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

Optimal. Leaf size=131 \[ -\frac{a^{10}}{2 x^2}-\frac{6 a^9 b}{x^{5/3}}-\frac{135 a^8 b^2}{4 x^{4/3}}-\frac{120 a^7 b^3}{x}-\frac{315 a^6 b^4}{x^{2/3}}-\frac{756 a^5 b^5}{\sqrt [3]{x}}+210 a^4 b^6 \log (x)+360 a^3 b^7 \sqrt [3]{x}+\frac{135}{2} a^2 b^8 x^{2/3}+10 a b^9 x+\frac{3}{4} b^{10} x^{4/3} \]

[Out]

-a^10/(2*x^2) - (6*a^9*b)/x^(5/3) - (135*a^8*b^2)/(4*x^(4/3)) - (120*a^7*b^3)/x
- (315*a^6*b^4)/x^(2/3) - (756*a^5*b^5)/x^(1/3) + 360*a^3*b^7*x^(1/3) + (135*a^2
*b^8*x^(2/3))/2 + 10*a*b^9*x + (3*b^10*x^(4/3))/4 + 210*a^4*b^6*Log[x]

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Rubi [A]  time = 0.178483, antiderivative size = 131, 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 \[ -\frac{a^{10}}{2 x^2}-\frac{6 a^9 b}{x^{5/3}}-\frac{135 a^8 b^2}{4 x^{4/3}}-\frac{120 a^7 b^3}{x}-\frac{315 a^6 b^4}{x^{2/3}}-\frac{756 a^5 b^5}{\sqrt [3]{x}}+210 a^4 b^6 \log (x)+360 a^3 b^7 \sqrt [3]{x}+\frac{135}{2} a^2 b^8 x^{2/3}+10 a b^9 x+\frac{3}{4} b^{10} x^{4/3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^(1/3))^10/x^3,x]

[Out]

-a^10/(2*x^2) - (6*a^9*b)/x^(5/3) - (135*a^8*b^2)/(4*x^(4/3)) - (120*a^7*b^3)/x
- (315*a^6*b^4)/x^(2/3) - (756*a^5*b^5)/x^(1/3) + 360*a^3*b^7*x^(1/3) + (135*a^2
*b^8*x^(2/3))/2 + 10*a*b^9*x + (3*b^10*x^(4/3))/4 + 210*a^4*b^6*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{10}}{2 x^{2}} - \frac{6 a^{9} b}{x^{\frac{5}{3}}} - \frac{135 a^{8} b^{2}}{4 x^{\frac{4}{3}}} - \frac{120 a^{7} b^{3}}{x} - \frac{315 a^{6} b^{4}}{x^{\frac{2}{3}}} - \frac{756 a^{5} b^{5}}{\sqrt [3]{x}} + 630 a^{4} b^{6} \log{\left (\sqrt [3]{x} \right )} + 360 a^{3} b^{7} \sqrt [3]{x} + 135 a^{2} b^{8} \int ^{\sqrt [3]{x}} x\, dx + 10 a b^{9} x + \frac{3 b^{10} x^{\frac{4}{3}}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**10/(2*x**2) - 6*a**9*b/x**(5/3) - 135*a**8*b**2/(4*x**(4/3)) - 120*a**7*b**3
/x - 315*a**6*b**4/x**(2/3) - 756*a**5*b**5/x**(1/3) + 630*a**4*b**6*log(x**(1/3
)) + 360*a**3*b**7*x**(1/3) + 135*a**2*b**8*Integral(x, (x, x**(1/3))) + 10*a*b*
*9*x + 3*b**10*x**(4/3)/4

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Mathematica [A]  time = 0.0475645, size = 130, normalized size = 0.99 \[ -\frac{2 a^{10}+24 a^9 b \sqrt [3]{x}+135 a^8 b^2 x^{2/3}+480 a^7 b^3 x+1260 a^6 b^4 x^{4/3}+3024 a^5 b^5 x^{5/3}-840 a^4 b^6 x^2 \log (x)-1440 a^3 b^7 x^{7/3}-270 a^2 b^8 x^{8/3}-40 a b^9 x^3-3 b^{10} x^{10/3}}{4 x^2} \]

Antiderivative was successfully verified.

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

[Out]

-(2*a^10 + 24*a^9*b*x^(1/3) + 135*a^8*b^2*x^(2/3) + 480*a^7*b^3*x + 1260*a^6*b^4
*x^(4/3) + 3024*a^5*b^5*x^(5/3) - 1440*a^3*b^7*x^(7/3) - 270*a^2*b^8*x^(8/3) - 4
0*a*b^9*x^3 - 3*b^10*x^(10/3) - 840*a^4*b^6*x^2*Log[x])/(4*x^2)

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Maple [A]  time = 0.014, size = 110, normalized size = 0.8 \[ -{\frac{{a}^{10}}{2\,{x}^{2}}}-6\,{\frac{{a}^{9}b}{{x}^{5/3}}}-{\frac{135\,{a}^{8}{b}^{2}}{4}{x}^{-{\frac{4}{3}}}}-120\,{\frac{{a}^{7}{b}^{3}}{x}}-315\,{\frac{{a}^{6}{b}^{4}}{{x}^{2/3}}}-756\,{\frac{{a}^{5}{b}^{5}}{\sqrt [3]{x}}}+360\,{a}^{3}{b}^{7}\sqrt [3]{x}+{\frac{135\,{a}^{2}{b}^{8}}{2}{x}^{{\frac{2}{3}}}}+10\,a{b}^{9}x+{\frac{3\,{b}^{10}}{4}{x}^{{\frac{4}{3}}}}+210\,{a}^{4}{b}^{6}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/3))^10/x^3,x)

