3.2333 \(\int \left (a+b \sqrt [3]{x}\right )^{15} x^4 \, dx\)

Optimal. Leaf size=217 \[ \frac{a^{15} x^5}{5}+\frac{45}{16} a^{14} b x^{16/3}+\frac{315}{17} a^{13} b^2 x^{17/3}+\frac{455}{6} a^{12} b^3 x^6+\frac{4095}{19} a^{11} b^4 x^{19/3}+\frac{9009}{20} a^{10} b^5 x^{20/3}+715 a^9 b^6 x^7+\frac{1755}{2} a^8 b^7 x^{22/3}+\frac{19305}{23} a^7 b^8 x^{23/3}+\frac{5005}{8} a^6 b^9 x^8+\frac{9009}{25} a^5 b^{10} x^{25/3}+\frac{315}{2} a^4 b^{11} x^{26/3}+\frac{455}{9} a^3 b^{12} x^9+\frac{45}{4} a^2 b^{13} x^{28/3}+\frac{45}{29} a b^{14} x^{29/3}+\frac{b^{15} x^{10}}{10} \]

[Out]

(a^15*x^5)/5 + (45*a^14*b*x^(16/3))/16 + (315*a^13*b^2*x^(17/3))/17 + (455*a^12*
b^3*x^6)/6 + (4095*a^11*b^4*x^(19/3))/19 + (9009*a^10*b^5*x^(20/3))/20 + 715*a^9
*b^6*x^7 + (1755*a^8*b^7*x^(22/3))/2 + (19305*a^7*b^8*x^(23/3))/23 + (5005*a^6*b
^9*x^8)/8 + (9009*a^5*b^10*x^(25/3))/25 + (315*a^4*b^11*x^(26/3))/2 + (455*a^3*b
^12*x^9)/9 + (45*a^2*b^13*x^(28/3))/4 + (45*a*b^14*x^(29/3))/29 + (b^15*x^10)/10

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Rubi [A]  time = 0.350772, antiderivative size = 217, 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 \[ \frac{a^{15} x^5}{5}+\frac{45}{16} a^{14} b x^{16/3}+\frac{315}{17} a^{13} b^2 x^{17/3}+\frac{455}{6} a^{12} b^3 x^6+\frac{4095}{19} a^{11} b^4 x^{19/3}+\frac{9009}{20} a^{10} b^5 x^{20/3}+715 a^9 b^6 x^7+\frac{1755}{2} a^8 b^7 x^{22/3}+\frac{19305}{23} a^7 b^8 x^{23/3}+\frac{5005}{8} a^6 b^9 x^8+\frac{9009}{25} a^5 b^{10} x^{25/3}+\frac{315}{2} a^4 b^{11} x^{26/3}+\frac{455}{9} a^3 b^{12} x^9+\frac{45}{4} a^2 b^{13} x^{28/3}+\frac{45}{29} a b^{14} x^{29/3}+\frac{b^{15} x^{10}}{10} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^(1/3))^15*x^4,x]

[Out]

(a^15*x^5)/5 + (45*a^14*b*x^(16/3))/16 + (315*a^13*b^2*x^(17/3))/17 + (455*a^12*
b^3*x^6)/6 + (4095*a^11*b^4*x^(19/3))/19 + (9009*a^10*b^5*x^(20/3))/20 + 715*a^9
*b^6*x^7 + (1755*a^8*b^7*x^(22/3))/2 + (19305*a^7*b^8*x^(23/3))/23 + (5005*a^6*b
^9*x^8)/8 + (9009*a^5*b^10*x^(25/3))/25 + (315*a^4*b^11*x^(26/3))/2 + (455*a^3*b
^12*x^9)/9 + (45*a^2*b^13*x^(28/3))/4 + (45*a*b^14*x^(29/3))/29 + (b^15*x^10)/10

