3.2339 \(\int (a+b \sqrt [3]{x})^{15} x^5 \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.171554, 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} \[ \frac{63}{4} a^{13} b^2 x^{20/3}+65 a^{12} b^3 x^7+\frac{4095}{22} a^{11} b^4 x^{22/3}+\frac{9009}{23} a^{10} b^5 x^{23/3}+\frac{5005}{8} a^9 b^6 x^8+\frac{3861}{5} a^8 b^7 x^{25/3}+\frac{1485}{2} a^7 b^8 x^{26/3}+\frac{5005}{9} a^6 b^9 x^9+\frac{1287}{4} a^5 b^{10} x^{28/3}+\frac{4095}{29} a^4 b^{11} x^{29/3}+\frac{91}{2} a^3 b^{12} x^{10}+\frac{315}{31} a^2 b^{13} x^{31/3}+\frac{45}{19} a^{14} b x^{19/3}+\frac{a^{15} x^6}{6}+\frac{45}{32} a b^{14} x^{32/3}+\frac{b^{15} x^{11}}{11} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))^15*x^5,x]

[Out]

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

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 \left (a+b \sqrt [3]{x}\right )^{15} x^5 \, dx &=3 \operatorname{Subst}\left (\int x^{17} (a+b x)^{15} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (a^{15} x^{17}+15 a^{14} b x^{18}+105 a^{13} b^2 x^{19}+455 a^{12} b^3 x^{20}+1365 a^{11} b^4 x^{21}+3003 a^{10} b^5 x^{22}+5005 a^9 b^6 x^{23}+6435 a^8 b^7 x^{24}+6435 a^7 b^8 x^{25}+5005 a^6 b^9 x^{26}+3003 a^5 b^{10} x^{27}+1365 a^4 b^{11} x^{28}+455 a^3 b^{12} x^{29}+105 a^2 b^{13} x^{30}+15 a b^{14} x^{31}+b^{15} x^{32}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{a^{15} x^6}{6}+\frac{45}{19} a^{14} b x^{19/3}+\frac{63}{4} a^{13} b^2 x^{20/3}+65 a^{12} b^3 x^7+\frac{4095}{22} a^{11} b^4 x^{22/3}+\frac{9009}{23} a^{10} b^5 x^{23/3}+\frac{5005}{8} a^9 b^6 x^8+\frac{3861}{5} a^8 b^7 x^{25/3}+\frac{1485}{2} a^7 b^8 x^{26/3}+\frac{5005}{9} a^6 b^9 x^9+\frac{1287}{4} a^5 b^{10} x^{28/3}+\frac{4095}{29} a^4 b^{11} x^{29/3}+\frac{91}{2} a^3 b^{12} x^{10}+\frac{315}{31} a^2 b^{13} x^{31/3}+\frac{45}{32} a b^{14} x^{32/3}+\frac{b^{15} x^{11}}{11}\\ \end{align*}

Mathematica [A]  time = 0.119015, size = 217, normalized size = 1. \[ \frac{63}{4} a^{13} b^2 x^{20/3}+65 a^{12} b^3 x^7+\frac{4095}{22} a^{11} b^4 x^{22/3}+\frac{9009}{23} a^{10} b^5 x^{23/3}+\frac{5005}{8} a^9 b^6 x^8+\frac{3861}{5} a^8 b^7 x^{25/3}+\frac{1485}{2} a^7 b^8 x^{26/3}+\frac{5005}{9} a^6 b^9 x^9+\frac{1287}{4} a^5 b^{10} x^{28/3}+\frac{4095}{29} a^4 b^{11} x^{29/3}+\frac{91}{2} a^3 b^{12} x^{10}+\frac{315}{31} a^2 b^{13} x^{31/3}+\frac{45}{19} a^{14} b x^{19/3}+\frac{a^{15} x^6}{6}+\frac{45}{32} a b^{14} x^{32/3}+\frac{b^{15} x^{11}}{11} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^15*x^5,x]

