3.2184 \(\int \frac{\left (a+b \sqrt{x}\right )^{15}}{x^{13}} \, dx\)

Optimal. Leaf size=220 \[ -\frac{b^8 \left (a+b \sqrt{x}\right )^{16}}{5883768 a^9 x^8}+\frac{2 b^7 \left (a+b \sqrt{x}\right )^{16}}{735471 a^8 x^{17/2}}-\frac{b^6 \left (a+b \sqrt{x}\right )^{16}}{43263 a^7 x^9}+\frac{2 b^5 \left (a+b \sqrt{x}\right )^{16}}{14421 a^6 x^{19/2}}-\frac{b^4 \left (a+b \sqrt{x}\right )^{16}}{1518 a^5 x^{10}}+\frac{2 b^3 \left (a+b \sqrt{x}\right )^{16}}{759 a^4 x^{21/2}}-\frac{7 b^2 \left (a+b \sqrt{x}\right )^{16}}{759 a^3 x^{11}}+\frac{2 b \left (a+b \sqrt{x}\right )^{16}}{69 a^2 x^{23/2}}-\frac{\left (a+b \sqrt{x}\right )^{16}}{12 a x^{12}} \]

[Out]

-(a + b*Sqrt[x])^16/(12*a*x^12) + (2*b*(a + b*Sqrt[x])^16)/(69*a^2*x^(23/2)) - (
7*b^2*(a + b*Sqrt[x])^16)/(759*a^3*x^11) + (2*b^3*(a + b*Sqrt[x])^16)/(759*a^4*x
^(21/2)) - (b^4*(a + b*Sqrt[x])^16)/(1518*a^5*x^10) + (2*b^5*(a + b*Sqrt[x])^16)
/(14421*a^6*x^(19/2)) - (b^6*(a + b*Sqrt[x])^16)/(43263*a^7*x^9) + (2*b^7*(a + b
*Sqrt[x])^16)/(735471*a^8*x^(17/2)) - (b^8*(a + b*Sqrt[x])^16)/(5883768*a^9*x^8)

_______________________________________________________________________________________

Rubi [A]  time = 0.287344, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{b^8 \left (a+b \sqrt{x}\right )^{16}}{5883768 a^9 x^8}+\frac{2 b^7 \left (a+b \sqrt{x}\right )^{16}}{735471 a^8 x^{17/2}}-\frac{b^6 \left (a+b \sqrt{x}\right )^{16}}{43263 a^7 x^9}+\frac{2 b^5 \left (a+b \sqrt{x}\right )^{16}}{14421 a^6 x^{19/2}}-\frac{b^4 \left (a+b \sqrt{x}\right )^{16}}{1518 a^5 x^{10}}+\frac{2 b^3 \left (a+b \sqrt{x}\right )^{16}}{759 a^4 x^{21/2}}-\frac{7 b^2 \left (a+b \sqrt{x}\right )^{16}}{759 a^3 x^{11}}+\frac{2 b \left (a+b \sqrt{x}\right )^{16}}{69 a^2 x^{23/2}}-\frac{\left (a+b \sqrt{x}\right )^{16}}{12 a x^{12}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^15/x^13,x]

[Out]

-(a + b*Sqrt[x])^16/(12*a*x^12) + (2*b*(a + b*Sqrt[x])^16)/(69*a^2*x^(23/2)) - (
7*b^2*(a + b*Sqrt[x])^16)/(759*a^3*x^11) + (2*b^3*(a + b*Sqrt[x])^16)/(759*a^4*x
^(21/2)) - (b^4*(a + b*Sqrt[x])^16)/(1518*a^5*x^10) + (2*b^5*(a + b*Sqrt[x])^16)
/(14421*a^6*x^(19/2)) - (b^6*(a + b*Sqrt[x])^16)/(43263*a^7*x^9) + (2*b^7*(a + b
*Sqrt[x])^16)/(735471*a^8*x^(17/2)) - (b^8*(a + b*Sqrt[x])^16)/(5883768*a^9*x^8)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 41.8581, size = 206, normalized size = 0.94 \[ - \frac{\left (a + b \sqrt{x}\right )^{16}}{12 a x^{12}} + \frac{2 b \left (a + b \sqrt{x}\right )^{16}}{69 a^{2} x^{\frac{23}{2}}} - \frac{7 b^{2} \left (a + b \sqrt{x}\right )^{16}}{759 a^{3} x^{11}} + \frac{2 b^{3} \left (a + b \sqrt{x}\right )^{16}}{759 a^{4} x^{\frac{21}{2}}} - \frac{b^{4} \left (a + b \sqrt{x}\right )^{16}}{1518 a^{5} x^{10}} + \frac{2 b^{5} \left (a + b \sqrt{x}\right )^{16}}{14421 a^{6} x^{\frac{19}{2}}} - \frac{b^{6} \left (a + b \sqrt{x}\right )^{16}}{43263 a^{7} x^{9}} + \frac{2 b^{7} \left (a + b \sqrt{x}\right )^{16}}{735471 a^{8} x^{\frac{17}{2}}} - \frac{b^{8} \left (a + b \sqrt{x}\right )^{16}}{5883768 a^{9} x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**(1/2))**15/x**13,x)