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Rubi in Sympy [A]  time = 61.9748, size = 218, normalized size = 1. \[ \frac{a^{15} x^{5}}{5} + \frac{45 a^{14} b x^{\frac{16}{3}}}{16} + \frac{315 a^{13} b^{2} x^{\frac{17}{3}}}{17} + \frac{455 a^{12} b^{3} x^{6}}{6} + \frac{4095 a^{11} b^{4} x^{\frac{19}{3}}}{19} + \frac{9009 a^{10} b^{5} x^{\frac{20}{3}}}{20} + 715 a^{9} b^{6} x^{7} + \frac{1755 a^{8} b^{7} x^{\frac{22}{3}}}{2} + \frac{19305 a^{7} b^{8} x^{\frac{23}{3}}}{23} + \frac{5005 a^{6} b^{9} x^{8}}{8} + \frac{9009 a^{5} b^{10} x^{\frac{25}{3}}}{25} + \frac{315 a^{4} b^{11} x^{\frac{26}{3}}}{2} + \frac{455 a^{3} b^{12} x^{9}}{9} + \frac{45 a^{2} b^{13} x^{\frac{28}{3}}}{4} + \frac{45 a b^{14} x^{\frac{29}{3}}}{29} + \frac{b^{15} x^{10}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**(1/3))**15*x**4,x)

[Out]

a**15*x**5/5 + 45*a**14*b*x**(16/3)/16 + 315*a**13*b**2*x**(17/3)/17 + 455*a**12
*b**3*x**6/6 + 4095*a**11*b**4*x**(19/3)/19 + 9009*a**10*b**5*x**(20/3)/20 + 715
*a**9*b**6*x**7 + 1755*a**8*b**7*x**(22/3)/2 + 19305*a**7*b**8*x**(23/3)/23 + 50
05*a**6*b**9*x**8/8 + 9009*a**5*b**10*x**(25/3)/25 + 315*a**4*b**11*x**(26/3)/2
+ 455*a**3*b**12*x**9/9 + 45*a**2*b**13*x**(28/3)/4 + 45*a*b**14*x**(29/3)/29 +
b**15*x**10/10

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Mathematica [A]  time = 0.030992, size = 217, normalized size = 1. \[ \frac{a^{15} x^5}{5}+\frac{45}{16} a^{14} b x^{16/3}+\frac{315}{17} a^{13} b^2 x^{17/3}+\frac{455}{6} a^{12} b^3 x^6+\frac{4095}{19} a^{11} b^4 x^{19/3}+\frac{9009}{20} a^{10} b^5 x^{20/3}+715 a^9 b^6 x^7+\frac{1755}{2} a^8 b^7 x^{22/3}+\frac{19305}{23} a^7 b^8 x^{23/3}+\frac{5005}{8} a^6 b^9 x^8+\frac{9009}{25} a^5 b^{10} x^{25/3}+\frac{315}{2} a^4 b^{11} x^{26/3}+\frac{455}{9} a^3 b^{12} x^9+\frac{45}{4} a^2 b^{13} x^{28/3}+\frac{45}{29} a b^{14} x^{29/3}+\frac{b^{15} x^{10}}{10} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^(1/3))^15*x^4,x]

[Out]

(a^15*x^5)/5 + (45*a^14*b*x^(16/3))/16 + (315*a^13*b^2*x^(17/3))/17 + (455*a^12*
b^3*x^6)/6 + (4095*a^11*b^4*x^(19/3))/19 + (9009*a^10*b^5*x^(20/3))/20 + 715*a^9
*b^6*x^7 + (1755*a^8*b^7*x^(22/3))/2 + (19305*a^7*b^8*x^(23/3))/23 + (5005*a^6*b
^9*x^8)/8 + (9009*a^5*b^10*x^(25/3))/25 + (315*a^4*b^11*x^(26/3))/2 + (455*a^3*b
^12*x^9)/9 + (45*a^2*b^13*x^(28/3))/4 + (45*a*b^14*x^(29/3))/29 + (b^15*x^10)/10

