3.2331 \(\int \frac{\left (a+b \sqrt [3]{x}\right )^{10}}{x^{10}} \, dx\)

Optimal. Leaf size=142 \[ -\frac{a^{10}}{9 x^9}-\frac{15 a^9 b}{13 x^{26/3}}-\frac{27 a^8 b^2}{5 x^{25/3}}-\frac{15 a^7 b^3}{x^8}-\frac{630 a^6 b^4}{23 x^{23/3}}-\frac{378 a^5 b^5}{11 x^{22/3}}-\frac{30 a^4 b^6}{x^7}-\frac{18 a^3 b^7}{x^{20/3}}-\frac{135 a^2 b^8}{19 x^{19/3}}-\frac{5 a b^9}{3 x^6}-\frac{3 b^{10}}{17 x^{17/3}} \]

[Out]

-a^10/(9*x^9) - (15*a^9*b)/(13*x^(26/3)) - (27*a^8*b^2)/(5*x^(25/3)) - (15*a^7*b
^3)/x^8 - (630*a^6*b^4)/(23*x^(23/3)) - (378*a^5*b^5)/(11*x^(22/3)) - (30*a^4*b^
6)/x^7 - (18*a^3*b^7)/x^(20/3) - (135*a^2*b^8)/(19*x^(19/3)) - (5*a*b^9)/(3*x^6)
- (3*b^10)/(17*x^(17/3))

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Rubi [A]  time = 0.182231, antiderivative size = 142, 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 \[ -\frac{a^{10}}{9 x^9}-\frac{15 a^9 b}{13 x^{26/3}}-\frac{27 a^8 b^2}{5 x^{25/3}}-\frac{15 a^7 b^3}{x^8}-\frac{630 a^6 b^4}{23 x^{23/3}}-\frac{378 a^5 b^5}{11 x^{22/3}}-\frac{30 a^4 b^6}{x^7}-\frac{18 a^3 b^7}{x^{20/3}}-\frac{135 a^2 b^8}{19 x^{19/3}}-\frac{5 a b^9}{3 x^6}-\frac{3 b^{10}}{17 x^{17/3}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^(1/3))^10/x^10,x]

[Out]

-a^10/(9*x^9) - (15*a^9*b)/(13*x^(26/3)) - (27*a^8*b^2)/(5*x^(25/3)) - (15*a^7*b
^3)/x^8 - (630*a^6*b^4)/(23*x^(23/3)) - (378*a^5*b^5)/(11*x^(22/3)) - (30*a^4*b^
6)/x^7 - (18*a^3*b^7)/x^(20/3) - (135*a^2*b^8)/(19*x^(19/3)) - (5*a*b^9)/(3*x^6)
- (3*b^10)/(17*x^(17/3))

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Rubi in Sympy [A]  time = 32.7075, size = 144, normalized size = 1.01 \[ - \frac{a^{10}}{9 x^{9}} - \frac{15 a^{9} b}{13 x^{\frac{26}{3}}} - \frac{27 a^{8} b^{2}}{5 x^{\frac{25}{3}}} - \frac{15 a^{7} b^{3}}{x^{8}} - \frac{630 a^{6} b^{4}}{23 x^{\frac{23}{3}}} - \frac{378 a^{5} b^{5}}{11 x^{\frac{22}{3}}} - \frac{30 a^{4} b^{6}}{x^{7}} - \frac{18 a^{3} b^{7}}{x^{\frac{20}{3}}} - \frac{135 a^{2} b^{8}}{19 x^{\frac{19}{3}}} - \frac{5 a b^{9}}{3 x^{6}} - \frac{3 b^{10}}{17 x^{\frac{17}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**10/(9*x**9) - 15*a**9*b/(13*x**(26/3)) - 27*a**8*b**2/(5*x**(25/3)) - 15*a**
7*b**3/x**8 - 630*a**6*b**4/(23*x**(23/3)) - 378*a**5*b**5/(11*x**(22/3)) - 30*a
**4*b**6/x**7 - 18*a**3*b**7/x**(20/3) - 135*a**2*b**8/(19*x**(19/3)) - 5*a*b**9
/(3*x**6) - 3*b**10/(17*x**(17/3))

