3.2190 \(\int \frac {(a+b \sqrt {x})^{15}}{x^{17}} \, dx\)

Optimal. Leaf size=207 \[ -\frac {a^{15}}{16 x^{16}}-\frac {30 a^{14} b}{31 x^{31/2}}-\frac {7 a^{13} b^2}{x^{15}}-\frac {910 a^{12} b^3}{29 x^{29/2}}-\frac {195 a^{11} b^4}{2 x^{14}}-\frac {2002 a^{10} b^5}{9 x^{27/2}}-\frac {385 a^9 b^6}{x^{13}}-\frac {2574 a^8 b^7}{5 x^{25/2}}-\frac {2145 a^7 b^8}{4 x^{12}}-\frac {10010 a^6 b^9}{23 x^{23/2}}-\frac {273 a^5 b^{10}}{x^{11}}-\frac {130 a^4 b^{11}}{x^{21/2}}-\frac {91 a^3 b^{12}}{2 x^{10}}-\frac {210 a^2 b^{13}}{19 x^{19/2}}-\frac {5 a b^{14}}{3 x^9}-\frac {2 b^{15}}{17 x^{17/2}} \]

[Out]

-1/16*a^15/x^16-30/31*a^14*b/x^(31/2)-7*a^13*b^2/x^15-910/29*a^12*b^3/x^(29/2)-195/2*a^11*b^4/x^14-2002/9*a^10
*b^5/x^(27/2)-385*a^9*b^6/x^13-2574/5*a^8*b^7/x^(25/2)-2145/4*a^7*b^8/x^12-10010/23*a^6*b^9/x^(23/2)-273*a^5*b
^10/x^11-130*a^4*b^11/x^(21/2)-91/2*a^3*b^12/x^10-210/19*a^2*b^13/x^(19/2)-5/3*a*b^14/x^9-2/17*b^15/x^(17/2)

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Rubi [A]  time = 0.11, antiderivative size = 207, normalized size of antiderivative = 1.00, 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} \[ -\frac {7 a^{13} b^2}{x^{15}}-\frac {910 a^{12} b^3}{29 x^{29/2}}-\frac {195 a^{11} b^4}{2 x^{14}}-\frac {2002 a^{10} b^5}{9 x^{27/2}}-\frac {385 a^9 b^6}{x^{13}}-\frac {2574 a^8 b^7}{5 x^{25/2}}-\frac {2145 a^7 b^8}{4 x^{12}}-\frac {10010 a^6 b^9}{23 x^{23/2}}-\frac {273 a^5 b^{10}}{x^{11}}-\frac {130 a^4 b^{11}}{x^{21/2}}-\frac {91 a^3 b^{12}}{2 x^{10}}-\frac {210 a^2 b^{13}}{19 x^{19/2}}-\frac {30 a^{14} b}{31 x^{31/2}}-\frac {a^{15}}{16 x^{16}}-\frac {5 a b^{14}}{3 x^9}-\frac {2 b^{15}}{17 x^{17/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^15/x^17,x]

[Out]

-a^15/(16*x^16) - (30*a^14*b)/(31*x^(31/2)) - (7*a^13*b^2)/x^15 - (910*a^12*b^3)/(29*x^(29/2)) - (195*a^11*b^4
)/(2*x^14) - (2002*a^10*b^5)/(9*x^(27/2)) - (385*a^9*b^6)/x^13 - (2574*a^8*b^7)/(5*x^(25/2)) - (2145*a^7*b^8)/
(4*x^12) - (10010*a^6*b^9)/(23*x^(23/2)) - (273*a^5*b^10)/x^11 - (130*a^4*b^11)/x^(21/2) - (91*a^3*b^12)/(2*x^
10) - (210*a^2*b^13)/(19*x^(19/2)) - (5*a*b^14)/(3*x^9) - (2*b^15)/(17*x^(17/2))

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])

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]]

