3.2343 \(\int (a+b \sqrt [3]{x})^{15} x \, dx\)

Optimal. Leaf size=122 \[ \frac{30 a^2 \left (a+b \sqrt [3]{x}\right )^{19}}{19 b^6}-\frac{5 a^3 \left (a+b \sqrt [3]{x}\right )^{18}}{3 b^6}+\frac{15 a^4 \left (a+b \sqrt [3]{x}\right )^{17}}{17 b^6}-\frac{3 a^5 \left (a+b \sqrt [3]{x}\right )^{16}}{16 b^6}+\frac{\left (a+b \sqrt [3]{x}\right )^{21}}{7 b^6}-\frac{3 a \left (a+b \sqrt [3]{x}\right )^{20}}{4 b^6} \]

[Out]

(-3*a^5*(a + b*x^(1/3))^16)/(16*b^6) + (15*a^4*(a + b*x^(1/3))^17)/(17*b^6) - (5*a^3*(a + b*x^(1/3))^18)/(3*b^
6) + (30*a^2*(a + b*x^(1/3))^19)/(19*b^6) - (3*a*(a + b*x^(1/3))^20)/(4*b^6) + (a + b*x^(1/3))^21/(7*b^6)

________________________________________________________________________________________

Rubi [A]  time = 0.075441, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{30 a^2 \left (a+b \sqrt [3]{x}\right )^{19}}{19 b^6}-\frac{5 a^3 \left (a+b \sqrt [3]{x}\right )^{18}}{3 b^6}+\frac{15 a^4 \left (a+b \sqrt [3]{x}\right )^{17}}{17 b^6}-\frac{3 a^5 \left (a+b \sqrt [3]{x}\right )^{16}}{16 b^6}+\frac{\left (a+b \sqrt [3]{x}\right )^{21}}{7 b^6}-\frac{3 a \left (a+b \sqrt [3]{x}\right )^{20}}{4 b^6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))^15*x,x]

[Out]

(-3*a^5*(a + b*x^(1/3))^16)/(16*b^6) + (15*a^4*(a + b*x^(1/3))^17)/(17*b^6) - (5*a^3*(a + b*x^(1/3))^18)/(3*b^
6) + (30*a^2*(a + b*x^(1/3))^19)/(19*b^6) - (3*a*(a + b*x^(1/3))^20)/(4*b^6) + (a + b*x^(1/3))^21/(7*b^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 \left (a+b \sqrt [3]{x}\right )^{15} x \, dx &=3 \operatorname{Subst}\left (\int x^5 (a+b x)^{15} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (-\frac{a^5 (a+b x)^{15}}{b^5}+\frac{5 a^4 (a+b x)^{16}}{b^5}-\frac{10 a^3 (a+b x)^{17}}{b^5}+\frac{10 a^2 (a+b x)^{18}}{b^5}-\frac{5 a (a+b x)^{19}}{b^5}+\frac{(a+b x)^{20}}{b^5}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{3 a^5 \left (a+b \sqrt [3]{x}\right )^{16}}{16 b^6}+\frac{15 a^4 \left (a+b \sqrt [3]{x}\right )^{17}}{17 b^6}-\frac{5 a^3 \left (a+b \sqrt [3]{x}\right )^{18}}{3 b^6}+\frac{30 a^2 \left (a+b \sqrt [3]{x}\right )^{19}}{19 b^6}-\frac{3 a \left (a+b \sqrt [3]{x}\right )^{20}}{4 b^6}+\frac{\left (a+b \sqrt [3]{x}\right )^{21}}{7 b^6}\\ \end{align*}

Mathematica [A]  time = 0.0610548, size = 76, normalized size = 0.62 \[ -\frac{\left (a+b \sqrt [3]{x}\right )^{16} \left (136 a^3 b^2 x^{2/3}-816 a^2 b^3 x-16 a^4 b \sqrt [3]{x}+a^5+3876 a b^4 x^{4/3}-15504 b^5 x^{5/3}\right )}{108528 b^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^15*x,x]

[Out]

-((a + b*x^(1/3))^16*(a^5 - 16*a^4*b*x^(1/3) + 136*a^3*b^2*x^(2/3) - 816*a^2*b^3*x + 3876*a*b^4*x^(4/3) - 1550
4*b^5*x^(5/3)))/(108528*b^6)