[Out]

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

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Maple [A]  time = 0.003, size = 168, normalized size = 0.8 \begin{align*}{\frac{{a}^{15}{x}^{6}}{6}}+{\frac{45\,{a}^{14}b}{19}{x}^{{\frac{19}{3}}}}+{\frac{63\,{a}^{13}{b}^{2}}{4}{x}^{{\frac{20}{3}}}}+65\,{a}^{12}{b}^{3}{x}^{7}+{\frac{4095\,{a}^{11}{b}^{4}}{22}{x}^{{\frac{22}{3}}}}+{\frac{9009\,{a}^{10}{b}^{5}}{23}{x}^{{\frac{23}{3}}}}+{\frac{5005\,{a}^{9}{b}^{6}{x}^{8}}{8}}+{\frac{3861\,{a}^{8}{b}^{7}}{5}{x}^{{\frac{25}{3}}}}+{\frac{1485\,{a}^{7}{b}^{8}}{2}{x}^{{\frac{26}{3}}}}+{\frac{5005\,{a}^{6}{b}^{9}{x}^{9}}{9}}+{\frac{1287\,{a}^{5}{b}^{10}}{4}{x}^{{\frac{28}{3}}}}+{\frac{4095\,{a}^{4}{b}^{11}}{29}{x}^{{\frac{29}{3}}}}+{\frac{91\,{a}^{3}{b}^{12}{x}^{10}}{2}}+{\frac{315\,{a}^{2}{b}^{13}}{31}{x}^{{\frac{31}{3}}}}+{\frac{45\,a{b}^{14}}{32}{x}^{{\frac{32}{3}}}}+{\frac{{b}^{15}{x}^{11}}{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/3))^15*x^5,x)

[Out]

1/6*a^15*x^6+45/19*a^14*b*x^(19/3)+63/4*a^13*b^2*x^(20/3)+65*a^12*b^3*x^7+4095/22*a^11*b^4*x^(22/3)+9009/23*a^
10*b^5*x^(23/3)+5005/8*a^9*b^6*x^8+3861/5*a^8*b^7*x^(25/3)+1485/2*a^7*b^8*x^(26/3)+5005/9*a^6*b^9*x^9+1287/4*a
^5*b^10*x^(28/3)+4095/29*a^4*b^11*x^(29/3)+91/2*a^3*b^12*x^10+315/31*a^2*b^13*x^(31/3)+45/32*a*b^14*x^(32/3)+1
/11*b^15*x^11

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Maxima [A]  time = 1.02869, size = 408, normalized size = 1.88 \begin{align*} \frac{{\left (b x^{\frac{1}{3}} + a\right )}^{33}}{11 \, b^{18}} - \frac{51 \,{\left (b x^{\frac{1}{3}} + a\right )}^{32} a}{32 \, b^{18}} + \frac{408 \,{\left (b x^{\frac{1}{3}} + a\right )}^{31} a^{2}}{31 \, b^{18}} - \frac{68 \,{\left (b x^{\frac{1}{3}} + a\right )}^{30} a^{3}}{b^{18}} + \frac{7140 \,{\left (b x^{\frac{1}{3}} + a\right )}^{29} a^{4}}{29 \, b^{18}} - \frac{663 \,{\left (b x^{\frac{1}{3}} + a\right )}^{28} a^{5}}{b^{18}} + \frac{12376 \,{\left (b x^{\frac{1}{3}} + a\right )}^{27} a^{6}}{9 \, b^{18}} - \frac{2244 \,{\left (b x^{\frac{1}{3}} + a\right )}^{26} a^{7}}{b^{18}} + \frac{14586 \,{\left (b x^{\frac{1}{3}} + a\right )}^{25} a^{8}}{5 \, b^{18}} - \frac{12155 \,{\left (b x^{\frac{1}{3}} + a\right )}^{24} a^{9}}{4 \, b^{18}} + \frac{58344 \,{\left (b x^{\frac{1}{3}} + a\right )}^{23} a^{10}}{23 \, b^{18}} - \frac{18564 \,{\left (b x^{\frac{1}{3}} + a\right )}^{22} a^{11}}{11 \, b^{18}} + \frac{884 \,{\left (b x^{\frac{1}{3}} + a\right )}^{21} a^{12}}{b^{18}} - \frac{357 \,{\left (b x^{\frac{1}{3}} + a\right )}^{20} a^{13}}{b^{18}} + \frac{2040 \,{\left (b x^{\frac{1}{3}} + a\right )}^{19} a^{14}}{19 \, b^{18}} - \frac{68 \,{\left (b x^{\frac{1}{3}} + a\right )}^{18} a^{15}}{3 \, b^{18}} + \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{17} a^{16}}{b^{18}} - \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{16} a^{17}}{16 \, b^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^15*x^5,x, algorithm="maxima")

