3.2341 \(\int (a+b \sqrt [3]{x})^{15} x^3 \, dx\)

Optimal. Leaf size=244 \[ \frac{33 a^2 \left (a+b \sqrt [3]{x}\right )^{25}}{5 b^{12}}-\frac{165 a^3 \left (a+b \sqrt [3]{x}\right )^{24}}{8 b^{12}}+\frac{990 a^4 \left (a+b \sqrt [3]{x}\right )^{23}}{23 b^{12}}-\frac{63 a^5 \left (a+b \sqrt [3]{x}\right )^{22}}{b^{12}}+\frac{66 a^6 \left (a+b \sqrt [3]{x}\right )^{21}}{b^{12}}-\frac{99 a^7 \left (a+b \sqrt [3]{x}\right )^{20}}{2 b^{12}}+\frac{495 a^8 \left (a+b \sqrt [3]{x}\right )^{19}}{19 b^{12}}-\frac{55 a^9 \left (a+b \sqrt [3]{x}\right )^{18}}{6 b^{12}}+\frac{33 a^{10} \left (a+b \sqrt [3]{x}\right )^{17}}{17 b^{12}}-\frac{3 a^{11} \left (a+b \sqrt [3]{x}\right )^{16}}{16 b^{12}}+\frac{\left (a+b \sqrt [3]{x}\right )^{27}}{9 b^{12}}-\frac{33 a \left (a+b \sqrt [3]{x}\right )^{26}}{26 b^{12}} \]

[Out]

(-3*a^11*(a + b*x^(1/3))^16)/(16*b^12) + (33*a^10*(a + b*x^(1/3))^17)/(17*b^12) - (55*a^9*(a + b*x^(1/3))^18)/
(6*b^12) + (495*a^8*(a + b*x^(1/3))^19)/(19*b^12) - (99*a^7*(a + b*x^(1/3))^20)/(2*b^12) + (66*a^6*(a + b*x^(1
/3))^21)/b^12 - (63*a^5*(a + b*x^(1/3))^22)/b^12 + (990*a^4*(a + b*x^(1/3))^23)/(23*b^12) - (165*a^3*(a + b*x^
(1/3))^24)/(8*b^12) + (33*a^2*(a + b*x^(1/3))^25)/(5*b^12) - (33*a*(a + b*x^(1/3))^26)/(26*b^12) + (a + b*x^(1
/3))^27/(9*b^12)

________________________________________________________________________________________

Rubi [A]  time = 0.135668, antiderivative size = 244, 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} \[ \frac{33 a^2 \left (a+b \sqrt [3]{x}\right )^{25}}{5 b^{12}}-\frac{165 a^3 \left (a+b \sqrt [3]{x}\right )^{24}}{8 b^{12}}+\frac{990 a^4 \left (a+b \sqrt [3]{x}\right )^{23}}{23 b^{12}}-\frac{63 a^5 \left (a+b \sqrt [3]{x}\right )^{22}}{b^{12}}+\frac{66 a^6 \left (a+b \sqrt [3]{x}\right )^{21}}{b^{12}}-\frac{99 a^7 \left (a+b \sqrt [3]{x}\right )^{20}}{2 b^{12}}+\frac{495 a^8 \left (a+b \sqrt [3]{x}\right )^{19}}{19 b^{12}}-\frac{55 a^9 \left (a+b \sqrt [3]{x}\right )^{18}}{6 b^{12}}+\frac{33 a^{10} \left (a+b \sqrt [3]{x}\right )^{17}}{17 b^{12}}-\frac{3 a^{11} \left (a+b \sqrt [3]{x}\right )^{16}}{16 b^{12}}+\frac{\left (a+b \sqrt [3]{x}\right )^{27}}{9 b^{12}}-\frac{33 a \left (a+b \sqrt [3]{x}\right )^{26}}{26 b^{12}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))^15*x^3,x]

[Out]

