3.2354 \(\int \frac{(a+b \sqrt [3]{x})^{15}}{x^{11}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.11242, 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, Rules used = {266, 43} \[ -\frac{45 a^{13} b^2}{4 x^{28/3}}-\frac{455 a^{12} b^3}{9 x^9}-\frac{315 a^{11} b^4}{2 x^{26/3}}-\frac{9009 a^{10} b^5}{25 x^{25/3}}-\frac{5005 a^9 b^6}{8 x^8}-\frac{19305 a^8 b^7}{23 x^{23/3}}-\frac{1755 a^7 b^8}{2 x^{22/3}}-\frac{715 a^6 b^9}{x^7}-\frac{9009 a^5 b^{10}}{20 x^{20/3}}-\frac{4095 a^4 b^{11}}{19 x^{19/3}}-\frac{455 a^3 b^{12}}{6 x^6}-\frac{315 a^2 b^{13}}{17 x^{17/3}}-\frac{45 a^{14} b}{29 x^{29/3}}-\frac{a^{15}}{10 x^{10}}-\frac{45 a b^{14}}{16 x^{16/3}}-\frac{b^{15}}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))^15/x^11,x]

[Out]

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

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^{11}} \, dx &=3 \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{31}} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{a^{15}}{x^{31}}+\frac{15 a^{14} b}{x^{30}}+\frac{105 a^{13} b^2}{x^{29}}+\frac{455 a^{12} b^3}{x^{28}}+\frac{1365 a^{11} b^4}{x^{27}}+\frac{3003 a^{10} b^5}{x^{26}}+\frac{5005 a^9 b^6}{x^{25}}+\frac{6435 a^8 b^7}{x^{24}}+\frac{6435 a^7 b^8}{x^{23}}+\frac{5005 a^6 b^9}{x^{22}}+\frac{3003 a^5 b^{10}}{x^{21}}+\frac{1365 a^4 b^{11}}{x^{20}}+\frac{455 a^3 b^{12}}{x^{19}}+\frac{105 a^2 b^{13}}{x^{18}}+\frac{15 a b^{14}}{x^{17}}+\frac{b^{15}}{x^{16}}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{a^{15}}{10 x^{10}}-\frac{45 a^{14} b}{29 x^{29/3}}-\frac{45 a^{13} b^2}{4 x^{28/3}}-\frac{455 a^{12} b^3}{9 x^9}-\frac{315 a^{11} b^4}{2 x^{26/3}}-\frac{9009 a^{10} b^5}{25 x^{25/3}}-\frac{5005 a^9 b^6}{8 x^8}-\frac{19305 a^8 b^7}{23 x^{23/3}}-\frac{1755 a^7 b^8}{2 x^{22/3}}-\frac{715 a^6 b^9}{x^7}-\frac{9009 a^5 b^{10}}{20 x^{20/3}}-\frac{4095 a^4 b^{11}}{19 x^{19/3}}-\frac{455 a^3 b^{12}}{6 x^6}-\frac{315 a^2 b^{13}}{17 x^{17/3}}-\frac{45 a b^{14}}{16 x^{16/3}}-\frac{b^{15}}{5 x^5}\\ \end{align*}

Mathematica [A]  time = 0.100803, size = 217, normalized size = 1. \[ -\frac{45 a^{13} b^2}{4 x^{28/3}}-\frac{455 a^{12} b^3}{9 x^9}-\frac{315 a^{11} b^4}{2 x^{26/3}}-\frac{9009 a^{10} b^5}{25 x^{25/3}}-\frac{5005 a^9 b^6}{8 x^8}-\frac{19305 a^8 b^7}{23 x^{23/3}}-\frac{1755 a^7 b^8}{2 x^{22/3}}-\frac{715 a^6 b^9}{x^7}-\frac{9009 a^5 b^{10}}{20 x^{20/3}}-\frac{4095 a^4 b^{11}}{19 x^{19/3}}-\frac{455 a^3 b^{12}}{6 x^6}-\frac{315 a^2 b^{13}}{17 x^{17/3}}-\frac{45 a^{14} b}{29 x^{29/3}}-\frac{a^{15}}{10 x^{10}}-\frac{45 a b^{14}}{16 x^{16/3}}-\frac{b^{15}}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^15/x^11,x]

[Out]

