3.2353 \(\int \frac{(a+b \sqrt [3]{x})^{15}}{x^{10}} \, dx\)

Optimal. Leaf size=215 \[ -\frac{63 a^{13} b^2}{5 x^{25/3}}-\frac{455 a^{12} b^3}{8 x^8}-\frac{4095 a^{11} b^4}{23 x^{23/3}}-\frac{819 a^{10} b^5}{2 x^{22/3}}-\frac{715 a^9 b^6}{x^7}-\frac{3861 a^8 b^7}{4 x^{20/3}}-\frac{19305 a^7 b^8}{19 x^{19/3}}-\frac{5005 a^6 b^9}{6 x^6}-\frac{9009 a^5 b^{10}}{17 x^{17/3}}-\frac{4095 a^4 b^{11}}{16 x^{16/3}}-\frac{91 a^3 b^{12}}{x^5}-\frac{45 a^2 b^{13}}{2 x^{14/3}}-\frac{45 a^{14} b}{26 x^{26/3}}-\frac{a^{15}}{9 x^9}-\frac{45 a b^{14}}{13 x^{13/3}}-\frac{b^{15}}{4 x^4} \]

[Out]

-a^15/(9*x^9) - (45*a^14*b)/(26*x^(26/3)) - (63*a^13*b^2)/(5*x^(25/3)) - (455*a^12*b^3)/(8*x^8) - (4095*a^11*b
^4)/(23*x^(23/3)) - (819*a^10*b^5)/(2*x^(22/3)) - (715*a^9*b^6)/x^7 - (3861*a^8*b^7)/(4*x^(20/3)) - (19305*a^7
*b^8)/(19*x^(19/3)) - (5005*a^6*b^9)/(6*x^6) - (9009*a^5*b^10)/(17*x^(17/3)) - (4095*a^4*b^11)/(16*x^(16/3)) -
(91*a^3*b^12)/x^5 - (45*a^2*b^13)/(2*x^(14/3)) - (45*a*b^14)/(13*x^(13/3)) - b^15/(4*x^4)

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Rubi [A]  time = 0.112684, antiderivative size = 215, 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} \[ -\frac{63 a^{13} b^2}{5 x^{25/3}}-\frac{455 a^{12} b^3}{8 x^8}-\frac{4095 a^{11} b^4}{23 x^{23/3}}-\frac{819 a^{10} b^5}{2 x^{22/3}}-\frac{715 a^9 b^6}{x^7}-\frac{3861 a^8 b^7}{4 x^{20/3}}-\frac{19305 a^7 b^8}{19 x^{19/3}}-\frac{5005 a^6 b^9}{6 x^6}-\frac{9009 a^5 b^{10}}{17 x^{17/3}}-\frac{4095 a^4 b^{11}}{16 x^{16/3}}-\frac{91 a^3 b^{12}}{x^5}-\frac{45 a^2 b^{13}}{2 x^{14/3}}-\frac{45 a^{14} b}{26 x^{26/3}}-\frac{a^{15}}{9 x^9}-\frac{45 a b^{14}}{13 x^{13/3}}-\frac{b^{15}}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))^15/x^10,x]

[Out]

-a^15/(9*x^9) - (45*a^14*b)/(26*x^(26/3)) - (63*a^13*b^2)/(5*x^(25/3)) - (455*a^12*b^3)/(8*x^8) - (4095*a^11*b
^4)/(23*x^(23/3)) - (819*a^10*b^5)/(2*x^(22/3)) - (715*a^9*b^6)/x^7 - (3861*a^8*b^7)/(4*x^(20/3)) - (19305*a^7
*b^8)/(19*x^(19/3)) - (5005*a^6*b^9)/(6*x^6) - (9009*a^5*b^10)/(17*x^(17/3)) - (4095*a^4*b^11)/(16*x^(16/3)) -
(91*a^3*b^12)/x^5 - (45*a^2*b^13)/(2*x^(14/3)) - (45*a*b^14)/(13*x^(13/3)) - b^15/(4*x^4)

