3.2172 \(\int (a+b \sqrt{x})^{15} x^2 \, dx\)

Optimal. Leaf size=122 \[ \frac{20 a^2 \left (a+b \sqrt{x}\right )^{19}}{19 b^6}-\frac{10 a^3 \left (a+b \sqrt{x}\right )^{18}}{9 b^6}+\frac{10 a^4 \left (a+b \sqrt{x}\right )^{17}}{17 b^6}-\frac{a^5 \left (a+b \sqrt{x}\right )^{16}}{8 b^6}+\frac{2 \left (a+b \sqrt{x}\right )^{21}}{21 b^6}-\frac{a \left (a+b \sqrt{x}\right )^{20}}{2 b^6} \]

[Out]

-(a^5*(a + b*Sqrt[x])^16)/(8*b^6) + (10*a^4*(a + b*Sqrt[x])^17)/(17*b^6) - (10*a^3*(a + b*Sqrt[x])^18)/(9*b^6)
+ (20*a^2*(a + b*Sqrt[x])^19)/(19*b^6) - (a*(a + b*Sqrt[x])^20)/(2*b^6) + (2*(a + b*Sqrt[x])^21)/(21*b^6)

________________________________________________________________________________________

Rubi [A]  time = 0.0736079, 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} \[ \frac{20 a^2 \left (a+b \sqrt{x}\right )^{19}}{19 b^6}-\frac{10 a^3 \left (a+b \sqrt{x}\right )^{18}}{9 b^6}+\frac{10 a^4 \left (a+b \sqrt{x}\right )^{17}}{17 b^6}-\frac{a^5 \left (a+b \sqrt{x}\right )^{16}}{8 b^6}+\frac{2 \left (a+b \sqrt{x}\right )^{21}}{21 b^6}-\frac{a \left (a+b \sqrt{x}\right )^{20}}{2 b^6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^15*x^2,x]

[Out]

-(a^5*(a + b*Sqrt[x])^16)/(8*b^6) + (10*a^4*(a + b*Sqrt[x])^17)/(17*b^6) - (10*a^3*(a + b*Sqrt[x])^18)/(9*b^6)
+ (20*a^2*(a + b*Sqrt[x])^19)/(19*b^6) - (a*(a + b*Sqrt[x])^20)/(2*b^6) + (2*(a + b*Sqrt[x])^21)/(21*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 )^{15} x^2 \, dx &=2 \operatorname{Subst}\left (\int x^5 (a+b x)^{15} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-\frac{a^5 (a+b x)^{15}}{b^5}+\frac{5 a^4 (a+b x)^{16}}{b^5}-\frac{10 a^3 (a+b x)^{17}}{b^5}+\frac{10 a^2 (a+b x)^{18}}{b^5}-\frac{5 a (a+b x)^{19}}{b^5}+\frac{(a+b x)^{20}}{b^5}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{a^5 \left (a+b \sqrt{x}\right )^{16}}{8 b^6}+\frac{10 a^4 \left (a+b \sqrt{x}\right )^{17}}{17 b^6}-\frac{10 a^3 \left (a+b \sqrt{x}\right )^{18}}{9 b^6}+\frac{20 a^2 \left (a+b \sqrt{x}\right )^{19}}{19 b^6}-\frac{a \left (a+b \sqrt{x}\right )^{20}}{2 b^6}+\frac{2 \left (a+b \sqrt{x}\right )^{21}}{21 b^6}\\ \end{align*}

Mathematica [A]  time = 0.0648546, size = 74, normalized size = 0.61 \[ -\frac{\left (a+b \sqrt{x}\right )^{16} \left (-816 a^2 b^3 x^{3/2}+136 a^3 b^2 x-16 a^4 b \sqrt{x}+a^5+3876 a b^4 x^2-15504 b^5 x^{5/2}\right )}{162792 b^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^15*x^2,x]

[Out]

-((a + b*Sqrt[x])^16*(a^5 - 16*a^4*b*Sqrt[x] + 136*a^3*b^2*x - 816*a^2*b^3*x^(3/2) + 3876*a*b^4*x^2 - 15504*b^
5*x^(5/2)))/(162792*b^6)