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 168, normalized size = 1.4 \begin{align*}{\frac{{b}^{15}{x}^{7}}{7}}+{\frac{9\,a{b}^{14}}{4}{x}^{{\frac{20}{3}}}}+{\frac{315\,{a}^{2}{b}^{13}}{19}{x}^{{\frac{19}{3}}}}+{\frac{455\,{a}^{3}{b}^{12}{x}^{6}}{6}}+{\frac{4095\,{a}^{4}{b}^{11}}{17}{x}^{{\frac{17}{3}}}}+{\frac{9009\,{a}^{5}{b}^{10}}{16}{x}^{{\frac{16}{3}}}}+1001\,{a}^{6}{b}^{9}{x}^{5}+{\frac{19305\,{a}^{7}{b}^{8}}{14}{x}^{{\frac{14}{3}}}}+1485\,{a}^{8}{b}^{7}{x}^{13/3}+{\frac{5005\,{x}^{4}{a}^{9}{b}^{6}}{4}}+819\,{a}^{10}{b}^{5}{x}^{11/3}+{\frac{819\,{a}^{11}{b}^{4}}{2}{x}^{{\frac{10}{3}}}}+{\frac{455\,{a}^{12}{b}^{3}{x}^{3}}{3}}+{\frac{315\,{a}^{13}{b}^{2}}{8}{x}^{{\frac{8}{3}}}}+{\frac{45\,{a}^{14}b}{7}{x}^{{\frac{7}{3}}}}+{\frac{{x}^{2}{a}^{15}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*b^15*x^7+9/4*a*b^14*x^(20/3)+315/19*a^2*b^13*x^(19/3)+455/6*a^3*b^12*x^6+4095/17*a^4*b^11*x^(17/3)+9009/16
*a^5*b^10*x^(16/3)+1001*a^6*b^9*x^5+19305/14*a^7*b^8*x^(14/3)+1485*a^8*b^7*x^(13/3)+5005/4*x^4*a^9*b^6+819*a^1
0*b^5*x^(11/3)+819/2*a^11*b^4*x^(10/3)+455/3*a^12*b^3*x^3+315/8*x^(8/3)*a^13*b^2+45/7*a^14*b*x^(7/3)+1/2*x^2*a
^15

________________________________________________________________________________________

Maxima [A]  time = 0.973937, size = 132, normalized size = 1.08 \begin{align*} \frac{{\left (b x^{\frac{1}{3}} + a\right )}^{21}}{7 \, b^{6}} - \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{20} a}{4 \, b^{6}} + \frac{30 \,{\left (b x^{\frac{1}{3}} + a\right )}^{19} a^{2}}{19 \, b^{6}} - \frac{5 \,{\left (b x^{\frac{1}{3}} + a\right )}^{18} a^{3}}{3 \, b^{6}} + \frac{15 \,{\left (b x^{\frac{1}{3}} + a\right )}^{17} a^{4}}{17 \, b^{6}} - \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{16} a^{5}}{16 \, b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*(b*x^(1/3) + a)^21/b^6 - 3/4*(b*x^(1/3) + a)^20*a/b^6 + 30/19*(b*x^(1/3) + a)^19*a^2/b^6 - 5/3*(b*x^(1/3)
+ a)^18*a^3/b^6 + 15/17*(b*x^(1/3) + a)^17*a^4/b^6 - 3/16*(b*x^(1/3) + a)^16*a^5/b^6

________________________________________________________________________________________

Fricas [A]  time = 1.39788, size = 467, normalized size = 3.83 \begin{align*} \frac{1}{7} \, b^{15} x^{7} + \frac{455}{6} \, a^{3} b^{12} x^{6} + 1001 \, a^{6} b^{9} x^{5} + \frac{5005}{4} \, a^{9} b^{6} x^{4} + \frac{455}{3} \, a^{12} b^{3} x^{3} + \frac{1}{2} \, a^{15} x^{2} + \frac{9}{952} \,{\left (238 \, a b^{14} x^{6} + 25480 \, a^{4} b^{11} x^{5} + 145860 \, a^{7} b^{8} x^{4} + 86632 \, a^{10} b^{5} x^{3} + 4165 \, a^{13} b^{2} x^{2}\right )} x^{\frac{2}{3}} + \frac{9}{2128} \,{\left (3920 \, a^{2} b^{13} x^{6} + 133133 \, a^{5} b^{10} x^{5} + 351120 \, a^{8} b^{7} x^{4} + 96824 \, a^{11} b^{4} x^{3} + 1520 \, a^{14} b x^{2}\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,x, algorithm="fricas")

[Out]

