3.2189 \(\int \frac{(a+b \sqrt{x})^{15}}{x^{16}} \, dx\)

Optimal. Leaf size=211 \[ -\frac{15 a^{13} b^2}{2 x^{14}}-\frac{910 a^{12} b^3}{27 x^{27/2}}-\frac{105 a^{11} b^4}{x^{13}}-\frac{6006 a^{10} b^5}{25 x^{25/2}}-\frac{5005 a^9 b^6}{12 x^{12}}-\frac{12870 a^8 b^7}{23 x^{23/2}}-\frac{585 a^7 b^8}{x^{11}}-\frac{1430 a^6 b^9}{3 x^{21/2}}-\frac{3003 a^5 b^{10}}{10 x^{10}}-\frac{2730 a^4 b^{11}}{19 x^{19/2}}-\frac{455 a^3 b^{12}}{9 x^9}-\frac{210 a^2 b^{13}}{17 x^{17/2}}-\frac{30 a^{14} b}{29 x^{29/2}}-\frac{a^{15}}{15 x^{15}}-\frac{15 a b^{14}}{8 x^8}-\frac{2 b^{15}}{15 x^{15/2}} \]

[Out]

-a^15/(15*x^15) - (30*a^14*b)/(29*x^(29/2)) - (15*a^13*b^2)/(2*x^14) - (910*a^12*b^3)/(27*x^(27/2)) - (105*a^1
1*b^4)/x^13 - (6006*a^10*b^5)/(25*x^(25/2)) - (5005*a^9*b^6)/(12*x^12) - (12870*a^8*b^7)/(23*x^(23/2)) - (585*
a^7*b^8)/x^11 - (1430*a^6*b^9)/(3*x^(21/2)) - (3003*a^5*b^10)/(10*x^10) - (2730*a^4*b^11)/(19*x^(19/2)) - (455
*a^3*b^12)/(9*x^9) - (210*a^2*b^13)/(17*x^(17/2)) - (15*a*b^14)/(8*x^8) - (2*b^15)/(15*x^(15/2))

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Rubi [A]  time = 0.112956, antiderivative size = 211, 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} \[ -\frac{15 a^{13} b^2}{2 x^{14}}-\frac{910 a^{12} b^3}{27 x^{27/2}}-\frac{105 a^{11} b^4}{x^{13}}-\frac{6006 a^{10} b^5}{25 x^{25/2}}-\frac{5005 a^9 b^6}{12 x^{12}}-\frac{12870 a^8 b^7}{23 x^{23/2}}-\frac{585 a^7 b^8}{x^{11}}-\frac{1430 a^6 b^9}{3 x^{21/2}}-\frac{3003 a^5 b^{10}}{10 x^{10}}-\frac{2730 a^4 b^{11}}{19 x^{19/2}}-\frac{455 a^3 b^{12}}{9 x^9}-\frac{210 a^2 b^{13}}{17 x^{17/2}}-\frac{30 a^{14} b}{29 x^{29/2}}-\frac{a^{15}}{15 x^{15}}-\frac{15 a b^{14}}{8 x^8}-\frac{2 b^{15}}{15 x^{15/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^15/x^16,x]

[Out]

-a^15/(15*x^15) - (30*a^14*b)/(29*x^(29/2)) - (15*a^13*b^2)/(2*x^14) - (910*a^12*b^3)/(27*x^(27/2)) - (105*a^1
1*b^4)/x^13 - (6006*a^10*b^5)/(25*x^(25/2)) - (5005*a^9*b^6)/(12*x^12) - (12870*a^8*b^7)/(23*x^(23/2)) - (585*
a^7*b^8)/x^11 - (1430*a^6*b^9)/(3*x^(21/2)) - (3003*a^5*b^10)/(10*x^10) - (2730*a^4*b^11)/(19*x^(19/2)) - (455
*a^3*b^12)/(9*x^9) - (210*a^2*b^13)/(17*x^(17/2)) - (15*a*b^14)/(8*x^8) - (2*b^15)/(15*x^(15/2))

