3.2171 \(\int (a+b \sqrt{x})^{15} x^3 \, dx\)

Optimal. Leaf size=162 \[ \frac{2 a^2 \left (a+b \sqrt{x}\right )^{21}}{b^8}-\frac{7 a^3 \left (a+b \sqrt{x}\right )^{20}}{2 b^8}+\frac{70 a^4 \left (a+b \sqrt{x}\right )^{19}}{19 b^8}-\frac{7 a^5 \left (a+b \sqrt{x}\right )^{18}}{3 b^8}+\frac{14 a^6 \left (a+b \sqrt{x}\right )^{17}}{17 b^8}-\frac{a^7 \left (a+b \sqrt{x}\right )^{16}}{8 b^8}+\frac{2 \left (a+b \sqrt{x}\right )^{23}}{23 b^8}-\frac{7 a \left (a+b \sqrt{x}\right )^{22}}{11 b^8} \]

[Out]

-(a^7*(a + b*Sqrt[x])^16)/(8*b^8) + (14*a^6*(a + b*Sqrt[x])^17)/(17*b^8) - (7*a^5*(a + b*Sqrt[x])^18)/(3*b^8)
+ (70*a^4*(a + b*Sqrt[x])^19)/(19*b^8) - (7*a^3*(a + b*Sqrt[x])^20)/(2*b^8) + (2*a^2*(a + b*Sqrt[x])^21)/b^8 -
(7*a*(a + b*Sqrt[x])^22)/(11*b^8) + (2*(a + b*Sqrt[x])^23)/(23*b^8)

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Rubi [A]  time = 0.0918026, antiderivative size = 162, 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} \[ \frac{2 a^2 \left (a+b \sqrt{x}\right )^{21}}{b^8}-\frac{7 a^3 \left (a+b \sqrt{x}\right )^{20}}{2 b^8}+\frac{70 a^4 \left (a+b \sqrt{x}\right )^{19}}{19 b^8}-\frac{7 a^5 \left (a+b \sqrt{x}\right )^{18}}{3 b^8}+\frac{14 a^6 \left (a+b \sqrt{x}\right )^{17}}{17 b^8}-\frac{a^7 \left (a+b \sqrt{x}\right )^{16}}{8 b^8}+\frac{2 \left (a+b \sqrt{x}\right )^{23}}{23 b^8}-\frac{7 a \left (a+b \sqrt{x}\right )^{22}}{11 b^8} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^15*x^3,x]

[Out]

-(a^7*(a + b*Sqrt[x])^16)/(8*b^8) + (14*a^6*(a + b*Sqrt[x])^17)/(17*b^8) - (7*a^5*(a + b*Sqrt[x])^18)/(3*b^8)
+ (70*a^4*(a + b*Sqrt[x])^19)/(19*b^8) - (7*a^3*(a + b*Sqrt[x])^20)/(2*b^8) + (2*a^2*(a + b*Sqrt[x])^21)/b^8 -
(7*a*(a + b*Sqrt[x])^22)/(11*b^8) + (2*(a + b*Sqrt[x])^23)/(23*b^8)

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^3 \, dx &=2 \operatorname{Subst}\left (\int x^7 (a+b x)^{15} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-\frac{a^7 (a+b x)^{15}}{b^7}+\frac{7 a^6 (a+b x)^{16}}{b^7}-\frac{21 a^5 (a+b x)^{17}}{b^7}+\frac{35 a^4 (a+b x)^{18}}{b^7}-\frac{35 a^3 (a+b x)^{19}}{b^7}+\frac{21 a^2 (a+b x)^{20}}{b^7}-\frac{7 a (a+b x)^{21}}{b^7}+\frac{(a+b x)^{22}}{b^7}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{a^7 \left (a+b \sqrt{x}\right )^{16}}{8 b^8}+\frac{14 a^6 \left (a+b \sqrt{x}\right )^{17}}{17 b^8}-\frac{7 a^5 \left (a+b \sqrt{x}\right )^{18}}{3 b^8}+\frac{70 a^4 \left (a+b \sqrt{x}\right )^{19}}{19 b^8}-\frac{7 a^3 \left (a+b \sqrt{x}\right )^{20}}{2 b^8}+\frac{2 a^2 \left (a+b \sqrt{x}\right )^{21}}{b^8}-\frac{7 a \left (a+b \sqrt{x}\right )^{22}}{11 b^8}+\frac{2 \left (a+b \sqrt{x}\right )^{23}}{23 b^8}\\ \end{align*}

Mathematica [A]  time = 0.0752743, size = 98, normalized size = 0.6 \[ -\frac{\left (a+b \sqrt{x}\right )^{16} \left (-816 a^4 b^3 x^{3/2}+3876 a^3 b^4 x^2-15504 a^2 b^5 x^{5/2}+136 a^5 b^2 x-16 a^6 b \sqrt{x}+a^7+54264 a b^6 x^3-170544 b^7 x^{7/2}\right )}{1961256 b^8} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^15*x^3,x]

[Out]