(-3*a^11*(a + b*x^(1/3))^16)/(16*b^12) + (33*a^10*(a + b*x^(1/3))^17)/(17*b^12) - (55*a^9*(a + b*x^(1/3))^18)/
(6*b^12) + (495*a^8*(a + b*x^(1/3))^19)/(19*b^12) - (99*a^7*(a + b*x^(1/3))^20)/(2*b^12) + (66*a^6*(a + b*x^(1
/3))^21)/b^12 - (63*a^5*(a + b*x^(1/3))^22)/b^12 + (990*a^4*(a + b*x^(1/3))^23)/(23*b^12) - (165*a^3*(a + b*x^
(1/3))^24)/(8*b^12) + (33*a^2*(a + b*x^(1/3))^25)/(5*b^12) - (33*a*(a + b*x^(1/3))^26)/(26*b^12) + (a + b*x^(1
/3))^27/(9*b^12)

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^3 \, dx &=3 \operatorname{Subst}\left (\int x^{11} (a+b x)^{15} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (-\frac{a^{11} (a+b x)^{15}}{b^{11}}+\frac{11 a^{10} (a+b x)^{16}}{b^{11}}-\frac{55 a^9 (a+b x)^{17}}{b^{11}}+\frac{165 a^8 (a+b x)^{18}}{b^{11}}-\frac{330 a^7 (a+b x)^{19}}{b^{11}}+\frac{462 a^6 (a+b x)^{20}}{b^{11}}-\frac{462 a^5 (a+b x)^{21}}{b^{11}}+\frac{330 a^4 (a+b x)^{22}}{b^{11}}-\frac{165 a^3 (a+b x)^{23}}{b^{11}}+\frac{55 a^2 (a+b x)^{24}}{b^{11}}-\frac{11 a (a+b x)^{25}}{b^{11}}+\frac{(a+b x)^{26}}{b^{11}}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{3 a^{11} \left (a+b \sqrt [3]{x}\right )^{16}}{16 b^{12}}+\frac{33 a^{10} \left (a+b \sqrt [3]{x}\right )^{17}}{17 b^{12}}-\frac{55 a^9 \left (a+b \sqrt [3]{x}\right )^{18}}{6 b^{12}}+\frac{495 a^8 \left (a+b \sqrt [3]{x}\right )^{19}}{19 b^{12}}-\frac{99 a^7 \left (a+b \sqrt [3]{x}\right )^{20}}{2 b^{12}}+\frac{66 a^6 \left (a+b \sqrt [3]{x}\right )^{21}}{b^{12}}-\frac{63 a^5 \left (a+b \sqrt [3]{x}\right )^{22}}{b^{12}}+\frac{990 a^4 \left (a+b \sqrt [3]{x}\right )^{23}}{23 b^{12}}-\frac{165 a^3 \left (a+b \sqrt [3]{x}\right )^{24}}{8 b^{12}}+\frac{33 a^2 \left (a+b \sqrt [3]{x}\right )^{25}}{5 b^{12}}-\frac{33 a \left (a+b \sqrt [3]{x}\right )^{26}}{26 b^{12}}+\frac{\left (a+b \sqrt [3]{x}\right )^{27}}{9 b^{12}}\\ \end{align*}

Mathematica [A]  time = 0.087597, size = 215, normalized size = 0.88 \[ \frac{45}{2} a^{13} b^2 x^{14/3}+91 a^{12} b^3 x^5+\frac{4095}{16} a^{11} b^4 x^{16/3}+\frac{9009}{17} a^{10} b^5 x^{17/3}+\frac{5005}{6} a^9 b^6 x^6+\frac{19305}{19} a^8 b^7 x^{19/3}+\frac{3861}{4} a^7 b^8 x^{20/3}+715 a^6 b^9 x^7+\frac{819}{2} a^5 b^{10} x^{22/3}+\frac{4095}{23} a^4 b^{11} x^{23/3}+\frac{455}{8} a^3 b^{12} x^8+\frac{63}{5} a^2 b^{13} x^{25/3}+\frac{45}{13} a^{14} b x^{13/3}+\frac{a^{15} x^4}{4}+\frac{45}{26} a b^{14} x^{26/3}+\frac{b^{15} x^9}{9} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^15*x^3,x]

[Out]

