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

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

[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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Rubi [A]  time = 0.12, antiderivative size = 217, normalized size of antiderivative = 1.00, 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 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])

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]]

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*}

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Mathematica [A]  time = 0.10, size = 217, normalized size = 1.00 \[ -\frac {a^{15}}{11 x^{11}}-\frac {45 a^{14} b}{32 x^{32/3}}-\frac {315 a^{13} b^2}{31 x^{31/3}}-\frac {91 a^{12} b^3}{2 x^{10}}-\frac {4095 a^{11} b^4}{29 x^{29/3}}-\frac {1287 a^{10} b^5}{4 x^{28/3}}-\frac {5005 a^9 b^6}{9 x^9}-\frac {1485 a^8 b^7}{2 x^{26/3}}-\frac {3861 a^7 b^8}{5 x^{25/3}}-\frac {5005 a^6 b^9}{8 x^8}-\frac {9009 a^5 b^{10}}{23 x^{23/3}}-\frac {4095 a^4 b^{11}}{22 x^{22/3}}-\frac {65 a^3 b^{12}}{x^7}-\frac {63 a^2 b^{13}}{4 x^{20/3}}-\frac {45 a b^{14}}{19 x^{19/3}}-\frac {b^{15}}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^15/x^12,x]

[Out]

-1/11*a^15/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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fricas [A]  time = 0.86, size = 169, normalized size = 0.78 \[ -\frac {1037158320 \, b^{15} x^{5} + 404491744800 \, a^{3} b^{12} x^{4} + 3893233043700 \, a^{6} b^{9} x^{3} + 3460651594400 \, a^{9} b^{6} x^{2} + 283144221360 \, a^{12} b^{3} x + 565722720 \, a^{15} + 432216 \, {\left (34100 \, a b^{14} x^{4} + 2679950 \, a^{4} b^{11} x^{3} + 11117964 \, a^{7} b^{8} x^{2} + 4632485 \, a^{10} b^{5} x + 146300 \, a^{13} b^{2}\right )} x^{\frac {2}{3}} + 2623995 \, {\left (37352 \, a^{2} b^{13} x^{4} + 928928 \, a^{5} b^{10} x^{3} + 1760880 \, a^{8} b^{7} x^{2} + 334880 \, a^{11} b^{4} x + 3335 \, a^{14} b\right )} x^{\frac {1}{3}}}{6222949920 \, x^{11}} \]

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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giac [A]  time = 0.20, size = 167, normalized size = 0.77 \[ -\frac {1037158320 \, b^{15} x^{5} + 14738565600 \, a b^{14} x^{\frac {14}{3}} + 98011461240 \, a^{2} b^{13} x^{\frac {13}{3}} + 404491744800 \, a^{3} b^{12} x^{4} + 1158317269200 \, a^{4} b^{11} x^{\frac {11}{3}} + 2437502427360 \, a^{5} b^{10} x^{\frac {10}{3}} + 3893233043700 \, a^{6} b^{9} x^{3} + 4805361928224 \, a^{7} b^{8} x^{\frac {8}{3}} + 4620540315600 \, a^{8} b^{7} x^{\frac {7}{3}} + 3460651594400 \, a^{9} b^{6} x^{2} + 2002234136760 \, a^{10} b^{5} x^{\frac {5}{3}} + 878723445600 \, a^{11} b^{4} x^{\frac {4}{3}} + 283144221360 \, a^{12} b^{3} x + 63233200800 \, a^{13} b^{2} x^{\frac {2}{3}} + 8751023325 \, a^{14} b x^{\frac {1}{3}} + 565722720 \, a^{15}}{6222949920 \, x^{11}} \]

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

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maple [A]  time = 0.01, size = 168, normalized size = 0.77 \[ -\frac {b^{15}}{6 x^{6}}-\frac {45 a \,b^{14}}{19 x^{\frac {19}{3}}}-\frac {63 a^{2} b^{13}}{4 x^{\frac {20}{3}}}-\frac {65 a^{3} b^{12}}{x^{7}}-\frac {4095 a^{4} b^{11}}{22 x^{\frac {22}{3}}}-\frac {9009 a^{5} b^{10}}{23 x^{\frac {23}{3}}}-\frac {5005 a^{6} b^{9}}{8 x^{8}}-\frac {3861 a^{7} b^{8}}{5 x^{\frac {25}{3}}}-\frac {1485 a^{8} b^{7}}{2 x^{\frac {26}{3}}}-\frac {5005 a^{9} b^{6}}{9 x^{9}}-\frac {1287 a^{10} b^{5}}{4 x^{\frac {28}{3}}}-\frac {4095 a^{11} b^{4}}{29 x^{\frac {29}{3}}}-\frac {91 a^{12} b^{3}}{2 x^{10}}-\frac {315 a^{13} b^{2}}{31 x^{\frac {31}{3}}}-\frac {45 a^{14} b}{32 x^{\frac {32}{3}}}-\frac {a^{15}}{11 x^{11}} \]

