3.2345 \(\int \frac{\left (a+b \sqrt [3]{x}\right )^{15}}{x^9} \, dx\)

Optimal. Leaf size=224 \[ -\frac{b^8 \left (a+b \sqrt [3]{x}\right )^{16}}{3922512 a^9 x^{16/3}}+\frac{b^7 \left (a+b \sqrt [3]{x}\right )^{16}}{245157 a^8 x^{17/3}}-\frac{b^6 \left (a+b \sqrt [3]{x}\right )^{16}}{28842 a^7 x^6}+\frac{b^5 \left (a+b \sqrt [3]{x}\right )^{16}}{4807 a^6 x^{19/3}}-\frac{b^4 \left (a+b \sqrt [3]{x}\right )^{16}}{1012 a^5 x^{20/3}}+\frac{b^3 \left (a+b \sqrt [3]{x}\right )^{16}}{253 a^4 x^7}-\frac{7 b^2 \left (a+b \sqrt [3]{x}\right )^{16}}{506 a^3 x^{22/3}}+\frac{b \left (a+b \sqrt [3]{x}\right )^{16}}{23 a^2 x^{23/3}}-\frac{\left (a+b \sqrt [3]{x}\right )^{16}}{8 a x^8} \]

[Out]

-(a + b*x^(1/3))^16/(8*a*x^8) + (b*(a + b*x^(1/3))^16)/(23*a^2*x^(23/3)) - (7*b^
2*(a + b*x^(1/3))^16)/(506*a^3*x^(22/3)) + (b^3*(a + b*x^(1/3))^16)/(253*a^4*x^7
) - (b^4*(a + b*x^(1/3))^16)/(1012*a^5*x^(20/3)) + (b^5*(a + b*x^(1/3))^16)/(480
7*a^6*x^(19/3)) - (b^6*(a + b*x^(1/3))^16)/(28842*a^7*x^6) + (b^7*(a + b*x^(1/3)
)^16)/(245157*a^8*x^(17/3)) - (b^8*(a + b*x^(1/3))^16)/(3922512*a^9*x^(16/3))

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Rubi [A]  time = 0.295676, antiderivative size = 224, 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 \[ -\frac{b^8 \left (a+b \sqrt [3]{x}\right )^{16}}{3922512 a^9 x^{16/3}}+\frac{b^7 \left (a+b \sqrt [3]{x}\right )^{16}}{245157 a^8 x^{17/3}}-\frac{b^6 \left (a+b \sqrt [3]{x}\right )^{16}}{28842 a^7 x^6}+\frac{b^5 \left (a+b \sqrt [3]{x}\right )^{16}}{4807 a^6 x^{19/3}}-\frac{b^4 \left (a+b \sqrt [3]{x}\right )^{16}}{1012 a^5 x^{20/3}}+\frac{b^3 \left (a+b \sqrt [3]{x}\right )^{16}}{253 a^4 x^7}-\frac{7 b^2 \left (a+b \sqrt [3]{x}\right )^{16}}{506 a^3 x^{22/3}}+\frac{b \left (a+b \sqrt [3]{x}\right )^{16}}{23 a^2 x^{23/3}}-\frac{\left (a+b \sqrt [3]{x}\right )^{16}}{8 a x^8} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^(1/3))^15/x^9,x]

[Out]

-(a + b*x^(1/3))^16/(8*a*x^8) + (b*(a + b*x^(1/3))^16)/(23*a^2*x^(23/3)) - (7*b^
2*(a + b*x^(1/3))^16)/(506*a^3*x^(22/3)) + (b^3*(a + b*x^(1/3))^16)/(253*a^4*x^7
) - (b^4*(a + b*x^(1/3))^16)/(1012*a^5*x^(20/3)) + (b^5*(a + b*x^(1/3))^16)/(480
7*a^6*x^(19/3)) - (b^6*(a + b*x^(1/3))^16)/(28842*a^7*x^6) + (b^7*(a + b*x^(1/3)
)^16)/(245157*a^8*x^(17/3)) - (b^8*(a + b*x^(1/3))^16)/(3922512*a^9*x^(16/3))