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Maple [A]  time = 0.004, size = 168, normalized size = 0.8 \[{\frac{{a}^{15}{x}^{5}}{5}}+{\frac{45\,{a}^{14}b}{16}{x}^{{\frac{16}{3}}}}+{\frac{315\,{a}^{13}{b}^{2}}{17}{x}^{{\frac{17}{3}}}}+{\frac{455\,{a}^{12}{b}^{3}{x}^{6}}{6}}+{\frac{4095\,{a}^{11}{b}^{4}}{19}{x}^{{\frac{19}{3}}}}+{\frac{9009\,{a}^{10}{b}^{5}}{20}{x}^{{\frac{20}{3}}}}+715\,{a}^{9}{b}^{6}{x}^{7}+{\frac{1755\,{a}^{8}{b}^{7}}{2}{x}^{{\frac{22}{3}}}}+{\frac{19305\,{a}^{7}{b}^{8}}{23}{x}^{{\frac{23}{3}}}}+{\frac{5005\,{a}^{6}{b}^{9}{x}^{8}}{8}}+{\frac{9009\,{a}^{5}{b}^{10}}{25}{x}^{{\frac{25}{3}}}}+{\frac{315\,{a}^{4}{b}^{11}}{2}{x}^{{\frac{26}{3}}}}+{\frac{455\,{a}^{3}{b}^{12}{x}^{9}}{9}}+{\frac{45\,{a}^{2}{b}^{13}}{4}{x}^{{\frac{28}{3}}}}+{\frac{45\,a{b}^{14}}{29}{x}^{{\frac{29}{3}}}}+{\frac{{b}^{15}{x}^{10}}{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/3))^15*x^4,x)

[Out]

1/5*a^15*x^5+45/16*a^14*b*x^(16/3)+315/17*a^13*b^2*x^(17/3)+455/6*a^12*b^3*x^6+4
095/19*a^11*b^4*x^(19/3)+9009/20*a^10*b^5*x^(20/3)+715*a^9*b^6*x^7+1755/2*a^8*b^
7*x^(22/3)+19305/23*a^7*b^8*x^(23/3)+5005/8*a^6*b^9*x^8+9009/25*a^5*b^10*x^(25/3
)+315/2*a^4*b^11*x^(26/3)+455/9*a^3*b^12*x^9+45/4*a^2*b^13*x^(28/3)+45/29*a*b^14
*x^(29/3)+1/10*b^15*x^10

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Maxima [A]  time = 1.44837, size = 339, normalized size = 1.56 \[ \frac{{\left (b x^{\frac{1}{3}} + a\right )}^{30}}{10 \, b^{15}} - \frac{42 \,{\left (b x^{\frac{1}{3}} + a\right )}^{29} a}{29 \, b^{15}} + \frac{39 \,{\left (b x^{\frac{1}{3}} + a\right )}^{28} a^{2}}{4 \, b^{15}} - \frac{364 \,{\left (b x^{\frac{1}{3}} + a\right )}^{27} a^{3}}{9 \, b^{15}} + \frac{231 \,{\left (b x^{\frac{1}{3}} + a\right )}^{26} a^{4}}{2 \, b^{15}} - \frac{6006 \,{\left (b x^{\frac{1}{3}} + a\right )}^{25} a^{5}}{25 \, b^{15}} + \frac{3003 \,{\left (b x^{\frac{1}{3}} + a\right )}^{24} a^{6}}{8 \, b^{15}} - \frac{10296 \,{\left (b x^{\frac{1}{3}} + a\right )}^{23} a^{7}}{23 \, b^{15}} + \frac{819 \,{\left (b x^{\frac{1}{3}} + a\right )}^{22} a^{8}}{2 \, b^{15}} - \frac{286 \,{\left (b x^{\frac{1}{3}} + a\right )}^{21} a^{9}}{b^{15}} + \frac{3003 \,{\left (b x^{\frac{1}{3}} + a\right )}^{20} a^{10}}{20 \, b^{15}} - \frac{1092 \,{\left (b x^{\frac{1}{3}} + a\right )}^{19} a^{11}}{19 \, b^{15}} + \frac{91 \,{\left (b x^{\frac{1}{3}} + a\right )}^{18} a^{12}}{6 \, b^{15}} - \frac{42 \,{\left (b x^{\frac{1}{3}} + a\right )}^{17} a^{13}}{17 \, b^{15}} + \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{16} a^{14}}{16 \, b^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^15*x^4,x, algorithm="maxima")

