3.2326 \(\int (a+b \sqrt [3]{x})^{10} x^2 \, dx\)

Optimal. Leaf size=179 \[ \frac{84 a^2 \left (a+b \sqrt [3]{x}\right )^{17}}{17 b^9}-\frac{21 a^3 \left (a+b \sqrt [3]{x}\right )^{16}}{2 b^9}+\frac{14 a^4 \left (a+b \sqrt [3]{x}\right )^{15}}{b^9}-\frac{12 a^5 \left (a+b \sqrt [3]{x}\right )^{14}}{b^9}+\frac{84 a^6 \left (a+b \sqrt [3]{x}\right )^{13}}{13 b^9}-\frac{2 a^7 \left (a+b \sqrt [3]{x}\right )^{12}}{b^9}+\frac{3 a^8 \left (a+b \sqrt [3]{x}\right )^{11}}{11 b^9}+\frac{3 \left (a+b \sqrt [3]{x}\right )^{19}}{19 b^9}-\frac{4 a \left (a+b \sqrt [3]{x}\right )^{18}}{3 b^9} \]

[Out]

(3*a^8*(a + b*x^(1/3))^11)/(11*b^9) - (2*a^7*(a + b*x^(1/3))^12)/b^9 + (84*a^6*(a + b*x^(1/3))^13)/(13*b^9) -
(12*a^5*(a + b*x^(1/3))^14)/b^9 + (14*a^4*(a + b*x^(1/3))^15)/b^9 - (21*a^3*(a + b*x^(1/3))^16)/(2*b^9) + (84*
a^2*(a + b*x^(1/3))^17)/(17*b^9) - (4*a*(a + b*x^(1/3))^18)/(3*b^9) + (3*(a + b*x^(1/3))^19)/(19*b^9)

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Rubi [A]  time = 0.0919547, antiderivative size = 179, 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} \[ \frac{84 a^2 \left (a+b \sqrt [3]{x}\right )^{17}}{17 b^9}-\frac{21 a^3 \left (a+b \sqrt [3]{x}\right )^{16}}{2 b^9}+\frac{14 a^4 \left (a+b \sqrt [3]{x}\right )^{15}}{b^9}-\frac{12 a^5 \left (a+b \sqrt [3]{x}\right )^{14}}{b^9}+\frac{84 a^6 \left (a+b \sqrt [3]{x}\right )^{13}}{13 b^9}-\frac{2 a^7 \left (a+b \sqrt [3]{x}\right )^{12}}{b^9}+\frac{3 a^8 \left (a+b \sqrt [3]{x}\right )^{11}}{11 b^9}+\frac{3 \left (a+b \sqrt [3]{x}\right )^{19}}{19 b^9}-\frac{4 a \left (a+b \sqrt [3]{x}\right )^{18}}{3 b^9} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))^10*x^2,x]

[Out]

(3*a^8*(a + b*x^(1/3))^11)/(11*b^9) - (2*a^7*(a + b*x^(1/3))^12)/b^9 + (84*a^6*(a + b*x^(1/3))^13)/(13*b^9) -
(12*a^5*(a + b*x^(1/3))^14)/b^9 + (14*a^4*(a + b*x^(1/3))^15)/b^9 - (21*a^3*(a + b*x^(1/3))^16)/(2*b^9) + (84*
a^2*(a + b*x^(1/3))^17)/(17*b^9) - (4*a*(a + b*x^(1/3))^18)/(3*b^9) + (3*(a + b*x^(1/3))^19)/(19*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 )^{10} x^2 \, dx &=3 \operatorname{Subst}\left (\int x^8 (a+b x)^{10} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{a^8 (a+b x)^{10}}{b^8}-\frac{8 a^7 (a+b x)^{11}}{b^8}+\frac{28 a^6 (a+b x)^{12}}{b^8}-\frac{56 a^5 (a+b x)^{13}}{b^8}+\frac{70 a^4 (a+b x)^{14}}{b^8}-\frac{56 a^3 (a+b x)^{15}}{b^8}+\frac{28 a^2 (a+b x)^{16}}{b^8}-\frac{8 a (a+b x)^{17}}{b^8}+\frac{(a+b x)^{18}}{b^8}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{3 a^8 \left (a+b \sqrt [3]{x}\right )^{11}}{11 b^9}-\frac{2 a^7 \left (a+b \sqrt [3]{x}\right )^{12}}{b^9}+\frac{84 a^6 \left (a+b \sqrt [3]{x}\right )^{13}}{13 b^9}-\frac{12 a^5 \left (a+b \sqrt [3]{x}\right )^{14}}{b^9}+\frac{14 a^4 \left (a+b \sqrt [3]{x}\right )^{15}}{b^9}-\frac{21 a^3 \left (a+b \sqrt [3]{x}\right )^{16}}{2 b^9}+\frac{84 a^2 \left (a+b \sqrt [3]{x}\right )^{17}}{17 b^9}-\frac{4 a \left (a+b \sqrt [3]{x}\right )^{18}}{3 b^9}+\frac{3 \left (a+b \sqrt [3]{x}\right )^{19}}{19 b^9}\\ \end{align*}