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^{10}} \, dx &=3 \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{28}} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{a^{15}}{x^{28}}+\frac{15 a^{14} b}{x^{27}}+\frac{105 a^{13} b^2}{x^{26}}+\frac{455 a^{12} b^3}{x^{25}}+\frac{1365 a^{11} b^4}{x^{24}}+\frac{3003 a^{10} b^5}{x^{23}}+\frac{5005 a^9 b^6}{x^{22}}+\frac{6435 a^8 b^7}{x^{21}}+\frac{6435 a^7 b^8}{x^{20}}+\frac{5005 a^6 b^9}{x^{19}}+\frac{3003 a^5 b^{10}}{x^{18}}+\frac{1365 a^4 b^{11}}{x^{17}}+\frac{455 a^3 b^{12}}{x^{16}}+\frac{105 a^2 b^{13}}{x^{15}}+\frac{15 a b^{14}}{x^{14}}+\frac{b^{15}}{x^{13}}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{a^{15}}{9 x^9}-\frac{45 a^{14} b}{26 x^{26/3}}-\frac{63 a^{13} b^2}{5 x^{25/3}}-\frac{455 a^{12} b^3}{8 x^8}-\frac{4095 a^{11} b^4}{23 x^{23/3}}-\frac{819 a^{10} b^5}{2 x^{22/3}}-\frac{715 a^9 b^6}{x^7}-\frac{3861 a^8 b^7}{4 x^{20/3}}-\frac{19305 a^7 b^8}{19 x^{19/3}}-\frac{5005 a^6 b^9}{6 x^6}-\frac{9009 a^5 b^{10}}{17 x^{17/3}}-\frac{4095 a^4 b^{11}}{16 x^{16/3}}-\frac{91 a^3 b^{12}}{x^5}-\frac{45 a^2 b^{13}}{2 x^{14/3}}-\frac{45 a b^{14}}{13 x^{13/3}}-\frac{b^{15}}{4 x^4}\\ \end{align*}

Mathematica [A]  time = 0.109709, size = 215, normalized size = 1. \[ -\frac{63 a^{13} b^2}{5 x^{25/3}}-\frac{455 a^{12} b^3}{8 x^8}-\frac{4095 a^{11} b^4}{23 x^{23/3}}-\frac{819 a^{10} b^5}{2 x^{22/3}}-\frac{715 a^9 b^6}{x^7}-\frac{3861 a^8 b^7}{4 x^{20/3}}-\frac{19305 a^7 b^8}{19 x^{19/3}}-\frac{5005 a^6 b^9}{6 x^6}-\frac{9009 a^5 b^{10}}{17 x^{17/3}}-\frac{4095 a^4 b^{11}}{16 x^{16/3}}-\frac{91 a^3 b^{12}}{x^5}-\frac{45 a^2 b^{13}}{2 x^{14/3}}-\frac{45 a^{14} b}{26 x^{26/3}}-\frac{a^{15}}{9 x^9}-\frac{45 a b^{14}}{13 x^{13/3}}-\frac{b^{15}}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^15/x^10,x]

[Out]

-a^15/(9*x^9) - (45*a^14*b)/(26*x^(26/3)) - (63*a^13*b^2)/(5*x^(25/3)) - (455*a^12*b^3)/(8*x^8) - (4095*a^11*b
^4)/(23*x^(23/3)) - (819*a^10*b^5)/(2*x^(22/3)) - (715*a^9*b^6)/x^7 - (3861*a^8*b^7)/(4*x^(20/3)) - (19305*a^7
*b^8)/(19*x^(19/3)) - (5005*a^6*b^9)/(6*x^6) - (9009*a^5*b^10)/(17*x^(17/3)) - (4095*a^4*b^11)/(16*x^(16/3)) -
(91*a^3*b^12)/x^5 - (45*a^2*b^13)/(2*x^(14/3)) - (45*a*b^14)/(13*x^(13/3)) - b^15/(4*x^4)

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Maple [A]  time = 0.008, size = 168, normalized size = 0.8 \begin{align*} -{\frac{{a}^{15}}{9\,{x}^{9}}}-{\frac{45\,{a}^{14}b}{26}{x}^{-{\frac{26}{3}}}}-{\frac{63\,{a}^{13}{b}^{2}}{5}{x}^{-{\frac{25}{3}}}}-{\frac{455\,{a}^{12}{b}^{3}}{8\,{x}^{8}}}-{\frac{4095\,{a}^{11}{b}^{4}}{23}{x}^{-{\frac{23}{3}}}}-{\frac{819\,{a}^{10}{b}^{5}}{2}{x}^{-{\frac{22}{3}}}}-715\,{\frac{{a}^{9}{b}^{6}}{{x}^{7}}}-{\frac{3861\,{a}^{8}{b}^{7}}{4}{x}^{-{\frac{20}{3}}}}-{\frac{19305\,{a}^{7}{b}^{8}}{19}{x}^{-{\frac{19}{3}}}}-{\frac{5005\,{a}^{6}{b}^{9}}{6\,{x}^{6}}}-{\frac{9009\,{a}^{5}{b}^{10}}{17}{x}^{-{\frac{17}{3}}}}-{\frac{4095\,{a}^{4}{b}^{11}}{16}{x}^{-{\frac{16}{3}}}}-91\,{\frac{{a}^{3}{b}^{12}}{{x}^{5}}}-{\frac{45\,{a}^{2}{b}^{13}}{2}{x}^{-{\frac{14}{3}}}}-{\frac{45\,a{b}^{14}}{13}{x}^{-{\frac{13}{3}}}}-{\frac{{b}^{15}}{4\,{x}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/3))^15/x^10,x)