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

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Maple [A]  time = 0.003, size = 168, normalized size = 0.7 \begin{align*}{\frac{{b}^{15}{x}^{9}}{9}}+{\frac{45\,a{b}^{14}}{26}{x}^{{\frac{26}{3}}}}+{\frac{63\,{a}^{2}{b}^{13}}{5}{x}^{{\frac{25}{3}}}}+{\frac{455\,{x}^{8}{a}^{3}{b}^{12}}{8}}+{\frac{4095\,{a}^{4}{b}^{11}}{23}{x}^{{\frac{23}{3}}}}+{\frac{819\,{a}^{5}{b}^{10}}{2}{x}^{{\frac{22}{3}}}}+715\,{a}^{6}{b}^{9}{x}^{7}+{\frac{3861\,{a}^{7}{b}^{8}}{4}{x}^{{\frac{20}{3}}}}+{\frac{19305\,{a}^{8}{b}^{7}}{19}{x}^{{\frac{19}{3}}}}+{\frac{5005\,{x}^{6}{a}^{9}{b}^{6}}{6}}+{\frac{9009\,{a}^{10}{b}^{5}}{17}{x}^{{\frac{17}{3}}}}+{\frac{4095\,{a}^{11}{b}^{4}}{16}{x}^{{\frac{16}{3}}}}+91\,{a}^{12}{b}^{3}{x}^{5}+{\frac{45\,{a}^{13}{b}^{2}}{2}{x}^{{\frac{14}{3}}}}+{\frac{45\,{a}^{14}b}{13}{x}^{{\frac{13}{3}}}}+{\frac{{x}^{4}{a}^{15}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.969073, size = 270, normalized size = 1.11 \begin{align*} \frac{{\left (b x^{\frac{1}{3}} + a\right )}^{27}}{9 \, b^{12}} - \frac{33 \,{\left (b x^{\frac{1}{3}} + a\right )}^{26} a}{26 \, b^{12}} + \frac{33 \,{\left (b x^{\frac{1}{3}} + a\right )}^{25} a^{2}}{5 \, b^{12}} - \frac{165 \,{\left (b x^{\frac{1}{3}} + a\right )}^{24} a^{3}}{8 \, b^{12}} + \frac{990 \,{\left (b x^{\frac{1}{3}} + a\right )}^{23} a^{4}}{23 \, b^{12}} - \frac{63 \,{\left (b x^{\frac{1}{3}} + a\right )}^{22} a^{5}}{b^{12}} + \frac{66 \,{\left (b x^{\frac{1}{3}} + a\right )}^{21} a^{6}}{b^{12}} - \frac{99 \,{\left (b x^{\frac{1}{3}} + a\right )}^{20} a^{7}}{2 \, b^{12}} + \frac{495 \,{\left (b x^{\frac{1}{3}} + a\right )}^{19} a^{8}}{19 \, b^{12}} - \frac{55 \,{\left (b x^{\frac{1}{3}} + a\right )}^{18} a^{9}}{6 \, b^{12}} + \frac{33 \,{\left (b x^{\frac{1}{3}} + a\right )}^{17} a^{10}}{17 \, b^{12}} - \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{16} a^{11}}{16 \, b^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*(b*x^(1/3) + a)^27/b^12 - 33/26*(b*x^(1/3) + a)^26*a/b^12 + 33/5*(b*x^(1/3) + a)^25*a^2/b^12 - 165/8*(b*x^
(1/3) + a)^24*a^3/b^12 + 990/23*(b*x^(1/3) + a)^23*a^4/b^12 - 63*(b*x^(1/3) + a)^22*a^5/b^12 + 66*(b*x^(1/3) +
a)^21*a^6/b^12 - 99/2*(b*x^(1/3) + a)^20*a^7/b^12 + 495/19*(b*x^(1/3) + a)^19*a^8/b^12 - 55/6*(b*x^(1/3) + a)
^18*a^9/b^12 + 33/17*(b*x^(1/3) + a)^17*a^10/b^12 - 3/16*(b*x^(1/3) + a)^16*a^11/b^12