[Out]

1/11*(b*x^(1/3) + a)^33/b^18 - 51/32*(b*x^(1/3) + a)^32*a/b^18 + 408/31*(b*x^(1/3) + a)^31*a^2/b^18 - 68*(b*x^
(1/3) + a)^30*a^3/b^18 + 7140/29*(b*x^(1/3) + a)^29*a^4/b^18 - 663*(b*x^(1/3) + a)^28*a^5/b^18 + 12376/9*(b*x^
(1/3) + a)^27*a^6/b^18 - 2244*(b*x^(1/3) + a)^26*a^7/b^18 + 14586/5*(b*x^(1/3) + a)^25*a^8/b^18 - 12155/4*(b*x
^(1/3) + a)^24*a^9/b^18 + 58344/23*(b*x^(1/3) + a)^23*a^10/b^18 - 18564/11*(b*x^(1/3) + a)^22*a^11/b^18 + 884*
(b*x^(1/3) + a)^21*a^12/b^18 - 357*(b*x^(1/3) + a)^20*a^13/b^18 + 2040/19*(b*x^(1/3) + a)^19*a^14/b^18 - 68/3*
(b*x^(1/3) + a)^18*a^15/b^18 + 3*(b*x^(1/3) + a)^17*a^16/b^18 - 3/16*(b*x^(1/3) + a)^16*a^17/b^18

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Fricas [A]  time = 1.44365, size = 494, normalized size = 2.28 \begin{align*} \frac{1}{11} \, b^{15} x^{11} + \frac{91}{2} \, a^{3} b^{12} x^{10} + \frac{5005}{9} \, a^{6} b^{9} x^{9} + \frac{5005}{8} \, a^{9} b^{6} x^{8} + 65 \, a^{12} b^{3} x^{7} + \frac{1}{6} \, a^{15} x^{6} + \frac{9}{21344} \,{\left (3335 \, a b^{14} x^{10} + 334880 \, a^{4} b^{11} x^{9} + 1760880 \, a^{7} b^{8} x^{8} + 928928 \, a^{10} b^{5} x^{7} + 37352 \, a^{13} b^{2} x^{6}\right )} x^{\frac{2}{3}} + \frac{9}{129580} \,{\left (146300 \, a^{2} b^{13} x^{10} + 4632485 \, a^{5} b^{10} x^{9} + 11117964 \, a^{8} b^{7} x^{8} + 2679950 \, a^{11} b^{4} x^{7} + 34100 \, a^{14} b x^{6}\right )} x^{\frac{1}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^15*x^5,x, algorithm="fricas")

[Out]

