3.2355 \(\int \frac{(a+b \sqrt [3]{x})^{15}}{x^{12}} \, dx\)

Optimal. Leaf size=217 \[ -\frac{315 a^{13} b^2}{31 x^{31/3}}-\frac{91 a^{12} b^3}{2 x^{10}}-\frac{4095 a^{11} b^4}{29 x^{29/3}}-\frac{1287 a^{10} b^5}{4 x^{28/3}}-\frac{5005 a^9 b^6}{9 x^9}-\frac{1485 a^8 b^7}{2 x^{26/3}}-\frac{3861 a^7 b^8}{5 x^{25/3}}-\frac{5005 a^6 b^9}{8 x^8}-\frac{9009 a^5 b^{10}}{23 x^{23/3}}-\frac{4095 a^4 b^{11}}{22 x^{22/3}}-\frac{65 a^3 b^{12}}{x^7}-\frac{63 a^2 b^{13}}{4 x^{20/3}}-\frac{45 a^{14} b}{32 x^{32/3}}-\frac{a^{15}}{11 x^{11}}-\frac{45 a b^{14}}{19 x^{19/3}}-\frac{b^{15}}{6 x^6} \]

[Out]

-a^15/(11*x^11) - (45*a^14*b)/(32*x^(32/3)) - (315*a^13*b^2)/(31*x^(31/3)) - (91*a^12*b^3)/(2*x^10) - (4095*a^
11*b^4)/(29*x^(29/3)) - (1287*a^10*b^5)/(4*x^(28/3)) - (5005*a^9*b^6)/(9*x^9) - (1485*a^8*b^7)/(2*x^(26/3)) -
(3861*a^7*b^8)/(5*x^(25/3)) - (5005*a^6*b^9)/(8*x^8) - (9009*a^5*b^10)/(23*x^(23/3)) - (4095*a^4*b^11)/(22*x^(
22/3)) - (65*a^3*b^12)/x^7 - (63*a^2*b^13)/(4*x^(20/3)) - (45*a*b^14)/(19*x^(19/3)) - b^15/(6*x^6)

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Rubi [A]  time = 0.118027, 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} \[ -\frac{315 a^{13} b^2}{31 x^{31/3}}-\frac{91 a^{12} b^3}{2 x^{10}}-\frac{4095 a^{11} b^4}{29 x^{29/3}}-\frac{1287 a^{10} b^5}{4 x^{28/3}}-\frac{5005 a^9 b^6}{9 x^9}-\frac{1485 a^8 b^7}{2 x^{26/3}}-\frac{3861 a^7 b^8}{5 x^{25/3}}-\frac{5005 a^6 b^9}{8 x^8}-\frac{9009 a^5 b^{10}}{23 x^{23/3}}-\frac{4095 a^4 b^{11}}{22 x^{22/3}}-\frac{65 a^3 b^{12}}{x^7}-\frac{63 a^2 b^{13}}{4 x^{20/3}}-\frac{45 a^{14} b}{32 x^{32/3}}-\frac{a^{15}}{11 x^{11}}-\frac{45 a b^{14}}{19 x^{19/3}}-\frac{b^{15}}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))^15/x^12,x]

[Out]

-a^15/(11*x^11) - (45*a^14*b)/(32*x^(32/3)) - (315*a^13*b^2)/(31*x^(31/3)) - (91*a^12*b^3)/(2*x^10) - (4095*a^
11*b^4)/(29*x^(29/3)) - (1287*a^10*b^5)/(4*x^(28/3)) - (5005*a^9*b^6)/(9*x^9) - (1485*a^8*b^7)/(2*x^(26/3)) -
(3861*a^7*b^8)/(5*x^(25/3)) - (5005*a^6*b^9)/(8*x^8) - (9009*a^5*b^10)/(23*x^(23/3)) - (4095*a^4*b^11)/(22*x^(
22/3)) - (65*a^3*b^12)/x^7 - (63*a^2*b^13)/(4*x^(20/3)) - (45*a*b^14)/(19*x^(19/3)) - b^15/(6*x^6)