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Mathematica [A]  time = 0.0510856, size = 142, normalized size = 1. \[ -\frac{a^{10}}{9 x^9}-\frac{15 a^9 b}{13 x^{26/3}}-\frac{27 a^8 b^2}{5 x^{25/3}}-\frac{15 a^7 b^3}{x^8}-\frac{630 a^6 b^4}{23 x^{23/3}}-\frac{378 a^5 b^5}{11 x^{22/3}}-\frac{30 a^4 b^6}{x^7}-\frac{18 a^3 b^7}{x^{20/3}}-\frac{135 a^2 b^8}{19 x^{19/3}}-\frac{5 a b^9}{3 x^6}-\frac{3 b^{10}}{17 x^{17/3}} \]

Antiderivative was successfully verified.

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

[Out]

-a^10/(9*x^9) - (15*a^9*b)/(13*x^(26/3)) - (27*a^8*b^2)/(5*x^(25/3)) - (15*a^7*b
^3)/x^8 - (630*a^6*b^4)/(23*x^(23/3)) - (378*a^5*b^5)/(11*x^(22/3)) - (30*a^4*b^
6)/x^7 - (18*a^3*b^7)/x^(20/3) - (135*a^2*b^8)/(19*x^(19/3)) - (5*a*b^9)/(3*x^6)
- (3*b^10)/(17*x^(17/3))

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Maple [A]  time = 0.011, size = 113, normalized size = 0.8 \[ -{\frac{{a}^{10}}{9\,{x}^{9}}}-{\frac{15\,{a}^{9}b}{13}{x}^{-{\frac{26}{3}}}}-{\frac{27\,{a}^{8}{b}^{2}}{5}{x}^{-{\frac{25}{3}}}}-15\,{\frac{{a}^{7}{b}^{3}}{{x}^{8}}}-{\frac{630\,{a}^{6}{b}^{4}}{23}{x}^{-{\frac{23}{3}}}}-{\frac{378\,{a}^{5}{b}^{5}}{11}{x}^{-{\frac{22}{3}}}}-30\,{\frac{{a}^{4}{b}^{6}}{{x}^{7}}}-18\,{{a}^{3}{b}^{7}{x}^{-{\frac{20}{3}}}}-{\frac{135\,{a}^{2}{b}^{8}}{19}{x}^{-{\frac{19}{3}}}}-{\frac{5\,a{b}^{9}}{3\,{x}^{6}}}-{\frac{3\,{b}^{10}}{17}{x}^{-{\frac{17}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/3))^10/x^10,x)

[Out]

-1/9*a^10/x^9-15/13*a^9*b/x^(26/3)-27/5*a^8*b^2/x^(25/3)-15*a^7*b^3/x^8-630/23*a
^6*b^4/x^(23/3)-378/11*a^5*b^5/x^(22/3)-30*a^4*b^6/x^7-18*a^3*b^7/x^(20/3)-135/1
9*a^2*b^8/x^(19/3)-5/3*a*b^9/x^6-3/17*b^10/x^(17/3)

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Maxima [A]  time = 1.44608, size = 151, normalized size = 1.06 \[ -\frac{8436285 \, b^{10} x^{\frac{10}{3}} + 79676025 \, a b^{9} x^{3} + 339671475 \, a^{2} b^{8} x^{\frac{8}{3}} + 860501070 \, a^{3} b^{7} x^{\frac{7}{3}} + 1434168450 \, a^{4} b^{6} x^{2} + 1642774770 \, a^{5} b^{5} x^{\frac{5}{3}} + 1309458150 \, a^{6} b^{4} x^{\frac{4}{3}} + 717084225 \, a^{7} b^{3} x + 258150321 \, a^{8} b^{2} x^{\frac{2}{3}} + 55160325 \, a^{9} b x^{\frac{1}{3}} + 5311735 \, a^{10}}{47805615 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/47805615*(8436285*b^10*x^(10/3) + 79676025*a*b^9*x^3 + 339671475*a^2*b^8*x^(8
/3) + 860501070*a^3*b^7*x^(7/3) + 1434168450*a^4*b^6*x^2 + 1642774770*a^5*b^5*x^
(5/3) + 1309458150*a^6*b^4*x^(4/3) + 717084225*a^7*b^3*x + 258150321*a^8*b^2*x^(
2/3) + 55160325*a^9*b*x^(1/3) + 5311735*a^10)/x^9