1/11*b^15*x^11 + 91/2*a^3*b^12*x^10 + 5005/9*a^6*b^9*x^9 + 5005/8*a^9*b^6*x^8 + 65*a^12*b^3*x^7 + 1/6*a^15*x^6
+ 9/21344*(3335*a*b^14*x^10 + 334880*a^4*b^11*x^9 + 1760880*a^7*b^8*x^8 + 928928*a^10*b^5*x^7 + 37352*a^13*b^
2*x^6)*x^(2/3) + 9/129580*(146300*a^2*b^13*x^10 + 4632485*a^5*b^10*x^9 + 11117964*a^8*b^7*x^8 + 2679950*a^11*b
^4*x^7 + 34100*a^14*b*x^6)*x^(1/3)

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Sympy [A]  time = 22.6872, size = 218, normalized size = 1. \begin{align*} \frac{a^{15} x^{6}}{6} + \frac{45 a^{14} b x^{\frac{19}{3}}}{19} + \frac{63 a^{13} b^{2} x^{\frac{20}{3}}}{4} + 65 a^{12} b^{3} x^{7} + \frac{4095 a^{11} b^{4} x^{\frac{22}{3}}}{22} + \frac{9009 a^{10} b^{5} x^{\frac{23}{3}}}{23} + \frac{5005 a^{9} b^{6} x^{8}}{8} + \frac{3861 a^{8} b^{7} x^{\frac{25}{3}}}{5} + \frac{1485 a^{7} b^{8} x^{\frac{26}{3}}}{2} + \frac{5005 a^{6} b^{9} x^{9}}{9} + \frac{1287 a^{5} b^{10} x^{\frac{28}{3}}}{4} + \frac{4095 a^{4} b^{11} x^{\frac{29}{3}}}{29} + \frac{91 a^{3} b^{12} x^{10}}{2} + \frac{315 a^{2} b^{13} x^{\frac{31}{3}}}{31} + \frac{45 a b^{14} x^{\frac{32}{3}}}{32} + \frac{b^{15} x^{11}}{11} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/3))**15*x**5,x)

[Out]

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

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Giac [A]  time = 1.19831, size = 225, normalized size = 1.04 \begin{align*} \frac{1}{11} \, b^{15} x^{11} + \frac{45}{32} \, a b^{14} x^{\frac{32}{3}} + \frac{315}{31} \, a^{2} b^{13} x^{\frac{31}{3}} + \frac{91}{2} \, a^{3} b^{12} x^{10} + \frac{4095}{29} \, a^{4} b^{11} x^{\frac{29}{3}} + \frac{1287}{4} \, a^{5} b^{10} x^{\frac{28}{3}} + \frac{5005}{9} \, a^{6} b^{9} x^{9} + \frac{1485}{2} \, a^{7} b^{8} x^{\frac{26}{3}} + \frac{3861}{5} \, a^{8} b^{7} x^{\frac{25}{3}} + \frac{5005}{8} \, a^{9} b^{6} x^{8} + \frac{9009}{23} \, a^{10} b^{5} x^{\frac{23}{3}} + \frac{4095}{22} \, a^{11} b^{4} x^{\frac{22}{3}} + 65 \, a^{12} b^{3} x^{7} + \frac{63}{4} \, a^{13} b^{2} x^{\frac{20}{3}} + \frac{45}{19} \, a^{14} b x^{\frac{19}{3}} + \frac{1}{6} \, a^{15} x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^15*x^5,x, algorithm="giac")

[Out]

1/11*b^15*x^11 + 45/32*a*b^14*x^(32/3) + 315/31*a^2*b^13*x^(31/3) + 91/2*a^3*b^12*x^10 + 4095/29*a^4*b^11*x^(2
9/3) + 1287/4*a^5*b^10*x^(28/3) + 5005/9*a^6*b^9*x^9 + 1485/2*a^7*b^8*x^(26/3) + 3861/5*a^8*b^7*x^(25/3) + 500
5/8*a^9*b^6*x^8 + 9009/23*a^10*b^5*x^(23/3) + 4095/22*a^11*b^4*x^(22/3) + 65*a^12*b^3*x^7 + 63/4*a^13*b^2*x^(2
0/3) + 45/19*a^14*b*x^(19/3) + 1/6*a^15*x^6