Rubi steps

\begin {align*} \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{17}} \, dx &=2 \operatorname {Subst}\left (\int \frac {(a+b x)^{15}}{x^{33}} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (\frac {a^{15}}{x^{33}}+\frac {15 a^{14} b}{x^{32}}+\frac {105 a^{13} b^2}{x^{31}}+\frac {455 a^{12} b^3}{x^{30}}+\frac {1365 a^{11} b^4}{x^{29}}+\frac {3003 a^{10} b^5}{x^{28}}+\frac {5005 a^9 b^6}{x^{27}}+\frac {6435 a^8 b^7}{x^{26}}+\frac {6435 a^7 b^8}{x^{25}}+\frac {5005 a^6 b^9}{x^{24}}+\frac {3003 a^5 b^{10}}{x^{23}}+\frac {1365 a^4 b^{11}}{x^{22}}+\frac {455 a^3 b^{12}}{x^{21}}+\frac {105 a^2 b^{13}}{x^{20}}+\frac {15 a b^{14}}{x^{19}}+\frac {b^{15}}{x^{18}}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {a^{15}}{16 x^{16}}-\frac {30 a^{14} b}{31 x^{31/2}}-\frac {7 a^{13} b^2}{x^{15}}-\frac {910 a^{12} b^3}{29 x^{29/2}}-\frac {195 a^{11} b^4}{2 x^{14}}-\frac {2002 a^{10} b^5}{9 x^{27/2}}-\frac {385 a^9 b^6}{x^{13}}-\frac {2574 a^8 b^7}{5 x^{25/2}}-\frac {2145 a^7 b^8}{4 x^{12}}-\frac {10010 a^6 b^9}{23 x^{23/2}}-\frac {273 a^5 b^{10}}{x^{11}}-\frac {130 a^4 b^{11}}{x^{21/2}}-\frac {91 a^3 b^{12}}{2 x^{10}}-\frac {210 a^2 b^{13}}{19 x^{19/2}}-\frac {5 a b^{14}}{3 x^9}-\frac {2 b^{15}}{17 x^{17/2}}\\ \end {align*}

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Mathematica [A]  time = 0.14, size = 207, normalized size = 1.00 \[ -\frac {a^{15}}{16 x^{16}}-\frac {30 a^{14} b}{31 x^{31/2}}-\frac {7 a^{13} b^2}{x^{15}}-\frac {910 a^{12} b^3}{29 x^{29/2}}-\frac {195 a^{11} b^4}{2 x^{14}}-\frac {2002 a^{10} b^5}{9 x^{27/2}}-\frac {385 a^9 b^6}{x^{13}}-\frac {2574 a^8 b^7}{5 x^{25/2}}-\frac {2145 a^7 b^8}{4 x^{12}}-\frac {10010 a^6 b^9}{23 x^{23/2}}-\frac {273 a^5 b^{10}}{x^{11}}-\frac {130 a^4 b^{11}}{x^{21/2}}-\frac {91 a^3 b^{12}}{2 x^{10}}-\frac {210 a^2 b^{13}}{19 x^{19/2}}-\frac {5 a b^{14}}{3 x^9}-\frac {2 b^{15}}{17 x^{17/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^15/x^17,x]

[Out]

-1/16*a^15/x^16 - (30*a^14*b)/(31*x^(31/2)) - (7*a^13*b^2)/x^15 - (910*a^12*b^3)/(29*x^(29/2)) - (195*a^11*b^4
)/(2*x^14) - (2002*a^10*b^5)/(9*x^(27/2)) - (385*a^9*b^6)/x^13 - (2574*a^8*b^7)/(5*x^(25/2)) - (2145*a^7*b^8)/
(4*x^12) - (10010*a^6*b^9)/(23*x^(23/2)) - (273*a^5*b^10)/x^11 - (130*a^4*b^11)/x^(21/2) - (91*a^3*b^12)/(2*x^
10) - (210*a^2*b^13)/(19*x^(19/2)) - (5*a*b^14)/(3*x^9) - (2*b^15)/(17*x^(17/2))