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Fricas [A]  time = 0.216495, size = 154, normalized size = 1.08 \[ -\frac{79676025 \, a b^{9} x^{3} + 1434168450 \, a^{4} b^{6} x^{2} + 717084225 \, a^{7} b^{3} x + 5311735 \, a^{10} + 1235169 \,{\left (275 \, a^{2} b^{8} x^{2} + 1330 \, a^{5} b^{5} x + 209 \, a^{8} b^{2}\right )} x^{\frac{2}{3}} + 28215 \,{\left (299 \, b^{10} x^{3} + 30498 \, a^{3} b^{7} x^{2} + 46410 \, a^{6} b^{4} x + 1955 \, a^{9} b\right )} x^{\frac{1}{3}}}{47805615 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/47805615*(79676025*a*b^9*x^3 + 1434168450*a^4*b^6*x^2 + 717084225*a^7*b^3*x +
5311735*a^10 + 1235169*(275*a^2*b^8*x^2 + 1330*a^5*b^5*x + 209*a^8*b^2)*x^(2/3)
+ 28215*(299*b^10*x^3 + 30498*a^3*b^7*x^2 + 46410*a^6*b^4*x + 1955*a^9*b)*x^(1/
3))/x^9

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Sympy [A]  time = 144.63, size = 144, normalized size = 1.01 \[ - \frac{a^{10}}{9 x^{9}} - \frac{15 a^{9} b}{13 x^{\frac{26}{3}}} - \frac{27 a^{8} b^{2}}{5 x^{\frac{25}{3}}} - \frac{15 a^{7} b^{3}}{x^{8}} - \frac{630 a^{6} b^{4}}{23 x^{\frac{23}{3}}} - \frac{378 a^{5} b^{5}}{11 x^{\frac{22}{3}}} - \frac{30 a^{4} b^{6}}{x^{7}} - \frac{18 a^{3} b^{7}}{x^{\frac{20}{3}}} - \frac{135 a^{2} b^{8}}{19 x^{\frac{19}{3}}} - \frac{5 a b^{9}}{3 x^{6}} - \frac{3 b^{10}}{17 x^{\frac{17}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**10/(9*x**9) - 15*a**9*b/(13*x**(26/3)) - 27*a**8*b**2/(5*x**(25/3)) - 15*a**
7*b**3/x**8 - 630*a**6*b**4/(23*x**(23/3)) - 378*a**5*b**5/(11*x**(22/3)) - 30*a
**4*b**6/x**7 - 18*a**3*b**7/x**(20/3) - 135*a**2*b**8/(19*x**(19/3)) - 5*a*b**9
/(3*x**6) - 3*b**10/(17*x**(17/3))

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GIAC/XCAS [A]  time = 0.220455, size = 151, normalized size = 1.06 \[ -\frac{8436285 \, b^{10} x^{\frac{10}{3}} + 79676025 \, a b^{9} x^{3} + 339671475 \, a^{2} b^{8} x^{\frac{8}{3}} + 860501070 \, a^{3} b^{7} x^{\frac{7}{3}} + 1434168450 \, a^{4} b^{6} x^{2} + 1642774770 \, a^{5} b^{5} x^{\frac{5}{3}} + 1309458150 \, a^{6} b^{4} x^{\frac{4}{3}} + 717084225 \, a^{7} b^{3} x + 258150321 \, a^{8} b^{2} x^{\frac{2}{3}} + 55160325 \, a^{9} b x^{\frac{1}{3}} + 5311735 \, a^{10}}{47805615 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/47805615*(8436285*b^10*x^(10/3) + 79676025*a*b^9*x^3 + 339671475*a^2*b^8*x^(8
/3) + 860501070*a^3*b^7*x^(7/3) + 1434168450*a^4*b^6*x^2 + 1642774770*a^5*b^5*x^
(5/3) + 1309458150*a^6*b^4*x^(4/3) + 717084225*a^7*b^3*x + 258150321*a^8*b^2*x^(
2/3) + 55160325*a^9*b*x^(1/3) + 5311735*a^10)/x^9