3.2188 \(\int \frac{(a+b \sqrt{x})^{15}}{x^{15}} \, dx\)

Optimal. Leaf size=211 \[ -\frac{105 a^{13} b^2}{13 x^{13}}-\frac{182 a^{12} b^3}{5 x^{25/2}}-\frac{455 a^{11} b^4}{4 x^{12}}-\frac{6006 a^{10} b^5}{23 x^{23/2}}-\frac{455 a^9 b^6}{x^{11}}-\frac{4290 a^8 b^7}{7 x^{21/2}}-\frac{1287 a^7 b^8}{2 x^{10}}-\frac{10010 a^6 b^9}{19 x^{19/2}}-\frac{1001 a^5 b^{10}}{3 x^9}-\frac{2730 a^4 b^{11}}{17 x^{17/2}}-\frac{455 a^3 b^{12}}{8 x^8}-\frac{14 a^2 b^{13}}{x^{15/2}}-\frac{10 a^{14} b}{9 x^{27/2}}-\frac{a^{15}}{14 x^{14}}-\frac{15 a b^{14}}{7 x^7}-\frac{2 b^{15}}{13 x^{13/2}} \]

[Out]

-a^15/(14*x^14) - (10*a^14*b)/(9*x^(27/2)) - (105*a^13*b^2)/(13*x^13) - (182*a^12*b^3)/(5*x^(25/2)) - (455*a^1
1*b^4)/(4*x^12) - (6006*a^10*b^5)/(23*x^(23/2)) - (455*a^9*b^6)/x^11 - (4290*a^8*b^7)/(7*x^(21/2)) - (1287*a^7
*b^8)/(2*x^10) - (10010*a^6*b^9)/(19*x^(19/2)) - (1001*a^5*b^10)/(3*x^9) - (2730*a^4*b^11)/(17*x^(17/2)) - (45
5*a^3*b^12)/(8*x^8) - (14*a^2*b^13)/x^(15/2) - (15*a*b^14)/(7*x^7) - (2*b^15)/(13*x^(13/2))

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Rubi [A]  time = 0.116386, 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} \[ -\frac{105 a^{13} b^2}{13 x^{13}}-\frac{182 a^{12} b^3}{5 x^{25/2}}-\frac{455 a^{11} b^4}{4 x^{12}}-\frac{6006 a^{10} b^5}{23 x^{23/2}}-\frac{455 a^9 b^6}{x^{11}}-\frac{4290 a^8 b^7}{7 x^{21/2}}-\frac{1287 a^7 b^8}{2 x^{10}}-\frac{10010 a^6 b^9}{19 x^{19/2}}-\frac{1001 a^5 b^{10}}{3 x^9}-\frac{2730 a^4 b^{11}}{17 x^{17/2}}-\frac{455 a^3 b^{12}}{8 x^8}-\frac{14 a^2 b^{13}}{x^{15/2}}-\frac{10 a^{14} b}{9 x^{27/2}}-\frac{a^{15}}{14 x^{14}}-\frac{15 a b^{14}}{7 x^7}-\frac{2 b^{15}}{13 x^{13/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^15/x^15,x]

[Out]

-a^15/(14*x^14) - (10*a^14*b)/(9*x^(27/2)) - (105*a^13*b^2)/(13*x^13) - (182*a^12*b^3)/(5*x^(25/2)) - (455*a^1
1*b^4)/(4*x^12) - (6006*a^10*b^5)/(23*x^(23/2)) - (455*a^9*b^6)/x^11 - (4290*a^8*b^7)/(7*x^(21/2)) - (1287*a^7
*b^8)/(2*x^10) - (10010*a^6*b^9)/(19*x^(19/2)) - (1001*a^5*b^10)/(3*x^9) - (2730*a^4*b^11)/(17*x^(17/2)) - (45
5*a^3*b^12)/(8*x^8) - (14*a^2*b^13)/x^(15/2) - (15*a*b^14)/(7*x^7) - (2*b^15)/(13*x^(13/2))