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fricas [A]  time = 0.97, size = 168, normalized size = 0.81 \[ -\frac {8014405200 \, a b^{14} x^{7} + 218793261960 \, a^{3} b^{12} x^{6} + 1312759571760 \, a^{5} b^{10} x^{5} + 2578634873100 \, a^{7} b^{8} x^{4} + 1851327601200 \, a^{9} b^{6} x^{3} + 468842704200 \, a^{11} b^{4} x^{2} + 33660501840 \, a^{13} b^{2} x + 300540195 \, a^{15} + 32 \, {\left (17678835 \, b^{15} x^{7} + 1660880025 \, a^{2} b^{13} x^{6} + 19535112675 \, a^{4} b^{11} x^{5} + 65400159825 \, a^{6} b^{9} x^{4} + 77359046193 \, a^{8} b^{7} x^{3} + 33426748355 \, a^{10} b^{5} x^{2} + 4715372025 \, a^{12} b^{3} x + 145422675 \, a^{14} b\right )} \sqrt {x}}{4808643120 \, x^{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^15/x^17,x, algorithm="fricas")

[Out]

-1/4808643120*(8014405200*a*b^14*x^7 + 218793261960*a^3*b^12*x^6 + 1312759571760*a^5*b^10*x^5 + 2578634873100*
a^7*b^8*x^4 + 1851327601200*a^9*b^6*x^3 + 468842704200*a^11*b^4*x^2 + 33660501840*a^13*b^2*x + 300540195*a^15
+ 32*(17678835*b^15*x^7 + 1660880025*a^2*b^13*x^6 + 19535112675*a^4*b^11*x^5 + 65400159825*a^6*b^9*x^4 + 77359
046193*a^8*b^7*x^3 + 33426748355*a^10*b^5*x^2 + 4715372025*a^12*b^3*x + 145422675*a^14*b)*sqrt(x))/x^16

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giac [A]  time = 0.18, size = 167, normalized size = 0.81 \[ -\frac {565722720 \, b^{15} x^{\frac {15}{2}} + 8014405200 \, a b^{14} x^{7} + 53148160800 \, a^{2} b^{13} x^{\frac {13}{2}} + 218793261960 \, a^{3} b^{12} x^{6} + 625123605600 \, a^{4} b^{11} x^{\frac {11}{2}} + 1312759571760 \, a^{5} b^{10} x^{5} + 2092805114400 \, a^{6} b^{9} x^{\frac {9}{2}} + 2578634873100 \, a^{7} b^{8} x^{4} + 2475489478176 \, a^{8} b^{7} x^{\frac {7}{2}} + 1851327601200 \, a^{9} b^{6} x^{3} + 1069655947360 \, a^{10} b^{5} x^{\frac {5}{2}} + 468842704200 \, a^{11} b^{4} x^{2} + 150891904800 \, a^{12} b^{3} x^{\frac {3}{2}} + 33660501840 \, a^{13} b^{2} x + 4653525600 \, a^{14} b \sqrt {x} + 300540195 \, a^{15}}{4808643120 \, x^{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^15/x^17,x, algorithm="giac")

[Out]

-1/4808643120*(565722720*b^15*x^(15/2) + 8014405200*a*b^14*x^7 + 53148160800*a^2*b^13*x^(13/2) + 218793261960*
a^3*b^12*x^6 + 625123605600*a^4*b^11*x^(11/2) + 1312759571760*a^5*b^10*x^5 + 2092805114400*a^6*b^9*x^(9/2) + 2
578634873100*a^7*b^8*x^4 + 2475489478176*a^8*b^7*x^(7/2) + 1851327601200*a^9*b^6*x^3 + 1069655947360*a^10*b^5*
x^(5/2) + 468842704200*a^11*b^4*x^2 + 150891904800*a^12*b^3*x^(3/2) + 33660501840*a^13*b^2*x + 4653525600*a^14
*b*sqrt(x) + 300540195*a^15)/x^16