[Out]

-1/9*a^15/x^9-45/26*a^14*b/x^(26/3)-63/5*a^13*b^2/x^(25/3)-455/8*a^12*b^3/x^8-4095/23*a^11*b^4/x^(23/3)-819/2*
a^10*b^5/x^(22/3)-715*a^9*b^6/x^7-3861/4*a^8*b^7/x^(20/3)-19305/19*a^7*b^8/x^(19/3)-5005/6*a^6*b^9/x^6-9009/17
*a^5*b^10/x^(17/3)-4095/16*a^4*b^11/x^(16/3)-91*a^3*b^12/x^5-45/2*a^2*b^13/x^(14/3)-45/13*a*b^14/x^(13/3)-1/4*
b^15/x^4

________________________________________________________________________________________

Maxima [A]  time = 0.991521, size = 225, normalized size = 1.05 \begin{align*} -\frac{17383860 \, b^{15} x^{5} + 240699600 \, a b^{14} x^{\frac{14}{3}} + 1564547400 \, a^{2} b^{13} x^{\frac{13}{3}} + 6327725040 \, a^{3} b^{12} x^{4} + 17796726675 \, a^{4} b^{11} x^{\frac{11}{3}} + 36849692880 \, a^{5} b^{10} x^{\frac{10}{3}} + 58004146200 \, a^{6} b^{9} x^{3} + 70651666800 \, a^{7} b^{8} x^{\frac{8}{3}} + 67119083460 \, a^{8} b^{7} x^{\frac{7}{3}} + 49717839600 \, a^{9} b^{6} x^{2} + 28474762680 \, a^{10} b^{5} x^{\frac{5}{3}} + 12380331600 \, a^{11} b^{4} x^{\frac{4}{3}} + 3954828150 \, a^{12} b^{3} x + 876146544 \, a^{13} b^{2} x^{\frac{2}{3}} + 120349800 \, a^{14} b x^{\frac{1}{3}} + 7726160 \, a^{15}}{69535440 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^15/x^10,x, algorithm="maxima")

[Out]

-1/69535440*(17383860*b^15*x^5 + 240699600*a*b^14*x^(14/3) + 1564547400*a^2*b^13*x^(13/3) + 6327725040*a^3*b^1
2*x^4 + 17796726675*a^4*b^11*x^(11/3) + 36849692880*a^5*b^10*x^(10/3) + 58004146200*a^6*b^9*x^3 + 70651666800*
a^7*b^8*x^(8/3) + 67119083460*a^8*b^7*x^(7/3) + 49717839600*a^9*b^6*x^2 + 28474762680*a^10*b^5*x^(5/3) + 12380
331600*a^11*b^4*x^(4/3) + 3954828150*a^12*b^3*x + 876146544*a^13*b^2*x^(2/3) + 120349800*a^14*b*x^(1/3) + 7726
160*a^15)/x^9

________________________________________________________________________________________

Fricas [A]  time = 1.66385, size = 520, normalized size = 2.42 \begin{align*} -\frac{17383860 \, b^{15} x^{5} + 6327725040 \, a^{3} b^{12} x^{4} + 58004146200 \, a^{6} b^{9} x^{3} + 49717839600 \, a^{9} b^{6} x^{2} + 3954828150 \, a^{12} b^{3} x + 7726160 \, a^{15} + 31671 \,{\left (7600 \, a b^{14} x^{4} + 561925 \, a^{4} b^{11} x^{3} + 2230800 \, a^{7} b^{8} x^{2} + 899080 \, a^{10} b^{5} x + 27664 \, a^{13} b^{2}\right )} x^{\frac{2}{3}} + 30780 \,{\left (50830 \, a^{2} b^{13} x^{4} + 1197196 \, a^{5} b^{10} x^{3} + 2180607 \, a^{8} b^{7} x^{2} + 402220 \, a^{11} b^{4} x + 3910 \, a^{14} b\right )} x^{\frac{1}{3}}}{69535440 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^15/x^10,x, algorithm="fricas")

[Out]