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 168, normalized size = 1.4 \begin{align*}{\frac{2\,{b}^{15}}{21}{x}^{{\frac{21}{2}}}}+{\frac{3\,{x}^{10}a{b}^{14}}{2}}+{\frac{210\,{a}^{2}{b}^{13}}{19}{x}^{{\frac{19}{2}}}}+{\frac{455\,{a}^{3}{b}^{12}{x}^{9}}{9}}+{\frac{2730\,{a}^{4}{b}^{11}}{17}{x}^{{\frac{17}{2}}}}+{\frac{3003\,{x}^{8}{a}^{5}{b}^{10}}{8}}+{\frac{2002\,{a}^{6}{b}^{9}}{3}{x}^{{\frac{15}{2}}}}+{\frac{6435\,{x}^{7}{a}^{7}{b}^{8}}{7}}+990\,{x}^{13/2}{a}^{8}{b}^{7}+{\frac{5005\,{x}^{6}{a}^{9}{b}^{6}}{6}}+546\,{x}^{11/2}{a}^{10}{b}^{5}+273\,{x}^{5}{a}^{11}{b}^{4}+{\frac{910\,{a}^{12}{b}^{3}}{9}{x}^{{\frac{9}{2}}}}+{\frac{105\,{x}^{4}{a}^{13}{b}^{2}}{4}}+{\frac{30\,{a}^{14}b}{7}{x}^{{\frac{7}{2}}}}+{\frac{{x}^{3}{a}^{15}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*x^(21/2)*b^15+3/2*x^10*a*b^14+210/19*x^(19/2)*a^2*b^13+455/9*a^3*b^12*x^9+2730/17*x^(17/2)*a^4*b^11+3003/
8*x^8*a^5*b^10+2002/3*x^(15/2)*a^6*b^9+6435/7*x^7*a^7*b^8+990*x^(13/2)*a^8*b^7+5005/6*x^6*a^9*b^6+546*x^(11/2)
*a^10*b^5+273*x^5*a^11*b^4+910/9*x^(9/2)*a^12*b^3+105/4*x^4*a^13*b^2+30/7*x^(7/2)*a^14*b+1/3*x^3*a^15

________________________________________________________________________________________

Maxima [A]  time = 0.964473, size = 132, normalized size = 1.08 \begin{align*} \frac{2 \,{\left (b \sqrt{x} + a\right )}^{21}}{21 \, b^{6}} - \frac{{\left (b \sqrt{x} + a\right )}^{20} a}{2 \, b^{6}} + \frac{20 \,{\left (b \sqrt{x} + a\right )}^{19} a^{2}}{19 \, b^{6}} - \frac{10 \,{\left (b \sqrt{x} + a\right )}^{18} a^{3}}{9 \, b^{6}} + \frac{10 \,{\left (b \sqrt{x} + a\right )}^{17} a^{4}}{17 \, b^{6}} - \frac{{\left (b \sqrt{x} + a\right )}^{16} a^{5}}{8 \, b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*(b*sqrt(x) + a)^21/b^6 - 1/2*(b*sqrt(x) + a)^20*a/b^6 + 20/19*(b*sqrt(x) + a)^19*a^2/b^6 - 10/9*(b*sqrt(x
) + a)^18*a^3/b^6 + 10/17*(b*sqrt(x) + a)^17*a^4/b^6 - 1/8*(b*sqrt(x) + a)^16*a^5/b^6

________________________________________________________________________________________

Fricas [A]  time = 1.30689, size = 467, normalized size = 3.83 \begin{align*} \frac{3}{2} \, a b^{14} x^{10} + \frac{455}{9} \, a^{3} b^{12} x^{9} + \frac{3003}{8} \, a^{5} b^{10} x^{8} + \frac{6435}{7} \, a^{7} b^{8} x^{7} + \frac{5005}{6} \, a^{9} b^{6} x^{6} + 273 \, a^{11} b^{4} x^{5} + \frac{105}{4} \, a^{13} b^{2} x^{4} + \frac{1}{3} \, a^{15} x^{3} + \frac{2}{20349} \,{\left (969 \, b^{15} x^{10} + 112455 \, a^{2} b^{13} x^{9} + 1633905 \, a^{4} b^{11} x^{8} + 6789783 \, a^{6} b^{9} x^{7} + 10072755 \, a^{8} b^{7} x^{6} + 5555277 \, a^{10} b^{5} x^{5} + 1028755 \, a^{12} b^{3} x^{4} + 43605 \, a^{14} 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))^15,x, algorithm="fricas")

[Out]

