3.2330 \(\int \frac{\left (a+b \sqrt [3]{x}\right )^{10}}{x^9} \, dx\)

Optimal. Leaf size=144 \[ -\frac{a^{10}}{8 x^8}-\frac{30 a^9 b}{23 x^{23/3}}-\frac{135 a^8 b^2}{22 x^{22/3}}-\frac{120 a^7 b^3}{7 x^7}-\frac{63 a^6 b^4}{2 x^{20/3}}-\frac{756 a^5 b^5}{19 x^{19/3}}-\frac{35 a^4 b^6}{x^6}-\frac{360 a^3 b^7}{17 x^{17/3}}-\frac{135 a^2 b^8}{16 x^{16/3}}-\frac{2 a b^9}{x^5}-\frac{3 b^{10}}{14 x^{14/3}} \]

[Out]

-a^10/(8*x^8) - (30*a^9*b)/(23*x^(23/3)) - (135*a^8*b^2)/(22*x^(22/3)) - (120*a^
7*b^3)/(7*x^7) - (63*a^6*b^4)/(2*x^(20/3)) - (756*a^5*b^5)/(19*x^(19/3)) - (35*a
^4*b^6)/x^6 - (360*a^3*b^7)/(17*x^(17/3)) - (135*a^2*b^8)/(16*x^(16/3)) - (2*a*b
^9)/x^5 - (3*b^10)/(14*x^(14/3))

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Rubi [A]  time = 0.182271, 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 \[ -\frac{a^{10}}{8 x^8}-\frac{30 a^9 b}{23 x^{23/3}}-\frac{135 a^8 b^2}{22 x^{22/3}}-\frac{120 a^7 b^3}{7 x^7}-\frac{63 a^6 b^4}{2 x^{20/3}}-\frac{756 a^5 b^5}{19 x^{19/3}}-\frac{35 a^4 b^6}{x^6}-\frac{360 a^3 b^7}{17 x^{17/3}}-\frac{135 a^2 b^8}{16 x^{16/3}}-\frac{2 a b^9}{x^5}-\frac{3 b^{10}}{14 x^{14/3}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^(1/3))^10/x^9,x]

[Out]

-a^10/(8*x^8) - (30*a^9*b)/(23*x^(23/3)) - (135*a^8*b^2)/(22*x^(22/3)) - (120*a^
7*b^3)/(7*x^7) - (63*a^6*b^4)/(2*x^(20/3)) - (756*a^5*b^5)/(19*x^(19/3)) - (35*a
^4*b^6)/x^6 - (360*a^3*b^7)/(17*x^(17/3)) - (135*a^2*b^8)/(16*x^(16/3)) - (2*a*b
^9)/x^5 - (3*b^10)/(14*x^(14/3))

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Rubi in Sympy [A]  time = 31.6797, size = 146, normalized size = 1.01 \[ - \frac{a^{10}}{8 x^{8}} - \frac{30 a^{9} b}{23 x^{\frac{23}{3}}} - \frac{135 a^{8} b^{2}}{22 x^{\frac{22}{3}}} - \frac{120 a^{7} b^{3}}{7 x^{7}} - \frac{63 a^{6} b^{4}}{2 x^{\frac{20}{3}}} - \frac{756 a^{5} b^{5}}{19 x^{\frac{19}{3}}} - \frac{35 a^{4} b^{6}}{x^{6}} - \frac{360 a^{3} b^{7}}{17 x^{\frac{17}{3}}} - \frac{135 a^{2} b^{8}}{16 x^{\frac{16}{3}}} - \frac{2 a b^{9}}{x^{5}} - \frac{3 b^{10}}{14 x^{\frac{14}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**(1/3))**10/x**9,x)

[Out]

-a**10/(8*x**8) - 30*a**9*b/(23*x**(23/3)) - 135*a**8*b**2/(22*x**(22/3)) - 120*
a**7*b**3/(7*x**7) - 63*a**6*b**4/(2*x**(20/3)) - 756*a**5*b**5/(19*x**(19/3)) -
35*a**4*b**6/x**6 - 360*a**3*b**7/(17*x**(17/3)) - 135*a**2*b**8/(16*x**(16/3))
- 2*a*b**9/x**5 - 3*b**10/(14*x**(14/3))