Mathematica [A]  time = 0.051242, size = 140, normalized size = 0.78 \[ \frac{135}{11} a^8 b^2 x^{11/3}+30 a^7 b^3 x^4+\frac{630}{13} a^6 b^4 x^{13/3}+54 a^5 b^5 x^{14/3}+42 a^4 b^6 x^5+\frac{45}{2} a^3 b^7 x^{16/3}+\frac{135}{17} a^2 b^8 x^{17/3}+3 a^9 b x^{10/3}+\frac{a^{10} x^3}{3}+\frac{5}{3} a b^9 x^6+\frac{3}{19} b^{10} x^{19/3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^10*x^2,x]

[Out]

(a^10*x^3)/3 + 3*a^9*b*x^(10/3) + (135*a^8*b^2*x^(11/3))/11 + 30*a^7*b^3*x^4 + (630*a^6*b^4*x^(13/3))/13 + 54*
a^5*b^5*x^(14/3) + 42*a^4*b^6*x^5 + (45*a^3*b^7*x^(16/3))/2 + (135*a^2*b^8*x^(17/3))/17 + (5*a*b^9*x^6)/3 + (3
*b^10*x^(19/3))/19

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Maple [A]  time = 0.002, size = 113, normalized size = 0.6 \begin{align*}{\frac{3\,{b}^{10}}{19}{x}^{{\frac{19}{3}}}}+{\frac{5\,a{b}^{9}{x}^{6}}{3}}+{\frac{135\,{a}^{2}{b}^{8}}{17}{x}^{{\frac{17}{3}}}}+{\frac{45\,{a}^{3}{b}^{7}}{2}{x}^{{\frac{16}{3}}}}+42\,{a}^{4}{b}^{6}{x}^{5}+54\,{a}^{5}{b}^{5}{x}^{14/3}+{\frac{630\,{a}^{6}{b}^{4}}{13}{x}^{{\frac{13}{3}}}}+30\,{a}^{7}{b}^{3}{x}^{4}+{\frac{135\,{a}^{8}{b}^{2}}{11}{x}^{{\frac{11}{3}}}}+3\,{a}^{9}b{x}^{10/3}+{\frac{{x}^{3}{a}^{10}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/3))^10*x^2,x)

[Out]

3/19*b^10*x^(19/3)+5/3*a*b^9*x^6+135/17*a^2*b^8*x^(17/3)+45/2*a^3*b^7*x^(16/3)+42*a^4*b^6*x^5+54*a^5*b^5*x^(14
/3)+630/13*a^6*b^4*x^(13/3)+30*a^7*b^3*x^4+135/11*a^8*b^2*x^(11/3)+3*a^9*b*x^(10/3)+1/3*x^3*a^10

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Maxima [A]  time = 0.970359, size = 201, normalized size = 1.12 \begin{align*} \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{19}}{19 \, b^{9}} - \frac{4 \,{\left (b x^{\frac{1}{3}} + a\right )}^{18} a}{3 \, b^{9}} + \frac{84 \,{\left (b x^{\frac{1}{3}} + a\right )}^{17} a^{2}}{17 \, b^{9}} - \frac{21 \,{\left (b x^{\frac{1}{3}} + a\right )}^{16} a^{3}}{2 \, b^{9}} + \frac{14 \,{\left (b x^{\frac{1}{3}} + a\right )}^{15} a^{4}}{b^{9}} - \frac{12 \,{\left (b x^{\frac{1}{3}} + a\right )}^{14} a^{5}}{b^{9}} + \frac{84 \,{\left (b x^{\frac{1}{3}} + a\right )}^{13} a^{6}}{13 \, b^{9}} - \frac{2 \,{\left (b x^{\frac{1}{3}} + a\right )}^{12} a^{7}}{b^{9}} + \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{11} a^{8}}{11 \, b^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^10*x^2,x, algorithm="maxima")

[Out]