1/7*b^15*x^7 + 455/6*a^3*b^12*x^6 + 1001*a^6*b^9*x^5 + 5005/4*a^9*b^6*x^4 + 455/3*a^12*b^3*x^3 + 1/2*a^15*x^2
+ 9/952*(238*a*b^14*x^6 + 25480*a^4*b^11*x^5 + 145860*a^7*b^8*x^4 + 86632*a^10*b^5*x^3 + 4165*a^13*b^2*x^2)*x^
(2/3) + 9/2128*(3920*a^2*b^13*x^6 + 133133*a^5*b^10*x^5 + 351120*a^8*b^7*x^4 + 96824*a^11*b^4*x^3 + 1520*a^14*
b*x^2)*x^(1/3)

________________________________________________________________________________________

Sympy [A]  time = 5.23246, size = 214, normalized size = 1.75 \begin{align*} \frac{a^{15} x^{2}}{2} + \frac{45 a^{14} b x^{\frac{7}{3}}}{7} + \frac{315 a^{13} b^{2} x^{\frac{8}{3}}}{8} + \frac{455 a^{12} b^{3} x^{3}}{3} + \frac{819 a^{11} b^{4} x^{\frac{10}{3}}}{2} + 819 a^{10} b^{5} x^{\frac{11}{3}} + \frac{5005 a^{9} b^{6} x^{4}}{4} + 1485 a^{8} b^{7} x^{\frac{13}{3}} + \frac{19305 a^{7} b^{8} x^{\frac{14}{3}}}{14} + 1001 a^{6} b^{9} x^{5} + \frac{9009 a^{5} b^{10} x^{\frac{16}{3}}}{16} + \frac{4095 a^{4} b^{11} x^{\frac{17}{3}}}{17} + \frac{455 a^{3} b^{12} x^{6}}{6} + \frac{315 a^{2} b^{13} x^{\frac{19}{3}}}{19} + \frac{9 a b^{14} x^{\frac{20}{3}}}{4} + \frac{b^{15} x^{7}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**15*x**2/2 + 45*a**14*b*x**(7/3)/7 + 315*a**13*b**2*x**(8/3)/8 + 455*a**12*b**3*x**3/3 + 819*a**11*b**4*x**(
10/3)/2 + 819*a**10*b**5*x**(11/3) + 5005*a**9*b**6*x**4/4 + 1485*a**8*b**7*x**(13/3) + 19305*a**7*b**8*x**(14
/3)/14 + 1001*a**6*b**9*x**5 + 9009*a**5*b**10*x**(16/3)/16 + 4095*a**4*b**11*x**(17/3)/17 + 455*a**3*b**12*x*
*6/6 + 315*a**2*b**13*x**(19/3)/19 + 9*a*b**14*x**(20/3)/4 + b**15*x**7/7

________________________________________________________________________________________

Giac [A]  time = 1.16061, size = 225, normalized size = 1.84 \begin{align*} \frac{1}{7} \, b^{15} x^{7} + \frac{9}{4} \, a b^{14} x^{\frac{20}{3}} + \frac{315}{19} \, a^{2} b^{13} x^{\frac{19}{3}} + \frac{455}{6} \, a^{3} b^{12} x^{6} + \frac{4095}{17} \, a^{4} b^{11} x^{\frac{17}{3}} + \frac{9009}{16} \, a^{5} b^{10} x^{\frac{16}{3}} + 1001 \, a^{6} b^{9} x^{5} + \frac{19305}{14} \, a^{7} b^{8} x^{\frac{14}{3}} + 1485 \, a^{8} b^{7} x^{\frac{13}{3}} + \frac{5005}{4} \, a^{9} b^{6} x^{4} + 819 \, a^{10} b^{5} x^{\frac{11}{3}} + \frac{819}{2} \, a^{11} b^{4} x^{\frac{10}{3}} + \frac{455}{3} \, a^{12} b^{3} x^{3} + \frac{315}{8} \, a^{13} b^{2} x^{\frac{8}{3}} + \frac{45}{7} \, a^{14} b x^{\frac{7}{3}} + \frac{1}{2} \, a^{15} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*b^15*x^7 + 9/4*a*b^14*x^(20/3) + 315/19*a^2*b^13*x^(19/3) + 455/6*a^3*b^12*x^6 + 4095/17*a^4*b^11*x^(17/3)
+ 9009/16*a^5*b^10*x^(16/3) + 1001*a^6*b^9*x^5 + 19305/14*a^7*b^8*x^(14/3) + 1485*a^8*b^7*x^(13/3) + 5005/4*a
^9*b^6*x^4 + 819*a^10*b^5*x^(11/3) + 819/2*a^11*b^4*x^(10/3) + 455/3*a^12*b^3*x^3 + 315/8*a^13*b^2*x^(8/3) + 4
5/7*a^14*b*x^(7/3) + 1/2*a^15*x^2