3.2342 \(\int (a+b \sqrt [3]{x})^{15} x^2 \, dx\)

Optimal. Leaf size=183 \[ \frac{42 a^2 \left (a+b \sqrt [3]{x}\right )^{22}}{11 b^9}-\frac{8 a^3 \left (a+b \sqrt [3]{x}\right )^{21}}{b^9}+\frac{21 a^4 \left (a+b \sqrt [3]{x}\right )^{20}}{2 b^9}-\frac{168 a^5 \left (a+b \sqrt [3]{x}\right )^{19}}{19 b^9}+\frac{14 a^6 \left (a+b \sqrt [3]{x}\right )^{18}}{3 b^9}-\frac{24 a^7 \left (a+b \sqrt [3]{x}\right )^{17}}{17 b^9}+\frac{3 a^8 \left (a+b \sqrt [3]{x}\right )^{16}}{16 b^9}+\frac{\left (a+b \sqrt [3]{x}\right )^{24}}{8 b^9}-\frac{24 a \left (a+b \sqrt [3]{x}\right )^{23}}{23 b^9} \]

[Out]

(3*a^8*(a + b*x^(1/3))^16)/(16*b^9) - (24*a^7*(a + b*x^(1/3))^17)/(17*b^9) + (14*a^6*(a + b*x^(1/3))^18)/(3*b^
9) - (168*a^5*(a + b*x^(1/3))^19)/(19*b^9) + (21*a^4*(a + b*x^(1/3))^20)/(2*b^9) - (8*a^3*(a + b*x^(1/3))^21)/
b^9 + (42*a^2*(a + b*x^(1/3))^22)/(11*b^9) - (24*a*(a + b*x^(1/3))^23)/(23*b^9) + (a + b*x^(1/3))^24/(8*b^9)

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Rubi [A]  time = 0.100791, antiderivative size = 183, 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} \[ \frac{42 a^2 \left (a+b \sqrt [3]{x}\right )^{22}}{11 b^9}-\frac{8 a^3 \left (a+b \sqrt [3]{x}\right )^{21}}{b^9}+\frac{21 a^4 \left (a+b \sqrt [3]{x}\right )^{20}}{2 b^9}-\frac{168 a^5 \left (a+b \sqrt [3]{x}\right )^{19}}{19 b^9}+\frac{14 a^6 \left (a+b \sqrt [3]{x}\right )^{18}}{3 b^9}-\frac{24 a^7 \left (a+b \sqrt [3]{x}\right )^{17}}{17 b^9}+\frac{3 a^8 \left (a+b \sqrt [3]{x}\right )^{16}}{16 b^9}+\frac{\left (a+b \sqrt [3]{x}\right )^{24}}{8 b^9}-\frac{24 a \left (a+b \sqrt [3]{x}\right )^{23}}{23 b^9} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))^15*x^2,x]

[Out]

(3*a^8*(a + b*x^(1/3))^16)/(16*b^9) - (24*a^7*(a + b*x^(1/3))^17)/(17*b^9) + (14*a^6*(a + b*x^(1/3))^18)/(3*b^
9) - (168*a^5*(a + b*x^(1/3))^19)/(19*b^9) + (21*a^4*(a + b*x^(1/3))^20)/(2*b^9) - (8*a^3*(a + b*x^(1/3))^21)/
b^9 + (42*a^2*(a + b*x^(1/3))^22)/(11*b^9) - (24*a*(a + b*x^(1/3))^23)/(23*b^9) + (a + b*x^(1/3))^24/(8*b^9)

