3.2349 \(\int \frac{(a+b \sqrt [3]{x})^{15}}{x^6} \, dx\)

Optimal. Leaf size=211 \[ -\frac{315 a^{13} b^2}{13 x^{13/3}}-\frac{455 a^{12} b^3}{4 x^4}-\frac{4095 a^{11} b^4}{11 x^{11/3}}-\frac{9009 a^{10} b^5}{10 x^{10/3}}-\frac{5005 a^9 b^6}{3 x^3}-\frac{19305 a^8 b^7}{8 x^{8/3}}-\frac{19305 a^7 b^8}{7 x^{7/3}}-\frac{5005 a^6 b^9}{2 x^2}-\frac{9009 a^5 b^{10}}{5 x^{5/3}}-\frac{4095 a^4 b^{11}}{4 x^{4/3}}-\frac{315 a^2 b^{13}}{2 x^{2/3}}-\frac{455 a^3 b^{12}}{x}-\frac{45 a^{14} b}{14 x^{14/3}}-\frac{a^{15}}{5 x^5}-\frac{45 a b^{14}}{\sqrt [3]{x}}+b^{15} \log (x) \]

[Out]

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

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Rubi [A]  time = 0.116477, antiderivative size = 211, 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} \[ -\frac{315 a^{13} b^2}{13 x^{13/3}}-\frac{455 a^{12} b^3}{4 x^4}-\frac{4095 a^{11} b^4}{11 x^{11/3}}-\frac{9009 a^{10} b^5}{10 x^{10/3}}-\frac{5005 a^9 b^6}{3 x^3}-\frac{19305 a^8 b^7}{8 x^{8/3}}-\frac{19305 a^7 b^8}{7 x^{7/3}}-\frac{5005 a^6 b^9}{2 x^2}-\frac{9009 a^5 b^{10}}{5 x^{5/3}}-\frac{4095 a^4 b^{11}}{4 x^{4/3}}-\frac{315 a^2 b^{13}}{2 x^{2/3}}-\frac{455 a^3 b^{12}}{x}-\frac{45 a^{14} b}{14 x^{14/3}}-\frac{a^{15}}{5 x^5}-\frac{45 a b^{14}}{\sqrt [3]{x}}+b^{15} \log (x) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))^15/x^6,x]

[Out]

-a^15/(5*x^5) - (45*a^14*b)/(14*x^(14/3)) - (315*a^13*b^2)/(13*x^(13/3)) - (455*a^12*b^3)/(4*x^4) - (4095*a^11
*b^4)/(11*x^(11/3)) - (9009*a^10*b^5)/(10*x^(10/3)) - (5005*a^9*b^6)/(3*x^3) - (19305*a^8*b^7)/(8*x^(8/3)) - (
19305*a^7*b^8)/(7*x^(7/3)) - (5005*a^6*b^9)/(2*x^2) - (9009*a^5*b^10)/(5*x^(5/3)) - (4095*a^4*b^11)/(4*x^(4/3)
) - (455*a^3*b^12)/x - (315*a^2*b^13)/(2*x^(2/3)) - (45*a*b^14)/x^(1/3) + b^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^6} \, dx &=3 \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{16}} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{a^{15}}{x^{16}}+\frac{15 a^{14} b}{x^{15}}+\frac{105 a^{13} b^2}{x^{14}}+\frac{455 a^{12} b^3}{x^{13}}+\frac{1365 a^{11} b^4}{x^{12}}+\frac{3003 a^{10} b^5}{x^{11}}+\frac{5005 a^9 b^6}{x^{10}}+\frac{6435 a^8 b^7}{x^9}+\frac{6435 a^7 b^8}{x^8}+\frac{5005 a^6 b^9}{x^7}+\frac{3003 a^5 b^{10}}{x^6}+\frac{1365 a^4 b^{11}}{x^5}+\frac{455 a^3 b^{12}}{x^4}+\frac{105 a^2 b^{13}}{x^3}+\frac{15 a b^{14}}{x^2}+\frac{b^{15}}{x}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{a^{15}}{5 x^5}-\frac{45 a^{14} b}{14 x^{14/3}}-\frac{315 a^{13} b^2}{13 x^{13/3}}-\frac{455 a^{12} b^3}{4 x^4}-\frac{4095 a^{11} b^4}{11 x^{11/3}}-\frac{9009 a^{10} b^5}{10 x^{10/3}}-\frac{5005 a^9 b^6}{3 x^3}-\frac{19305 a^8 b^7}{8 x^{8/3}}-\frac{19305 a^7 b^8}{7 x^{7/3}}-\frac{5005 a^6 b^9}{2 x^2}-\frac{9009 a^5 b^{10}}{5 x^{5/3}}-\frac{4095 a^4 b^{11}}{4 x^{4/3}}-\frac{455 a^3 b^{12}}{x}-\frac{315 a^2 b^{13}}{2 x^{2/3}}-\frac{45 a b^{14}}{\sqrt [3]{x}}+b^{15} \log (x)\\ \end{align*}