-((a + b*Sqrt[x])^16*(a^7 - 16*a^6*b*Sqrt[x] + 136*a^5*b^2*x - 816*a^4*b^3*x^(3/2) + 3876*a^3*b^4*x^2 - 15504*
a^2*b^5*x^(5/2) + 54264*a*b^6*x^3 - 170544*b^7*x^(7/2)))/(1961256*b^8)

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Maple [A]  time = 0.003, size = 168, normalized size = 1. \begin{align*}{\frac{2\,{b}^{15}}{23}{x}^{{\frac{23}{2}}}}+{\frac{15\,{x}^{11}a{b}^{14}}{11}}+10\,{x}^{21/2}{a}^{2}{b}^{13}+{\frac{91\,{a}^{3}{b}^{12}{x}^{10}}{2}}+{\frac{2730\,{a}^{4}{b}^{11}}{19}{x}^{{\frac{19}{2}}}}+{\frac{1001\,{x}^{9}{a}^{5}{b}^{10}}{3}}+{\frac{10010\,{a}^{6}{b}^{9}}{17}{x}^{{\frac{17}{2}}}}+{\frac{6435\,{x}^{8}{a}^{7}{b}^{8}}{8}}+858\,{x}^{15/2}{a}^{8}{b}^{7}+715\,{a}^{9}{b}^{6}{x}^{7}+462\,{x}^{13/2}{a}^{10}{b}^{5}+{\frac{455\,{x}^{6}{a}^{11}{b}^{4}}{2}}+{\frac{910\,{a}^{12}{b}^{3}}{11}{x}^{{\frac{11}{2}}}}+21\,{x}^{5}{a}^{13}{b}^{2}+{\frac{10\,{a}^{14}b}{3}{x}^{{\frac{9}{2}}}}+{\frac{{x}^{4}{a}^{15}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a+b*x^(1/2))^15,x)

[Out]

2/23*x^(23/2)*b^15+15/11*x^11*a*b^14+10*x^(21/2)*a^2*b^13+91/2*a^3*b^12*x^10+2730/19*x^(19/2)*a^4*b^11+1001/3*
x^9*a^5*b^10+10010/17*x^(17/2)*a^6*b^9+6435/8*x^8*a^7*b^8+858*x^(15/2)*a^8*b^7+715*a^9*b^6*x^7+462*x^(13/2)*a^
10*b^5+455/2*x^6*a^11*b^4+910/11*x^(11/2)*a^12*b^3+21*x^5*a^13*b^2+10/3*x^(9/2)*a^14*b+1/4*x^4*a^15

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Maxima [A]  time = 0.961047, size = 178, normalized size = 1.1 \begin{align*} \frac{2 \,{\left (b \sqrt{x} + a\right )}^{23}}{23 \, b^{8}} - \frac{7 \,{\left (b \sqrt{x} + a\right )}^{22} a}{11 \, b^{8}} + \frac{2 \,{\left (b \sqrt{x} + a\right )}^{21} a^{2}}{b^{8}} - \frac{7 \,{\left (b \sqrt{x} + a\right )}^{20} a^{3}}{2 \, b^{8}} + \frac{70 \,{\left (b \sqrt{x} + a\right )}^{19} a^{4}}{19 \, b^{8}} - \frac{7 \,{\left (b \sqrt{x} + a\right )}^{18} a^{5}}{3 \, b^{8}} + \frac{14 \,{\left (b \sqrt{x} + a\right )}^{17} a^{6}}{17 \, b^{8}} - \frac{{\left (b \sqrt{x} + a\right )}^{16} a^{7}}{8 \, b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*x^(1/2))^15,x, algorithm="maxima")

[Out]

2/23*(b*sqrt(x) + a)^23/b^8 - 7/11*(b*sqrt(x) + a)^22*a/b^8 + 2*(b*sqrt(x) + a)^21*a^2/b^8 - 7/2*(b*sqrt(x) +
a)^20*a^3/b^8 + 70/19*(b*sqrt(x) + a)^19*a^4/b^8 - 7/3*(b*sqrt(x) + a)^18*a^5/b^8 + 14/17*(b*sqrt(x) + a)^17*a
^6/b^8 - 1/8*(b*sqrt(x) + a)^16*a^7/b^8

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Fricas [A]  time = 1.3559, size = 479, normalized size = 2.96 \begin{align*} \frac{15}{11} \, a b^{14} x^{11} + \frac{91}{2} \, a^{3} b^{12} x^{10} + \frac{1001}{3} \, a^{5} b^{10} x^{9} + \frac{6435}{8} \, a^{7} b^{8} x^{8} + 715 \, a^{9} b^{6} x^{7} + \frac{455}{2} \, a^{11} b^{4} x^{6} + 21 \, a^{13} b^{2} x^{5} + \frac{1}{4} \, a^{15} x^{4} + \frac{2}{245157} \,{\left (10659 \, b^{15} x^{11} + 1225785 \, a^{2} b^{13} x^{10} + 17612595 \, a^{4} b^{11} x^{9} + 72177105 \, a^{6} b^{9} x^{8} + 105172353 \, a^{8} b^{7} x^{7} + 56631267 \, a^{10} b^{5} x^{6} + 10140585 \, a^{12} b^{3} x^{5} + 408595 \, a^{14} b x^{4}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*x^(1/2))^15,x, algorithm="fricas")