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Mathematica [A]  time = 0.0500089, size = 144, normalized size = 1. \[ -\frac{a^{10}}{8 x^8}-\frac{30 a^9 b}{23 x^{23/3}}-\frac{135 a^8 b^2}{22 x^{22/3}}-\frac{120 a^7 b^3}{7 x^7}-\frac{63 a^6 b^4}{2 x^{20/3}}-\frac{756 a^5 b^5}{19 x^{19/3}}-\frac{35 a^4 b^6}{x^6}-\frac{360 a^3 b^7}{17 x^{17/3}}-\frac{135 a^2 b^8}{16 x^{16/3}}-\frac{2 a b^9}{x^5}-\frac{3 b^{10}}{14 x^{14/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^(1/3))^10/x^9,x]

[Out]

-a^10/(8*x^8) - (30*a^9*b)/(23*x^(23/3)) - (135*a^8*b^2)/(22*x^(22/3)) - (120*a^
7*b^3)/(7*x^7) - (63*a^6*b^4)/(2*x^(20/3)) - (756*a^5*b^5)/(19*x^(19/3)) - (35*a
^4*b^6)/x^6 - (360*a^3*b^7)/(17*x^(17/3)) - (135*a^2*b^8)/(16*x^(16/3)) - (2*a*b
^9)/x^5 - (3*b^10)/(14*x^(14/3))

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Maple [A]  time = 0.011, size = 113, normalized size = 0.8 \[ -{\frac{{a}^{10}}{8\,{x}^{8}}}-{\frac{30\,{a}^{9}b}{23}{x}^{-{\frac{23}{3}}}}-{\frac{135\,{a}^{8}{b}^{2}}{22}{x}^{-{\frac{22}{3}}}}-{\frac{120\,{a}^{7}{b}^{3}}{7\,{x}^{7}}}-{\frac{63\,{a}^{6}{b}^{4}}{2}{x}^{-{\frac{20}{3}}}}-{\frac{756\,{a}^{5}{b}^{5}}{19}{x}^{-{\frac{19}{3}}}}-35\,{\frac{{a}^{4}{b}^{6}}{{x}^{6}}}-{\frac{360\,{a}^{3}{b}^{7}}{17}{x}^{-{\frac{17}{3}}}}-{\frac{135\,{a}^{2}{b}^{8}}{16}{x}^{-{\frac{16}{3}}}}-2\,{\frac{a{b}^{9}}{{x}^{5}}}-{\frac{3\,{b}^{10}}{14}{x}^{-{\frac{14}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/3))^10/x^9,x)

[Out]

-1/8*a^10/x^8-30/23*a^9*b/x^(23/3)-135/22*a^8*b^2/x^(22/3)-120/7*a^7*b^3/x^7-63/
2*a^6*b^4/x^(20/3)-756/19*a^5*b^5/x^(19/3)-35*a^4*b^6/x^6-360/17*a^3*b^7/x^(17/3
)-135/16*a^2*b^8/x^(16/3)-2*a*b^9/x^5-3/14*b^10/x^(14/3)

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Maxima [A]  time = 1.44629, size = 151, normalized size = 1.05 \[ -\frac{1961256 \, b^{10} x^{\frac{10}{3}} + 18305056 \, a b^{9} x^{3} + 77224455 \, a^{2} b^{8} x^{\frac{8}{3}} + 193818240 \, a^{3} b^{7} x^{\frac{7}{3}} + 320338480 \, a^{4} b^{6} x^{2} + 364174272 \, a^{5} b^{5} x^{\frac{5}{3}} + 288304632 \, a^{6} b^{4} x^{\frac{4}{3}} + 156900480 \, a^{7} b^{3} x + 56163240 \, a^{8} b^{2} x^{\frac{2}{3}} + 11938080 \, a^{9} b x^{\frac{1}{3}} + 1144066 \, a^{10}}{9152528 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^10/x^9,x, algorithm="maxima")

[Out]

-1/9152528*(1961256*b^10*x^(10/3) + 18305056*a*b^9*x^3 + 77224455*a^2*b^8*x^(8/3
) + 193818240*a^3*b^7*x^(7/3) + 320338480*a^4*b^6*x^2 + 364174272*a^5*b^5*x^(5/3
) + 288304632*a^6*b^4*x^(4/3) + 156900480*a^7*b^3*x + 56163240*a^8*b^2*x^(2/3) +
11938080*a^9*b*x^(1/3) + 1144066*a^10)/x^8