[Out]

-(a + b*sqrt(x))**16/(12*a*x**12) + 2*b*(a + b*sqrt(x))**16/(69*a**2*x**(23/2))
- 7*b**2*(a + b*sqrt(x))**16/(759*a**3*x**11) + 2*b**3*(a + b*sqrt(x))**16/(759*
a**4*x**(21/2)) - b**4*(a + b*sqrt(x))**16/(1518*a**5*x**10) + 2*b**5*(a + b*sqr
t(x))**16/(14421*a**6*x**(19/2)) - b**6*(a + b*sqrt(x))**16/(43263*a**7*x**9) +
2*b**7*(a + b*sqrt(x))**16/(735471*a**8*x**(17/2)) - b**8*(a + b*sqrt(x))**16/(5
883768*a**9*x**8)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0698245, size = 209, normalized size = 0.95 \[ -\frac{a^{15}}{12 x^{12}}-\frac{30 a^{14} b}{23 x^{23/2}}-\frac{105 a^{13} b^2}{11 x^{11}}-\frac{130 a^{12} b^3}{3 x^{21/2}}-\frac{273 a^{11} b^4}{2 x^{10}}-\frac{6006 a^{10} b^5}{19 x^{19/2}}-\frac{5005 a^9 b^6}{9 x^9}-\frac{12870 a^8 b^7}{17 x^{17/2}}-\frac{6435 a^7 b^8}{8 x^8}-\frac{2002 a^6 b^9}{3 x^{15/2}}-\frac{429 a^5 b^{10}}{x^7}-\frac{210 a^4 b^{11}}{x^{13/2}}-\frac{455 a^3 b^{12}}{6 x^6}-\frac{210 a^2 b^{13}}{11 x^{11/2}}-\frac{3 a b^{14}}{x^5}-\frac{2 b^{15}}{9 x^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*Sqrt[x])^15/x^13,x]

[Out]

-a^15/(12*x^12) - (30*a^14*b)/(23*x^(23/2)) - (105*a^13*b^2)/(11*x^11) - (130*a^
12*b^3)/(3*x^(21/2)) - (273*a^11*b^4)/(2*x^10) - (6006*a^10*b^5)/(19*x^(19/2)) -
(5005*a^9*b^6)/(9*x^9) - (12870*a^8*b^7)/(17*x^(17/2)) - (6435*a^7*b^8)/(8*x^8)
- (2002*a^6*b^9)/(3*x^(15/2)) - (429*a^5*b^10)/x^7 - (210*a^4*b^11)/x^(13/2) -
(455*a^3*b^12)/(6*x^6) - (210*a^2*b^13)/(11*x^(11/2)) - (3*a*b^14)/x^5 - (2*b^15
)/(9*x^(9/2))

_______________________________________________________________________________________

Maple [A]  time = 0.005, size = 168, normalized size = 0.8 \[ -{\frac{2\,{b}^{15}}{9}{x}^{-{\frac{9}{2}}}}-3\,{\frac{a{b}^{14}}{{x}^{5}}}-{\frac{210\,{a}^{2}{b}^{13}}{11}{x}^{-{\frac{11}{2}}}}-{\frac{455\,{a}^{3}{b}^{12}}{6\,{x}^{6}}}-210\,{\frac{{a}^{4}{b}^{11}}{{x}^{13/2}}}-429\,{\frac{{a}^{5}{b}^{10}}{{x}^{7}}}-{\frac{2002\,{a}^{6}{b}^{9}}{3}{x}^{-{\frac{15}{2}}}}-{\frac{6435\,{a}^{7}{b}^{8}}{8\,{x}^{8}}}-{\frac{12870\,{a}^{8}{b}^{7}}{17}{x}^{-{\frac{17}{2}}}}-{\frac{5005\,{a}^{9}{b}^{6}}{9\,{x}^{9}}}-{\frac{6006\,{a}^{10}{b}^{5}}{19}{x}^{-{\frac{19}{2}}}}-{\frac{273\,{a}^{11}{b}^{4}}{2\,{x}^{10}}}-{\frac{130\,{a}^{12}{b}^{3}}{3}{x}^{-{\frac{21}{2}}}}-{\frac{105\,{a}^{13}{b}^{2}}{11\,{x}^{11}}}-{\frac{30\,{a}^{14}b}{23}{x}^{-{\frac{23}{2}}}}-{\frac{{a}^{15}}{12\,{x}^{12}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/2))^15/x^13,x)