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^{12}} \, dx &=3 \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{34}} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{a^{15}}{x^{34}}+\frac{15 a^{14} b}{x^{33}}+\frac{105 a^{13} b^2}{x^{32}}+\frac{455 a^{12} b^3}{x^{31}}+\frac{1365 a^{11} b^4}{x^{30}}+\frac{3003 a^{10} b^5}{x^{29}}+\frac{5005 a^9 b^6}{x^{28}}+\frac{6435 a^8 b^7}{x^{27}}+\frac{6435 a^7 b^8}{x^{26}}+\frac{5005 a^6 b^9}{x^{25}}+\frac{3003 a^5 b^{10}}{x^{24}}+\frac{1365 a^4 b^{11}}{x^{23}}+\frac{455 a^3 b^{12}}{x^{22}}+\frac{105 a^2 b^{13}}{x^{21}}+\frac{15 a b^{14}}{x^{20}}+\frac{b^{15}}{x^{19}}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{a^{15}}{11 x^{11}}-\frac{45 a^{14} b}{32 x^{32/3}}-\frac{315 a^{13} b^2}{31 x^{31/3}}-\frac{91 a^{12} b^3}{2 x^{10}}-\frac{4095 a^{11} b^4}{29 x^{29/3}}-\frac{1287 a^{10} b^5}{4 x^{28/3}}-\frac{5005 a^9 b^6}{9 x^9}-\frac{1485 a^8 b^7}{2 x^{26/3}}-\frac{3861 a^7 b^8}{5 x^{25/3}}-\frac{5005 a^6 b^9}{8 x^8}-\frac{9009 a^5 b^{10}}{23 x^{23/3}}-\frac{4095 a^4 b^{11}}{22 x^{22/3}}-\frac{65 a^3 b^{12}}{x^7}-\frac{63 a^2 b^{13}}{4 x^{20/3}}-\frac{45 a b^{14}}{19 x^{19/3}}-\frac{b^{15}}{6 x^6}\\ \end{align*}

Mathematica [A]  time = 0.100448, size = 217, normalized size = 1. \[ -\frac{315 a^{13} b^2}{31 x^{31/3}}-\frac{91 a^{12} b^3}{2 x^{10}}-\frac{4095 a^{11} b^4}{29 x^{29/3}}-\frac{1287 a^{10} b^5}{4 x^{28/3}}-\frac{5005 a^9 b^6}{9 x^9}-\frac{1485 a^8 b^7}{2 x^{26/3}}-\frac{3861 a^7 b^8}{5 x^{25/3}}-\frac{5005 a^6 b^9}{8 x^8}-\frac{9009 a^5 b^{10}}{23 x^{23/3}}-\frac{4095 a^4 b^{11}}{22 x^{22/3}}-\frac{65 a^3 b^{12}}{x^7}-\frac{63 a^2 b^{13}}{4 x^{20/3}}-\frac{45 a^{14} b}{32 x^{32/3}}-\frac{a^{15}}{11 x^{11}}-\frac{45 a b^{14}}{19 x^{19/3}}-\frac{b^{15}}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^15/x^12,x]

[Out]

-a^15/(11*x^11) - (45*a^14*b)/(32*x^(32/3)) - (315*a^13*b^2)/(31*x^(31/3)) - (91*a^12*b^3)/(2*x^10) - (4095*a^
11*b^4)/(29*x^(29/3)) - (1287*a^10*b^5)/(4*x^(28/3)) - (5005*a^9*b^6)/(9*x^9) - (1485*a^8*b^7)/(2*x^(26/3)) -
(3861*a^7*b^8)/(5*x^(25/3)) - (5005*a^6*b^9)/(8*x^8) - (9009*a^5*b^10)/(23*x^(23/3)) - (4095*a^4*b^11)/(22*x^(
22/3)) - (65*a^3*b^12)/x^7 - (63*a^2*b^13)/(4*x^(20/3)) - (45*a*b^14)/(19*x^(19/3)) - b^15/(6*x^6)

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Maple [A]  time = 0.01, size = 168, normalized size = 0.8 \begin{align*} -{\frac{{a}^{15}}{11\,{x}^{11}}}-{\frac{45\,{a}^{14}b}{32}{x}^{-{\frac{32}{3}}}}-{\frac{315\,{a}^{13}{b}^{2}}{31}{x}^{-{\frac{31}{3}}}}-{\frac{91\,{a}^{12}{b}^{3}}{2\,{x}^{10}}}-{\frac{4095\,{a}^{11}{b}^{4}}{29}{x}^{-{\frac{29}{3}}}}-{\frac{1287\,{a}^{10}{b}^{5}}{4}{x}^{-{\frac{28}{3}}}}-{\frac{5005\,{a}^{9}{b}^{6}}{9\,{x}^{9}}}-{\frac{1485\,{a}^{8}{b}^{7}}{2}{x}^{-{\frac{26}{3}}}}-{\frac{3861\,{a}^{7}{b}^{8}}{5}{x}^{-{\frac{25}{3}}}}-{\frac{5005\,{a}^{6}{b}^{9}}{8\,{x}^{8}}}-{\frac{9009\,{a}^{5}{b}^{10}}{23}{x}^{-{\frac{23}{3}}}}-{\frac{4095\,{a}^{4}{b}^{11}}{22}{x}^{-{\frac{22}{3}}}}-65\,{\frac{{a}^{3}{b}^{12}}{{x}^{7}}}-{\frac{63\,{a}^{2}{b}^{13}}{4}{x}^{-{\frac{20}{3}}}}-{\frac{45\,a{b}^{14}}{19}{x}^{-{\frac{19}{3}}}}-{\frac{{b}^{15}}{6\,{x}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/3))^15/x^12,x)

