3.2324 \(\int (a+b \sqrt [3]{x})^{10} x^4 \, dx\)

Optimal. Leaf size=144 \[ \frac{135}{17} a^8 b^2 x^{17/3}+20 a^7 b^3 x^6+\frac{630}{19} a^6 b^4 x^{19/3}+\frac{189}{5} a^5 b^5 x^{20/3}+30 a^4 b^6 x^7+\frac{180}{11} a^3 b^7 x^{22/3}+\frac{135}{23} a^2 b^8 x^{23/3}+\frac{15}{8} a^9 b x^{16/3}+\frac{a^{10} x^5}{5}+\frac{5}{4} a b^9 x^8+\frac{3}{25} b^{10} x^{25/3} \]

[Out]

(a^10*x^5)/5 + (15*a^9*b*x^(16/3))/8 + (135*a^8*b^2*x^(17/3))/17 + 20*a^7*b^3*x^6 + (630*a^6*b^4*x^(19/3))/19
+ (189*a^5*b^5*x^(20/3))/5 + 30*a^4*b^6*x^7 + (180*a^3*b^7*x^(22/3))/11 + (135*a^2*b^8*x^(23/3))/23 + (5*a*b^9
*x^8)/4 + (3*b^10*x^(25/3))/25

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Rubi [A]  time = 0.0952417, antiderivative size = 144, 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} \[ \frac{135}{17} a^8 b^2 x^{17/3}+20 a^7 b^3 x^6+\frac{630}{19} a^6 b^4 x^{19/3}+\frac{189}{5} a^5 b^5 x^{20/3}+30 a^4 b^6 x^7+\frac{180}{11} a^3 b^7 x^{22/3}+\frac{135}{23} a^2 b^8 x^{23/3}+\frac{15}{8} a^9 b x^{16/3}+\frac{a^{10} x^5}{5}+\frac{5}{4} a b^9 x^8+\frac{3}{25} b^{10} x^{25/3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))^10*x^4,x]

[Out]

(a^10*x^5)/5 + (15*a^9*b*x^(16/3))/8 + (135*a^8*b^2*x^(17/3))/17 + 20*a^7*b^3*x^6 + (630*a^6*b^4*x^(19/3))/19
+ (189*a^5*b^5*x^(20/3))/5 + 30*a^4*b^6*x^7 + (180*a^3*b^7*x^(22/3))/11 + (135*a^2*b^8*x^(23/3))/23 + (5*a*b^9
*x^8)/4 + (3*b^10*x^(25/3))/25

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^4 \, dx &=3 \operatorname{Subst}\left (\int x^{14} (a+b x)^{10} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (a^{10} x^{14}+10 a^9 b x^{15}+45 a^8 b^2 x^{16}+120 a^7 b^3 x^{17}+210 a^6 b^4 x^{18}+252 a^5 b^5 x^{19}+210 a^4 b^6 x^{20}+120 a^3 b^7 x^{21}+45 a^2 b^8 x^{22}+10 a b^9 x^{23}+b^{10} x^{24}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{a^{10} x^5}{5}+\frac{15}{8} a^9 b x^{16/3}+\frac{135}{17} a^8 b^2 x^{17/3}+20 a^7 b^3 x^6+\frac{630}{19} a^6 b^4 x^{19/3}+\frac{189}{5} a^5 b^5 x^{20/3}+30 a^4 b^6 x^7+\frac{180}{11} a^3 b^7 x^{22/3}+\frac{135}{23} a^2 b^8 x^{23/3}+\frac{5}{4} a b^9 x^8+\frac{3}{25} b^{10} x^{25/3}\\ \end{align*}

Mathematica [A]  time = 0.0722272, size = 144, normalized size = 1. \[ \frac{135}{17} a^8 b^2 x^{17/3}+20 a^7 b^3 x^6+\frac{630}{19} a^6 b^4 x^{19/3}+\frac{189}{5} a^5 b^5 x^{20/3}+30 a^4 b^6 x^7+\frac{180}{11} a^3 b^7 x^{22/3}+\frac{135}{23} a^2 b^8 x^{23/3}+\frac{15}{8} a^9 b x^{16/3}+\frac{a^{10} x^5}{5}+\frac{5}{4} a b^9 x^8+\frac{3}{25} b^{10} x^{25/3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^10*x^4,x]

[Out]

(a^10*x^5)/5 + (15*a^9*b*x^(16/3))/8 + (135*a^8*b^2*x^(17/3))/17 + 20*a^7*b^3*x^6 + (630*a^6*b^4*x^(19/3))/19
+ (189*a^5*b^5*x^(20/3))/5 + 30*a^4*b^6*x^7 + (180*a^3*b^7*x^(22/3))/11 + (135*a^2*b^8*x^(23/3))/23 + (5*a*b^9
*x^8)/4 + (3*b^10*x^(25/3))/25