[Out]

-2/9*b^15/x^(9/2)-3*a*b^14/x^5-210/11*a^2*b^13/x^(11/2)-455/6*a^3*b^12/x^6-210*a
^4*b^11/x^(13/2)-429*a^5*b^10/x^7-2002/3*a^6*b^9/x^(15/2)-6435/8*a^7*b^8/x^8-128
70/17*a^8*b^7/x^(17/2)-5005/9*a^9*b^6/x^9-6006/19*a^10*b^5/x^(19/2)-273/2*a^11*b
^4/x^10-130/3*a^12*b^3/x^(21/2)-105/11*a^13*b^2/x^11-30/23*a^14*b/x^(23/2)-1/12*
a^15/x^12

_______________________________________________________________________________________

Maxima [A]  time = 1.43073, size = 225, normalized size = 1.02 \[ -\frac{1307504 \, b^{15} x^{\frac{15}{2}} + 17651304 \, a b^{14} x^{7} + 112326480 \, a^{2} b^{13} x^{\frac{13}{2}} + 446185740 \, a^{3} b^{12} x^{6} + 1235591280 \, a^{4} b^{11} x^{\frac{11}{2}} + 2524136472 \, a^{5} b^{10} x^{5} + 3926434512 \, a^{6} b^{9} x^{\frac{9}{2}} + 4732755885 \, a^{7} b^{8} x^{4} + 4454358480 \, a^{8} b^{7} x^{\frac{7}{2}} + 3272028760 \, a^{9} b^{6} x^{3} + 1859890032 \, a^{10} b^{5} x^{\frac{5}{2}} + 803134332 \, a^{11} b^{4} x^{2} + 254963280 \, a^{12} b^{3} x^{\frac{3}{2}} + 56163240 \, a^{13} b^{2} x + 7674480 \, a^{14} b \sqrt{x} + 490314 \, a^{15}}{5883768 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^15/x^13,x, algorithm="maxima")

[Out]

-1/5883768*(1307504*b^15*x^(15/2) + 17651304*a*b^14*x^7 + 112326480*a^2*b^13*x^(
13/2) + 446185740*a^3*b^12*x^6 + 1235591280*a^4*b^11*x^(11/2) + 2524136472*a^5*b
^10*x^5 + 3926434512*a^6*b^9*x^(9/2) + 4732755885*a^7*b^8*x^4 + 4454358480*a^8*b
^7*x^(7/2) + 3272028760*a^9*b^6*x^3 + 1859890032*a^10*b^5*x^(5/2) + 803134332*a^
11*b^4*x^2 + 254963280*a^12*b^3*x^(3/2) + 56163240*a^13*b^2*x + 7674480*a^14*b*s
qrt(x) + 490314*a^15)/x^12

_______________________________________________________________________________________

Fricas [A]  time = 0.240932, size = 227, normalized size = 1.03 \[ -\frac{17651304 \, a b^{14} x^{7} + 446185740 \, a^{3} b^{12} x^{6} + 2524136472 \, a^{5} b^{10} x^{5} + 4732755885 \, a^{7} b^{8} x^{4} + 3272028760 \, a^{9} b^{6} x^{3} + 803134332 \, a^{11} b^{4} x^{2} + 56163240 \, a^{13} b^{2} x + 490314 \, a^{15} + 16 \,{\left (81719 \, b^{15} x^{7} + 7020405 \, a^{2} b^{13} x^{6} + 77224455 \, a^{4} b^{11} x^{5} + 245402157 \, a^{6} b^{9} x^{4} + 278397405 \, a^{8} b^{7} x^{3} + 116243127 \, a^{10} b^{5} x^{2} + 15935205 \, a^{12} b^{3} x + 479655 \, a^{14} b\right )} \sqrt{x}}{5883768 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^15/x^13,x, algorithm="fricas")

[Out]

