3.2338 \(\int \frac{\left (a+b \sqrt [3]{x}\right )^{15}}{x} \, dx\)

Optimal. Leaf size=209 \[ a^{15} \log (x)+45 a^{14} b \sqrt [3]{x}+\frac{315}{2} a^{13} b^2 x^{2/3}+455 a^{12} b^3 x+\frac{4095}{4} a^{11} b^4 x^{4/3}+\frac{9009}{5} a^{10} b^5 x^{5/3}+\frac{5005}{2} a^9 b^6 x^2+\frac{19305}{7} a^8 b^7 x^{7/3}+\frac{19305}{8} a^7 b^8 x^{8/3}+\frac{5005}{3} a^6 b^9 x^3+\frac{9009}{10} a^5 b^{10} x^{10/3}+\frac{4095}{11} a^4 b^{11} x^{11/3}+\frac{455}{4} a^3 b^{12} x^4+\frac{315}{13} a^2 b^{13} x^{13/3}+\frac{45}{14} a b^{14} x^{14/3}+\frac{b^{15} x^5}{5} \]

[Out]

45*a^14*b*x^(1/3) + (315*a^13*b^2*x^(2/3))/2 + 455*a^12*b^3*x + (4095*a^11*b^4*x
^(4/3))/4 + (9009*a^10*b^5*x^(5/3))/5 + (5005*a^9*b^6*x^2)/2 + (19305*a^8*b^7*x^
(7/3))/7 + (19305*a^7*b^8*x^(8/3))/8 + (5005*a^6*b^9*x^3)/3 + (9009*a^5*b^10*x^(
10/3))/10 + (4095*a^4*b^11*x^(11/3))/11 + (455*a^3*b^12*x^4)/4 + (315*a^2*b^13*x
^(13/3))/13 + (45*a*b^14*x^(14/3))/14 + (b^15*x^5)/5 + a^15*Log[x]

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Rubi [A]  time = 0.243086, antiderivative size = 209, 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 \[ a^{15} \log (x)+45 a^{14} b \sqrt [3]{x}+\frac{315}{2} a^{13} b^2 x^{2/3}+455 a^{12} b^3 x+\frac{4095}{4} a^{11} b^4 x^{4/3}+\frac{9009}{5} a^{10} b^5 x^{5/3}+\frac{5005}{2} a^9 b^6 x^2+\frac{19305}{7} a^8 b^7 x^{7/3}+\frac{19305}{8} a^7 b^8 x^{8/3}+\frac{5005}{3} a^6 b^9 x^3+\frac{9009}{10} a^5 b^{10} x^{10/3}+\frac{4095}{11} a^4 b^{11} x^{11/3}+\frac{455}{4} a^3 b^{12} x^4+\frac{315}{13} a^2 b^{13} x^{13/3}+\frac{45}{14} a b^{14} x^{14/3}+\frac{b^{15} x^5}{5} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^(1/3))^15/x,x]

[Out]

45*a^14*b*x^(1/3) + (315*a^13*b^2*x^(2/3))/2 + 455*a^12*b^3*x + (4095*a^11*b^4*x
^(4/3))/4 + (9009*a^10*b^5*x^(5/3))/5 + (5005*a^9*b^6*x^2)/2 + (19305*a^8*b^7*x^
(7/3))/7 + (19305*a^7*b^8*x^(8/3))/8 + (5005*a^6*b^9*x^3)/3 + (9009*a^5*b^10*x^(
10/3))/10 + (4095*a^4*b^11*x^(11/3))/11 + (455*a^3*b^12*x^4)/4 + (315*a^2*b^13*x
^(13/3))/13 + (45*a*b^14*x^(14/3))/14 + (b^15*x^5)/5 + a^15*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ 3 a^{15} \log{\left (\sqrt [3]{x} \right )} + 45 a^{14} b \sqrt [3]{x} + 315 a^{13} b^{2} \int ^{\sqrt [3]{x}} x\, dx + 455 a^{12} b^{3} x + \frac{4095 a^{11} b^{4} x^{\frac{4}{3}}}{4} + \frac{9009 a^{10} b^{5} x^{\frac{5}{3}}}{5} + \frac{5005 a^{9} b^{6} x^{2}}{2} + \frac{19305 a^{8} b^{7} x^{\frac{7}{3}}}{7} + \frac{19305 a^{7} b^{8} x^{\frac{8}{3}}}{8} + \frac{5005 a^{6} b^{9} x^{3}}{3} + \frac{9009 a^{5} b^{10} x^{\frac{10}{3}}}{10} + \frac{4095 a^{4} b^{11} x^{\frac{11}{3}}}{11} + \frac{455 a^{3} b^{12} x^{4}}{4} + \frac{315 a^{2} b^{13} x^{\frac{13}{3}}}{13} + \frac{45 a b^{14} x^{\frac{14}{3}}}{14} + \frac{b^{15} x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*a**15*log(x**(1/3)) + 45*a**14*b*x**(1/3) + 315*a**13*b**2*Integral(x, (x, x**
(1/3))) + 455*a**12*b**3*x + 4095*a**11*b**4*x**(4/3)/4 + 9009*a**10*b**5*x**(5/
3)/5 + 5005*a**9*b**6*x**2/2 + 19305*a**8*b**7*x**(7/3)/7 + 19305*a**7*b**8*x**(
8/3)/8 + 5005*a**6*b**9*x**3/3 + 9009*a**5*b**10*x**(10/3)/10 + 4095*a**4*b**11*
x**(11/3)/11 + 455*a**3*b**12*x**4/4 + 315*a**2*b**13*x**(13/3)/13 + 45*a*b**14*
x**(14/3)/14 + b**15*x**5/5