3/2*a*b^14*x^10 + 455/9*a^3*b^12*x^9 + 3003/8*a^5*b^10*x^8 + 6435/7*a^7*b^8*x^7 + 5005/6*a^9*b^6*x^6 + 273*a^1
1*b^4*x^5 + 105/4*a^13*b^2*x^4 + 1/3*a^15*x^3 + 2/20349*(969*b^15*x^10 + 112455*a^2*b^13*x^9 + 1633905*a^4*b^1
1*x^8 + 6789783*a^6*b^9*x^7 + 10072755*a^8*b^7*x^6 + 5555277*a^10*b^5*x^5 + 1028755*a^12*b^3*x^4 + 43605*a^14*
b*x^3)*sqrt(x)

________________________________________________________________________________________

Sympy [A]  time = 5.29331, size = 212, normalized size = 1.74 \begin{align*} \frac{a^{15} x^{3}}{3} + \frac{30 a^{14} b x^{\frac{7}{2}}}{7} + \frac{105 a^{13} b^{2} x^{4}}{4} + \frac{910 a^{12} b^{3} x^{\frac{9}{2}}}{9} + 273 a^{11} b^{4} x^{5} + 546 a^{10} b^{5} x^{\frac{11}{2}} + \frac{5005 a^{9} b^{6} x^{6}}{6} + 990 a^{8} b^{7} x^{\frac{13}{2}} + \frac{6435 a^{7} b^{8} x^{7}}{7} + \frac{2002 a^{6} b^{9} x^{\frac{15}{2}}}{3} + \frac{3003 a^{5} b^{10} x^{8}}{8} + \frac{2730 a^{4} b^{11} x^{\frac{17}{2}}}{17} + \frac{455 a^{3} b^{12} x^{9}}{9} + \frac{210 a^{2} b^{13} x^{\frac{19}{2}}}{19} + \frac{3 a b^{14} x^{10}}{2} + \frac{2 b^{15} x^{\frac{21}{2}}}{21} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**15*x**3/3 + 30*a**14*b*x**(7/2)/7 + 105*a**13*b**2*x**4/4 + 910*a**12*b**3*x**(9/2)/9 + 273*a**11*b**4*x**5
+ 546*a**10*b**5*x**(11/2) + 5005*a**9*b**6*x**6/6 + 990*a**8*b**7*x**(13/2) + 6435*a**7*b**8*x**7/7 + 2002*a
**6*b**9*x**(15/2)/3 + 3003*a**5*b**10*x**8/8 + 2730*a**4*b**11*x**(17/2)/17 + 455*a**3*b**12*x**9/9 + 210*a**
2*b**13*x**(19/2)/19 + 3*a*b**14*x**10/2 + 2*b**15*x**(21/2)/21

________________________________________________________________________________________

Giac [A]  time = 1.09752, size = 225, normalized size = 1.84 \begin{align*} \frac{2}{21} \, b^{15} x^{\frac{21}{2}} + \frac{3}{2} \, a b^{14} x^{10} + \frac{210}{19} \, a^{2} b^{13} x^{\frac{19}{2}} + \frac{455}{9} \, a^{3} b^{12} x^{9} + \frac{2730}{17} \, a^{4} b^{11} x^{\frac{17}{2}} + \frac{3003}{8} \, a^{5} b^{10} x^{8} + \frac{2002}{3} \, a^{6} b^{9} x^{\frac{15}{2}} + \frac{6435}{7} \, a^{7} b^{8} x^{7} + 990 \, a^{8} b^{7} x^{\frac{13}{2}} + \frac{5005}{6} \, a^{9} b^{6} x^{6} + 546 \, a^{10} b^{5} x^{\frac{11}{2}} + 273 \, a^{11} b^{4} x^{5} + \frac{910}{9} \, a^{12} b^{3} x^{\frac{9}{2}} + \frac{105}{4} \, a^{13} b^{2} x^{4} + \frac{30}{7} \, a^{14} b x^{\frac{7}{2}} + \frac{1}{3} \, a^{15} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*b^15*x^(21/2) + 3/2*a*b^14*x^10 + 210/19*a^2*b^13*x^(19/2) + 455/9*a^3*b^12*x^9 + 2730/17*a^4*b^11*x^(17/
2) + 3003/8*a^5*b^10*x^8 + 2002/3*a^6*b^9*x^(15/2) + 6435/7*a^7*b^8*x^7 + 990*a^8*b^7*x^(13/2) + 5005/6*a^9*b^
6*x^6 + 546*a^10*b^5*x^(11/2) + 273*a^11*b^4*x^5 + 910/9*a^12*b^3*x^(9/2) + 105/4*a^13*b^2*x^4 + 30/7*a^14*b*x
^(7/2) + 1/3*a^15*x^3