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

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Maple [A]  time = 0.009, size = 168, normalized size = 0.8 \begin{align*} -{\frac{{a}^{15}}{10\,{x}^{10}}}-{\frac{45\,{a}^{14}b}{29}{x}^{-{\frac{29}{3}}}}-{\frac{45\,{a}^{13}{b}^{2}}{4}{x}^{-{\frac{28}{3}}}}-{\frac{455\,{a}^{12}{b}^{3}}{9\,{x}^{9}}}-{\frac{315\,{a}^{11}{b}^{4}}{2}{x}^{-{\frac{26}{3}}}}-{\frac{9009\,{a}^{10}{b}^{5}}{25}{x}^{-{\frac{25}{3}}}}-{\frac{5005\,{a}^{9}{b}^{6}}{8\,{x}^{8}}}-{\frac{19305\,{a}^{8}{b}^{7}}{23}{x}^{-{\frac{23}{3}}}}-{\frac{1755\,{a}^{7}{b}^{8}}{2}{x}^{-{\frac{22}{3}}}}-715\,{\frac{{a}^{6}{b}^{9}}{{x}^{7}}}-{\frac{9009\,{a}^{5}{b}^{10}}{20}{x}^{-{\frac{20}{3}}}}-{\frac{4095\,{a}^{4}{b}^{11}}{19}{x}^{-{\frac{19}{3}}}}-{\frac{455\,{a}^{3}{b}^{12}}{6\,{x}^{6}}}-{\frac{315\,{a}^{2}{b}^{13}}{17}{x}^{-{\frac{17}{3}}}}-{\frac{45\,a{b}^{14}}{16}{x}^{-{\frac{16}{3}}}}-{\frac{{b}^{15}}{5\,{x}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/3))^15/x^11,x)

[Out]

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

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Maxima [A]  time = 0.981255, size = 225, normalized size = 1.04 \begin{align*} -\frac{155117520 \, b^{15} x^{5} + 2181340125 \, a b^{14} x^{\frac{14}{3}} + 14371182000 \, a^{2} b^{13} x^{\frac{13}{3}} + 58815393000 \, a^{3} b^{12} x^{4} + 167159538000 \, a^{4} b^{11} x^{\frac{11}{3}} + 349363434420 \, a^{5} b^{10} x^{\frac{10}{3}} + 554545134000 \, a^{6} b^{9} x^{3} + 680578119000 \, a^{7} b^{8} x^{\frac{8}{3}} + 650987766000 \, a^{8} b^{7} x^{\frac{7}{3}} + 485226992250 \, a^{9} b^{6} x^{2} + 279490747536 \, a^{10} b^{5} x^{\frac{5}{3}} + 122155047000 \, a^{11} b^{4} x^{\frac{4}{3}} + 39210262000 \, a^{12} b^{3} x + 8725360500 \, a^{13} b^{2} x^{\frac{2}{3}} + 1203498000 \, a^{14} b x^{\frac{1}{3}} + 77558760 \, a^{15}}{775587600 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^15/x^11,x, algorithm="maxima")

[Out]

-1/775587600*(155117520*b^15*x^5 + 2181340125*a*b^14*x^(14/3) + 14371182000*a^2*b^13*x^(13/3) + 58815393000*a^
3*b^12*x^4 + 167159538000*a^4*b^11*x^(11/3) + 349363434420*a^5*b^10*x^(10/3) + 554545134000*a^6*b^9*x^3 + 6805
78119000*a^7*b^8*x^(8/3) + 650987766000*a^8*b^7*x^(7/3) + 485226992250*a^9*b^6*x^2 + 279490747536*a^10*b^5*x^(
5/3) + 122155047000*a^11*b^4*x^(4/3) + 39210262000*a^12*b^3*x + 8725360500*a^13*b^2*x^(2/3) + 1203498000*a^14*
b*x^(1/3) + 77558760*a^15)/x^10

________________________________________________________________________________________

Fricas [A]  time = 1.49144, size = 536, normalized size = 2.47 \begin{align*} -\frac{155117520 \, b^{15} x^{5} + 58815393000 \, a^{3} b^{12} x^{4} + 554545134000 \, a^{6} b^{9} x^{3} + 485226992250 \, a^{9} b^{6} x^{2} + 39210262000 \, a^{12} b^{3} x + 77558760 \, a^{15} + 918459 \,{\left (2375 \, a b^{14} x^{4} + 182000 \, a^{4} b^{11} x^{3} + 741000 \, a^{7} b^{8} x^{2} + 304304 \, a^{10} b^{5} x + 9500 \, a^{13} b^{2}\right )} x^{\frac{2}{3}} + 30780 \,{\left (466900 \, a^{2} b^{13} x^{4} + 11350339 \, a^{5} b^{10} x^{3} + 21149700 \, a^{8} b^{7} x^{2} + 3968650 \, a^{11} b^{4} x + 39100 \, a^{14} b\right )} x^{\frac{1}{3}}}{775587600 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^15/x^11,x, algorithm="fricas")