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Maple [A]  time = 0.003, size = 113, normalized size = 0.8 \begin{align*}{\frac{{a}^{10}{x}^{5}}{5}}+{\frac{15\,{a}^{9}b}{8}{x}^{{\frac{16}{3}}}}+{\frac{135\,{a}^{8}{b}^{2}}{17}{x}^{{\frac{17}{3}}}}+20\,{a}^{7}{b}^{3}{x}^{6}+{\frac{630\,{a}^{6}{b}^{4}}{19}{x}^{{\frac{19}{3}}}}+{\frac{189\,{a}^{5}{b}^{5}}{5}{x}^{{\frac{20}{3}}}}+30\,{a}^{4}{b}^{6}{x}^{7}+{\frac{180\,{a}^{3}{b}^{7}}{11}{x}^{{\frac{22}{3}}}}+{\frac{135\,{a}^{2}{b}^{8}}{23}{x}^{{\frac{23}{3}}}}+{\frac{5\,a{b}^{9}{x}^{8}}{4}}+{\frac{3\,{b}^{10}}{25}{x}^{{\frac{25}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/3))^10*x^4,x)

[Out]

1/5*a^10*x^5+15/8*a^9*b*x^(16/3)+135/17*a^8*b^2*x^(17/3)+20*a^7*b^3*x^6+630/19*a^6*b^4*x^(19/3)+189/5*a^5*b^5*
x^(20/3)+30*a^4*b^6*x^7+180/11*a^3*b^7*x^(22/3)+135/23*a^2*b^8*x^(23/3)+5/4*a*b^9*x^8+3/25*b^10*x^(25/3)

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Maxima [B]  time = 1.01009, size = 339, normalized size = 2.35 \begin{align*} \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{25}}{25 \, b^{15}} - \frac{7 \,{\left (b x^{\frac{1}{3}} + a\right )}^{24} a}{4 \, b^{15}} + \frac{273 \,{\left (b x^{\frac{1}{3}} + a\right )}^{23} a^{2}}{23 \, b^{15}} - \frac{546 \,{\left (b x^{\frac{1}{3}} + a\right )}^{22} a^{3}}{11 \, b^{15}} + \frac{143 \,{\left (b x^{\frac{1}{3}} + a\right )}^{21} a^{4}}{b^{15}} - \frac{3003 \,{\left (b x^{\frac{1}{3}} + a\right )}^{20} a^{5}}{10 \, b^{15}} + \frac{9009 \,{\left (b x^{\frac{1}{3}} + a\right )}^{19} a^{6}}{19 \, b^{15}} - \frac{572 \,{\left (b x^{\frac{1}{3}} + a\right )}^{18} a^{7}}{b^{15}} + \frac{9009 \,{\left (b x^{\frac{1}{3}} + a\right )}^{17} a^{8}}{17 \, b^{15}} - \frac{3003 \,{\left (b x^{\frac{1}{3}} + a\right )}^{16} a^{9}}{8 \, b^{15}} + \frac{1001 \,{\left (b x^{\frac{1}{3}} + a\right )}^{15} a^{10}}{5 \, b^{15}} - \frac{78 \,{\left (b x^{\frac{1}{3}} + a\right )}^{14} a^{11}}{b^{15}} + \frac{21 \,{\left (b x^{\frac{1}{3}} + a\right )}^{13} a^{12}}{b^{15}} - \frac{7 \,{\left (b x^{\frac{1}{3}} + a\right )}^{12} a^{13}}{2 \, b^{15}} + \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{11} a^{14}}{11 \, b^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^10*x^4,x, algorithm="maxima")

[Out]

3/25*(b*x^(1/3) + a)^25/b^15 - 7/4*(b*x^(1/3) + a)^24*a/b^15 + 273/23*(b*x^(1/3) + a)^23*a^2/b^15 - 546/11*(b*
x^(1/3) + a)^22*a^3/b^15 + 143*(b*x^(1/3) + a)^21*a^4/b^15 - 3003/10*(b*x^(1/3) + a)^20*a^5/b^15 + 9009/19*(b*
x^(1/3) + a)^19*a^6/b^15 - 572*(b*x^(1/3) + a)^18*a^7/b^15 + 9009/17*(b*x^(1/3) + a)^17*a^8/b^15 - 3003/8*(b*x
^(1/3) + a)^16*a^9/b^15 + 1001/5*(b*x^(1/3) + a)^15*a^10/b^15 - 78*(b*x^(1/3) + a)^14*a^11/b^15 + 21*(b*x^(1/3
) + a)^13*a^12/b^15 - 7/2*(b*x^(1/3) + a)^12*a^13/b^15 + 3/11*(b*x^(1/3) + a)^11*a^14/b^15