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 \frac{\left (a+b \sqrt{x}\right )^{15}}{x^{16}} \, dx &=2 \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{31}} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{a^{15}}{x^{31}}+\frac{15 a^{14} b}{x^{30}}+\frac{105 a^{13} b^2}{x^{29}}+\frac{455 a^{12} b^3}{x^{28}}+\frac{1365 a^{11} b^4}{x^{27}}+\frac{3003 a^{10} b^5}{x^{26}}+\frac{5005 a^9 b^6}{x^{25}}+\frac{6435 a^8 b^7}{x^{24}}+\frac{6435 a^7 b^8}{x^{23}}+\frac{5005 a^6 b^9}{x^{22}}+\frac{3003 a^5 b^{10}}{x^{21}}+\frac{1365 a^4 b^{11}}{x^{20}}+\frac{455 a^3 b^{12}}{x^{19}}+\frac{105 a^2 b^{13}}{x^{18}}+\frac{15 a b^{14}}{x^{17}}+\frac{b^{15}}{x^{16}}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{a^{15}}{15 x^{15}}-\frac{30 a^{14} b}{29 x^{29/2}}-\frac{15 a^{13} b^2}{2 x^{14}}-\frac{910 a^{12} b^3}{27 x^{27/2}}-\frac{105 a^{11} b^4}{x^{13}}-\frac{6006 a^{10} b^5}{25 x^{25/2}}-\frac{5005 a^9 b^6}{12 x^{12}}-\frac{12870 a^8 b^7}{23 x^{23/2}}-\frac{585 a^7 b^8}{x^{11}}-\frac{1430 a^6 b^9}{3 x^{21/2}}-\frac{3003 a^5 b^{10}}{10 x^{10}}-\frac{2730 a^4 b^{11}}{19 x^{19/2}}-\frac{455 a^3 b^{12}}{9 x^9}-\frac{210 a^2 b^{13}}{17 x^{17/2}}-\frac{15 a b^{14}}{8 x^8}-\frac{2 b^{15}}{15 x^{15/2}}\\ \end{align*}

Mathematica [A]  time = 0.101826, size = 211, normalized size = 1. \[ -\frac{15 a^{13} b^2}{2 x^{14}}-\frac{910 a^{12} b^3}{27 x^{27/2}}-\frac{105 a^{11} b^4}{x^{13}}-\frac{6006 a^{10} b^5}{25 x^{25/2}}-\frac{5005 a^9 b^6}{12 x^{12}}-\frac{12870 a^8 b^7}{23 x^{23/2}}-\frac{585 a^7 b^8}{x^{11}}-\frac{1430 a^6 b^9}{3 x^{21/2}}-\frac{3003 a^5 b^{10}}{10 x^{10}}-\frac{2730 a^4 b^{11}}{19 x^{19/2}}-\frac{455 a^3 b^{12}}{9 x^9}-\frac{210 a^2 b^{13}}{17 x^{17/2}}-\frac{30 a^{14} b}{29 x^{29/2}}-\frac{a^{15}}{15 x^{15}}-\frac{15 a b^{14}}{8 x^8}-\frac{2 b^{15}}{15 x^{15/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^15/x^16,x]

[Out]

-a^15/(15*x^15) - (30*a^14*b)/(29*x^(29/2)) - (15*a^13*b^2)/(2*x^14) - (910*a^12*b^3)/(27*x^(27/2)) - (105*a^1
1*b^4)/x^13 - (6006*a^10*b^5)/(25*x^(25/2)) - (5005*a^9*b^6)/(12*x^12) - (12870*a^8*b^7)/(23*x^(23/2)) - (585*
a^7*b^8)/x^11 - (1430*a^6*b^9)/(3*x^(21/2)) - (3003*a^5*b^10)/(10*x^10) - (2730*a^4*b^11)/(19*x^(19/2)) - (455
*a^3*b^12)/(9*x^9) - (210*a^2*b^13)/(17*x^(17/2)) - (15*a*b^14)/(8*x^8) - (2*b^15)/(15*x^(15/2))

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Maple [A]  time = 0.004, size = 168, normalized size = 0.8 \begin{align*} -{\frac{{a}^{15}}{15\,{x}^{15}}}-{\frac{30\,{a}^{14}b}{29}{x}^{-{\frac{29}{2}}}}-{\frac{15\,{a}^{13}{b}^{2}}{2\,{x}^{14}}}-{\frac{910\,{a}^{12}{b}^{3}}{27}{x}^{-{\frac{27}{2}}}}-105\,{\frac{{a}^{11}{b}^{4}}{{x}^{13}}}-{\frac{6006\,{a}^{10}{b}^{5}}{25}{x}^{-{\frac{25}{2}}}}-{\frac{5005\,{a}^{9}{b}^{6}}{12\,{x}^{12}}}-{\frac{12870\,{a}^{8}{b}^{7}}{23}{x}^{-{\frac{23}{2}}}}-585\,{\frac{{a}^{7}{b}^{8}}{{x}^{11}}}-{\frac{1430\,{a}^{6}{b}^{9}}{3}{x}^{-{\frac{21}{2}}}}-{\frac{3003\,{a}^{5}{b}^{10}}{10\,{x}^{10}}}-{\frac{2730\,{a}^{4}{b}^{11}}{19}{x}^{-{\frac{19}{2}}}}-{\frac{455\,{a}^{3}{b}^{12}}{9\,{x}^{9}}}-{\frac{210\,{a}^{2}{b}^{13}}{17}{x}^{-{\frac{17}{2}}}}-{\frac{15\,a{b}^{14}}{8\,{x}^{8}}}-{\frac{2\,{b}^{15}}{15}{x}^{-{\frac{15}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/2))^15/x^16,x)