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Rubi in Sympy [A]  time = 42.1918, size = 202, normalized size = 0.9 \[ - \frac{\left (a + b \sqrt [3]{x}\right )^{16}}{8 a x^{8}} + \frac{b \left (a + b \sqrt [3]{x}\right )^{16}}{23 a^{2} x^{\frac{23}{3}}} - \frac{7 b^{2} \left (a + b \sqrt [3]{x}\right )^{16}}{506 a^{3} x^{\frac{22}{3}}} + \frac{b^{3} \left (a + b \sqrt [3]{x}\right )^{16}}{253 a^{4} x^{7}} - \frac{b^{4} \left (a + b \sqrt [3]{x}\right )^{16}}{1012 a^{5} x^{\frac{20}{3}}} + \frac{b^{5} \left (a + b \sqrt [3]{x}\right )^{16}}{4807 a^{6} x^{\frac{19}{3}}} - \frac{b^{6} \left (a + b \sqrt [3]{x}\right )^{16}}{28842 a^{7} x^{6}} + \frac{b^{7} \left (a + b \sqrt [3]{x}\right )^{16}}{245157 a^{8} x^{\frac{17}{3}}} - \frac{b^{8} \left (a + b \sqrt [3]{x}\right )^{16}}{3922512 a^{9} x^{\frac{16}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**(1/3))**15/x**9,x)

[Out]

-(a + b*x**(1/3))**16/(8*a*x**8) + b*(a + b*x**(1/3))**16/(23*a**2*x**(23/3)) -
7*b**2*(a + b*x**(1/3))**16/(506*a**3*x**(22/3)) + b**3*(a + b*x**(1/3))**16/(25
3*a**4*x**7) - b**4*(a + b*x**(1/3))**16/(1012*a**5*x**(20/3)) + b**5*(a + b*x**
(1/3))**16/(4807*a**6*x**(19/3)) - b**6*(a + b*x**(1/3))**16/(28842*a**7*x**6) +
b**7*(a + b*x**(1/3))**16/(245157*a**8*x**(17/3)) - b**8*(a + b*x**(1/3))**16/(
3922512*a**9*x**(16/3))

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Mathematica [A]  time = 0.0744744, size = 213, normalized size = 0.95 \[ -\frac{a^{15}}{8 x^8}-\frac{45 a^{14} b}{23 x^{23/3}}-\frac{315 a^{13} b^2}{22 x^{22/3}}-\frac{65 a^{12} b^3}{x^7}-\frac{819 a^{11} b^4}{4 x^{20/3}}-\frac{9009 a^{10} b^5}{19 x^{19/3}}-\frac{5005 a^9 b^6}{6 x^6}-\frac{19305 a^8 b^7}{17 x^{17/3}}-\frac{19305 a^7 b^8}{16 x^{16/3}}-\frac{1001 a^6 b^9}{x^5}-\frac{1287 a^5 b^{10}}{2 x^{14/3}}-\frac{315 a^4 b^{11}}{x^{13/3}}-\frac{455 a^3 b^{12}}{4 x^4}-\frac{315 a^2 b^{13}}{11 x^{11/3}}-\frac{9 a b^{14}}{2 x^{10/3}}-\frac{b^{15}}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^(1/3))^15/x^9,x]

[Out]

-a^15/(8*x^8) - (45*a^14*b)/(23*x^(23/3)) - (315*a^13*b^2)/(22*x^(22/3)) - (65*a
^12*b^3)/x^7 - (819*a^11*b^4)/(4*x^(20/3)) - (9009*a^10*b^5)/(19*x^(19/3)) - (50
05*a^9*b^6)/(6*x^6) - (19305*a^8*b^7)/(17*x^(17/3)) - (19305*a^7*b^8)/(16*x^(16/
3)) - (1001*a^6*b^9)/x^5 - (1287*a^5*b^10)/(2*x^(14/3)) - (315*a^4*b^11)/x^(13/3
) - (455*a^3*b^12)/(4*x^4) - (315*a^2*b^13)/(11*x^(11/3)) - (9*a*b^14)/(2*x^(10/
3)) - b^15/(3*x^3)

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Maple [A]  time = 0.011, size = 168, normalized size = 0.8 \[ -{\frac{819\,{a}^{11}{b}^{4}}{4}{x}^{-{\frac{20}{3}}}}-{\frac{455\,{a}^{3}{b}^{12}}{4\,{x}^{4}}}-1001\,{\frac{{a}^{6}{b}^{9}}{{x}^{5}}}-{\frac{19305\,{a}^{7}{b}^{8}}{16}{x}^{-{\frac{16}{3}}}}-{\frac{9009\,{a}^{10}{b}^{5}}{19}{x}^{-{\frac{19}{3}}}}-315\,{\frac{{a}^{4}{b}^{11}}{{x}^{13/3}}}-{\frac{45\,{a}^{14}b}{23}{x}^{-{\frac{23}{3}}}}-{\frac{9\,a{b}^{14}}{2}{x}^{-{\frac{10}{3}}}}-{\frac{19305\,{a}^{8}{b}^{7}}{17}{x}^{-{\frac{17}{3}}}}-{\frac{315\,{a}^{2}{b}^{13}}{11}{x}^{-{\frac{11}{3}}}}-{\frac{1287\,{a}^{5}{b}^{10}}{2}{x}^{-{\frac{14}{3}}}}-{\frac{{b}^{15}}{3\,{x}^{3}}}-{\frac{315\,{a}^{13}{b}^{2}}{22}{x}^{-{\frac{22}{3}}}}-{\frac{5005\,{a}^{9}{b}^{6}}{6\,{x}^{6}}}-{\frac{{a}^{15}}{8\,{x}^{8}}}-65\,{\frac{{a}^{12}{b}^{3}}{{x}^{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/3))^15/x^9,x)

