3.2322 \(\int \frac{\left (a+b \sqrt [3]{x}\right )^{10}}{x} \, dx\)

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

[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 + (13
5*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 = 0.157414, 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 \[ a^{10} \log (x)+30 a^9 b \sqrt [3]{x}+\frac{135}{2} a^8 b^2 x^{2/3}+120 a^7 b^3 x+\frac{315}{2} a^6 b^4 x^{4/3}+\frac{756}{5} a^5 b^5 x^{5/3}+105 a^4 b^6 x^2+\frac{360}{7} a^3 b^7 x^{7/3}+\frac{135}{8} a^2 b^8 x^{8/3}+\frac{10}{3} a b^9 x^3+\frac{3}{10} b^{10} x^{10/3} \]

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 + (13
5*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 in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ 3 a^{10} \log{\left (\sqrt [3]{x} \right )} + 30 a^{9} b \sqrt [3]{x} + 135 a^{8} b^{2} \int ^{\sqrt [3]{x}} x\, dx + 120 a^{7} b^{3} x + \frac{315 a^{6} b^{4} x^{\frac{4}{3}}}{2} + \frac{756 a^{5} b^{5} x^{\frac{5}{3}}}{5} + 105 a^{4} b^{6} x^{2} + \frac{360 a^{3} b^{7} x^{\frac{7}{3}}}{7} + \frac{135 a^{2} b^{8} x^{\frac{8}{3}}}{8} + \frac{10 a b^{9} x^{3}}{3} + \frac{3 b^{10} x^{\frac{10}{3}}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**(1/3))**10/x,x)

[Out]

3*a**10*log(x**(1/3)) + 30*a**9*b*x**(1/3) + 135*a**8*b**2*Integral(x, (x, x**(1
/3))) + 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

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Mathematica [A]  time = 0.0321525, size = 136, normalized size = 1. \[ a^{10} \log (x)+30 a^9 b \sqrt [3]{x}+\frac{135}{2} a^8 b^2 x^{2/3}+120 a^7 b^3 x+\frac{315}{2} a^6 b^4 x^{4/3}+\frac{756}{5} a^5 b^5 x^{5/3}+105 a^4 b^6 x^2+\frac{360}{7} a^3 b^7 x^{7/3}+\frac{135}{8} a^2 b^8 x^{8/3}+\frac{10}{3} a b^9 x^3+\frac{3}{10} b^{10} x^{10/3} \]

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 + (13
5*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 = 0.006, size = 109, normalized size = 0.8 \[ 30\,{a}^{9}b\sqrt [3]{x}+{\frac{135\,{a}^{8}{b}^{2}}{2}{x}^{{\frac{2}{3}}}}+120\,{a}^{7}{b}^{3}x+{\frac{315\,{a}^{6}{b}^{4}}{2}{x}^{{\frac{4}{3}}}}+{\frac{756\,{a}^{5}{b}^{5}}{5}{x}^{{\frac{5}{3}}}}+105\,{a}^{4}{b}^{6}{x}^{2}+{\frac{360\,{a}^{3}{b}^{7}}{7}{x}^{{\frac{7}{3}}}}+{\frac{135\,{a}^{2}{b}^{8}}{8}{x}^{{\frac{8}{3}}}}+{\frac{10\,a{b}^{9}{x}^{3}}{3}}+{\frac{3\,{b}^{10}}{10}{x}^{{\frac{10}{3}}}}+{a}^{10}\ln \left ( x \right ) \]

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 = 1.44564, size = 146, normalized size = 1.07 \[ \frac{3}{10} \, b^{10} x^{\frac{10}{3}} + \frac{10}{3} \, a b^{9} x^{3} + \frac{135}{8} \, a^{2} b^{8} x^{\frac{8}{3}} + \frac{360}{7} \, a^{3} b^{7} x^{\frac{7}{3}} + 105 \, a^{4} b^{6} x^{2} + \frac{756}{5} \, a^{5} b^{5} x^{\frac{5}{3}} + \frac{315}{2} \, a^{6} b^{4} x^{\frac{4}{3}} + 120 \, a^{7} b^{3} x + a^{10} \log \left (x\right ) + \frac{135}{2} \, a^{8} b^{2} x^{\frac{2}{3}} + 30 \, a^{9} b x^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^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 = 0.244942, size = 153, normalized size = 1.12 \[ \frac{10}{3} \, a b^{9} x^{3} + 105 \, a^{4} b^{6} x^{2} + 120 \, a^{7} b^{3} x + 3 \, a^{10} \log \left (x^{\frac{1}{3}}\right ) + \frac{27}{40} \,{\left (25 \, a^{2} b^{8} x^{2} + 224 \, a^{5} b^{5} x + 100 \, a^{8} b^{2}\right )} x^{\frac{2}{3}} + \frac{3}{70} \,{\left (7 \, b^{10} x^{3} + 1200 \, a^{3} b^{7} x^{2} + 3675 \, a^{6} b^{4} x + 700 \, a^{9} b\right )} x^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^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 = 10.1803, size = 139, normalized size = 1.02 \[ a^{10} \log{\left (x \right )} + 30 a^{9} b \sqrt [3]{x} + \frac{135 a^{8} b^{2} x^{\frac{2}{3}}}{2} + 120 a^{7} b^{3} x + \frac{315 a^{6} b^{4} x^{\frac{4}{3}}}{2} + \frac{756 a^{5} b^{5} x^{\frac{5}{3}}}{5} + 105 a^{4} b^{6} x^{2} + \frac{360 a^{3} b^{7} x^{\frac{7}{3}}}{7} + \frac{135 a^{2} b^{8} x^{\frac{8}{3}}}{8} + \frac{10 a b^{9} x^{3}}{3} + \frac{3 b^{10} x^{\frac{10}{3}}}{10} \]

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 + 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

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GIAC/XCAS [A]  time = 0.22748, size = 147, normalized size = 1.08 \[ \frac{3}{10} \, b^{10} x^{\frac{10}{3}} + \frac{10}{3} \, a b^{9} x^{3} + \frac{135}{8} \, a^{2} b^{8} x^{\frac{8}{3}} + \frac{360}{7} \, a^{3} b^{7} x^{\frac{7}{3}} + 105 \, a^{4} b^{6} x^{2} + \frac{756}{5} \, a^{5} b^{5} x^{\frac{5}{3}} + \frac{315}{2} \, a^{6} b^{4} x^{\frac{4}{3}} + 120 \, a^{7} b^{3} x + a^{10}{\rm ln}\left ({\left | x \right |}\right ) + \frac{135}{2} \, a^{8} b^{2} x^{\frac{2}{3}} + 30 \, a^{9} b x^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^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*ln(abs(x)) + 135/2*a^8*b^2*x^(2/3) + 30*a^9*b*x^(1/3)