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^2 \, dx &=3 \operatorname{Subst}\left (\int x^8 (a+b x)^{15} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{a^8 (a+b x)^{15}}{b^8}-\frac{8 a^7 (a+b x)^{16}}{b^8}+\frac{28 a^6 (a+b x)^{17}}{b^8}-\frac{56 a^5 (a+b x)^{18}}{b^8}+\frac{70 a^4 (a+b x)^{19}}{b^8}-\frac{56 a^3 (a+b x)^{20}}{b^8}+\frac{28 a^2 (a+b x)^{21}}{b^8}-\frac{8 a (a+b x)^{22}}{b^8}+\frac{(a+b x)^{23}}{b^8}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{3 a^8 \left (a+b \sqrt [3]{x}\right )^{16}}{16 b^9}-\frac{24 a^7 \left (a+b \sqrt [3]{x}\right )^{17}}{17 b^9}+\frac{14 a^6 \left (a+b \sqrt [3]{x}\right )^{18}}{3 b^9}-\frac{168 a^5 \left (a+b \sqrt [3]{x}\right )^{19}}{19 b^9}+\frac{21 a^4 \left (a+b \sqrt [3]{x}\right )^{20}}{2 b^9}-\frac{8 a^3 \left (a+b \sqrt [3]{x}\right )^{21}}{b^9}+\frac{42 a^2 \left (a+b \sqrt [3]{x}\right )^{22}}{11 b^9}-\frac{24 a \left (a+b \sqrt [3]{x}\right )^{23}}{23 b^9}+\frac{\left (a+b \sqrt [3]{x}\right )^{24}}{8 b^9}\\ \end{align*}

Mathematica [A]  time = 0.0773708, size = 113, normalized size = 0.62 \[ \frac{\left (a+b \sqrt [3]{x}\right )^{16} \left (136 a^6 b^2 x^{2/3}+3876 a^4 b^4 x^{4/3}-15504 a^3 b^5 x^{5/3}+54264 a^2 b^6 x^2-816 a^5 b^3 x-16 a^7 b \sqrt [3]{x}+a^8-170544 a b^7 x^{7/3}+490314 b^8 x^{8/3}\right )}{3922512 b^9} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^15*x^2,x]

[Out]

((a + b*x^(1/3))^16*(a^8 - 16*a^7*b*x^(1/3) + 136*a^6*b^2*x^(2/3) - 816*a^5*b^3*x + 3876*a^4*b^4*x^(4/3) - 155
04*a^3*b^5*x^(5/3) + 54264*a^2*b^6*x^2 - 170544*a*b^7*x^(7/3) + 490314*b^8*x^(8/3)))/(3922512*b^9)

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Maple [A]  time = 0.004, size = 168, normalized size = 0.9 \begin{align*}{\frac{{b}^{15}{x}^{8}}{8}}+{\frac{45\,a{b}^{14}}{23}{x}^{{\frac{23}{3}}}}+{\frac{315\,{a}^{2}{b}^{13}}{22}{x}^{{\frac{22}{3}}}}+65\,{x}^{7}{a}^{3}{b}^{12}+{\frac{819\,{a}^{4}{b}^{11}}{4}{x}^{{\frac{20}{3}}}}+{\frac{9009\,{a}^{5}{b}^{10}}{19}{x}^{{\frac{19}{3}}}}+{\frac{5005\,{a}^{6}{b}^{9}{x}^{6}}{6}}+{\frac{19305\,{a}^{7}{b}^{8}}{17}{x}^{{\frac{17}{3}}}}+{\frac{19305\,{a}^{8}{b}^{7}}{16}{x}^{{\frac{16}{3}}}}+1001\,{x}^{5}{a}^{9}{b}^{6}+{\frac{1287\,{a}^{10}{b}^{5}}{2}{x}^{{\frac{14}{3}}}}+315\,{a}^{11}{b}^{4}{x}^{13/3}+{\frac{455\,{a}^{12}{b}^{3}{x}^{4}}{4}}+{\frac{315\,{a}^{13}{b}^{2}}{11}{x}^{{\frac{11}{3}}}}+{\frac{9\,{a}^{14}b}{2}{x}^{{\frac{10}{3}}}}+{\frac{{x}^{3}{a}^{15}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/3))^15*x^2,x)

[Out]

1/8*b^15*x^8+45/23*a*b^14*x^(23/3)+315/22*a^2*b^13*x^(22/3)+65*x^7*a^3*b^12+819/4*a^4*b^11*x^(20/3)+9009/19*a^
5*b^10*x^(19/3)+5005/6*a^6*b^9*x^6+19305/17*a^7*b^8*x^(17/3)+19305/16*a^8*b^7*x^(16/3)+1001*x^5*a^9*b^6+1287/2
*a^10*b^5*x^(14/3)+315*a^11*b^4*x^(13/3)+455/4*a^12*b^3*x^4+315/11*a^13*b^2*x^(11/3)+9/2*a^14*b*x^(10/3)+1/3*x
^3*a^15

