3.2327 \(\int \frac{\left (a+b \sqrt [3]{x}\right )^{10}}{x^6} \, dx\)

Optimal. Leaf size=122 \[ -\frac{b^4 \left (a+b \sqrt [3]{x}\right )^{11}}{5005 a^5 x^{11/3}}+\frac{b^3 \left (a+b \sqrt [3]{x}\right )^{11}}{455 a^4 x^4}-\frac{6 b^2 \left (a+b \sqrt [3]{x}\right )^{11}}{455 a^3 x^{13/3}}+\frac{2 b \left (a+b \sqrt [3]{x}\right )^{11}}{35 a^2 x^{14/3}}-\frac{\left (a+b \sqrt [3]{x}\right )^{11}}{5 a x^5} \]

[Out]

-(a + b*x^(1/3))^11/(5*a*x^5) + (2*b*(a + b*x^(1/3))^11)/(35*a^2*x^(14/3)) - (6*
b^2*(a + b*x^(1/3))^11)/(455*a^3*x^(13/3)) + (b^3*(a + b*x^(1/3))^11)/(455*a^4*x
^4) - (b^4*(a + b*x^(1/3))^11)/(5005*a^5*x^(11/3))

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Rubi [A]  time = 0.13323, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{b^4 \left (a+b \sqrt [3]{x}\right )^{11}}{5005 a^5 x^{11/3}}+\frac{b^3 \left (a+b \sqrt [3]{x}\right )^{11}}{455 a^4 x^4}-\frac{6 b^2 \left (a+b \sqrt [3]{x}\right )^{11}}{455 a^3 x^{13/3}}+\frac{2 b \left (a+b \sqrt [3]{x}\right )^{11}}{35 a^2 x^{14/3}}-\frac{\left (a+b \sqrt [3]{x}\right )^{11}}{5 a x^5} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^(1/3))^10/x^6,x]

[Out]

-(a + b*x^(1/3))^11/(5*a*x^5) + (2*b*(a + b*x^(1/3))^11)/(35*a^2*x^(14/3)) - (6*
b^2*(a + b*x^(1/3))^11)/(455*a^3*x^(13/3)) + (b^3*(a + b*x^(1/3))^11)/(455*a^4*x
^4) - (b^4*(a + b*x^(1/3))^11)/(5005*a^5*x^(11/3))

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Rubi in Sympy [A]  time = 16.1106, size = 110, normalized size = 0.9 \[ - \frac{\left (a + b \sqrt [3]{x}\right )^{11}}{5 a x^{5}} + \frac{2 b \left (a + b \sqrt [3]{x}\right )^{11}}{35 a^{2} x^{\frac{14}{3}}} - \frac{6 b^{2} \left (a + b \sqrt [3]{x}\right )^{11}}{455 a^{3} x^{\frac{13}{3}}} + \frac{b^{3} \left (a + b \sqrt [3]{x}\right )^{11}}{455 a^{4} x^{4}} - \frac{b^{4} \left (a + b \sqrt [3]{x}\right )^{11}}{5005 a^{5} x^{\frac{11}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**(1/3))**10/x**6,x)

[Out]

-(a + b*x**(1/3))**11/(5*a*x**5) + 2*b*(a + b*x**(1/3))**11/(35*a**2*x**(14/3))
- 6*b**2*(a + b*x**(1/3))**11/(455*a**3*x**(13/3)) + b**3*(a + b*x**(1/3))**11/(
455*a**4*x**4) - b**4*(a + b*x**(1/3))**11/(5005*a**5*x**(11/3))

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Mathematica [A]  time = 0.054367, size = 140, normalized size = 1.15 \[ -\frac{a^{10}}{5 x^5}-\frac{15 a^9 b}{7 x^{14/3}}-\frac{135 a^8 b^2}{13 x^{13/3}}-\frac{30 a^7 b^3}{x^4}-\frac{630 a^6 b^4}{11 x^{11/3}}-\frac{378 a^5 b^5}{5 x^{10/3}}-\frac{70 a^4 b^6}{x^3}-\frac{45 a^3 b^7}{x^{8/3}}-\frac{135 a^2 b^8}{7 x^{7/3}}-\frac{5 a b^9}{x^2}-\frac{3 b^{10}}{5 x^{5/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^(1/3))^10/x^6,x]

