3.2329 \(\int \frac{(a+b \sqrt [3]{x})^{10}}{x} \, dx\)

Optimal. Leaf size=136 \[ \frac{135}{2} a^8 b^2 x^{2/3}+\frac{315}{2} a^6 b^4 x^{4/3}+\frac{756}{5} a^5 b^5 x^{5/3}+105 a^4 b^6 x^2+\frac{360}{7} a^3 b^7 x^{7/3}+\frac{135}{8} a^2 b^8 x^{8/3}+120 a^7 b^3 x+30 a^9 b \sqrt [3]{x}+a^{10} \log (x)+\frac{10}{3} a b^9 x^3+\frac{3}{10} b^{10} x^{10/3} \]

[Out]

30*a^9*b*x^(1/3) + (135*a^8*b^2*x^(2/3))/2 + 120*a^7*b^3*x + (315*a^6*b^4*x^(4/3))/2 + (756*a^5*b^5*x^(5/3))/5
+ 105*a^4*b^6*x^2 + (360*a^3*b^7*x^(7/3))/7 + (135*a^2*b^8*x^(8/3))/8 + (10*a*b^9*x^3)/3 + (3*b^10*x^(10/3))/
10 + a^10*Log[x]

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Rubi [A]  time = 0.0655307, antiderivative size = 136, 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} \[ \frac{135}{2} a^8 b^2 x^{2/3}+\frac{315}{2} a^6 b^4 x^{4/3}+\frac{756}{5} a^5 b^5 x^{5/3}+105 a^4 b^6 x^2+\frac{360}{7} a^3 b^7 x^{7/3}+\frac{135}{8} a^2 b^8 x^{8/3}+120 a^7 b^3 x+30 a^9 b \sqrt [3]{x}+a^{10} \log (x)+\frac{10}{3} a b^9 x^3+\frac{3}{10} b^{10} x^{10/3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))^10/x,x]

[Out]

30*a^9*b*x^(1/3) + (135*a^8*b^2*x^(2/3))/2 + 120*a^7*b^3*x + (315*a^6*b^4*x^(4/3))/2 + (756*a^5*b^5*x^(5/3))/5
+ 105*a^4*b^6*x^2 + (360*a^3*b^7*x^(7/3))/7 + (135*a^2*b^8*x^(8/3))/8 + (10*a*b^9*x^3)/3 + (3*b^10*x^(10/3))/
10 + a^10*Log[x]

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 \frac{\left (a+b \sqrt [3]{x}\right )^{10}}{x} \, dx &=3 \operatorname{Subst}\left (\int \frac{(a+b x)^{10}}{x} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (10 a^9 b+\frac{a^{10}}{x}+45 a^8 b^2 x+120 a^7 b^3 x^2+210 a^6 b^4 x^3+252 a^5 b^5 x^4+210 a^4 b^6 x^5+120 a^3 b^7 x^6+45 a^2 b^8 x^7+10 a b^9 x^8+b^{10} x^9\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=30 a^9 b \sqrt [3]{x}+\frac{135}{2} a^8 b^2 x^{2/3}+120 a^7 b^3 x+\frac{315}{2} a^6 b^4 x^{4/3}+\frac{756}{5} a^5 b^5 x^{5/3}+105 a^4 b^6 x^2+\frac{360}{7} a^3 b^7 x^{7/3}+\frac{135}{8} a^2 b^8 x^{8/3}+\frac{10}{3} a b^9 x^3+\frac{3}{10} b^{10} x^{10/3}+a^{10} \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0420488, size = 136, normalized size = 1. \[ \frac{135}{2} a^8 b^2 x^{2/3}+\frac{315}{2} a^6 b^4 x^{4/3}+\frac{756}{5} a^5 b^5 x^{5/3}+105 a^4 b^6 x^2+\frac{360}{7} a^3 b^7 x^{7/3}+\frac{135}{8} a^2 b^8 x^{8/3}+120 a^7 b^3 x+30 a^9 b \sqrt [3]{x}+a^{10} \log (x)+\frac{10}{3} a b^9 x^3+\frac{3}{10} b^{10} x^{10/3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^10/x,x]

[Out]

30*a^9*b*x^(1/3) + (135*a^8*b^2*x^(2/3))/2 + 120*a^7*b^3*x + (315*a^6*b^4*x^(4/3))/2 + (756*a^5*b^5*x^(5/3))/5
+ 105*a^4*b^6*x^2 + (360*a^3*b^7*x^(7/3))/7 + (135*a^2*b^8*x^(8/3))/8 + (10*a*b^9*x^3)/3 + (3*b^10*x^(10/3))/
10 + a^10*Log[x]

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Maple [A]  time = 0.003, size = 109, normalized size = 0.8 \begin{align*} 30\,{a}^{9}b\sqrt [3]{x}+{\frac{135\,{a}^{8}{b}^{2}}{2}{x}^{{\frac{2}{3}}}}+120\,{a}^{7}{b}^{3}x+{\frac{315\,{a}^{6}{b}^{4}}{2}{x}^{{\frac{4}{3}}}}+{\frac{756\,{a}^{5}{b}^{5}}{5}{x}^{{\frac{5}{3}}}}+105\,{a}^{4}{b}^{6}{x}^{2}+{\frac{360\,{a}^{3}{b}^{7}}{7}{x}^{{\frac{7}{3}}}}+{\frac{135\,{a}^{2}{b}^{8}}{8}{x}^{{\frac{8}{3}}}}+{\frac{10\,a{b}^{9}{x}^{3}}{3}}+{\frac{3\,{b}^{10}}{10}{x}^{{\frac{10}{3}}}}+{a}^{10}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

