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

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

[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))/1
3 + (45*a*b^14*x^(14/3))/14 + (b^15*x^5)/5 + a^15*Log[x]

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Rubi [A]  time = 0.105216, 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, Rules used = {266, 43} \[ \frac{315}{2} a^{13} b^2 x^{2/3}+\frac{4095}{4} a^{11} b^4 x^{4/3}+\frac{9009}{5} a^{10} b^5 x^{5/3}+\frac{5005}{2} a^9 b^6 x^2+\frac{19305}{7} a^8 b^7 x^{7/3}+\frac{19305}{8} a^7 b^8 x^{8/3}+\frac{5005}{3} a^6 b^9 x^3+\frac{9009}{10} a^5 b^{10} x^{10/3}+\frac{4095}{11} a^4 b^{11} x^{11/3}+\frac{455}{4} a^3 b^{12} x^4+\frac{315}{13} a^2 b^{13} x^{13/3}+455 a^{12} b^3 x+45 a^{14} b \sqrt [3]{x}+a^{15} \log (x)+\frac{45}{14} a b^{14} x^{14/3}+\frac{b^{15} x^5}{5} \]

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))/1
3 + (45*a*b^14*x^(14/3))/14 + (b^15*x^5)/5 + a^15*Log[x]

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]]

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])

Rubi steps

\begin{align*} \int \frac{\left (a+b \sqrt [3]{x}\right )^{15}}{x} \, dx &=3 \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (15 a^{14} b+\frac{a^{15}}{x}+105 a^{13} b^2 x+455 a^{12} b^3 x^2+1365 a^{11} b^4 x^3+3003 a^{10} b^5 x^4+5005 a^9 b^6 x^5+6435 a^8 b^7 x^6+6435 a^7 b^8 x^7+5005 a^6 b^9 x^8+3003 a^5 b^{10} x^9+1365 a^4 b^{11} x^{10}+455 a^3 b^{12} x^{11}+105 a^2 b^{13} x^{12}+15 a b^{14} x^{13}+b^{15} x^{14}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=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}+a^{15} \log (x)\\ \end{align*}

Mathematica [A]  time = 0.070759, size = 209, normalized size = 1. \[ \frac{315}{2} a^{13} b^2 x^{2/3}+\frac{4095}{4} a^{11} b^4 x^{4/3}+\frac{9009}{5} a^{10} b^5 x^{5/3}+\frac{5005}{2} a^9 b^6 x^2+\frac{19305}{7} a^8 b^7 x^{7/3}+\frac{19305}{8} a^7 b^8 x^{8/3}+\frac{5005}{3} a^6 b^9 x^3+\frac{9009}{10} a^5 b^{10} x^{10/3}+\frac{4095}{11} a^4 b^{11} x^{11/3}+\frac{455}{4} a^3 b^{12} x^4+\frac{315}{13} a^2 b^{13} x^{13/3}+455 a^{12} b^3 x+45 a^{14} b \sqrt [3]{x}+a^{15} \log (x)+\frac{45}{14} a b^{14} x^{14/3}+\frac{b^{15} x^5}{5} \]

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))/1
3 + (45*a*b^14*x^(14/3))/14 + (b^15*x^5)/5 + a^15*Log[x]

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Maple [A]  time = 0.003, size = 164, normalized size = 0.8 \begin{align*} 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 ) \end{align*}

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 = 0.976381, size = 220, normalized size = 1.05 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^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 + 9009/5*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 = 1.47114, size = 466, normalized size = 2.23 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^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 + 455*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 = 7.25159, size = 212, normalized size = 1.01 \begin{align*} 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} \end{align*}

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 [A]  time = 1.17589, size = 221, normalized size = 1.06 \begin{align*} \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 ({\left | x \right |}\right ) + \frac{315}{2} \, a^{13} b^{2} x^{\frac{2}{3}} + 45 \, a^{14} b x^{\frac{1}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^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 + 9009/5*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(abs(x)) + 315/2
*a^13*b^2*x^(2/3) + 45*a^14*b*x^(1/3)