[Out]

-1/2*a^10/x^2-6*a^9*b/x^(5/3)-135/4*a^8*b^2/x^(4/3)-120*a^7*b^3/x-315*a^6*b^4/x^
(2/3)-756*a^5*b^5/x^(1/3)+360*a^3*b^7*x^(1/3)+135/2*a^2*b^8*x^(2/3)+10*a*b^9*x+3
/4*b^10*x^(4/3)+210*a^4*b^6*ln(x)

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Maxima [A]  time = 1.43768, size = 149, normalized size = 1.14 \[ \frac{3}{4} \, b^{10} x^{\frac{4}{3}} + 10 \, a b^{9} x + 210 \, a^{4} b^{6} \log \left (x\right ) + \frac{135}{2} \, a^{2} b^{8} x^{\frac{2}{3}} + 360 \, a^{3} b^{7} x^{\frac{1}{3}} - \frac{3024 \, a^{5} b^{5} x^{\frac{5}{3}} + 1260 \, a^{6} b^{4} x^{\frac{4}{3}} + 480 \, a^{7} b^{3} x + 135 \, a^{8} b^{2} x^{\frac{2}{3}} + 24 \, a^{9} b x^{\frac{1}{3}} + 2 \, a^{10}}{4 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/4*b^10*x^(4/3) + 10*a*b^9*x + 210*a^4*b^6*log(x) + 135/2*a^2*b^8*x^(2/3) + 360
*a^3*b^7*x^(1/3) - 1/4*(3024*a^5*b^5*x^(5/3) + 1260*a^6*b^4*x^(4/3) + 480*a^7*b^
3*x + 135*a^8*b^2*x^(2/3) + 24*a^9*b*x^(1/3) + 2*a^10)/x^2

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Fricas [A]  time = 0.241707, size = 158, normalized size = 1.21 \[ \frac{40 \, a b^{9} x^{3} + 2520 \, a^{4} b^{6} x^{2} \log \left (x^{\frac{1}{3}}\right ) - 480 \, a^{7} b^{3} x - 2 \, a^{10} + 27 \,{\left (10 \, a^{2} b^{8} x^{2} - 112 \, a^{5} b^{5} x - 5 \, a^{8} b^{2}\right )} x^{\frac{2}{3}} + 3 \,{\left (b^{10} x^{3} + 480 \, a^{3} b^{7} x^{2} - 420 \, a^{6} b^{4} x - 8 \, a^{9} b\right )} x^{\frac{1}{3}}}{4 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^10/x^3,x, algorithm="fricas")

[Out]

1/4*(40*a*b^9*x^3 + 2520*a^4*b^6*x^2*log(x^(1/3)) - 480*a^7*b^3*x - 2*a^10 + 27*
(10*a^2*b^8*x^2 - 112*a^5*b^5*x - 5*a^8*b^2)*x^(2/3) + 3*(b^10*x^3 + 480*a^3*b^7
*x^2 - 420*a^6*b^4*x - 8*a^9*b)*x^(1/3))/x^2

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Sympy [A]  time = 10.1997, size = 133, normalized size = 1.02 \[ - \frac{a^{10}}{2 x^{2}} - \frac{6 a^{9} b}{x^{\frac{5}{3}}} - \frac{135 a^{8} b^{2}}{4 x^{\frac{4}{3}}} - \frac{120 a^{7} b^{3}}{x} - \frac{315 a^{6} b^{4}}{x^{\frac{2}{3}}} - \frac{756 a^{5} b^{5}}{\sqrt [3]{x}} + 210 a^{4} b^{6} \log{\left (x \right )} + 360 a^{3} b^{7} \sqrt [3]{x} + \frac{135 a^{2} b^{8} x^{\frac{2}{3}}}{2} + 10 a b^{9} x + \frac{3 b^{10} x^{\frac{4}{3}}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**10/(2*x**2) - 6*a**9*b/x**(5/3) - 135*a**8*b**2/(4*x**(4/3)) - 120*a**7*b**3
/x - 315*a**6*b**4/x**(2/3) - 756*a**5*b**5/x**(1/3) + 210*a**4*b**6*log(x) + 36
0*a**3*b**7*x**(1/3) + 135*a**2*b**8*x**(2/3)/2 + 10*a*b**9*x + 3*b**10*x**(4/3)
/4

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GIAC/XCAS [A]  time = 0.236443, size = 150, normalized size = 1.15 \[ \frac{3}{4} \, b^{10} x^{\frac{4}{3}} + 10 \, a b^{9} x + 210 \, a^{4} b^{6}{\rm ln}\left ({\left | x \right |}\right ) + \frac{135}{2} \, a^{2} b^{8} x^{\frac{2}{3}} + 360 \, a^{3} b^{7} x^{\frac{1}{3}} - \frac{3024 \, a^{5} b^{5} x^{\frac{5}{3}} + 1260 \, a^{6} b^{4} x^{\frac{4}{3}} + 480 \, a^{7} b^{3} x + 135 \, a^{8} b^{2} x^{\frac{2}{3}} + 24 \, a^{9} b x^{\frac{1}{3}} + 2 \, a^{10}}{4 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/4*b^10*x^(4/3) + 10*a*b^9*x + 210*a^4*b^6*ln(abs(x)) + 135/2*a^2*b^8*x^(2/3) +
360*a^3*b^7*x^(1/3) - 1/4*(3024*a^5*b^5*x^(5/3) + 1260*a^6*b^4*x^(4/3) + 480*a^
7*b^3*x + 135*a^8*b^2*x^(2/3) + 24*a^9*b*x^(1/3) + 2*a^10)/x^2