{18 \,{\left (b \sqrt{x} + a\right )}^{17} a^{8}}{17 \, b^{10}} - \frac{{\left (b \sqrt{x} + a\right )}^{16} a^{9}}{8 \, b^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*x^(1/2))^15,x, algorithm="maxima")

[Out]

2/25*(b*sqrt(x) + a)^25/b^10 - 3/4*(b*sqrt(x) + a)^24*a/b^10 + 72/23*(b*sqrt(x) + a)^23*a^2/b^10 - 84/11*(b*sq
rt(x) + a)^22*a^3/b^10 + 12*(b*sqrt(x) + a)^21*a^4/b^10 - 63/5*(b*sqrt(x) + a)^20*a^5/b^10 + 168/19*(b*sqrt(x)
+ a)^19*a^6/b^10 - 4*(b*sqrt(x) + a)^18*a^7/b^10 + 18/17*(b*sqrt(x) + a)^17*a^8/b^10 - 1/8*(b*sqrt(x) + a)^16
*a^9/b^10

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Fricas [A]  time = 1.26305, size = 490, normalized size = 2.43 \begin{align*} \frac{5}{4} \, a b^{14} x^{12} + \frac{455}{11} \, a^{3} b^{12} x^{11} + \frac{3003}{10} \, a^{5} b^{10} x^{10} + 715 \, a^{7} b^{8} x^{9} + \frac{5005}{8} \, a^{9} b^{6} x^{8} + 195 \, a^{11} b^{4} x^{7} + \frac{35}{2} \, a^{13} b^{2} x^{6} + \frac{1}{5} \, a^{15} x^{5} + \frac{2}{2042975} \,{\left (81719 \, b^{15} x^{12} + 9326625 \, a^{2} b^{13} x^{11} + 132793375 \, a^{4} b^{11} x^{10} + 538162625 \, a^{6} b^{9} x^{9} + 773326125 \, a^{8} b^{7} x^{8} + 409003595 \, a^{10} b^{5} x^{7} + 71504125 \, a^{12} b^{3} x^{6} + 2785875 \, a^{14} 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))^15,x, algorithm="fricas")

[Out]

5/4*a*b^14*x^12 + 455/11*a^3*b^12*x^11 + 3003/10*a^5*b^10*x^10 + 715*a^7*b^8*x^9 + 5005/8*a^9*b^6*x^8 + 195*a^
11*b^4*x^7 + 35/2*a^13*b^2*x^6 + 1/5*a^15*x^5 + 2/2042975*(81719*b^15*x^12 + 9326625*a^2*b^13*x^11 + 132793375
*a^4*b^11*x^10 + 538162625*a^6*b^9*x^9 + 773326125*a^8*b^7*x^8 + 409003595*a^10*b^5*x^7 + 71504125*a^12*b^3*x^
6 + 2785875*a^14*b*x^5)*sqrt(x)

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Sympy [A]  time = 6.78106, size = 211, normalized size = 1.04 \begin{align*} \frac{a^{15} x^{5}}{5} + \frac{30 a^{14} b x^{\frac{11}{2}}}{11} + \frac{35 a^{13} b^{2} x^{6}}{2} + 70 a^{12} b^{3} x^{\frac{13}{2}} + 195 a^{11} b^{4} x^{7} + \frac{2002 a^{10} b^{5} x^{\frac{15}{2}}}{5} + \frac{5005 a^{9} b^{6} x^{8}}{8} + \frac{12870 a^{8} b^{7} x^{\frac{17}{2}}}{17} + 715 a^{7} b^{8} x^{9} + \frac{10010 a^{6} b^{9} x^{\frac{19}{2}}}{19} + \frac{3003 a^{5} b^{10} x^{10}}{10} + 130 a^{4} b^{11} x^{\frac{21}{2}} + \frac{455 a^{3} b^{12} x^{11}}{11} + \frac{210 a^{2} b^{13} x^{\frac{23}{2}}}{23} + \frac{5 a b^{14} x^{12}}{4} + \frac{2 b^{15} x^{\frac{25}{2}}}{25} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(a+b*x**(1/2))**15,x)

[Out]

a**15*x**5/5 + 30*a**14*b*x**(11/2)/11 + 35*a**13*b**2*x**6/2 + 70*a**12*b**3*x**(13/2) + 195*a**11*b**4*x**7
+ 2002*a**10*b**5*x**(15/2)/5 + 5005*a**9*b**6*x**8/8 + 12870*a**8*b**7*x**(17/2)/17 + 715*a**7*b**8*x**9 + 10
010*a**6*b**9*x**(19/2)/19 + 3003*a**5*b**10*x**10/10 + 130*a**4*b**11*x**(21/2) + 455*a**3*b**12*x**11/11 + 2
10*a**2*b**13*x**(23/2)/23 + 5*a*b**14*x**12/4 + 2*b**15*x**(25/2)/25

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Giac [A]  time = 1.12385, size = 225, normalized size = 1.11 \begin{align*} \frac{2}{25} \, b^{15} x^{\frac{25}{2}} + \frac{5}{4} \, a b^{14} x^{12} + \frac{210}{23} \, a^{2} b^{13} x^{\frac{23}{2}} + \frac{455}{11} \, a^{3} b^{12} x^{11} + 130 \, a^{4} b^{11} x^{\frac{21}{2}} + \frac{3003}{10} \, a^{5} b^{10} x^{10} + \frac{10010}{19} \, a^{6} b^{9} x^{\frac{19}{2}} + 715 \, a^{7} b^{8} x^{9} + \frac{12870}{17} \, a^{8} b^{7} x^{\frac{17}{2}} + \frac{5005}{8} \, a^{9} b^{6} x^{8} + \frac{2002}{5} \, a^{10} b^{5} x^{\frac{15}{2}} + 195 \, a^{11} b^{4} x^{7} + 70 \, a^{12} b^{3} x^{\frac{13}{2}} + \frac{35}{2} \, a^{13} b^{2} x^{6} + \frac{30}{11} \, a^{14} b x^{\frac{11}{2}} + \frac{1}{5} \, a^{15} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*x^(1/2))^15,x, algorithm="giac")

[Out]

2/25*b^15*x^(25/2) + 5/4*a*b^14*x^12 + 210/23*a^2*b^13*x^(23/2) + 455/11*a^3*b^12*x^11 + 130*a^4*b^11*x^(21/2)
+ 3003/10*a^5*b^10*x^10 + 10010/19*a^6*b^9*x^(19/2) + 715*a^7*b^8*x^9 + 12870/17*a^8*b^7*x^(17/2) + 5005/8*a^
9*b^6*x^8 + 2002/5*a^10*b^5*x^(15/2) + 195*a^11*b^4*x^7 + 70*a^12*b^3*x^(13/2) + 35/2*a^13*b^2*x^6 + 30/11*a^1
4*b*x^(11/2) + 1/5*a^15*x^5