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maple [A]  time = 0.00, size = 168, normalized size = 0.81 \[ -\frac {2 b^{15}}{17 x^{\frac {17}{2}}}-\frac {5 a \,b^{14}}{3 x^{9}}-\frac {210 a^{2} b^{13}}{19 x^{\frac {19}{2}}}-\frac {91 a^{3} b^{12}}{2 x^{10}}-\frac {130 a^{4} b^{11}}{x^{\frac {21}{2}}}-\frac {273 a^{5} b^{10}}{x^{11}}-\frac {10010 a^{6} b^{9}}{23 x^{\frac {23}{2}}}-\frac {2145 a^{7} b^{8}}{4 x^{12}}-\frac {2574 a^{8} b^{7}}{5 x^{\frac {25}{2}}}-\frac {385 a^{9} b^{6}}{x^{13}}-\frac {2002 a^{10} b^{5}}{9 x^{\frac {27}{2}}}-\frac {195 a^{11} b^{4}}{2 x^{14}}-\frac {910 a^{12} b^{3}}{29 x^{\frac {29}{2}}}-\frac {7 a^{13} b^{2}}{x^{15}}-\frac {30 a^{14} b}{31 x^{\frac {31}{2}}}-\frac {a^{15}}{16 x^{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/2))^15/x^17,x)

[Out]

-1/16*a^15/x^16-30/31*a^14*b/x^(31/2)-7*a^13*b^2/x^15-910/29*a^12*b^3/x^(29/2)-195/2*a^11*b^4/x^14-2002/9*a^10
*b^5/x^(27/2)-385*a^9*b^6/x^13-2574/5*a^8*b^7/x^(25/2)-2145/4*a^7*b^8/x^12-10010/23*a^6*b^9/x^(23/2)-273*a^5*b
^10/x^11-130*a^4*b^11/x^(21/2)-91/2*a^3*b^12/x^10-210/19*a^2*b^13/x^(19/2)-5/3*a*b^14/x^9-2/17*b^15/x^(17/2)

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maxima [A]  time = 0.87, size = 167, normalized size = 0.81 \[ -\frac {565722720 \, b^{15} x^{\frac {15}{2}} + 8014405200 \, a b^{14} x^{7} + 53148160800 \, a^{2} b^{13} x^{\frac {13}{2}} + 218793261960 \, a^{3} b^{12} x^{6} + 625123605600 \, a^{4} b^{11} x^{\frac {11}{2}} + 1312759571760 \, a^{5} b^{10} x^{5} + 2092805114400 \, a^{6} b^{9} x^{\frac {9}{2}} + 2578634873100 \, a^{7} b^{8} x^{4} + 2475489478176 \, a^{8} b^{7} x^{\frac {7}{2}} + 1851327601200 \, a^{9} b^{6} x^{3} + 1069655947360 \, a^{10} b^{5} x^{\frac {5}{2}} + 468842704200 \, a^{11} b^{4} x^{2} + 150891904800 \, a^{12} b^{3} x^{\frac {3}{2}} + 33660501840 \, a^{13} b^{2} x + 4653525600 \, a^{14} b \sqrt {x} + 300540195 \, a^{15}}{4808643120 \, x^{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^15/x^17,x, algorithm="maxima")

[Out]

-1/4808643120*(565722720*b^15*x^(15/2) + 8014405200*a*b^14*x^7 + 53148160800*a^2*b^13*x^(13/2) + 218793261960*
a^3*b^12*x^6 + 625123605600*a^4*b^11*x^(11/2) + 1312759571760*a^5*b^10*x^5 + 2092805114400*a^6*b^9*x^(9/2) + 2
578634873100*a^7*b^8*x^4 + 2475489478176*a^8*b^7*x^(7/2) + 1851327601200*a^9*b^6*x^3 + 1069655947360*a^10*b^5*
x^(5/2) + 468842704200*a^11*b^4*x^2 + 150891904800*a^12*b^3*x^(3/2) + 33660501840*a^13*b^2*x + 4653525600*a^14
*b*sqrt(x) + 300540195*a^15)/x^16