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Maxima [A]  time = 0.974624, size = 201, normalized size = 1.1 \begin{align*} \frac{{\left (b x^{\frac{1}{3}} + a\right )}^{24}}{8 \, b^{9}} - \frac{24 \,{\left (b x^{\frac{1}{3}} + a\right )}^{23} a}{23 \, b^{9}} + \frac{42 \,{\left (b x^{\frac{1}{3}} + a\right )}^{22} a^{2}}{11 \, b^{9}} - \frac{8 \,{\left (b x^{\frac{1}{3}} + a\right )}^{21} a^{3}}{b^{9}} + \frac{21 \,{\left (b x^{\frac{1}{3}} + a\right )}^{20} a^{4}}{2 \, b^{9}} - \frac{168 \,{\left (b x^{\frac{1}{3}} + a\right )}^{19} a^{5}}{19 \, b^{9}} + \frac{14 \,{\left (b x^{\frac{1}{3}} + a\right )}^{18} a^{6}}{3 \, b^{9}} - \frac{24 \,{\left (b x^{\frac{1}{3}} + a\right )}^{17} a^{7}}{17 \, b^{9}} + \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{16} a^{8}}{16 \, b^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^15*x^2,x, algorithm="maxima")

[Out]

1/8*(b*x^(1/3) + a)^24/b^9 - 24/23*(b*x^(1/3) + a)^23*a/b^9 + 42/11*(b*x^(1/3) + a)^22*a^2/b^9 - 8*(b*x^(1/3)
+ a)^21*a^3/b^9 + 21/2*(b*x^(1/3) + a)^20*a^4/b^9 - 168/19*(b*x^(1/3) + a)^19*a^5/b^9 + 14/3*(b*x^(1/3) + a)^1
8*a^6/b^9 - 24/17*(b*x^(1/3) + a)^17*a^7/b^9 + 3/16*(b*x^(1/3) + a)^16*a^8/b^9

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Fricas [A]  time = 1.5263, size = 475, normalized size = 2.6 \begin{align*} \frac{1}{8} \, b^{15} x^{8} + 65 \, a^{3} b^{12} x^{7} + \frac{5005}{6} \, a^{6} b^{9} x^{6} + 1001 \, a^{9} b^{6} x^{5} + \frac{455}{4} \, a^{12} b^{3} x^{4} + \frac{1}{3} \, a^{15} x^{3} + \frac{9}{17204} \,{\left (3740 \, a b^{14} x^{7} + 391391 \, a^{4} b^{11} x^{6} + 2170740 \, a^{7} b^{8} x^{5} + 1230086 \, a^{10} b^{5} x^{4} + 54740 \, a^{13} b^{2} x^{3}\right )} x^{\frac{2}{3}} + \frac{9}{3344} \,{\left (5320 \, a^{2} b^{13} x^{7} + 176176 \, a^{5} b^{10} x^{6} + 448305 \, a^{8} b^{7} x^{5} + 117040 \, a^{11} b^{4} x^{4} + 1672 \, a^{14} b x^{3}\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^2,x, algorithm="fricas")

[Out]

1/8*b^15*x^8 + 65*a^3*b^12*x^7 + 5005/6*a^6*b^9*x^6 + 1001*a^9*b^6*x^5 + 455/4*a^12*b^3*x^4 + 1/3*a^15*x^3 + 9
/17204*(3740*a*b^14*x^7 + 391391*a^4*b^11*x^6 + 2170740*a^7*b^8*x^5 + 1230086*a^10*b^5*x^4 + 54740*a^13*b^2*x^
3)*x^(2/3) + 9/3344*(5320*a^2*b^13*x^7 + 176176*a^5*b^10*x^6 + 448305*a^8*b^7*x^5 + 117040*a^11*b^4*x^4 + 1672
*a^14*b*x^3)*x^(1/3)