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Fricas [A]  time = 1.50719, size = 478, normalized size = 1.96 \begin{align*} \frac{1}{9} \, b^{15} x^{9} + \frac{455}{8} \, a^{3} b^{12} x^{8} + 715 \, a^{6} b^{9} x^{7} + \frac{5005}{6} \, a^{9} b^{6} x^{6} + 91 \, a^{12} b^{3} x^{5} + \frac{1}{4} \, a^{15} x^{4} + \frac{9}{20332} \,{\left (3910 \, a b^{14} x^{8} + 402220 \, a^{4} b^{11} x^{7} + 2180607 \, a^{7} b^{8} x^{6} + 1197196 \, a^{10} b^{5} x^{5} + 50830 \, a^{13} b^{2} x^{4}\right )} x^{\frac{2}{3}} + \frac{9}{19760} \,{\left (27664 \, a^{2} b^{13} x^{8} + 899080 \, a^{5} b^{10} x^{7} + 2230800 \, a^{8} b^{7} x^{6} + 561925 \, a^{11} b^{4} x^{5} + 7600 \, a^{14} b x^{4}\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^3,x, algorithm="fricas")

[Out]

1/9*b^15*x^9 + 455/8*a^3*b^12*x^8 + 715*a^6*b^9*x^7 + 5005/6*a^9*b^6*x^6 + 91*a^12*b^3*x^5 + 1/4*a^15*x^4 + 9/
20332*(3910*a*b^14*x^8 + 402220*a^4*b^11*x^7 + 2180607*a^7*b^8*x^6 + 1197196*a^10*b^5*x^5 + 50830*a^13*b^2*x^4
)*x^(2/3) + 9/19760*(27664*a^2*b^13*x^8 + 899080*a^5*b^10*x^7 + 2230800*a^8*b^7*x^6 + 561925*a^11*b^4*x^5 + 76
00*a^14*b*x^4)*x^(1/3)

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Sympy [A]  time = 9.10894, size = 216, normalized size = 0.89 \begin{align*} \frac{a^{15} x^{4}}{4} + \frac{45 a^{14} b x^{\frac{13}{3}}}{13} + \frac{45 a^{13} b^{2} x^{\frac{14}{3}}}{2} + 91 a^{12} b^{3} x^{5} + \frac{4095 a^{11} b^{4} x^{\frac{16}{3}}}{16} + \frac{9009 a^{10} b^{5} x^{\frac{17}{3}}}{17} + \frac{5005 a^{9} b^{6} x^{6}}{6} + \frac{19305 a^{8} b^{7} x^{\frac{19}{3}}}{19} + \frac{3861 a^{7} b^{8} x^{\frac{20}{3}}}{4} + 715 a^{6} b^{9} x^{7} + \frac{819 a^{5} b^{10} x^{\frac{22}{3}}}{2} + \frac{4095 a^{4} b^{11} x^{\frac{23}{3}}}{23} + \frac{455 a^{3} b^{12} x^{8}}{8} + \frac{63 a^{2} b^{13} x^{\frac{25}{3}}}{5} + \frac{45 a b^{14} x^{\frac{26}{3}}}{26} + \frac{b^{15} x^{9}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.1921, size = 225, normalized size = 0.92 \begin{align*} \frac{1}{9} \, b^{15} x^{9} + \frac{45}{26} \, a b^{14} x^{\frac{26}{3}} + \frac{63}{5} \, a^{2} b^{13} x^{\frac{25}{3}} + \frac{455}{8} \, a^{3} b^{12} x^{8} + \frac{4095}{23} \, a^{4} b^{11} x^{\frac{23}{3}} + \frac{819}{2} \, a^{5} b^{10} x^{\frac{22}{3}} + 715 \, a^{6} b^{9} x^{7} + \frac{3861}{4} \, a^{7} b^{8} x^{\frac{20}{3}} + \frac{19305}{19} \, a^{8} b^{7} x^{\frac{19}{3}} + \frac{5005}{6} \, a^{9} b^{6} x^{6} + \frac{9009}{17} \, a^{10} b^{5} x^{\frac{17}{3}} + \frac{4095}{16} \, a^{11} b^{4} x^{\frac{16}{3}} + 91 \, a^{12} b^{3} x^{5} + \frac{45}{2} \, a^{13} b^{2} x^{\frac{14}{3}} + \frac{45}{13} \, a^{14} b x^{\frac{13}{3}} + \frac{1}{4} \, a^{15} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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