[Out]

-1/11*a^15/x^11-45/32*a^14*b/x^(32/3)-315/31*a^13*b^2/x^(31/3)-91/2*a^12*b^3/x^10-4095/29*a^11*b^4/x^(29/3)-12
87/4*a^10*b^5/x^(28/3)-5005/9*a^9*b^6/x^9-1485/2*a^8*b^7/x^(26/3)-3861/5*a^7*b^8/x^(25/3)-5005/8*a^6*b^9/x^8-9
009/23*a^5*b^10/x^(23/3)-4095/22*a^4*b^11/x^(22/3)-65*a^3*b^12/x^7-63/4*a^2*b^13/x^(20/3)-45/19*a*b^14/x^(19/3
)-1/6*b^15/x^6

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Maxima [A]  time = 0.979562, size = 225, normalized size = 1.04 \begin{align*} -\frac{1037158320 \, b^{15} x^{5} + 14738565600 \, a b^{14} x^{\frac{14}{3}} + 98011461240 \, a^{2} b^{13} x^{\frac{13}{3}} + 404491744800 \, a^{3} b^{12} x^{4} + 1158317269200 \, a^{4} b^{11} x^{\frac{11}{3}} + 2437502427360 \, a^{5} b^{10} x^{\frac{10}{3}} + 3893233043700 \, a^{6} b^{9} x^{3} + 4805361928224 \, a^{7} b^{8} x^{\frac{8}{3}} + 4620540315600 \, a^{8} b^{7} x^{\frac{7}{3}} + 3460651594400 \, a^{9} b^{6} x^{2} + 2002234136760 \, a^{10} b^{5} x^{\frac{5}{3}} + 878723445600 \, a^{11} b^{4} x^{\frac{4}{3}} + 283144221360 \, a^{12} b^{3} x + 63233200800 \, a^{13} b^{2} x^{\frac{2}{3}} + 8751023325 \, a^{14} b x^{\frac{1}{3}} + 565722720 \, a^{15}}{6222949920 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^15/x^12,x, algorithm="maxima")

[Out]

-1/6222949920*(1037158320*b^15*x^5 + 14738565600*a*b^14*x^(14/3) + 98011461240*a^2*b^13*x^(13/3) + 40449174480
0*a^3*b^12*x^4 + 1158317269200*a^4*b^11*x^(11/3) + 2437502427360*a^5*b^10*x^(10/3) + 3893233043700*a^6*b^9*x^3
+ 4805361928224*a^7*b^8*x^(8/3) + 4620540315600*a^8*b^7*x^(7/3) + 3460651594400*a^9*b^6*x^2 + 2002234136760*a
^10*b^5*x^(5/3) + 878723445600*a^11*b^4*x^(4/3) + 283144221360*a^12*b^3*x + 63233200800*a^13*b^2*x^(2/3) + 875
1023325*a^14*b*x^(1/3) + 565722720*a^15)/x^11

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Fricas [A]  time = 1.48353, size = 549, normalized size = 2.53 \begin{align*} -\frac{1037158320 \, b^{15} x^{5} + 404491744800 \, a^{3} b^{12} x^{4} + 3893233043700 \, a^{6} b^{9} x^{3} + 3460651594400 \, a^{9} b^{6} x^{2} + 283144221360 \, a^{12} b^{3} x + 565722720 \, a^{15} + 432216 \,{\left (34100 \, a b^{14} x^{4} + 2679950 \, a^{4} b^{11} x^{3} + 11117964 \, a^{7} b^{8} x^{2} + 4632485 \, a^{10} b^{5} x + 146300 \, a^{13} b^{2}\right )} x^{\frac{2}{3}} + 2623995 \,{\left (37352 \, a^{2} b^{13} x^{4} + 928928 \, a^{5} b^{10} x^{3} + 1760880 \, a^{8} b^{7} x^{2} + 334880 \, a^{11} b^{4} x + 3335 \, a^{14} b\right )} x^{\frac{1}{3}}}{6222949920 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^15/x^12,x, algorithm="fricas")

[Out]

-1/6222949920*(1037158320*b^15*x^5 + 404491744800*a^3*b^12*x^4 + 3893233043700*a^6*b^9*x^3 + 3460651594400*a^9
*b^6*x^2 + 283144221360*a^12*b^3*x + 565722720*a^15 + 432216*(34100*a*b^14*x^4 + 2679950*a^4*b^11*x^3 + 111179
64*a^7*b^8*x^2 + 4632485*a^10*b^5*x + 146300*a^13*b^2)*x^(2/3) + 2623995*(37352*a^2*b^13*x^4 + 928928*a^5*b^10
*x^3 + 1760880*a^8*b^7*x^2 + 334880*a^11*b^4*x + 3335*a^14*b)*x^(1/3))/x^11