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Sympy [A]  time = 7.01117, size = 214, normalized size = 1.17 \begin{align*} \frac{a^{15} x^{3}}{3} + \frac{9 a^{14} b x^{\frac{10}{3}}}{2} + \frac{315 a^{13} b^{2} x^{\frac{11}{3}}}{11} + \frac{455 a^{12} b^{3} x^{4}}{4} + 315 a^{11} b^{4} x^{\frac{13}{3}} + \frac{1287 a^{10} b^{5} x^{\frac{14}{3}}}{2} + 1001 a^{9} b^{6} x^{5} + \frac{19305 a^{8} b^{7} x^{\frac{16}{3}}}{16} + \frac{19305 a^{7} b^{8} x^{\frac{17}{3}}}{17} + \frac{5005 a^{6} b^{9} x^{6}}{6} + \frac{9009 a^{5} b^{10} x^{\frac{19}{3}}}{19} + \frac{819 a^{4} b^{11} x^{\frac{20}{3}}}{4} + 65 a^{3} b^{12} x^{7} + \frac{315 a^{2} b^{13} x^{\frac{22}{3}}}{22} + \frac{45 a b^{14} x^{\frac{23}{3}}}{23} + \frac{b^{15} x^{8}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/3))**15*x**2,x)

[Out]

a**15*x**3/3 + 9*a**14*b*x**(10/3)/2 + 315*a**13*b**2*x**(11/3)/11 + 455*a**12*b**3*x**4/4 + 315*a**11*b**4*x*
*(13/3) + 1287*a**10*b**5*x**(14/3)/2 + 1001*a**9*b**6*x**5 + 19305*a**8*b**7*x**(16/3)/16 + 19305*a**7*b**8*x
**(17/3)/17 + 5005*a**6*b**9*x**6/6 + 9009*a**5*b**10*x**(19/3)/19 + 819*a**4*b**11*x**(20/3)/4 + 65*a**3*b**1
2*x**7 + 315*a**2*b**13*x**(22/3)/22 + 45*a*b**14*x**(23/3)/23 + b**15*x**8/8

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Giac [A]  time = 1.17352, size = 225, normalized size = 1.23 \begin{align*} \frac{1}{8} \, b^{15} x^{8} + \frac{45}{23} \, a b^{14} x^{\frac{23}{3}} + \frac{315}{22} \, a^{2} b^{13} x^{\frac{22}{3}} + 65 \, a^{3} b^{12} x^{7} + \frac{819}{4} \, a^{4} b^{11} x^{\frac{20}{3}} + \frac{9009}{19} \, a^{5} b^{10} x^{\frac{19}{3}} + \frac{5005}{6} \, a^{6} b^{9} x^{6} + \frac{19305}{17} \, a^{7} b^{8} x^{\frac{17}{3}} + \frac{19305}{16} \, a^{8} b^{7} x^{\frac{16}{3}} + 1001 \, a^{9} b^{6} x^{5} + \frac{1287}{2} \, a^{10} b^{5} x^{\frac{14}{3}} + 315 \, a^{11} b^{4} x^{\frac{13}{3}} + \frac{455}{4} \, a^{12} b^{3} x^{4} + \frac{315}{11} \, a^{13} b^{2} x^{\frac{11}{3}} + \frac{9}{2} \, a^{14} b x^{\frac{10}{3}} + \frac{1}{3} \, a^{15} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^15*x^2,x, algorithm="giac")

[Out]

1/8*b^15*x^8 + 45/23*a*b^14*x^(23/3) + 315/22*a^2*b^13*x^(22/3) + 65*a^3*b^12*x^7 + 819/4*a^4*b^11*x^(20/3) +
9009/19*a^5*b^10*x^(19/3) + 5005/6*a^6*b^9*x^6 + 19305/17*a^7*b^8*x^(17/3) + 19305/16*a^8*b^7*x^(16/3) + 1001*
a^9*b^6*x^5 + 1287/2*a^10*b^5*x^(14/3) + 315*a^11*b^4*x^(13/3) + 455/4*a^12*b^3*x^4 + 315/11*a^13*b^2*x^(11/3)
+ 9/2*a^14*b*x^(10/3) + 1/3*a^15*x^3