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Mathematica [A]  time = 0.0471303, size = 209, normalized size = 1. \[ a^{15} \log (x)+45 a^{14} b \sqrt [3]{x}+\frac{315}{2} a^{13} b^2 x^{2/3}+455 a^{12} b^3 x+\frac{4095}{4} a^{11} b^4 x^{4/3}+\frac{9009}{5} a^{10} b^5 x^{5/3}+\frac{5005}{2} a^9 b^6 x^2+\frac{19305}{7} a^8 b^7 x^{7/3}+\frac{19305}{8} a^7 b^8 x^{8/3}+\frac{5005}{3} a^6 b^9 x^3+\frac{9009}{10} a^5 b^{10} x^{10/3}+\frac{4095}{11} a^4 b^{11} x^{11/3}+\frac{455}{4} a^3 b^{12} x^4+\frac{315}{13} a^2 b^{13} x^{13/3}+\frac{45}{14} a b^{14} x^{14/3}+\frac{b^{15} x^5}{5} \]

Antiderivative was successfully verified.

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

[Out]

45*a^14*b*x^(1/3) + (315*a^13*b^2*x^(2/3))/2 + 455*a^12*b^3*x + (4095*a^11*b^4*x
^(4/3))/4 + (9009*a^10*b^5*x^(5/3))/5 + (5005*a^9*b^6*x^2)/2 + (19305*a^8*b^7*x^
(7/3))/7 + (19305*a^7*b^8*x^(8/3))/8 + (5005*a^6*b^9*x^3)/3 + (9009*a^5*b^10*x^(
10/3))/10 + (4095*a^4*b^11*x^(11/3))/11 + (455*a^3*b^12*x^4)/4 + (315*a^2*b^13*x
^(13/3))/13 + (45*a*b^14*x^(14/3))/14 + (b^15*x^5)/5 + a^15*Log[x]

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Maple [A]  time = 0.006, size = 164, normalized size = 0.8 \[ 45\,{a}^{14}b\sqrt [3]{x}+{\frac{315\,{a}^{13}{b}^{2}}{2}{x}^{{\frac{2}{3}}}}+455\,{a}^{12}{b}^{3}x+{\frac{4095\,{a}^{11}{b}^{4}}{4}{x}^{{\frac{4}{3}}}}+{\frac{9009\,{a}^{10}{b}^{5}}{5}{x}^{{\frac{5}{3}}}}+{\frac{5005\,{a}^{9}{b}^{6}{x}^{2}}{2}}+{\frac{19305\,{a}^{8}{b}^{7}}{7}{x}^{{\frac{7}{3}}}}+{\frac{19305\,{a}^{7}{b}^{8}}{8}{x}^{{\frac{8}{3}}}}+{\frac{5005\,{a}^{6}{b}^{9}{x}^{3}}{3}}+{\frac{9009\,{a}^{5}{b}^{10}}{10}{x}^{{\frac{10}{3}}}}+{\frac{4095\,{a}^{4}{b}^{11}}{11}{x}^{{\frac{11}{3}}}}+{\frac{455\,{a}^{3}{b}^{12}{x}^{4}}{4}}+{\frac{315\,{a}^{2}{b}^{13}}{13}{x}^{{\frac{13}{3}}}}+{\frac{45\,a{b}^{14}}{14}{x}^{{\frac{14}{3}}}}+{\frac{{b}^{15}{x}^{5}}{5}}+{a}^{15}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/3))^15/x,x)