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mupad [B]  time = 0.20, size = 167, normalized size = 0.81 \[ -\frac {\frac {a^{15}}{16}+\frac {2\,b^{15}\,x^{15/2}}{17}+7\,a^{13}\,b^2\,x+\frac {30\,a^{14}\,b\,\sqrt {x}}{31}+\frac {5\,a\,b^{14}\,x^7}{3}+\frac {195\,a^{11}\,b^4\,x^2}{2}+385\,a^9\,b^6\,x^3+\frac {2145\,a^7\,b^8\,x^4}{4}+273\,a^5\,b^{10}\,x^5+\frac {910\,a^{12}\,b^3\,x^{3/2}}{29}+\frac {91\,a^3\,b^{12}\,x^6}{2}+\frac {2002\,a^{10}\,b^5\,x^{5/2}}{9}+\frac {2574\,a^8\,b^7\,x^{7/2}}{5}+\frac {10010\,a^6\,b^9\,x^{9/2}}{23}+130\,a^4\,b^{11}\,x^{11/2}+\frac {210\,a^2\,b^{13}\,x^{13/2}}{19}}{x^{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^(1/2))^15/x^17,x)

[Out]

-(a^15/16 + (2*b^15*x^(15/2))/17 + 7*a^13*b^2*x + (30*a^14*b*x^(1/2))/31 + (5*a*b^14*x^7)/3 + (195*a^11*b^4*x^
2)/2 + 385*a^9*b^6*x^3 + (2145*a^7*b^8*x^4)/4 + 273*a^5*b^10*x^5 + (910*a^12*b^3*x^(3/2))/29 + (91*a^3*b^12*x^
6)/2 + (2002*a^10*b^5*x^(5/2))/9 + (2574*a^8*b^7*x^(7/2))/5 + (10010*a^6*b^9*x^(9/2))/23 + 130*a^4*b^11*x^(11/
2) + (210*a^2*b^13*x^(13/2))/19)/x^16

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sympy [A]  time = 51.04, size = 212, normalized size = 1.02 \[ - \frac {a^{15}}{16 x^{16}} - \frac {30 a^{14} b}{31 x^{\frac {31}{2}}} - \frac {7 a^{13} b^{2}}{x^{15}} - \frac {910 a^{12} b^{3}}{29 x^{\frac {29}{2}}} - \frac {195 a^{11} b^{4}}{2 x^{14}} - \frac {2002 a^{10} b^{5}}{9 x^{\frac {27}{2}}} - \frac {385 a^{9} b^{6}}{x^{13}} - \frac {2574 a^{8} b^{7}}{5 x^{\frac {25}{2}}} - \frac {2145 a^{7} b^{8}}{4 x^{12}} - \frac {10010 a^{6} b^{9}}{23 x^{\frac {23}{2}}} - \frac {273 a^{5} b^{10}}{x^{11}} - \frac {130 a^{4} b^{11}}{x^{\frac {21}{2}}} - \frac {91 a^{3} b^{12}}{2 x^{10}} - \frac {210 a^{2} b^{13}}{19 x^{\frac {19}{2}}} - \frac {5 a b^{14}}{3 x^{9}} - \frac {2 b^{15}}{17 x^{\frac {17}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/2))**15/x**17,x)

[Out]

-a**15/(16*x**16) - 30*a**14*b/(31*x**(31/2)) - 7*a**13*b**2/x**15 - 910*a**12*b**3/(29*x**(29/2)) - 195*a**11
*b**4/(2*x**14) - 2002*a**10*b**5/(9*x**(27/2)) - 385*a**9*b**6/x**13 - 2574*a**8*b**7/(5*x**(25/2)) - 2145*a*
*7*b**8/(4*x**12) - 10010*a**6*b**9/(23*x**(23/2)) - 273*a**5*b**10/x**11 - 130*a**4*b**11/x**(21/2) - 91*a**3
*b**12/(2*x**10) - 210*a**2*b**13/(19*x**(19/2)) - 5*a*b**14/(3*x**9) - 2*b**15/(17*x**(17/2))

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