[Out]

-819/4*a^11*b^4/x^(20/3)-455/4*a^3*b^12/x^4-1001*a^6*b^9/x^5-19305/16*a^7*b^8/x^
(16/3)-9009/19*a^10*b^5/x^(19/3)-315*a^4*b^11/x^(13/3)-45/23*a^14*b/x^(23/3)-9/2
*a*b^14/x^(10/3)-19305/17*a^8*b^7/x^(17/3)-315/11*a^2*b^13/x^(11/3)-1287/2*a^5*b
^10/x^(14/3)-1/3*b^15/x^3-315/22*a^13*b^2/x^(22/3)-5005/6*a^9*b^6/x^6-1/8*a^15/x
^8-65*a^12*b^3/x^7

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Maxima [A]  time = 1.44404, size = 225, normalized size = 1. \[ -\frac{1307504 \, b^{15} x^{5} + 17651304 \, a b^{14} x^{\frac{14}{3}} + 112326480 \, a^{2} b^{13} x^{\frac{13}{3}} + 446185740 \, a^{3} b^{12} x^{4} + 1235591280 \, a^{4} b^{11} x^{\frac{11}{3}} + 2524136472 \, a^{5} b^{10} x^{\frac{10}{3}} + 3926434512 \, a^{6} b^{9} x^{3} + 4732755885 \, a^{7} b^{8} x^{\frac{8}{3}} + 4454358480 \, a^{8} b^{7} x^{\frac{7}{3}} + 3272028760 \, a^{9} b^{6} x^{2} + 1859890032 \, a^{10} b^{5} x^{\frac{5}{3}} + 803134332 \, a^{11} b^{4} x^{\frac{4}{3}} + 254963280 \, a^{12} b^{3} x + 56163240 \, a^{13} b^{2} x^{\frac{2}{3}} + 7674480 \, a^{14} b x^{\frac{1}{3}} + 490314 \, a^{15}}{3922512 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^15/x^9,x, algorithm="maxima")

[Out]

-1/3922512*(1307504*b^15*x^5 + 17651304*a*b^14*x^(14/3) + 112326480*a^2*b^13*x^(
13/3) + 446185740*a^3*b^12*x^4 + 1235591280*a^4*b^11*x^(11/3) + 2524136472*a^5*b
^10*x^(10/3) + 3926434512*a^6*b^9*x^3 + 4732755885*a^7*b^8*x^(8/3) + 4454358480*
a^8*b^7*x^(7/3) + 3272028760*a^9*b^6*x^2 + 1859890032*a^10*b^5*x^(5/3) + 8031343
32*a^11*b^4*x^(4/3) + 254963280*a^12*b^3*x + 56163240*a^13*b^2*x^(2/3) + 7674480
*a^14*b*x^(1/3) + 490314*a^15)/x^8

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Fricas [A]  time = 0.218407, size = 228, normalized size = 1.02 \[ -\frac{1307504 \, b^{15} x^{5} + 446185740 \, a^{3} b^{12} x^{4} + 3926434512 \, a^{6} b^{9} x^{3} + 3272028760 \, a^{9} b^{6} x^{2} + 254963280 \, a^{12} b^{3} x + 490314 \, a^{15} + 10557 \,{\left (1672 \, a b^{14} x^{4} + 117040 \, a^{4} b^{11} x^{3} + 448305 \, a^{7} b^{8} x^{2} + 176176 \, a^{10} b^{5} x + 5320 \, a^{13} b^{2}\right )} x^{\frac{2}{3}} + 2052 \,{\left (54740 \, a^{2} b^{13} x^{4} + 1230086 \, a^{5} b^{10} x^{3} + 2170740 \, a^{8} b^{7} x^{2} + 391391 \, a^{11} b^{4} x + 3740 \, a^{14} b\right )} x^{\frac{1}{3}}}{3922512 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^15/x^9,x, algorithm="fricas")

[Out]