[Out]

45*a^14*b*x^(1/3)+315/2*a^13*b^2*x^(2/3)+455*a^12*b^3*x+4095/4*a^11*b^4*x^(4/3)+
9009/5*a^10*b^5*x^(5/3)+5005/2*a^9*b^6*x^2+19305/7*a^8*b^7*x^(7/3)+19305/8*a^7*b
^8*x^(8/3)+5005/3*a^6*b^9*x^3+9009/10*a^5*b^10*x^(10/3)+4095/11*a^4*b^11*x^(11/3
)+455/4*a^3*b^12*x^4+315/13*a^2*b^13*x^(13/3)+45/14*a*b^14*x^(14/3)+1/5*b^15*x^5
+a^15*ln(x)

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Maxima [A]  time = 1.41987, size = 220, normalized size = 1.05 \[ \frac{1}{5} \, b^{15} x^{5} + \frac{45}{14} \, a b^{14} x^{\frac{14}{3}} + \frac{315}{13} \, a^{2} b^{13} x^{\frac{13}{3}} + \frac{455}{4} \, a^{3} b^{12} x^{4} + \frac{4095}{11} \, a^{4} b^{11} x^{\frac{11}{3}} + \frac{9009}{10} \, a^{5} b^{10} x^{\frac{10}{3}} + \frac{5005}{3} \, a^{6} b^{9} x^{3} + \frac{19305}{8} \, a^{7} b^{8} x^{\frac{8}{3}} + \frac{19305}{7} \, a^{8} b^{7} x^{\frac{7}{3}} + \frac{5005}{2} \, a^{9} b^{6} x^{2} + \frac{9009}{5} \, a^{10} b^{5} x^{\frac{5}{3}} + \frac{4095}{4} \, a^{11} b^{4} x^{\frac{4}{3}} + 455 \, a^{12} b^{3} x + a^{15} \log \left (x\right ) + \frac{315}{2} \, a^{13} b^{2} x^{\frac{2}{3}} + 45 \, a^{14} b x^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*b^15*x^5 + 45/14*a*b^14*x^(14/3) + 315/13*a^2*b^13*x^(13/3) + 455/4*a^3*b^12
*x^4 + 4095/11*a^4*b^11*x^(11/3) + 9009/10*a^5*b^10*x^(10/3) + 5005/3*a^6*b^9*x^
3 + 19305/8*a^7*b^8*x^(8/3) + 19305/7*a^8*b^7*x^(7/3) + 5005/2*a^9*b^6*x^2 + 900
9/5*a^10*b^5*x^(5/3) + 4095/4*a^11*b^4*x^(4/3) + 455*a^12*b^3*x + a^15*log(x) +
315/2*a^13*b^2*x^(2/3) + 45*a^14*b*x^(1/3)

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Fricas [A]  time = 0.22125, size = 227, normalized size = 1.09 \[ \frac{1}{5} \, b^{15} x^{5} + \frac{455}{4} \, a^{3} b^{12} x^{4} + \frac{5005}{3} \, a^{6} b^{9} x^{3} + \frac{5005}{2} \, a^{9} b^{6} x^{2} + 455 \, a^{12} b^{3} x + 3 \, a^{15} \log \left (x^{\frac{1}{3}}\right ) + \frac{9}{3080} \,{\left (1100 \, a b^{14} x^{4} + 127400 \, a^{4} b^{11} x^{3} + 825825 \, a^{7} b^{8} x^{2} + 616616 \, a^{10} b^{5} x + 53900 \, a^{13} b^{2}\right )} x^{\frac{2}{3}} + \frac{9}{1820} \,{\left (4900 \, a^{2} b^{13} x^{4} + 182182 \, a^{5} b^{10} x^{3} + 557700 \, a^{8} b^{7} x^{2} + 207025 \, a^{11} b^{4} x + 9100 \, a^{14} b\right )} x^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*b^15*x^5 + 455/4*a^3*b^12*x^4 + 5005/3*a^6*b^9*x^3 + 5005/2*a^9*b^6*x^2 + 45
5*a^12*b^3*x + 3*a^15*log(x^(1/3)) + 9/3080*(1100*a*b^14*x^4 + 127400*a^4*b^11*x
^3 + 825825*a^7*b^8*x^2 + 616616*a^10*b^5*x + 53900*a^13*b^2)*x^(2/3) + 9/1820*(
4900*a^2*b^13*x^4 + 182182*a^5*b^10*x^3 + 557700*a^8*b^7*x^2 + 207025*a^11*b^4*x
+ 9100*a^14*b)*x^(1/3)