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Fricas [A]  time = 0.218366, size = 154, normalized size = 1.07 \[ -\frac{18305056 \, a b^{9} x^{3} + 320338480 \, a^{4} b^{6} x^{2} + 156900480 \, a^{7} b^{3} x + 1144066 \, a^{10} + 73899 \,{\left (1045 \, a^{2} b^{8} x^{2} + 4928 \, a^{5} b^{5} x + 760 \, a^{8} b^{2}\right )} x^{\frac{2}{3}} + 5016 \,{\left (391 \, b^{10} x^{3} + 38640 \, a^{3} b^{7} x^{2} + 57477 \, a^{6} b^{4} x + 2380 \, a^{9} b\right )} x^{\frac{1}{3}}}{9152528 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^10/x^9,x, algorithm="fricas")

[Out]

-1/9152528*(18305056*a*b^9*x^3 + 320338480*a^4*b^6*x^2 + 156900480*a^7*b^3*x + 1
144066*a^10 + 73899*(1045*a^2*b^8*x^2 + 4928*a^5*b^5*x + 760*a^8*b^2)*x^(2/3) +
5016*(391*b^10*x^3 + 38640*a^3*b^7*x^2 + 57477*a^6*b^4*x + 2380*a^9*b)*x^(1/3))/
x^8

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Sympy [A]  time = 90.58, size = 146, normalized size = 1.01 \[ - \frac{a^{10}}{8 x^{8}} - \frac{30 a^{9} b}{23 x^{\frac{23}{3}}} - \frac{135 a^{8} b^{2}}{22 x^{\frac{22}{3}}} - \frac{120 a^{7} b^{3}}{7 x^{7}} - \frac{63 a^{6} b^{4}}{2 x^{\frac{20}{3}}} - \frac{756 a^{5} b^{5}}{19 x^{\frac{19}{3}}} - \frac{35 a^{4} b^{6}}{x^{6}} - \frac{360 a^{3} b^{7}}{17 x^{\frac{17}{3}}} - \frac{135 a^{2} b^{8}}{16 x^{\frac{16}{3}}} - \frac{2 a b^{9}}{x^{5}} - \frac{3 b^{10}}{14 x^{\frac{14}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**(1/3))**10/x**9,x)

[Out]

-a**10/(8*x**8) - 30*a**9*b/(23*x**(23/3)) - 135*a**8*b**2/(22*x**(22/3)) - 120*
a**7*b**3/(7*x**7) - 63*a**6*b**4/(2*x**(20/3)) - 756*a**5*b**5/(19*x**(19/3)) -
35*a**4*b**6/x**6 - 360*a**3*b**7/(17*x**(17/3)) - 135*a**2*b**8/(16*x**(16/3))
- 2*a*b**9/x**5 - 3*b**10/(14*x**(14/3))

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GIAC/XCAS [A]  time = 0.221964, size = 151, normalized size = 1.05 \[ -\frac{1961256 \, b^{10} x^{\frac{10}{3}} + 18305056 \, a b^{9} x^{3} + 77224455 \, a^{2} b^{8} x^{\frac{8}{3}} + 193818240 \, a^{3} b^{7} x^{\frac{7}{3}} + 320338480 \, a^{4} b^{6} x^{2} + 364174272 \, a^{5} b^{5} x^{\frac{5}{3}} + 288304632 \, a^{6} b^{4} x^{\frac{4}{3}} + 156900480 \, a^{7} b^{3} x + 56163240 \, a^{8} b^{2} x^{\frac{2}{3}} + 11938080 \, a^{9} b x^{\frac{1}{3}} + 1144066 \, a^{10}}{9152528 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^10/x^9,x, algorithm="giac")

[Out]

-1/9152528*(1961256*b^10*x^(10/3) + 18305056*a*b^9*x^3 + 77224455*a^2*b^8*x^(8/3
) + 193818240*a^3*b^7*x^(7/3) + 320338480*a^4*b^6*x^2 + 364174272*a^5*b^5*x^(5/3
) + 288304632*a^6*b^4*x^(4/3) + 156900480*a^7*b^3*x + 56163240*a^8*b^2*x^(2/3) +
11938080*a^9*b*x^(1/3) + 1144066*a^10)/x^8