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^{15}} \, dx &=2 \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{29}} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{a^{15}}{x^{29}}+\frac{15 a^{14} b}{x^{28}}+\frac{105 a^{13} b^2}{x^{27}}+\frac{455 a^{12} b^3}{x^{26}}+\frac{1365 a^{11} b^4}{x^{25}}+\frac{3003 a^{10} b^5}{x^{24}}+\frac{5005 a^9 b^6}{x^{23}}+\frac{6435 a^8 b^7}{x^{22}}+\frac{6435 a^7 b^8}{x^{21}}+\frac{5005 a^6 b^9}{x^{20}}+\frac{3003 a^5 b^{10}}{x^{19}}+\frac{1365 a^4 b^{11}}{x^{18}}+\frac{455 a^3 b^{12}}{x^{17}}+\frac{105 a^2 b^{13}}{x^{16}}+\frac{15 a b^{14}}{x^{15}}+\frac{b^{15}}{x^{14}}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{a^{15}}{14 x^{14}}-\frac{10 a^{14} b}{9 x^{27/2}}-\frac{105 a^{13} b^2}{13 x^{13}}-\frac{182 a^{12} b^3}{5 x^{25/2}}-\frac{455 a^{11} b^4}{4 x^{12}}-\frac{6006 a^{10} b^5}{23 x^{23/2}}-\frac{455 a^9 b^6}{x^{11}}-\frac{4290 a^8 b^7}{7 x^{21/2}}-\frac{1287 a^7 b^8}{2 x^{10}}-\frac{10010 a^6 b^9}{19 x^{19/2}}-\frac{1001 a^5 b^{10}}{3 x^9}-\frac{2730 a^4 b^{11}}{17 x^{17/2}}-\frac{455 a^3 b^{12}}{8 x^8}-\frac{14 a^2 b^{13}}{x^{15/2}}-\frac{15 a b^{14}}{7 x^7}-\frac{2 b^{15}}{13 x^{13/2}}\\ \end{align*}

Mathematica [A]  time = 0.100854, size = 211, normalized size = 1. \[ -\frac{105 a^{13} b^2}{13 x^{13}}-\frac{182 a^{12} b^3}{5 x^{25/2}}-\frac{455 a^{11} b^4}{4 x^{12}}-\frac{6006 a^{10} b^5}{23 x^{23/2}}-\frac{455 a^9 b^6}{x^{11}}-\frac{4290 a^8 b^7}{7 x^{21/2}}-\frac{1287 a^7 b^8}{2 x^{10}}-\frac{10010 a^6 b^9}{19 x^{19/2}}-\frac{1001 a^5 b^{10}}{3 x^9}-\frac{2730 a^4 b^{11}}{17 x^{17/2}}-\frac{455 a^3 b^{12}}{8 x^8}-\frac{14 a^2 b^{13}}{x^{15/2}}-\frac{10 a^{14} b}{9 x^{27/2}}-\frac{a^{15}}{14 x^{14}}-\frac{15 a b^{14}}{7 x^7}-\frac{2 b^{15}}{13 x^{13/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^15/x^15,x]

[Out]

-a^15/(14*x^14) - (10*a^14*b)/(9*x^(27/2)) - (105*a^13*b^2)/(13*x^13) - (182*a^12*b^3)/(5*x^(25/2)) - (455*a^1
1*b^4)/(4*x^12) - (6006*a^10*b^5)/(23*x^(23/2)) - (455*a^9*b^6)/x^11 - (4290*a^8*b^7)/(7*x^(21/2)) - (1287*a^7
*b^8)/(2*x^10) - (10010*a^6*b^9)/(19*x^(19/2)) - (1001*a^5*b^10)/(3*x^9) - (2730*a^4*b^11)/(17*x^(17/2)) - (45
5*a^3*b^12)/(8*x^8) - (14*a^2*b^13)/x^(15/2) - (15*a*b^14)/(7*x^7) - (2*b^15)/(13*x^(13/2))