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Fricas [A]  time = 1.53168, size = 312, normalized size = 2.17 \begin{align*} \frac{5}{4} \, a b^{9} x^{8} + 30 \, a^{4} b^{6} x^{7} + 20 \, a^{7} b^{3} x^{6} + \frac{1}{5} \, a^{10} x^{5} + \frac{27}{1955} \,{\left (425 \, a^{2} b^{8} x^{7} + 2737 \, a^{5} b^{5} x^{6} + 575 \, a^{8} b^{2} x^{5}\right )} x^{\frac{2}{3}} + \frac{3}{41800} \,{\left (1672 \, b^{10} x^{8} + 228000 \, a^{3} b^{7} x^{7} + 462000 \, a^{6} b^{4} x^{6} + 26125 \, a^{9} b x^{5}\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^4,x, algorithm="fricas")

[Out]

5/4*a*b^9*x^8 + 30*a^4*b^6*x^7 + 20*a^7*b^3*x^6 + 1/5*a^10*x^5 + 27/1955*(425*a^2*b^8*x^7 + 2737*a^5*b^5*x^6 +
575*a^8*b^2*x^5)*x^(2/3) + 3/41800*(1672*b^10*x^8 + 228000*a^3*b^7*x^7 + 462000*a^6*b^4*x^6 + 26125*a^9*b*x^5
)*x^(1/3)

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Sympy [A]  time = 8.66252, size = 144, normalized size = 1. \begin{align*} \frac{a^{10} x^{5}}{5} + \frac{15 a^{9} b x^{\frac{16}{3}}}{8} + \frac{135 a^{8} b^{2} x^{\frac{17}{3}}}{17} + 20 a^{7} b^{3} x^{6} + \frac{630 a^{6} b^{4} x^{\frac{19}{3}}}{19} + \frac{189 a^{5} b^{5} x^{\frac{20}{3}}}{5} + 30 a^{4} b^{6} x^{7} + \frac{180 a^{3} b^{7} x^{\frac{22}{3}}}{11} + \frac{135 a^{2} b^{8} x^{\frac{23}{3}}}{23} + \frac{5 a b^{9} x^{8}}{4} + \frac{3 b^{10} x^{\frac{25}{3}}}{25} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/3))**10*x**4,x)

[Out]

a**10*x**5/5 + 15*a**9*b*x**(16/3)/8 + 135*a**8*b**2*x**(17/3)/17 + 20*a**7*b**3*x**6 + 630*a**6*b**4*x**(19/3
)/19 + 189*a**5*b**5*x**(20/3)/5 + 30*a**4*b**6*x**7 + 180*a**3*b**7*x**(22/3)/11 + 135*a**2*b**8*x**(23/3)/23
+ 5*a*b**9*x**8/4 + 3*b**10*x**(25/3)/25

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Giac [A]  time = 1.20333, size = 151, normalized size = 1.05 \begin{align*} \frac{3}{25} \, b^{10} x^{\frac{25}{3}} + \frac{5}{4} \, a b^{9} x^{8} + \frac{135}{23} \, a^{2} b^{8} x^{\frac{23}{3}} + \frac{180}{11} \, a^{3} b^{7} x^{\frac{22}{3}} + 30 \, a^{4} b^{6} x^{7} + \frac{189}{5} \, a^{5} b^{5} x^{\frac{20}{3}} + \frac{630}{19} \, a^{6} b^{4} x^{\frac{19}{3}} + 20 \, a^{7} b^{3} x^{6} + \frac{135}{17} \, a^{8} b^{2} x^{\frac{17}{3}} + \frac{15}{8} \, a^{9} b x^{\frac{16}{3}} + \frac{1}{5} \, a^{10} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^10*x^4,x, algorithm="giac")

[Out]

3/25*b^10*x^(25/3) + 5/4*a*b^9*x^8 + 135/23*a^2*b^8*x^(23/3) + 180/11*a^3*b^7*x^(22/3) + 30*a^4*b^6*x^7 + 189/
5*a^5*b^5*x^(20/3) + 630/19*a^6*b^4*x^(19/3) + 20*a^7*b^3*x^6 + 135/17*a^8*b^2*x^(17/3) + 15/8*a^9*b*x^(16/3)
+ 1/5*a^10*x^5