-1/5883768*(17651304*a*b^14*x^7 + 446185740*a^3*b^12*x^6 + 2524136472*a^5*b^10*x
^5 + 4732755885*a^7*b^8*x^4 + 3272028760*a^9*b^6*x^3 + 803134332*a^11*b^4*x^2 +
56163240*a^13*b^2*x + 490314*a^15 + 16*(81719*b^15*x^7 + 7020405*a^2*b^13*x^6 +
77224455*a^4*b^11*x^5 + 245402157*a^6*b^9*x^4 + 278397405*a^8*b^7*x^3 + 11624312
7*a^10*b^5*x^2 + 15935205*a^12*b^3*x + 479655*a^14*b)*sqrt(x))/x^12

_______________________________________________________________________________________

Sympy [A]  time = 81.9025, size = 214, normalized size = 0.97 \[ - \frac{a^{15}}{12 x^{12}} - \frac{30 a^{14} b}{23 x^{\frac{23}{2}}} - \frac{105 a^{13} b^{2}}{11 x^{11}} - \frac{130 a^{12} b^{3}}{3 x^{\frac{21}{2}}} - \frac{273 a^{11} b^{4}}{2 x^{10}} - \frac{6006 a^{10} b^{5}}{19 x^{\frac{19}{2}}} - \frac{5005 a^{9} b^{6}}{9 x^{9}} - \frac{12870 a^{8} b^{7}}{17 x^{\frac{17}{2}}} - \frac{6435 a^{7} b^{8}}{8 x^{8}} - \frac{2002 a^{6} b^{9}}{3 x^{\frac{15}{2}}} - \frac{429 a^{5} b^{10}}{x^{7}} - \frac{210 a^{4} b^{11}}{x^{\frac{13}{2}}} - \frac{455 a^{3} b^{12}}{6 x^{6}} - \frac{210 a^{2} b^{13}}{11 x^{\frac{11}{2}}} - \frac{3 a b^{14}}{x^{5}} - \frac{2 b^{15}}{9 x^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**(1/2))**15/x**13,x)

[Out]

-a**15/(12*x**12) - 30*a**14*b/(23*x**(23/2)) - 105*a**13*b**2/(11*x**11) - 130*
a**12*b**3/(3*x**(21/2)) - 273*a**11*b**4/(2*x**10) - 6006*a**10*b**5/(19*x**(19
/2)) - 5005*a**9*b**6/(9*x**9) - 12870*a**8*b**7/(17*x**(17/2)) - 6435*a**7*b**8
/(8*x**8) - 2002*a**6*b**9/(3*x**(15/2)) - 429*a**5*b**10/x**7 - 210*a**4*b**11/
x**(13/2) - 455*a**3*b**12/(6*x**6) - 210*a**2*b**13/(11*x**(11/2)) - 3*a*b**14/
x**5 - 2*b**15/(9*x**(9/2))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.224153, size = 225, normalized size = 1.02 \[ -\frac{1307504 \, b^{15} x^{\frac{15}{2}} + 17651304 \, a b^{14} x^{7} + 112326480 \, a^{2} b^{13} x^{\frac{13}{2}} + 446185740 \, a^{3} b^{12} x^{6} + 1235591280 \, a^{4} b^{11} x^{\frac{11}{2}} + 2524136472 \, a^{5} b^{10} x^{5} + 3926434512 \, a^{6} b^{9} x^{\frac{9}{2}} + 4732755885 \, a^{7} b^{8} x^{4} + 4454358480 \, a^{8} b^{7} x^{\frac{7}{2}} + 3272028760 \, a^{9} b^{6} x^{3} + 1859890032 \, a^{10} b^{5} x^{\frac{5}{2}} + 803134332 \, a^{11} b^{4} x^{2} + 254963280 \, a^{12} b^{3} x^{\frac{3}{2}} + 56163240 \, a^{13} b^{2} x + 7674480 \, a^{14} b \sqrt{x} + 490314 \, a^{15}}{5883768 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^15/x^13,x, algorithm="giac")

[Out]

-1/5883768*(1307504*b^15*x^(15/2) + 17651304*a*b^14*x^7 + 112326480*a^2*b^13*x^(
13/2) + 446185740*a^3*b^12*x^6 + 1235591280*a^4*b^11*x^(11/2) + 2524136472*a^5*b
^10*x^5 + 3926434512*a^6*b^9*x^(9/2) + 4732755885*a^7*b^8*x^4 + 4454358480*a^8*b
^7*x^(7/2) + 3272028760*a^9*b^6*x^3 + 1859890032*a^10*b^5*x^(5/2) + 803134332*a^
11*b^4*x^2 + 254963280*a^12*b^3*x^(3/2) + 56163240*a^13*b^2*x + 7674480*a^14*b*s
qrt(x) + 490314*a^15)/x^12