30*a^9*b*x^(1/3)+135/2*a^8*b^2*x^(2/3)+120*a^7*b^3*x+315/2*a^6*b^4*x^(4/3)+756/5*a^5*b^5*x^(5/3)+105*a^4*b^6*x
^2+360/7*a^3*b^7*x^(7/3)+135/8*a^2*b^8*x^(8/3)+10/3*a*b^9*x^3+3/10*b^10*x^(10/3)+a^10*ln(x)

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Maxima [A]  time = 0.980391, size = 146, normalized size = 1.07 \begin{align*} \frac{3}{10} \, b^{10} x^{\frac{10}{3}} + \frac{10}{3} \, a b^{9} x^{3} + \frac{135}{8} \, a^{2} b^{8} x^{\frac{8}{3}} + \frac{360}{7} \, a^{3} b^{7} x^{\frac{7}{3}} + 105 \, a^{4} b^{6} x^{2} + \frac{756}{5} \, a^{5} b^{5} x^{\frac{5}{3}} + \frac{315}{2} \, a^{6} b^{4} x^{\frac{4}{3}} + 120 \, a^{7} b^{3} x + a^{10} \log \left (x\right ) + \frac{135}{2} \, a^{8} b^{2} x^{\frac{2}{3}} + 30 \, a^{9} b 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,x, algorithm="maxima")

[Out]

3/10*b^10*x^(10/3) + 10/3*a*b^9*x^3 + 135/8*a^2*b^8*x^(8/3) + 360/7*a^3*b^7*x^(7/3) + 105*a^4*b^6*x^2 + 756/5*
a^5*b^5*x^(5/3) + 315/2*a^6*b^4*x^(4/3) + 120*a^7*b^3*x + a^10*log(x) + 135/2*a^8*b^2*x^(2/3) + 30*a^9*b*x^(1/
3)

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Fricas [A]  time = 1.49208, size = 285, normalized size = 2.1 \begin{align*} \frac{10}{3} \, a b^{9} x^{3} + 105 \, a^{4} b^{6} x^{2} + 120 \, a^{7} b^{3} x + 3 \, a^{10} \log \left (x^{\frac{1}{3}}\right ) + \frac{27}{40} \,{\left (25 \, a^{2} b^{8} x^{2} + 224 \, a^{5} b^{5} x + 100 \, a^{8} b^{2}\right )} x^{\frac{2}{3}} + \frac{3}{70} \,{\left (7 \, b^{10} x^{3} + 1200 \, a^{3} b^{7} x^{2} + 3675 \, a^{6} b^{4} x + 700 \, a^{9} b\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,x, algorithm="fricas")

[Out]

10/3*a*b^9*x^3 + 105*a^4*b^6*x^2 + 120*a^7*b^3*x + 3*a^10*log(x^(1/3)) + 27/40*(25*a^2*b^8*x^2 + 224*a^5*b^5*x
+ 100*a^8*b^2)*x^(2/3) + 3/70*(7*b^10*x^3 + 1200*a^3*b^7*x^2 + 3675*a^6*b^4*x + 700*a^9*b)*x^(1/3)

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Sympy [A]  time = 55.8137, size = 139, normalized size = 1.02 \begin{align*} a^{10} \log{\left (x \right )} + 30 a^{9} b \sqrt [3]{x} + \frac{135 a^{8} b^{2} x^{\frac{2}{3}}}{2} + 120 a^{7} b^{3} x + \frac{315 a^{6} b^{4} x^{\frac{4}{3}}}{2} + \frac{756 a^{5} b^{5} x^{\frac{5}{3}}}{5} + 105 a^{4} b^{6} x^{2} + \frac{360 a^{3} b^{7} x^{\frac{7}{3}}}{7} + \frac{135 a^{2} b^{8} x^{\frac{8}{3}}}{8} + \frac{10 a b^{9} x^{3}}{3} + \frac{3 b^{10} x^{\frac{10}{3}}}{10} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**10*log(x) + 30*a**9*b*x**(1/3) + 135*a**8*b**2*x**(2/3)/2 + 120*a**7*b**3*x + 315*a**6*b**4*x**(4/3)/2 + 75
6*a**5*b**5*x**(5/3)/5 + 105*a**4*b**6*x**2 + 360*a**3*b**7*x**(7/3)/7 + 135*a**2*b**8*x**(8/3)/8 + 10*a*b**9*
x**3/3 + 3*b**10*x**(10/3)/10

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Giac [A]  time = 1.16296, size = 147, normalized size = 1.08 \begin{align*} \frac{3}{10} \, b^{10} x^{\frac{10}{3}} + \frac{10}{3} \, a b^{9} x^{3} + \frac{135}{8} \, a^{2} b^{8} x^{\frac{8}{3}} + \frac{360}{7} \, a^{3} b^{7} x^{\frac{7}{3}} + 105 \, a^{4} b^{6} x^{2} + \frac{756}{5} \, a^{5} b^{5} x^{\frac{5}{3}} + \frac{315}{2} \, a^{6} b^{4} x^{\frac{4}{3}} + 120 \, a^{7} b^{3} x + a^{10} \log \left ({\left | x \right |}\right ) + \frac{135}{2} \, a^{8} b^{2} x^{\frac{2}{3}} + 30 \, a^{9} b 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,x, algorithm="giac")

[Out]

3/10*b^10*x^(10/3) + 10/3*a*b^9*x^3 + 135/8*a^2*b^8*x^(8/3) + 360/7*a^3*b^7*x^(7/3) + 105*a^4*b^6*x^2 + 756/5*
a^5*b^5*x^(5/3) + 315/2*a^6*b^4*x^(4/3) + 120*a^7*b^3*x + a^10*log(abs(x)) + 135/2*a^8*b^2*x^(2/3) + 30*a^9*b*
x^(1/3)