[Out]

-1/15*a^15/x^15-30/29*a^14*b/x^(29/2)-15/2*a^13*b^2/x^14-910/27*a^12*b^3/x^(27/2)-105*a^11*b^4/x^13-6006/25*a^
10*b^5/x^(25/2)-5005/12*a^9*b^6/x^12-12870/23*a^8*b^7/x^(23/2)-585*a^7*b^8/x^11-1430/3*a^6*b^9/x^(21/2)-3003/1
0*a^5*b^10/x^10-2730/19*a^4*b^11/x^(19/2)-455/9*a^3*b^12/x^9-210/17*a^2*b^13/x^(17/2)-15/8*a*b^14/x^8-2/15*b^1
5/x^(15/2)

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Maxima [A]  time = 1.00038, size = 225, normalized size = 1.07 \begin{align*} -\frac{155117520 \, b^{15} x^{\frac{15}{2}} + 2181340125 \, a b^{14} x^{7} + 14371182000 \, a^{2} b^{13} x^{\frac{13}{2}} + 58815393000 \, a^{3} b^{12} x^{6} + 167159538000 \, a^{4} b^{11} x^{\frac{11}{2}} + 349363434420 \, a^{5} b^{10} x^{5} + 554545134000 \, a^{6} b^{9} x^{\frac{9}{2}} + 680578119000 \, a^{7} b^{8} x^{4} + 650987766000 \, a^{8} b^{7} x^{\frac{7}{2}} + 485226992250 \, a^{9} b^{6} x^{3} + 279490747536 \, a^{10} b^{5} x^{\frac{5}{2}} + 122155047000 \, a^{11} b^{4} x^{2} + 39210262000 \, a^{12} b^{3} x^{\frac{3}{2}} + 8725360500 \, a^{13} b^{2} x + 1203498000 \, a^{14} b \sqrt{x} + 77558760 \, a^{15}}{1163381400 \, x^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^15/x^16,x, algorithm="maxima")

[Out]

-1/1163381400*(155117520*b^15*x^(15/2) + 2181340125*a*b^14*x^7 + 14371182000*a^2*b^13*x^(13/2) + 58815393000*a
^3*b^12*x^6 + 167159538000*a^4*b^11*x^(11/2) + 349363434420*a^5*b^10*x^5 + 554545134000*a^6*b^9*x^(9/2) + 6805
78119000*a^7*b^8*x^4 + 650987766000*a^8*b^7*x^(7/2) + 485226992250*a^9*b^6*x^3 + 279490747536*a^10*b^5*x^(5/2)
+ 122155047000*a^11*b^4*x^2 + 39210262000*a^12*b^3*x^(3/2) + 8725360500*a^13*b^2*x + 1203498000*a^14*b*sqrt(x
) + 77558760*a^15)/x^15

________________________________________________________________________________________

Fricas [A]  time = 1.27011, size = 575, normalized size = 2.73 \begin{align*} -\frac{2181340125 \, a b^{14} x^{7} + 58815393000 \, a^{3} b^{12} x^{6} + 349363434420 \, a^{5} b^{10} x^{5} + 680578119000 \, a^{7} b^{8} x^{4} + 485226992250 \, a^{9} b^{6} x^{3} + 122155047000 \, a^{11} b^{4} x^{2} + 8725360500 \, a^{13} b^{2} x + 77558760 \, a^{15} + 16 \,{\left (9694845 \, b^{15} x^{7} + 898198875 \, a^{2} b^{13} x^{6} + 10447471125 \, a^{4} b^{11} x^{5} + 34659070875 \, a^{6} b^{9} x^{4} + 40686735375 \, a^{8} b^{7} x^{3} + 17468171721 \, a^{10} b^{5} x^{2} + 2450641375 \, a^{12} b^{3} x + 75218625 \, a^{14} b\right )} \sqrt{x}}{1163381400 \, x^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^15/x^16,x, algorithm="fricas")

[Out]