[Out]

-1/775587600*(155117520*b^15*x^5 + 58815393000*a^3*b^12*x^4 + 554545134000*a^6*b^9*x^3 + 485226992250*a^9*b^6*
x^2 + 39210262000*a^12*b^3*x + 77558760*a^15 + 918459*(2375*a*b^14*x^4 + 182000*a^4*b^11*x^3 + 741000*a^7*b^8*
x^2 + 304304*a^10*b^5*x + 9500*a^13*b^2)*x^(2/3) + 30780*(466900*a^2*b^13*x^4 + 11350339*a^5*b^10*x^3 + 211497
00*a^8*b^7*x^2 + 3968650*a^11*b^4*x + 39100*a^14*b)*x^(1/3))/x^10

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Sympy [A]  time = 62.8371, size = 219, normalized size = 1.01 \begin{align*} - \frac{a^{15}}{10 x^{10}} - \frac{45 a^{14} b}{29 x^{\frac{29}{3}}} - \frac{45 a^{13} b^{2}}{4 x^{\frac{28}{3}}} - \frac{455 a^{12} b^{3}}{9 x^{9}} - \frac{315 a^{11} b^{4}}{2 x^{\frac{26}{3}}} - \frac{9009 a^{10} b^{5}}{25 x^{\frac{25}{3}}} - \frac{5005 a^{9} b^{6}}{8 x^{8}} - \frac{19305 a^{8} b^{7}}{23 x^{\frac{23}{3}}} - \frac{1755 a^{7} b^{8}}{2 x^{\frac{22}{3}}} - \frac{715 a^{6} b^{9}}{x^{7}} - \frac{9009 a^{5} b^{10}}{20 x^{\frac{20}{3}}} - \frac{4095 a^{4} b^{11}}{19 x^{\frac{19}{3}}} - \frac{455 a^{3} b^{12}}{6 x^{6}} - \frac{315 a^{2} b^{13}}{17 x^{\frac{17}{3}}} - \frac{45 a b^{14}}{16 x^{\frac{16}{3}}} - \frac{b^{15}}{5 x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/3))**15/x**11,x)

[Out]

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

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Giac [A]  time = 1.15582, size = 225, normalized size = 1.04 \begin{align*} -\frac{155117520 \, b^{15} x^{5} + 2181340125 \, a b^{14} x^{\frac{14}{3}} + 14371182000 \, a^{2} b^{13} x^{\frac{13}{3}} + 58815393000 \, a^{3} b^{12} x^{4} + 167159538000 \, a^{4} b^{11} x^{\frac{11}{3}} + 349363434420 \, a^{5} b^{10} x^{\frac{10}{3}} + 554545134000 \, a^{6} b^{9} x^{3} + 680578119000 \, a^{7} b^{8} x^{\frac{8}{3}} + 650987766000 \, a^{8} b^{7} x^{\frac{7}{3}} + 485226992250 \, a^{9} b^{6} x^{2} + 279490747536 \, a^{10} b^{5} x^{\frac{5}{3}} + 122155047000 \, a^{11} b^{4} x^{\frac{4}{3}} + 39210262000 \, a^{12} b^{3} x + 8725360500 \, a^{13} b^{2} x^{\frac{2}{3}} + 1203498000 \, a^{14} b x^{\frac{1}{3}} + 77558760 \, a^{15}}{775587600 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^15/x^11,x, algorithm="giac")

[Out]

-1/775587600*(155117520*b^15*x^5 + 2181340125*a*b^14*x^(14/3) + 14371182000*a^2*b^13*x^(13/3) + 58815393000*a^
3*b^12*x^4 + 167159538000*a^4*b^11*x^(11/3) + 349363434420*a^5*b^10*x^(10/3) + 554545134000*a^6*b^9*x^3 + 6805
78119000*a^7*b^8*x^(8/3) + 650987766000*a^8*b^7*x^(7/3) + 485226992250*a^9*b^6*x^2 + 279490747536*a^10*b^5*x^(
5/3) + 122155047000*a^11*b^4*x^(4/3) + 39210262000*a^12*b^3*x + 8725360500*a^13*b^2*x^(2/3) + 1203498000*a^14*
b*x^(1/3) + 77558760*a^15)/x^10