-1/3922512*(1307504*b^15*x^5 + 446185740*a^3*b^12*x^4 + 3926434512*a^6*b^9*x^3 +
3272028760*a^9*b^6*x^2 + 254963280*a^12*b^3*x + 490314*a^15 + 10557*(1672*a*b^1
4*x^4 + 117040*a^4*b^11*x^3 + 448305*a^7*b^8*x^2 + 176176*a^10*b^5*x + 5320*a^13
*b^2)*x^(2/3) + 2052*(54740*a^2*b^13*x^4 + 1230086*a^5*b^10*x^3 + 2170740*a^8*b^
7*x^2 + 391391*a^11*b^4*x + 3740*a^14*b)*x^(1/3))/x^8

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Sympy [A]  time = 93.4596, size = 216, normalized size = 0.96 \[ - \frac{a^{15}}{8 x^{8}} - \frac{45 a^{14} b}{23 x^{\frac{23}{3}}} - \frac{315 a^{13} b^{2}}{22 x^{\frac{22}{3}}} - \frac{65 a^{12} b^{3}}{x^{7}} - \frac{819 a^{11} b^{4}}{4 x^{\frac{20}{3}}} - \frac{9009 a^{10} b^{5}}{19 x^{\frac{19}{3}}} - \frac{5005 a^{9} b^{6}}{6 x^{6}} - \frac{19305 a^{8} b^{7}}{17 x^{\frac{17}{3}}} - \frac{19305 a^{7} b^{8}}{16 x^{\frac{16}{3}}} - \frac{1001 a^{6} b^{9}}{x^{5}} - \frac{1287 a^{5} b^{10}}{2 x^{\frac{14}{3}}} - \frac{315 a^{4} b^{11}}{x^{\frac{13}{3}}} - \frac{455 a^{3} b^{12}}{4 x^{4}} - \frac{315 a^{2} b^{13}}{11 x^{\frac{11}{3}}} - \frac{9 a b^{14}}{2 x^{\frac{10}{3}}} - \frac{b^{15}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**(1/3))**15/x**9,x)

[Out]

-a**15/(8*x**8) - 45*a**14*b/(23*x**(23/3)) - 315*a**13*b**2/(22*x**(22/3)) - 65
*a**12*b**3/x**7 - 819*a**11*b**4/(4*x**(20/3)) - 9009*a**10*b**5/(19*x**(19/3))
- 5005*a**9*b**6/(6*x**6) - 19305*a**8*b**7/(17*x**(17/3)) - 19305*a**7*b**8/(1
6*x**(16/3)) - 1001*a**6*b**9/x**5 - 1287*a**5*b**10/(2*x**(14/3)) - 315*a**4*b*
*11/x**(13/3) - 455*a**3*b**12/(4*x**4) - 315*a**2*b**13/(11*x**(11/3)) - 9*a*b*
*14/(2*x**(10/3)) - b**15/(3*x**3)

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GIAC/XCAS [A]  time = 0.221664, size = 225, normalized size = 1. \[ -\frac{1307504 \, b^{15} x^{5} + 17651304 \, a b^{14} x^{\frac{14}{3}} + 112326480 \, a^{2} b^{13} x^{\frac{13}{3}} + 446185740 \, a^{3} b^{12} x^{4} + 1235591280 \, a^{4} b^{11} x^{\frac{11}{3}} + 2524136472 \, a^{5} b^{10} x^{\frac{10}{3}} + 3926434512 \, a^{6} b^{9} x^{3} + 4732755885 \, a^{7} b^{8} x^{\frac{8}{3}} + 4454358480 \, a^{8} b^{7} x^{\frac{7}{3}} + 3272028760 \, a^{9} b^{6} x^{2} + 1859890032 \, a^{10} b^{5} x^{\frac{5}{3}} + 803134332 \, a^{11} b^{4} x^{\frac{4}{3}} + 254963280 \, a^{12} b^{3} x + 56163240 \, a^{13} b^{2} x^{\frac{2}{3}} + 7674480 \, a^{14} b x^{\frac{1}{3}} + 490314 \, a^{15}}{3922512 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^15/x^9,x, algorithm="giac")

[Out]

-1/3922512*(1307504*b^15*x^5 + 17651304*a*b^14*x^(14/3) + 112326480*a^2*b^13*x^(
13/3) + 446185740*a^3*b^12*x^4 + 1235591280*a^4*b^11*x^(11/3) + 2524136472*a^5*b
^10*x^(10/3) + 3926434512*a^6*b^9*x^3 + 4732755885*a^7*b^8*x^(8/3) + 4454358480*
a^8*b^7*x^(7/3) + 3272028760*a^9*b^6*x^2 + 1859890032*a^10*b^5*x^(5/3) + 8031343
32*a^11*b^4*x^(4/3) + 254963280*a^12*b^3*x + 56163240*a^13*b^2*x^(2/3) + 7674480
*a^14*b*x^(1/3) + 490314*a^15)/x^8