-1/1163381400*(2181340125*a*b^14*x^7 + 58815393000*a^3*b^12*x^6 + 349363434420*a^5*b^10*x^5 + 680578119000*a^7
*b^8*x^4 + 485226992250*a^9*b^6*x^3 + 122155047000*a^11*b^4*x^2 + 8725360500*a^13*b^2*x + 77558760*a^15 + 16*(
9694845*b^15*x^7 + 898198875*a^2*b^13*x^6 + 10447471125*a^4*b^11*x^5 + 34659070875*a^6*b^9*x^4 + 40686735375*a
^8*b^7*x^3 + 17468171721*a^10*b^5*x^2 + 2450641375*a^12*b^3*x + 75218625*a^14*b)*sqrt(x))/x^15

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Sympy [A]  time = 41.9702, size = 216, normalized size = 1.02 \begin{align*} - \frac{a^{15}}{15 x^{15}} - \frac{30 a^{14} b}{29 x^{\frac{29}{2}}} - \frac{15 a^{13} b^{2}}{2 x^{14}} - \frac{910 a^{12} b^{3}}{27 x^{\frac{27}{2}}} - \frac{105 a^{11} b^{4}}{x^{13}} - \frac{6006 a^{10} b^{5}}{25 x^{\frac{25}{2}}} - \frac{5005 a^{9} b^{6}}{12 x^{12}} - \frac{12870 a^{8} b^{7}}{23 x^{\frac{23}{2}}} - \frac{585 a^{7} b^{8}}{x^{11}} - \frac{1430 a^{6} b^{9}}{3 x^{\frac{21}{2}}} - \frac{3003 a^{5} b^{10}}{10 x^{10}} - \frac{2730 a^{4} b^{11}}{19 x^{\frac{19}{2}}} - \frac{455 a^{3} b^{12}}{9 x^{9}} - \frac{210 a^{2} b^{13}}{17 x^{\frac{17}{2}}} - \frac{15 a b^{14}}{8 x^{8}} - \frac{2 b^{15}}{15 x^{\frac{15}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/2))**15/x**16,x)

[Out]

-a**15/(15*x**15) - 30*a**14*b/(29*x**(29/2)) - 15*a**13*b**2/(2*x**14) - 910*a**12*b**3/(27*x**(27/2)) - 105*
a**11*b**4/x**13 - 6006*a**10*b**5/(25*x**(25/2)) - 5005*a**9*b**6/(12*x**12) - 12870*a**8*b**7/(23*x**(23/2))
- 585*a**7*b**8/x**11 - 1430*a**6*b**9/(3*x**(21/2)) - 3003*a**5*b**10/(10*x**10) - 2730*a**4*b**11/(19*x**(1
9/2)) - 455*a**3*b**12/(9*x**9) - 210*a**2*b**13/(17*x**(17/2)) - 15*a*b**14/(8*x**8) - 2*b**15/(15*x**(15/2))

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Giac [A]  time = 1.12428, size = 225, normalized size = 1.07 \begin{align*} -\frac{155117520 \, b^{15} x^{\frac{15}{2}} + 2181340125 \, a b^{14} x^{7} + 14371182000 \, a^{2} b^{13} x^{\frac{13}{2}} + 58815393000 \, a^{3} b^{12} x^{6} + 167159538000 \, a^{4} b^{11} x^{\frac{11}{2}} + 349363434420 \, a^{5} b^{10} x^{5} + 554545134000 \, a^{6} b^{9} x^{\frac{9}{2}} + 680578119000 \, a^{7} b^{8} x^{4} + 650987766000 \, a^{8} b^{7} x^{\frac{7}{2}} + 485226992250 \, a^{9} b^{6} x^{3} + 279490747536 \, a^{10} b^{5} x^{\frac{5}{2}} + 122155047000 \, a^{11} b^{4} x^{2} + 39210262000 \, a^{12} b^{3} x^{\frac{3}{2}} + 8725360500 \, a^{13} b^{2} x + 1203498000 \, a^{14} b \sqrt{x} + 77558760 \, a^{15}}{1163381400 \, x^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^15/x^16,x, algorithm="giac")

[Out]

-1/1163381400*(155117520*b^15*x^(15/2) + 2181340125*a*b^14*x^7 + 14371182000*a^2*b^13*x^(13/2) + 58815393000*a
^3*b^12*x^6 + 167159538000*a^4*b^11*x^(11/2) + 349363434420*a^5*b^10*x^5 + 554545134000*a^6*b^9*x^(9/2) + 6805
78119000*a^7*b^8*x^4 + 650987766000*a^8*b^7*x^(7/2) + 485226992250*a^9*b^6*x^3 + 279490747536*a^10*b^5*x^(5/2)
+ 122155047000*a^11*b^4*x^2 + 39210262000*a^12*b^3*x^(3/2) + 8725360500*a^13*b^2*x + 1203498000*a^14*b*sqrt(x
) + 77558760*a^15)/x^15