[Out]

-a^10/(5*x^5) - (15*a^9*b)/(7*x^(14/3)) - (135*a^8*b^2)/(13*x^(13/3)) - (30*a^7*
b^3)/x^4 - (630*a^6*b^4)/(11*x^(11/3)) - (378*a^5*b^5)/(5*x^(10/3)) - (70*a^4*b^
6)/x^3 - (45*a^3*b^7)/x^(8/3) - (135*a^2*b^8)/(7*x^(7/3)) - (5*a*b^9)/x^2 - (3*b
^10)/(5*x^(5/3))

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Maple [A]  time = 0.01, size = 113, normalized size = 0.9 \[ -30\,{\frac{{a}^{7}{b}^{3}}{{x}^{4}}}-{\frac{{a}^{10}}{5\,{x}^{5}}}-{\frac{135\,{a}^{8}{b}^{2}}{13}{x}^{-{\frac{13}{3}}}}-5\,{\frac{a{b}^{9}}{{x}^{2}}}-{\frac{378\,{a}^{5}{b}^{5}}{5}{x}^{-{\frac{10}{3}}}}-45\,{\frac{{a}^{3}{b}^{7}}{{x}^{8/3}}}-{\frac{630\,{a}^{6}{b}^{4}}{11}{x}^{-{\frac{11}{3}}}}-{\frac{15\,{a}^{9}b}{7}{x}^{-{\frac{14}{3}}}}-70\,{\frac{{a}^{4}{b}^{6}}{{x}^{3}}}-{\frac{3\,{b}^{10}}{5}{x}^{-{\frac{5}{3}}}}-{\frac{135\,{a}^{2}{b}^{8}}{7}{x}^{-{\frac{7}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/3))^10/x^6,x)

[Out]

-30*a^7*b^3/x^4-1/5*a^10/x^5-135/13*a^8*b^2/x^(13/3)-5*a*b^9/x^2-378/5*a^5*b^5/x
^(10/3)-45*a^3*b^7/x^(8/3)-630/11*a^6*b^4/x^(11/3)-15/7*a^9*b/x^(14/3)-70*a^4*b^
6/x^3-3/5*b^10/x^(5/3)-135/7*a^2*b^8/x^(7/3)

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Maxima [A]  time = 1.4443, size = 151, normalized size = 1.24 \[ -\frac{3003 \, b^{10} x^{\frac{10}{3}} + 25025 \, a b^{9} x^{3} + 96525 \, a^{2} b^{8} x^{\frac{8}{3}} + 225225 \, a^{3} b^{7} x^{\frac{7}{3}} + 350350 \, a^{4} b^{6} x^{2} + 378378 \, a^{5} b^{5} x^{\frac{5}{3}} + 286650 \, a^{6} b^{4} x^{\frac{4}{3}} + 150150 \, a^{7} b^{3} x + 51975 \, a^{8} b^{2} x^{\frac{2}{3}} + 10725 \, a^{9} b x^{\frac{1}{3}} + 1001 \, a^{10}}{5005 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^10/x^6,x, algorithm="maxima")

[Out]

-1/5005*(3003*b^10*x^(10/3) + 25025*a*b^9*x^3 + 96525*a^2*b^8*x^(8/3) + 225225*a
^3*b^7*x^(7/3) + 350350*a^4*b^6*x^2 + 378378*a^5*b^5*x^(5/3) + 286650*a^6*b^4*x^
(4/3) + 150150*a^7*b^3*x + 51975*a^8*b^2*x^(2/3) + 10725*a^9*b*x^(1/3) + 1001*a^
10)/x^5