3/19*(b*x^(1/3) + a)^19/b^9 - 4/3*(b*x^(1/3) + a)^18*a/b^9 + 84/17*(b*x^(1/3) + a)^17*a^2/b^9 - 21/2*(b*x^(1/3
) + a)^16*a^3/b^9 + 14*(b*x^(1/3) + a)^15*a^4/b^9 - 12*(b*x^(1/3) + a)^14*a^5/b^9 + 84/13*(b*x^(1/3) + a)^13*a
^6/b^9 - 2*(b*x^(1/3) + a)^12*a^7/b^9 + 3/11*(b*x^(1/3) + a)^11*a^8/b^9

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Fricas [A]  time = 1.42361, size = 293, normalized size = 1.64 \begin{align*} \frac{5}{3} \, a b^{9} x^{6} + 42 \, a^{4} b^{6} x^{5} + 30 \, a^{7} b^{3} x^{4} + \frac{1}{3} \, a^{10} x^{3} + \frac{27}{187} \,{\left (55 \, a^{2} b^{8} x^{5} + 374 \, a^{5} b^{5} x^{4} + 85 \, a^{8} b^{2} x^{3}\right )} x^{\frac{2}{3}} + \frac{3}{494} \,{\left (26 \, b^{10} x^{6} + 3705 \, a^{3} b^{7} x^{5} + 7980 \, a^{6} b^{4} x^{4} + 494 \, a^{9} 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))^10*x^2,x, algorithm="fricas")

[Out]

5/3*a*b^9*x^6 + 42*a^4*b^6*x^5 + 30*a^7*b^3*x^4 + 1/3*a^10*x^3 + 27/187*(55*a^2*b^8*x^5 + 374*a^5*b^5*x^4 + 85
*a^8*b^2*x^3)*x^(2/3) + 3/494*(26*b^10*x^6 + 3705*a^3*b^7*x^5 + 7980*a^6*b^4*x^4 + 494*a^9*b*x^3)*x^(1/3)

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Sympy [A]  time = 3.9175, size = 141, normalized size = 0.79 \begin{align*} \frac{a^{10} x^{3}}{3} + 3 a^{9} b x^{\frac{10}{3}} + \frac{135 a^{8} b^{2} x^{\frac{11}{3}}}{11} + 30 a^{7} b^{3} x^{4} + \frac{630 a^{6} b^{4} x^{\frac{13}{3}}}{13} + 54 a^{5} b^{5} x^{\frac{14}{3}} + 42 a^{4} b^{6} x^{5} + \frac{45 a^{3} b^{7} x^{\frac{16}{3}}}{2} + \frac{135 a^{2} b^{8} x^{\frac{17}{3}}}{17} + \frac{5 a b^{9} x^{6}}{3} + \frac{3 b^{10} x^{\frac{19}{3}}}{19} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/3))**10*x**2,x)

[Out]

a**10*x**3/3 + 3*a**9*b*x**(10/3) + 135*a**8*b**2*x**(11/3)/11 + 30*a**7*b**3*x**4 + 630*a**6*b**4*x**(13/3)/1
3 + 54*a**5*b**5*x**(14/3) + 42*a**4*b**6*x**5 + 45*a**3*b**7*x**(16/3)/2 + 135*a**2*b**8*x**(17/3)/17 + 5*a*b
**9*x**6/3 + 3*b**10*x**(19/3)/19

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Giac [A]  time = 1.21116, size = 151, normalized size = 0.84 \begin{align*} \frac{3}{19} \, b^{10} x^{\frac{19}{3}} + \frac{5}{3} \, a b^{9} x^{6} + \frac{135}{17} \, a^{2} b^{8} x^{\frac{17}{3}} + \frac{45}{2} \, a^{3} b^{7} x^{\frac{16}{3}} + 42 \, a^{4} b^{6} x^{5} + 54 \, a^{5} b^{5} x^{\frac{14}{3}} + \frac{630}{13} \, a^{6} b^{4} x^{\frac{13}{3}} + 30 \, a^{7} b^{3} x^{4} + \frac{135}{11} \, a^{8} b^{2} x^{\frac{11}{3}} + 3 \, a^{9} b x^{\frac{10}{3}} + \frac{1}{3} \, a^{10} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^10*x^2,x, algorithm="giac")

[Out]

3/19*b^10*x^(19/3) + 5/3*a*b^9*x^6 + 135/17*a^2*b^8*x^(17/3) + 45/2*a^3*b^7*x^(16/3) + 42*a^4*b^6*x^5 + 54*a^5
*b^5*x^(14/3) + 630/13*a^6*b^4*x^(13/3) + 30*a^7*b^3*x^4 + 135/11*a^8*b^2*x^(11/3) + 3*a^9*b*x^(10/3) + 1/3*a^
10*x^3