[Out]

15/11*a*b^14*x^11 + 91/2*a^3*b^12*x^10 + 1001/3*a^5*b^10*x^9 + 6435/8*a^7*b^8*x^8 + 715*a^9*b^6*x^7 + 455/2*a^
11*b^4*x^6 + 21*a^13*b^2*x^5 + 1/4*a^15*x^4 + 2/245157*(10659*b^15*x^11 + 1225785*a^2*b^13*x^10 + 17612595*a^4
*b^11*x^9 + 72177105*a^6*b^9*x^8 + 105172353*a^8*b^7*x^7 + 56631267*a^10*b^5*x^6 + 10140585*a^12*b^3*x^5 + 408
595*a^14*b*x^4)*sqrt(x)

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Sympy [A]  time = 5.81943, size = 209, normalized size = 1.29 \begin{align*} \frac{a^{15} x^{4}}{4} + \frac{10 a^{14} b x^{\frac{9}{2}}}{3} + 21 a^{13} b^{2} x^{5} + \frac{910 a^{12} b^{3} x^{\frac{11}{2}}}{11} + \frac{455 a^{11} b^{4} x^{6}}{2} + 462 a^{10} b^{5} x^{\frac{13}{2}} + 715 a^{9} b^{6} x^{7} + 858 a^{8} b^{7} x^{\frac{15}{2}} + \frac{6435 a^{7} b^{8} x^{8}}{8} + \frac{10010 a^{6} b^{9} x^{\frac{17}{2}}}{17} + \frac{1001 a^{5} b^{10} x^{9}}{3} + \frac{2730 a^{4} b^{11} x^{\frac{19}{2}}}{19} + \frac{91 a^{3} b^{12} x^{10}}{2} + 10 a^{2} b^{13} x^{\frac{21}{2}} + \frac{15 a b^{14} x^{11}}{11} + \frac{2 b^{15} x^{\frac{23}{2}}}{23} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(a+b*x**(1/2))**15,x)

[Out]

a**15*x**4/4 + 10*a**14*b*x**(9/2)/3 + 21*a**13*b**2*x**5 + 910*a**12*b**3*x**(11/2)/11 + 455*a**11*b**4*x**6/
2 + 462*a**10*b**5*x**(13/2) + 715*a**9*b**6*x**7 + 858*a**8*b**7*x**(15/2) + 6435*a**7*b**8*x**8/8 + 10010*a*
*6*b**9*x**(17/2)/17 + 1001*a**5*b**10*x**9/3 + 2730*a**4*b**11*x**(19/2)/19 + 91*a**3*b**12*x**10/2 + 10*a**2
*b**13*x**(21/2) + 15*a*b**14*x**11/11 + 2*b**15*x**(23/2)/23

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Giac [A]  time = 1.11742, size = 225, normalized size = 1.39 \begin{align*} \frac{2}{23} \, b^{15} x^{\frac{23}{2}} + \frac{15}{11} \, a b^{14} x^{11} + 10 \, a^{2} b^{13} x^{\frac{21}{2}} + \frac{91}{2} \, a^{3} b^{12} x^{10} + \frac{2730}{19} \, a^{4} b^{11} x^{\frac{19}{2}} + \frac{1001}{3} \, a^{5} b^{10} x^{9} + \frac{10010}{17} \, a^{6} b^{9} x^{\frac{17}{2}} + \frac{6435}{8} \, a^{7} b^{8} x^{8} + 858 \, a^{8} b^{7} x^{\frac{15}{2}} + 715 \, a^{9} b^{6} x^{7} + 462 \, a^{10} b^{5} x^{\frac{13}{2}} + \frac{455}{2} \, a^{11} b^{4} x^{6} + \frac{910}{11} \, a^{12} b^{3} x^{\frac{11}{2}} + 21 \, a^{13} b^{2} x^{5} + \frac{10}{3} \, a^{14} b x^{\frac{9}{2}} + \frac{1}{4} \, a^{15} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*x^(1/2))^15,x, algorithm="giac")

[Out]

2/23*b^15*x^(23/2) + 15/11*a*b^14*x^11 + 10*a^2*b^13*x^(21/2) + 91/2*a^3*b^12*x^10 + 2730/19*a^4*b^11*x^(19/2)
+ 1001/3*a^5*b^10*x^9 + 10010/17*a^6*b^9*x^(17/2) + 6435/8*a^7*b^8*x^8 + 858*a^8*b^7*x^(15/2) + 715*a^9*b^6*x
^7 + 462*a^10*b^5*x^(13/2) + 455/2*a^11*b^4*x^6 + 910/11*a^12*b^3*x^(11/2) + 21*a^13*b^2*x^5 + 10/3*a^14*b*x^(
9/2) + 1/4*a^15*x^4