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Sympy [A]  time = 87.2359, size = 219, normalized size = 1.01 \begin{align*} - \frac{a^{15}}{11 x^{11}} - \frac{45 a^{14} b}{32 x^{\frac{32}{3}}} - \frac{315 a^{13} b^{2}}{31 x^{\frac{31}{3}}} - \frac{91 a^{12} b^{3}}{2 x^{10}} - \frac{4095 a^{11} b^{4}}{29 x^{\frac{29}{3}}} - \frac{1287 a^{10} b^{5}}{4 x^{\frac{28}{3}}} - \frac{5005 a^{9} b^{6}}{9 x^{9}} - \frac{1485 a^{8} b^{7}}{2 x^{\frac{26}{3}}} - \frac{3861 a^{7} b^{8}}{5 x^{\frac{25}{3}}} - \frac{5005 a^{6} b^{9}}{8 x^{8}} - \frac{9009 a^{5} b^{10}}{23 x^{\frac{23}{3}}} - \frac{4095 a^{4} b^{11}}{22 x^{\frac{22}{3}}} - \frac{65 a^{3} b^{12}}{x^{7}} - \frac{63 a^{2} b^{13}}{4 x^{\frac{20}{3}}} - \frac{45 a b^{14}}{19 x^{\frac{19}{3}}} - \frac{b^{15}}{6 x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/3))**15/x**12,x)

[Out]

-a**15/(11*x**11) - 45*a**14*b/(32*x**(32/3)) - 315*a**13*b**2/(31*x**(31/3)) - 91*a**12*b**3/(2*x**10) - 4095
*a**11*b**4/(29*x**(29/3)) - 1287*a**10*b**5/(4*x**(28/3)) - 5005*a**9*b**6/(9*x**9) - 1485*a**8*b**7/(2*x**(2
6/3)) - 3861*a**7*b**8/(5*x**(25/3)) - 5005*a**6*b**9/(8*x**8) - 9009*a**5*b**10/(23*x**(23/3)) - 4095*a**4*b*
*11/(22*x**(22/3)) - 65*a**3*b**12/x**7 - 63*a**2*b**13/(4*x**(20/3)) - 45*a*b**14/(19*x**(19/3)) - b**15/(6*x
**6)

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Giac [A]  time = 1.17774, size = 225, normalized size = 1.04 \begin{align*} -\frac{1037158320 \, b^{15} x^{5} + 14738565600 \, a b^{14} x^{\frac{14}{3}} + 98011461240 \, a^{2} b^{13} x^{\frac{13}{3}} + 404491744800 \, a^{3} b^{12} x^{4} + 1158317269200 \, a^{4} b^{11} x^{\frac{11}{3}} + 2437502427360 \, a^{5} b^{10} x^{\frac{10}{3}} + 3893233043700 \, a^{6} b^{9} x^{3} + 4805361928224 \, a^{7} b^{8} x^{\frac{8}{3}} + 4620540315600 \, a^{8} b^{7} x^{\frac{7}{3}} + 3460651594400 \, a^{9} b^{6} x^{2} + 2002234136760 \, a^{10} b^{5} x^{\frac{5}{3}} + 878723445600 \, a^{11} b^{4} x^{\frac{4}{3}} + 283144221360 \, a^{12} b^{3} x + 63233200800 \, a^{13} b^{2} x^{\frac{2}{3}} + 8751023325 \, a^{14} b x^{\frac{1}{3}} + 565722720 \, a^{15}}{6222949920 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^15/x^12,x, algorithm="giac")

[Out]

-1/6222949920*(1037158320*b^15*x^5 + 14738565600*a*b^14*x^(14/3) + 98011461240*a^2*b^13*x^(13/3) + 40449174480
0*a^3*b^12*x^4 + 1158317269200*a^4*b^11*x^(11/3) + 2437502427360*a^5*b^10*x^(10/3) + 3893233043700*a^6*b^9*x^3
+ 4805361928224*a^7*b^8*x^(8/3) + 4620540315600*a^8*b^7*x^(7/3) + 3460651594400*a^9*b^6*x^2 + 2002234136760*a
^10*b^5*x^(5/3) + 878723445600*a^11*b^4*x^(4/3) + 283144221360*a^12*b^3*x + 63233200800*a^13*b^2*x^(2/3) + 875
1023325*a^14*b*x^(1/3) + 565722720*a^15)/x^11