Mathematica [A]  time = 0.13511, size = 211, normalized size = 1. \[ -\frac{315 a^{13} b^2}{13 x^{13/3}}-\frac{455 a^{12} b^3}{4 x^4}-\frac{4095 a^{11} b^4}{11 x^{11/3}}-\frac{9009 a^{10} b^5}{10 x^{10/3}}-\frac{5005 a^9 b^6}{3 x^3}-\frac{19305 a^8 b^7}{8 x^{8/3}}-\frac{19305 a^7 b^8}{7 x^{7/3}}-\frac{5005 a^6 b^9}{2 x^2}-\frac{9009 a^5 b^{10}}{5 x^{5/3}}-\frac{4095 a^4 b^{11}}{4 x^{4/3}}-\frac{315 a^2 b^{13}}{2 x^{2/3}}-\frac{455 a^3 b^{12}}{x}-\frac{45 a^{14} b}{14 x^{14/3}}-\frac{a^{15}}{5 x^5}-\frac{45 a b^{14}}{\sqrt [3]{x}}+b^{15} \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^15/x^6,x]

[Out]

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

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Maple [A]  time = 0.01, size = 166, normalized size = 0.8 \begin{align*} -{\frac{{a}^{15}}{5\,{x}^{5}}}-{\frac{45\,{a}^{14}b}{14}{x}^{-{\frac{14}{3}}}}-{\frac{315\,{a}^{13}{b}^{2}}{13}{x}^{-{\frac{13}{3}}}}-{\frac{455\,{a}^{12}{b}^{3}}{4\,{x}^{4}}}-{\frac{4095\,{a}^{11}{b}^{4}}{11}{x}^{-{\frac{11}{3}}}}-{\frac{9009\,{a}^{10}{b}^{5}}{10}{x}^{-{\frac{10}{3}}}}-{\frac{5005\,{a}^{9}{b}^{6}}{3\,{x}^{3}}}-{\frac{19305\,{a}^{8}{b}^{7}}{8}{x}^{-{\frac{8}{3}}}}-{\frac{19305\,{a}^{7}{b}^{8}}{7}{x}^{-{\frac{7}{3}}}}-{\frac{5005\,{a}^{6}{b}^{9}}{2\,{x}^{2}}}-{\frac{9009\,{a}^{5}{b}^{10}}{5}{x}^{-{\frac{5}{3}}}}-{\frac{4095\,{a}^{4}{b}^{11}}{4}{x}^{-{\frac{4}{3}}}}-455\,{\frac{{a}^{3}{b}^{12}}{x}}-{\frac{315\,{a}^{2}{b}^{13}}{2}{x}^{-{\frac{2}{3}}}}-45\,{\frac{a{b}^{14}}{\sqrt [3]{x}}}+{b}^{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^6,x)

[Out]