-1/69535440*(17383860*b^15*x^5 + 6327725040*a^3*b^12*x^4 + 58004146200*a^6*b^9*x^3 + 49717839600*a^9*b^6*x^2 +
3954828150*a^12*b^3*x + 7726160*a^15 + 31671*(7600*a*b^14*x^4 + 561925*a^4*b^11*x^3 + 2230800*a^7*b^8*x^2 + 8
99080*a^10*b^5*x + 27664*a^13*b^2)*x^(2/3) + 30780*(50830*a^2*b^13*x^4 + 1197196*a^5*b^10*x^3 + 2180607*a^8*b^
7*x^2 + 402220*a^11*b^4*x + 3910*a^14*b)*x^(1/3))/x^9

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Sympy [A]  time = 44.7693, size = 218, normalized size = 1.01 \begin{align*} - \frac{a^{15}}{9 x^{9}} - \frac{45 a^{14} b}{26 x^{\frac{26}{3}}} - \frac{63 a^{13} b^{2}}{5 x^{\frac{25}{3}}} - \frac{455 a^{12} b^{3}}{8 x^{8}} - \frac{4095 a^{11} b^{4}}{23 x^{\frac{23}{3}}} - \frac{819 a^{10} b^{5}}{2 x^{\frac{22}{3}}} - \frac{715 a^{9} b^{6}}{x^{7}} - \frac{3861 a^{8} b^{7}}{4 x^{\frac{20}{3}}} - \frac{19305 a^{7} b^{8}}{19 x^{\frac{19}{3}}} - \frac{5005 a^{6} b^{9}}{6 x^{6}} - \frac{9009 a^{5} b^{10}}{17 x^{\frac{17}{3}}} - \frac{4095 a^{4} b^{11}}{16 x^{\frac{16}{3}}} - \frac{91 a^{3} b^{12}}{x^{5}} - \frac{45 a^{2} b^{13}}{2 x^{\frac{14}{3}}} - \frac{45 a b^{14}}{13 x^{\frac{13}{3}}} - \frac{b^{15}}{4 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/3))**15/x**10,x)

[Out]

-a**15/(9*x**9) - 45*a**14*b/(26*x**(26/3)) - 63*a**13*b**2/(5*x**(25/3)) - 455*a**12*b**3/(8*x**8) - 4095*a**
11*b**4/(23*x**(23/3)) - 819*a**10*b**5/(2*x**(22/3)) - 715*a**9*b**6/x**7 - 3861*a**8*b**7/(4*x**(20/3)) - 19
305*a**7*b**8/(19*x**(19/3)) - 5005*a**6*b**9/(6*x**6) - 9009*a**5*b**10/(17*x**(17/3)) - 4095*a**4*b**11/(16*
x**(16/3)) - 91*a**3*b**12/x**5 - 45*a**2*b**13/(2*x**(14/3)) - 45*a*b**14/(13*x**(13/3)) - b**15/(4*x**4)

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Giac [A]  time = 1.21269, size = 225, normalized size = 1.05 \begin{align*} -\frac{17383860 \, b^{15} x^{5} + 240699600 \, a b^{14} x^{\frac{14}{3}} + 1564547400 \, a^{2} b^{13} x^{\frac{13}{3}} + 6327725040 \, a^{3} b^{12} x^{4} + 17796726675 \, a^{4} b^{11} x^{\frac{11}{3}} + 36849692880 \, a^{5} b^{10} x^{\frac{10}{3}} + 58004146200 \, a^{6} b^{9} x^{3} + 70651666800 \, a^{7} b^{8} x^{\frac{8}{3}} + 67119083460 \, a^{8} b^{7} x^{\frac{7}{3}} + 49717839600 \, a^{9} b^{6} x^{2} + 28474762680 \, a^{10} b^{5} x^{\frac{5}{3}} + 12380331600 \, a^{11} b^{4} x^{\frac{4}{3}} + 3954828150 \, a^{12} b^{3} x + 876146544 \, a^{13} b^{2} x^{\frac{2}{3}} + 120349800 \, a^{14} b x^{\frac{1}{3}} + 7726160 \, a^{15}}{69535440 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^15/x^10,x, algorithm="giac")

[Out]

-1/69535440*(17383860*b^15*x^5 + 240699600*a*b^14*x^(14/3) + 1564547400*a^2*b^13*x^(13/3) + 6327725040*a^3*b^1
2*x^4 + 17796726675*a^4*b^11*x^(11/3) + 36849692880*a^5*b^10*x^(10/3) + 58004146200*a^6*b^9*x^3 + 70651666800*
a^7*b^8*x^(8/3) + 67119083460*a^8*b^7*x^(7/3) + 49717839600*a^9*b^6*x^2 + 28474762680*a^10*b^5*x^(5/3) + 12380
331600*a^11*b^4*x^(4/3) + 3954828150*a^12*b^3*x + 876146544*a^13*b^2*x^(2/3) + 120349800*a^14*b*x^(1/3) + 7726
160*a^15)/x^9