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Fricas [A]  time = 0.217515, size = 154, normalized size = 1.26 \[ -\frac{25025 \, a b^{9} x^{3} + 350350 \, a^{4} b^{6} x^{2} + 150150 \, a^{7} b^{3} x + 1001 \, a^{10} + 297 \,{\left (325 \, a^{2} b^{8} x^{2} + 1274 \, a^{5} b^{5} x + 175 \, a^{8} b^{2}\right )} x^{\frac{2}{3}} + 39 \,{\left (77 \, b^{10} x^{3} + 5775 \, a^{3} b^{7} x^{2} + 7350 \, a^{6} b^{4} x + 275 \, a^{9} b\right )} x^{\frac{1}{3}}}{5005 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^10/x^6,x, algorithm="fricas")

[Out]

-1/5005*(25025*a*b^9*x^3 + 350350*a^4*b^6*x^2 + 150150*a^7*b^3*x + 1001*a^10 + 2
97*(325*a^2*b^8*x^2 + 1274*a^5*b^5*x + 175*a^8*b^2)*x^(2/3) + 39*(77*b^10*x^3 +
5775*a^3*b^7*x^2 + 7350*a^6*b^4*x + 275*a^9*b)*x^(1/3))/x^5

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Sympy [A]  time = 26.5369, size = 143, normalized size = 1.17 \[ - \frac{a^{10}}{5 x^{5}} - \frac{15 a^{9} b}{7 x^{\frac{14}{3}}} - \frac{135 a^{8} b^{2}}{13 x^{\frac{13}{3}}} - \frac{30 a^{7} b^{3}}{x^{4}} - \frac{630 a^{6} b^{4}}{11 x^{\frac{11}{3}}} - \frac{378 a^{5} b^{5}}{5 x^{\frac{10}{3}}} - \frac{70 a^{4} b^{6}}{x^{3}} - \frac{45 a^{3} b^{7}}{x^{\frac{8}{3}}} - \frac{135 a^{2} b^{8}}{7 x^{\frac{7}{3}}} - \frac{5 a b^{9}}{x^{2}} - \frac{3 b^{10}}{5 x^{\frac{5}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**(1/3))**10/x**6,x)

[Out]

-a**10/(5*x**5) - 15*a**9*b/(7*x**(14/3)) - 135*a**8*b**2/(13*x**(13/3)) - 30*a*
*7*b**3/x**4 - 630*a**6*b**4/(11*x**(11/3)) - 378*a**5*b**5/(5*x**(10/3)) - 70*a
**4*b**6/x**3 - 45*a**3*b**7/x**(8/3) - 135*a**2*b**8/(7*x**(7/3)) - 5*a*b**9/x*
*2 - 3*b**10/(5*x**(5/3))

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GIAC/XCAS [A]  time = 0.2618, size = 151, normalized size = 1.24 \[ -\frac{3003 \, b^{10} x^{\frac{10}{3}} + 25025 \, a b^{9} x^{3} + 96525 \, a^{2} b^{8} x^{\frac{8}{3}} + 225225 \, a^{3} b^{7} x^{\frac{7}{3}} + 350350 \, a^{4} b^{6} x^{2} + 378378 \, a^{5} b^{5} x^{\frac{5}{3}} + 286650 \, a^{6} b^{4} x^{\frac{4}{3}} + 150150 \, a^{7} b^{3} x + 51975 \, a^{8} b^{2} x^{\frac{2}{3}} + 10725 \, a^{9} b x^{\frac{1}{3}} + 1001 \, a^{10}}{5005 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^10/x^6,x, algorithm="giac")

[Out]

-1/5005*(3003*b^10*x^(10/3) + 25025*a*b^9*x^3 + 96525*a^2*b^8*x^(8/3) + 225225*a
^3*b^7*x^(7/3) + 350350*a^4*b^6*x^2 + 378378*a^5*b^5*x^(5/3) + 286650*a^6*b^4*x^
(4/3) + 150150*a^7*b^3*x + 51975*a^8*b^2*x^(2/3) + 10725*a^9*b*x^(1/3) + 1001*a^
10)/x^5