-1/5*a^15/x^5-45/14*a^14*b/x^(14/3)-315/13*a^13*b^2/x^(13/3)-455/4*a^12*b^3/x^4-4095/11*a^11*b^4/x^(11/3)-9009
/10*a^10*b^5/x^(10/3)-5005/3*a^9*b^6/x^3-19305/8*a^8*b^7/x^(8/3)-19305/7*a^7*b^8/x^(7/3)-5005/2*a^6*b^9/x^2-90
09/5*a^5*b^10/x^(5/3)-4095/4*a^4*b^11/x^(4/3)-455*a^3*b^12/x-315/2*a^2*b^13/x^(2/3)-45*a*b^14/x^(1/3)+b^15*ln(
x)

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Maxima [A]  time = 0.97445, size = 224, normalized size = 1.06 \begin{align*} b^{15} \log \left (x\right ) - \frac{5405400 \, a b^{14} x^{\frac{14}{3}} + 18918900 \, a^{2} b^{13} x^{\frac{13}{3}} + 54654600 \, a^{3} b^{12} x^{4} + 122972850 \, a^{4} b^{11} x^{\frac{11}{3}} + 216432216 \, a^{5} b^{10} x^{\frac{10}{3}} + 300600300 \, a^{6} b^{9} x^{3} + 331273800 \, a^{7} b^{8} x^{\frac{8}{3}} + 289864575 \, a^{8} b^{7} x^{\frac{7}{3}} + 200400200 \, a^{9} b^{6} x^{2} + 108216108 \, a^{10} b^{5} x^{\frac{5}{3}} + 44717400 \, a^{11} b^{4} x^{\frac{4}{3}} + 13663650 \, a^{12} b^{3} x + 2910600 \, a^{13} b^{2} x^{\frac{2}{3}} + 386100 \, a^{14} b x^{\frac{1}{3}} + 24024 \, a^{15}}{120120 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^15/x^6,x, algorithm="maxima")

[Out]

b^15*log(x) - 1/120120*(5405400*a*b^14*x^(14/3) + 18918900*a^2*b^13*x^(13/3) + 54654600*a^3*b^12*x^4 + 1229728
50*a^4*b^11*x^(11/3) + 216432216*a^5*b^10*x^(10/3) + 300600300*a^6*b^9*x^3 + 331273800*a^7*b^8*x^(8/3) + 28986
4575*a^8*b^7*x^(7/3) + 200400200*a^9*b^6*x^2 + 108216108*a^10*b^5*x^(5/3) + 44717400*a^11*b^4*x^(4/3) + 136636
50*a^12*b^3*x + 2910600*a^13*b^2*x^(2/3) + 386100*a^14*b*x^(1/3) + 24024*a^15)/x^5

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Fricas [A]  time = 1.54402, size = 506, normalized size = 2.4 \begin{align*} \frac{360360 \, b^{15} x^{5} \log \left (x^{\frac{1}{3}}\right ) - 54654600 \, a^{3} b^{12} x^{4} - 300600300 \, a^{6} b^{9} x^{3} - 200400200 \, a^{9} b^{6} x^{2} - 13663650 \, a^{12} b^{3} x - 24024 \, a^{15} - 594 \,{\left (9100 \, a b^{14} x^{4} + 207025 \, a^{4} b^{11} x^{3} + 557700 \, a^{7} b^{8} x^{2} + 182182 \, a^{10} b^{5} x + 4900 \, a^{13} b^{2}\right )} x^{\frac{2}{3}} - 351 \,{\left (53900 \, a^{2} b^{13} x^{4} + 616616 \, a^{5} b^{10} x^{3} + 825825 \, a^{8} b^{7} x^{2} + 127400 \, a^{11} b^{4} x + 1100 \, a^{14} b\right )} x^{\frac{1}{3}}}{120120 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^15/x^6,x, algorithm="fricas")

[Out]