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Maple [A]  time = 0.004, size = 168, normalized size = 0.8 \begin{align*} -{\frac{{a}^{15}}{14\,{x}^{14}}}-{\frac{10\,{a}^{14}b}{9}{x}^{-{\frac{27}{2}}}}-{\frac{105\,{a}^{13}{b}^{2}}{13\,{x}^{13}}}-{\frac{182\,{a}^{12}{b}^{3}}{5}{x}^{-{\frac{25}{2}}}}-{\frac{455\,{a}^{11}{b}^{4}}{4\,{x}^{12}}}-{\frac{6006\,{a}^{10}{b}^{5}}{23}{x}^{-{\frac{23}{2}}}}-455\,{\frac{{a}^{9}{b}^{6}}{{x}^{11}}}-{\frac{4290\,{a}^{8}{b}^{7}}{7}{x}^{-{\frac{21}{2}}}}-{\frac{1287\,{a}^{7}{b}^{8}}{2\,{x}^{10}}}-{\frac{10010\,{a}^{6}{b}^{9}}{19}{x}^{-{\frac{19}{2}}}}-{\frac{1001\,{a}^{5}{b}^{10}}{3\,{x}^{9}}}-{\frac{2730\,{a}^{4}{b}^{11}}{17}{x}^{-{\frac{17}{2}}}}-{\frac{455\,{a}^{3}{b}^{12}}{8\,{x}^{8}}}-14\,{\frac{{a}^{2}{b}^{13}}{{x}^{15/2}}}-{\frac{15\,a{b}^{14}}{7\,{x}^{7}}}-{\frac{2\,{b}^{15}}{13}{x}^{-{\frac{13}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/14*a^15/x^14-10/9*a^14*b/x^(27/2)-105/13*a^13*b^2/x^13-182/5*a^12*b^3/x^(25/2)-455/4*a^11*b^4/x^12-6006/23*
a^10*b^5/x^(23/2)-455*a^9*b^6/x^11-4290/7*a^8*b^7/x^(21/2)-1287/2*a^7*b^8/x^10-10010/19*a^6*b^9/x^(19/2)-1001/
3*a^5*b^10/x^9-2730/17*a^4*b^11/x^(17/2)-455/8*a^3*b^12/x^8-14*a^2*b^13/x^(15/2)-15/7*a*b^14/x^7-2/13*b^15/x^(
13/2)

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Maxima [A]  time = 0.995123, size = 225, normalized size = 1.07 \begin{align*} -\frac{37442160 \, b^{15} x^{\frac{15}{2}} + 521515800 \, a b^{14} x^{7} + 3407236560 \, a^{2} b^{13} x^{\frac{13}{2}} + 13841898525 \, a^{3} b^{12} x^{6} + 39083007600 \, a^{4} b^{11} x^{\frac{11}{2}} + 81205804680 \, a^{5} b^{10} x^{5} + 128219691600 \, a^{6} b^{9} x^{\frac{9}{2}} + 156611194740 \, a^{7} b^{8} x^{4} + 149153518800 \, a^{8} b^{7} x^{\frac{7}{2}} + 110735188200 \, a^{9} b^{6} x^{3} + 63552368880 \, a^{10} b^{5} x^{\frac{5}{2}} + 27683797050 \, a^{11} b^{4} x^{2} + 8858815056 \, a^{12} b^{3} x^{\frac{3}{2}} + 1965713400 \, a^{13} b^{2} x + 270415600 \, a^{14} b \sqrt{x} + 17383860 \, a^{15}}{243374040 \, x^{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/243374040*(37442160*b^15*x^(15/2) + 521515800*a*b^14*x^7 + 3407236560*a^2*b^13*x^(13/2) + 13841898525*a^3*b
^12*x^6 + 39083007600*a^4*b^11*x^(11/2) + 81205804680*a^5*b^10*x^5 + 128219691600*a^6*b^9*x^(9/2) + 1566111947
40*a^7*b^8*x^4 + 149153518800*a^8*b^7*x^(7/2) + 110735188200*a^9*b^6*x^3 + 63552368880*a^10*b^5*x^(5/2) + 2768
3797050*a^11*b^4*x^2 + 8858815056*a^12*b^3*x^(3/2) + 1965713400*a^13*b^2*x + 270415600*a^14*b*sqrt(x) + 173838
60*a^15)/x^14

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Fricas [A]  time = 1.26863, size = 563, normalized size = 2.67 \begin{align*} -\frac{521515800 \, a b^{14} x^{7} + 13841898525 \, a^{3} b^{12} x^{6} + 81205804680 \, a^{5} b^{10} x^{5} + 156611194740 \, a^{7} b^{8} x^{4} + 110735188200 \, a^{9} b^{6} x^{3} + 27683797050 \, a^{11} b^{4} x^{2} + 1965713400 \, a^{13} b^{2} x + 17383860 \, a^{15} + 16 \,{\left (2340135 \, b^{15} x^{7} + 212952285 \, a^{2} b^{13} x^{6} + 2442687975 \, a^{4} b^{11} x^{5} + 8013730725 \, a^{6} b^{9} x^{4} + 9322094925 \, a^{8} b^{7} x^{3} + 3972023055 \, a^{10} b^{5} x^{2} + 553675941 \, a^{12} b^{3} x + 16900975 \, a^{14} b\right )} \sqrt{x}}{243374040 \, x^{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/243374040*(521515800*a*b^14*x^7 + 13841898525*a^3*b^12*x^6 + 81205804680*a^5*b^10*x^5 + 156611194740*a^7*b^
8*x^4 + 110735188200*a^9*b^6*x^3 + 27683797050*a^11*b^4*x^2 + 1965713400*a^13*b^2*x + 17383860*a^15 + 16*(2340
135*b^15*x^7 + 212952285*a^2*b^13*x^6 + 2442687975*a^4*b^11*x^5 + 8013730725*a^6*b^9*x^4 + 9322094925*a^8*b^7*
x^3 + 3972023055*a^10*b^5*x^2 + 553675941*a^12*b^3*x + 16900975*a^14*b)*sqrt(x))/x^14