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Sympy [A]  time = 17.1838, size = 212, normalized size = 1.01 \[ a^{15} \log{\left (x \right )} + 45 a^{14} b \sqrt [3]{x} + \frac{315 a^{13} b^{2} x^{\frac{2}{3}}}{2} + 455 a^{12} b^{3} x + \frac{4095 a^{11} b^{4} x^{\frac{4}{3}}}{4} + \frac{9009 a^{10} b^{5} x^{\frac{5}{3}}}{5} + \frac{5005 a^{9} b^{6} x^{2}}{2} + \frac{19305 a^{8} b^{7} x^{\frac{7}{3}}}{7} + \frac{19305 a^{7} b^{8} x^{\frac{8}{3}}}{8} + \frac{5005 a^{6} b^{9} x^{3}}{3} + \frac{9009 a^{5} b^{10} x^{\frac{10}{3}}}{10} + \frac{4095 a^{4} b^{11} x^{\frac{11}{3}}}{11} + \frac{455 a^{3} b^{12} x^{4}}{4} + \frac{315 a^{2} b^{13} x^{\frac{13}{3}}}{13} + \frac{45 a b^{14} x^{\frac{14}{3}}}{14} + \frac{b^{15} x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**15*log(x) + 45*a**14*b*x**(1/3) + 315*a**13*b**2*x**(2/3)/2 + 455*a**12*b**3*
x + 4095*a**11*b**4*x**(4/3)/4 + 9009*a**10*b**5*x**(5/3)/5 + 5005*a**9*b**6*x**
2/2 + 19305*a**8*b**7*x**(7/3)/7 + 19305*a**7*b**8*x**(8/3)/8 + 5005*a**6*b**9*x
**3/3 + 9009*a**5*b**10*x**(10/3)/10 + 4095*a**4*b**11*x**(11/3)/11 + 455*a**3*b
**12*x**4/4 + 315*a**2*b**13*x**(13/3)/13 + 45*a*b**14*x**(14/3)/14 + b**15*x**5
/5

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GIAC/XCAS [A]  time = 0.224964, size = 221, normalized size = 1.06 \[ \frac{1}{5} \, b^{15} x^{5} + \frac{45}{14} \, a b^{14} x^{\frac{14}{3}} + \frac{315}{13} \, a^{2} b^{13} x^{\frac{13}{3}} + \frac{455}{4} \, a^{3} b^{12} x^{4} + \frac{4095}{11} \, a^{4} b^{11} x^{\frac{11}{3}} + \frac{9009}{10} \, a^{5} b^{10} x^{\frac{10}{3}} + \frac{5005}{3} \, a^{6} b^{9} x^{3} + \frac{19305}{8} \, a^{7} b^{8} x^{\frac{8}{3}} + \frac{19305}{7} \, a^{8} b^{7} x^{\frac{7}{3}} + \frac{5005}{2} \, a^{9} b^{6} x^{2} + \frac{9009}{5} \, a^{10} b^{5} x^{\frac{5}{3}} + \frac{4095}{4} \, a^{11} b^{4} x^{\frac{4}{3}} + 455 \, a^{12} b^{3} x + a^{15}{\rm ln}\left ({\left | x \right |}\right ) + \frac{315}{2} \, a^{13} b^{2} x^{\frac{2}{3}} + 45 \, a^{14} b x^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*b^15*x^5 + 45/14*a*b^14*x^(14/3) + 315/13*a^2*b^13*x^(13/3) + 455/4*a^3*b^12
*x^4 + 4095/11*a^4*b^11*x^(11/3) + 9009/10*a^5*b^10*x^(10/3) + 5005/3*a^6*b^9*x^
3 + 19305/8*a^7*b^8*x^(8/3) + 19305/7*a^8*b^7*x^(7/3) + 5005/2*a^9*b^6*x^2 + 900
9/5*a^10*b^5*x^(5/3) + 4095/4*a^11*b^4*x^(4/3) + 455*a^12*b^3*x + a^15*ln(abs(x)
) + 315/2*a^13*b^2*x^(2/3) + 45*a^14*b*x^(1/3)