[Out]

1/10*(b*x^(1/3) + a)^30/b^15 - 42/29*(b*x^(1/3) + a)^29*a/b^15 + 39/4*(b*x^(1/3)
+ a)^28*a^2/b^15 - 364/9*(b*x^(1/3) + a)^27*a^3/b^15 + 231/2*(b*x^(1/3) + a)^26
*a^4/b^15 - 6006/25*(b*x^(1/3) + a)^25*a^5/b^15 + 3003/8*(b*x^(1/3) + a)^24*a^6/
b^15 - 10296/23*(b*x^(1/3) + a)^23*a^7/b^15 + 819/2*(b*x^(1/3) + a)^22*a^8/b^15
- 286*(b*x^(1/3) + a)^21*a^9/b^15 + 3003/20*(b*x^(1/3) + a)^20*a^10/b^15 - 1092/
19*(b*x^(1/3) + a)^19*a^11/b^15 + 91/6*(b*x^(1/3) + a)^18*a^12/b^15 - 42/17*(b*x
^(1/3) + a)^17*a^13/b^15 + 3/16*(b*x^(1/3) + a)^16*a^14/b^15

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Fricas [A]  time = 0.215154, size = 242, normalized size = 1.12 \[ \frac{1}{10} \, b^{15} x^{10} + \frac{455}{9} \, a^{3} b^{12} x^{9} + \frac{5005}{8} \, a^{6} b^{9} x^{8} + 715 \, a^{9} b^{6} x^{7} + \frac{455}{6} \, a^{12} b^{3} x^{6} + \frac{1}{5} \, a^{15} x^{5} + \frac{9}{226780} \,{\left (39100 \, a b^{14} x^{9} + 3968650 \, a^{4} b^{11} x^{8} + 21149700 \, a^{7} b^{8} x^{7} + 11350339 \, a^{10} b^{5} x^{6} + 466900 \, a^{13} b^{2} x^{5}\right )} x^{\frac{2}{3}} + \frac{9}{7600} \,{\left (9500 \, a^{2} b^{13} x^{9} + 304304 \, a^{5} b^{10} x^{8} + 741000 \, a^{8} b^{7} x^{7} + 182000 \, a^{11} b^{4} x^{6} + 2375 \, a^{14} b x^{5}\right )} x^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^15*x^4,x, algorithm="fricas")

[Out]

1/10*b^15*x^10 + 455/9*a^3*b^12*x^9 + 5005/8*a^6*b^9*x^8 + 715*a^9*b^6*x^7 + 455
/6*a^12*b^3*x^6 + 1/5*a^15*x^5 + 9/226780*(39100*a*b^14*x^9 + 3968650*a^4*b^11*x
^8 + 21149700*a^7*b^8*x^7 + 11350339*a^10*b^5*x^6 + 466900*a^13*b^2*x^5)*x^(2/3)
+ 9/7600*(9500*a^2*b^13*x^9 + 304304*a^5*b^10*x^8 + 741000*a^8*b^7*x^7 + 182000
*a^11*b^4*x^6 + 2375*a^14*b*x^5)*x^(1/3)