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Sympy [A]  time = 41.9276, size = 216, normalized size = 1.02 \begin{align*} - \frac{a^{15}}{14 x^{14}} - \frac{10 a^{14} b}{9 x^{\frac{27}{2}}} - \frac{105 a^{13} b^{2}}{13 x^{13}} - \frac{182 a^{12} b^{3}}{5 x^{\frac{25}{2}}} - \frac{455 a^{11} b^{4}}{4 x^{12}} - \frac{6006 a^{10} b^{5}}{23 x^{\frac{23}{2}}} - \frac{455 a^{9} b^{6}}{x^{11}} - \frac{4290 a^{8} b^{7}}{7 x^{\frac{21}{2}}} - \frac{1287 a^{7} b^{8}}{2 x^{10}} - \frac{10010 a^{6} b^{9}}{19 x^{\frac{19}{2}}} - \frac{1001 a^{5} b^{10}}{3 x^{9}} - \frac{2730 a^{4} b^{11}}{17 x^{\frac{17}{2}}} - \frac{455 a^{3} b^{12}}{8 x^{8}} - \frac{14 a^{2} b^{13}}{x^{\frac{15}{2}}} - \frac{15 a b^{14}}{7 x^{7}} - \frac{2 b^{15}}{13 x^{\frac{13}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**15/(14*x**14) - 10*a**14*b/(9*x**(27/2)) - 105*a**13*b**2/(13*x**13) - 182*a**12*b**3/(5*x**(25/2)) - 455*
a**11*b**4/(4*x**12) - 6006*a**10*b**5/(23*x**(23/2)) - 455*a**9*b**6/x**11 - 4290*a**8*b**7/(7*x**(21/2)) - 1
287*a**7*b**8/(2*x**10) - 10010*a**6*b**9/(19*x**(19/2)) - 1001*a**5*b**10/(3*x**9) - 2730*a**4*b**11/(17*x**(
17/2)) - 455*a**3*b**12/(8*x**8) - 14*a**2*b**13/x**(15/2) - 15*a*b**14/(7*x**7) - 2*b**15/(13*x**(13/2))

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Giac [A]  time = 1.11295, size = 225, normalized size = 1.07 \begin{align*} -\frac{37442160 \, b^{15} x^{\frac{15}{2}} + 521515800 \, a b^{14} x^{7} + 3407236560 \, a^{2} b^{13} x^{\frac{13}{2}} + 13841898525 \, a^{3} b^{12} x^{6} + 39083007600 \, a^{4} b^{11} x^{\frac{11}{2}} + 81205804680 \, a^{5} b^{10} x^{5} + 128219691600 \, a^{6} b^{9} x^{\frac{9}{2}} + 156611194740 \, a^{7} b^{8} x^{4} + 149153518800 \, a^{8} b^{7} x^{\frac{7}{2}} + 110735188200 \, a^{9} b^{6} x^{3} + 63552368880 \, a^{10} b^{5} x^{\frac{5}{2}} + 27683797050 \, a^{11} b^{4} x^{2} + 8858815056 \, a^{12} b^{3} x^{\frac{3}{2}} + 1965713400 \, a^{13} b^{2} x + 270415600 \, a^{14} b \sqrt{x} + 17383860 \, a^{15}}{243374040 \, x^{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/243374040*(37442160*b^15*x^(15/2) + 521515800*a*b^14*x^7 + 3407236560*a^2*b^13*x^(13/2) + 13841898525*a^3*b
^12*x^6 + 39083007600*a^4*b^11*x^(11/2) + 81205804680*a^5*b^10*x^5 + 128219691600*a^6*b^9*x^(9/2) + 1566111947
40*a^7*b^8*x^4 + 149153518800*a^8*b^7*x^(7/2) + 110735188200*a^9*b^6*x^3 + 63552368880*a^10*b^5*x^(5/2) + 2768
3797050*a^11*b^4*x^2 + 8858815056*a^12*b^3*x^(3/2) + 1965713400*a^13*b^2*x + 270415600*a^14*b*sqrt(x) + 173838
60*a^15)/x^14