1/120120*(360360*b^15*x^5*log(x^(1/3)) - 54654600*a^3*b^12*x^4 - 300600300*a^6*b^9*x^3 - 200400200*a^9*b^6*x^2
- 13663650*a^12*b^3*x - 24024*a^15 - 594*(9100*a*b^14*x^4 + 207025*a^4*b^11*x^3 + 557700*a^7*b^8*x^2 + 182182
*a^10*b^5*x + 4900*a^13*b^2)*x^(2/3) - 351*(53900*a^2*b^13*x^4 + 616616*a^5*b^10*x^3 + 825825*a^8*b^7*x^2 + 12
7400*a^11*b^4*x + 1100*a^14*b)*x^(1/3))/x^5

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Sympy [A]  time = 9.60066, size = 212, normalized size = 1. \begin{align*} - \frac{a^{15}}{5 x^{5}} - \frac{45 a^{14} b}{14 x^{\frac{14}{3}}} - \frac{315 a^{13} b^{2}}{13 x^{\frac{13}{3}}} - \frac{455 a^{12} b^{3}}{4 x^{4}} - \frac{4095 a^{11} b^{4}}{11 x^{\frac{11}{3}}} - \frac{9009 a^{10} b^{5}}{10 x^{\frac{10}{3}}} - \frac{5005 a^{9} b^{6}}{3 x^{3}} - \frac{19305 a^{8} b^{7}}{8 x^{\frac{8}{3}}} - \frac{19305 a^{7} b^{8}}{7 x^{\frac{7}{3}}} - \frac{5005 a^{6} b^{9}}{2 x^{2}} - \frac{9009 a^{5} b^{10}}{5 x^{\frac{5}{3}}} - \frac{4095 a^{4} b^{11}}{4 x^{\frac{4}{3}}} - \frac{455 a^{3} b^{12}}{x} - \frac{315 a^{2} b^{13}}{2 x^{\frac{2}{3}}} - \frac{45 a b^{14}}{\sqrt [3]{x}} + b^{15} \log{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/3))**15/x**6,x)

[Out]

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

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Giac [A]  time = 1.17931, size = 225, normalized size = 1.07 \begin{align*} b^{15} \log \left ({\left | x \right |}\right ) - \frac{5405400 \, a b^{14} x^{\frac{14}{3}} + 18918900 \, a^{2} b^{13} x^{\frac{13}{3}} + 54654600 \, a^{3} b^{12} x^{4} + 122972850 \, a^{4} b^{11} x^{\frac{11}{3}} + 216432216 \, a^{5} b^{10} x^{\frac{10}{3}} + 300600300 \, a^{6} b^{9} x^{3} + 331273800 \, a^{7} b^{8} x^{\frac{8}{3}} + 289864575 \, a^{8} b^{7} x^{\frac{7}{3}} + 200400200 \, a^{9} b^{6} x^{2} + 108216108 \, a^{10} b^{5} x^{\frac{5}{3}} + 44717400 \, a^{11} b^{4} x^{\frac{4}{3}} + 13663650 \, a^{12} b^{3} x + 2910600 \, a^{13} b^{2} x^{\frac{2}{3}} + 386100 \, a^{14} b x^{\frac{1}{3}} + 24024 \, a^{15}}{120120 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^15/x^6,x, algorithm="giac")

[Out]

b^15*log(abs(x)) - 1/120120*(5405400*a*b^14*x^(14/3) + 18918900*a^2*b^13*x^(13/3) + 54654600*a^3*b^12*x^4 + 12
2972850*a^4*b^11*x^(11/3) + 216432216*a^5*b^10*x^(10/3) + 300600300*a^6*b^9*x^3 + 331273800*a^7*b^8*x^(8/3) +
289864575*a^8*b^7*x^(7/3) + 200400200*a^9*b^6*x^2 + 108216108*a^10*b^5*x^(5/3) + 44717400*a^11*b^4*x^(4/3) + 1
3663650*a^12*b^3*x + 2910600*a^13*b^2*x^(2/3) + 386100*a^14*b*x^(1/3) + 24024*a^15)/x^5