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Sympy [A]  time = 40.2561, size = 218, normalized size = 1. \[ \frac{a^{15} x^{5}}{5} + \frac{45 a^{14} b x^{\frac{16}{3}}}{16} + \frac{315 a^{13} b^{2} x^{\frac{17}{3}}}{17} + \frac{455 a^{12} b^{3} x^{6}}{6} + \frac{4095 a^{11} b^{4} x^{\frac{19}{3}}}{19} + \frac{9009 a^{10} b^{5} x^{\frac{20}{3}}}{20} + 715 a^{9} b^{6} x^{7} + \frac{1755 a^{8} b^{7} x^{\frac{22}{3}}}{2} + \frac{19305 a^{7} b^{8} x^{\frac{23}{3}}}{23} + \frac{5005 a^{6} b^{9} x^{8}}{8} + \frac{9009 a^{5} b^{10} x^{\frac{25}{3}}}{25} + \frac{315 a^{4} b^{11} x^{\frac{26}{3}}}{2} + \frac{455 a^{3} b^{12} x^{9}}{9} + \frac{45 a^{2} b^{13} x^{\frac{28}{3}}}{4} + \frac{45 a b^{14} x^{\frac{29}{3}}}{29} + \frac{b^{15} x^{10}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**(1/3))**15*x**4,x)

[Out]

a**15*x**5/5 + 45*a**14*b*x**(16/3)/16 + 315*a**13*b**2*x**(17/3)/17 + 455*a**12
*b**3*x**6/6 + 4095*a**11*b**4*x**(19/3)/19 + 9009*a**10*b**5*x**(20/3)/20 + 715
*a**9*b**6*x**7 + 1755*a**8*b**7*x**(22/3)/2 + 19305*a**7*b**8*x**(23/3)/23 + 50
05*a**6*b**9*x**8/8 + 9009*a**5*b**10*x**(25/3)/25 + 315*a**4*b**11*x**(26/3)/2
+ 455*a**3*b**12*x**9/9 + 45*a**2*b**13*x**(28/3)/4 + 45*a*b**14*x**(29/3)/29 +
b**15*x**10/10

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GIAC/XCAS [A]  time = 0.220065, size = 225, normalized size = 1.04 \[ \frac{1}{10} \, b^{15} x^{10} + \frac{45}{29} \, a b^{14} x^{\frac{29}{3}} + \frac{45}{4} \, a^{2} b^{13} x^{\frac{28}{3}} + \frac{455}{9} \, a^{3} b^{12} x^{9} + \frac{315}{2} \, a^{4} b^{11} x^{\frac{26}{3}} + \frac{9009}{25} \, a^{5} b^{10} x^{\frac{25}{3}} + \frac{5005}{8} \, a^{6} b^{9} x^{8} + \frac{19305}{23} \, a^{7} b^{8} x^{\frac{23}{3}} + \frac{1755}{2} \, a^{8} b^{7} x^{\frac{22}{3}} + 715 \, a^{9} b^{6} x^{7} + \frac{9009}{20} \, a^{10} b^{5} x^{\frac{20}{3}} + \frac{4095}{19} \, a^{11} b^{4} x^{\frac{19}{3}} + \frac{455}{6} \, a^{12} b^{3} x^{6} + \frac{315}{17} \, a^{13} b^{2} x^{\frac{17}{3}} + \frac{45}{16} \, a^{14} b x^{\frac{16}{3}} + \frac{1}{5} \, a^{15} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^15*x^4,x, algorithm="giac")

[Out]

1/10*b^15*x^10 + 45/29*a*b^14*x^(29/3) + 45/4*a^2*b^13*x^(28/3) + 455/9*a^3*b^12
*x^9 + 315/2*a^4*b^11*x^(26/3) + 9009/25*a^5*b^10*x^(25/3) + 5005/8*a^6*b^9*x^8
+ 19305/23*a^7*b^8*x^(23/3) + 1755/2*a^8*b^7*x^(22/3) + 715*a^9*b^6*x^7 + 9009/2
0*a^10*b^5*x^(20/3) + 4095/19*a^11*b^4*x^(19/3) + 455/6*a^12*b^3*x^6 + 315/17*a^
13*b^2*x^(17/3) + 45/16*a^14*b*x^(16/3) + 1/5*a^15*x^5