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 = 0.88, size = 167, normalized size = 0.77 \[ -\frac {1037158320 \, b^{15} x^{5} + 14738565600 \, a b^{14} x^{\frac {14}{3}} + 98011461240 \, a^{2} b^{13} x^{\frac {13}{3}} + 404491744800 \, a^{3} b^{12} x^{4} + 1158317269200 \, a^{4} b^{11} x^{\frac {11}{3}} + 2437502427360 \, a^{5} b^{10} x^{\frac {10}{3}} + 3893233043700 \, a^{6} b^{9} x^{3} + 4805361928224 \, a^{7} b^{8} x^{\frac {8}{3}} + 4620540315600 \, a^{8} b^{7} x^{\frac {7}{3}} + 3460651594400 \, a^{9} b^{6} x^{2} + 2002234136760 \, a^{10} b^{5} x^{\frac {5}{3}} + 878723445600 \, a^{11} b^{4} x^{\frac {4}{3}} + 283144221360 \, a^{12} b^{3} x + 63233200800 \, a^{13} b^{2} x^{\frac {2}{3}} + 8751023325 \, a^{14} b x^{\frac {1}{3}} + 565722720 \, a^{15}}{6222949920 \, x^{11}} \]

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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mupad [B]  time = 1.28, size = 167, normalized size = 0.77 \[ -\frac {\frac {a^{15}}{11}+\frac {b^{15}\,x^5}{6}+\frac {91\,a^{12}\,b^3\,x}{2}+\frac {45\,a^{14}\,b\,x^{1/3}}{32}+\frac {45\,a\,b^{14}\,x^{14/3}}{19}+\frac {5005\,a^9\,b^6\,x^2}{9}+\frac {5005\,a^6\,b^9\,x^3}{8}+65\,a^3\,b^{12}\,x^4+\frac {315\,a^{13}\,b^2\,x^{2/3}}{31}+\frac {4095\,a^{11}\,b^4\,x^{4/3}}{29}+\frac {1287\,a^{10}\,b^5\,x^{5/3}}{4}+\frac {1485\,a^8\,b^7\,x^{7/3}}{2}+\frac {3861\,a^7\,b^8\,x^{8/3}}{5}+\frac {9009\,a^5\,b^{10}\,x^{10/3}}{23}+\frac {4095\,a^4\,b^{11}\,x^{11/3}}{22}+\frac {63\,a^2\,b^{13}\,x^{13/3}}{4}}{x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^(1/3))^15/x^12,x)

[Out]

-(a^15/11 + (b^15*x^5)/6 + (91*a^12*b^3*x)/2 + (45*a^14*b*x^(1/3))/32 + (45*a*b^14*x^(14/3))/19 + (5005*a^9*b^
6*x^2)/9 + (5005*a^6*b^9*x^3)/8 + 65*a^3*b^12*x^4 + (315*a^13*b^2*x^(2/3))/31 + (4095*a^11*b^4*x^(4/3))/29 + (
1287*a^10*b^5*x^(5/3))/4 + (1485*a^8*b^7*x^(7/3))/2 + (3861*a^7*b^8*x^(8/3))/5 + (9009*a^5*b^10*x^(10/3))/23 +
(4095*a^4*b^11*x^(11/3))/22 + (63*a^2*b^13*x^(13/3))/4)/x^11

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sympy [A]  time = 57.16, size = 219, normalized size = 1.01 \[ - \frac {a^{15}}{11 x^{11}} - \frac {45 a^{14} b}{32 x^{\frac {32}{3}}} - \frac {315 a^{13} b^{2}}{31 x^{\frac {31}{3}}} - \frac {91 a^{12} b^{3}}{2 x^{10}} - \frac {4095 a^{11} b^{4}}{29 x^{\frac {29}{3}}} - \frac {1287 a^{10} b^{5}}{4 x^{\frac {28}{3}}} - \frac {5005 a^{9} b^{6}}{9 x^{9}} - \frac {1485 a^{8} b^{7}}{2 x^{\frac {26}{3}}} - \frac {3861 a^{7} b^{8}}{5 x^{\frac {25}{3}}} - \frac {5005 a^{6} b^{9}}{8 x^{8}} - \frac {9009 a^{5} b^{10}}{23 x^{\frac {23}{3}}} - \frac {4095 a^{4} b^{11}}{22 x^{\frac {22}{3}}} - \frac {65 a^{3} b^{12}}{x^{7}} - \frac {63 a^{2} b^{13}}{4 x^{\frac {20}{3}}} - \frac {45 a b^{14}}{19 x^{\frac {19}{3}}} - \frac {b^{15}}{6 x^{6}} \]

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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