3.2163 \(\int \frac{\left (a+b \sqrt{x}\right )^{10}}{x^8} \, dx\)

Optimal. Leaf size=96 \[ \frac{b^3 \left (a+b \sqrt{x}\right )^{11}}{2002 a^4 x^{11/2}}-\frac{b^2 \left (a+b \sqrt{x}\right )^{11}}{182 a^3 x^6}+\frac{3 b \left (a+b \sqrt{x}\right )^{11}}{91 a^2 x^{13/2}}-\frac{\left (a+b \sqrt{x}\right )^{11}}{7 a x^7} \]

[Out]

-(a + b*Sqrt[x])^11/(7*a*x^7) + (3*b*(a + b*Sqrt[x])^11)/(91*a^2*x^(13/2)) - (b^
2*(a + b*Sqrt[x])^11)/(182*a^3*x^6) + (b^3*(a + b*Sqrt[x])^11)/(2002*a^4*x^(11/2
))

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Rubi [A]  time = 0.106826, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{b^3 \left (a+b \sqrt{x}\right )^{11}}{2002 a^4 x^{11/2}}-\frac{b^2 \left (a+b \sqrt{x}\right )^{11}}{182 a^3 x^6}+\frac{3 b \left (a+b \sqrt{x}\right )^{11}}{91 a^2 x^{13/2}}-\frac{\left (a+b \sqrt{x}\right )^{11}}{7 a x^7} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^10/x^8,x]

[Out]

-(a + b*Sqrt[x])^11/(7*a*x^7) + (3*b*(a + b*Sqrt[x])^11)/(91*a^2*x^(13/2)) - (b^
2*(a + b*Sqrt[x])^11)/(182*a^3*x^6) + (b^3*(a + b*Sqrt[x])^11)/(2002*a^4*x^(11/2
))

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Rubi in Sympy [A]  time = 12.058, size = 85, normalized size = 0.89 \[ - \frac{\left (a + b \sqrt{x}\right )^{11}}{7 a x^{7}} + \frac{3 b \left (a + b \sqrt{x}\right )^{11}}{91 a^{2} x^{\frac{13}{2}}} - \frac{b^{2} \left (a + b \sqrt{x}\right )^{11}}{182 a^{3} x^{6}} + \frac{b^{3} \left (a + b \sqrt{x}\right )^{11}}{2002 a^{4} x^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**(1/2))**10/x**8,x)

[Out]

-(a + b*sqrt(x))**11/(7*a*x**7) + 3*b*(a + b*sqrt(x))**11/(91*a**2*x**(13/2)) -
b**2*(a + b*sqrt(x))**11/(182*a**3*x**6) + b**3*(a + b*sqrt(x))**11/(2002*a**4*x
**(11/2))

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Mathematica [A]  time = 0.0444066, size = 124, normalized size = 1.29 \[ -\frac{286 a^{10}+3080 a^9 b \sqrt{x}+15015 a^8 b^2 x+43680 a^7 b^3 x^{3/2}+84084 a^6 b^4 x^2+112112 a^5 b^5 x^{5/2}+105105 a^4 b^6 x^3+68640 a^3 b^7 x^{7/2}+30030 a^2 b^8 x^4+8008 a b^9 x^{9/2}+1001 b^{10} x^5}{2002 x^7} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*Sqrt[x])^10/x^8,x]

[Out]

-(286*a^10 + 3080*a^9*b*Sqrt[x] + 15015*a^8*b^2*x + 43680*a^7*b^3*x^(3/2) + 8408
4*a^6*b^4*x^2 + 112112*a^5*b^5*x^(5/2) + 105105*a^4*b^6*x^3 + 68640*a^3*b^7*x^(7
/2) + 30030*a^2*b^8*x^4 + 8008*a*b^9*x^(9/2) + 1001*b^10*x^5)/(2002*x^7)

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Maple [A]  time = 0.004, size = 113, normalized size = 1.2 \[ -{\frac{{b}^{10}}{2\,{x}^{2}}}-4\,{\frac{a{b}^{9}}{{x}^{5/2}}}-15\,{\frac{{a}^{2}{b}^{8}}{{x}^{3}}}-{\frac{240\,{a}^{3}{b}^{7}}{7}{x}^{-{\frac{7}{2}}}}-{\frac{105\,{a}^{4}{b}^{6}}{2\,{x}^{4}}}-56\,{\frac{{a}^{5}{b}^{5}}{{x}^{9/2}}}-42\,{\frac{{a}^{6}{b}^{4}}{{x}^{5}}}-{\frac{240\,{a}^{7}{b}^{3}}{11}{x}^{-{\frac{11}{2}}}}-{\frac{15\,{a}^{8}{b}^{2}}{2\,{x}^{6}}}-{\frac{20\,{a}^{9}b}{13}{x}^{-{\frac{13}{2}}}}-{\frac{{a}^{10}}{7\,{x}^{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/2))^10/x^8,x)

[Out]

-1/2*b^10/x^2-4*a*b^9/x^(5/2)-15*a^2*b^8/x^3-240/7*a^3*b^7/x^(7/2)-105/2*a^4*b^6
/x^4-56*a^5*b^5/x^(9/2)-42*a^6*b^4/x^5-240/11*a^7*b^3/x^(11/2)-15/2*a^8*b^2/x^6-
20/13*a^9*b/x^(13/2)-1/7*a^10/x^7

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Maxima [A]  time = 1.44842, size = 151, normalized size = 1.57 \[ -\frac{1001 \, b^{10} x^{5} + 8008 \, a b^{9} x^{\frac{9}{2}} + 30030 \, a^{2} b^{8} x^{4} + 68640 \, a^{3} b^{7} x^{\frac{7}{2}} + 105105 \, a^{4} b^{6} x^{3} + 112112 \, a^{5} b^{5} x^{\frac{5}{2}} + 84084 \, a^{6} b^{4} x^{2} + 43680 \, a^{7} b^{3} x^{\frac{3}{2}} + 15015 \, a^{8} b^{2} x + 3080 \, a^{9} b \sqrt{x} + 286 \, a^{10}}{2002 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^10/x^8,x, algorithm="maxima")

[Out]

-1/2002*(1001*b^10*x^5 + 8008*a*b^9*x^(9/2) + 30030*a^2*b^8*x^4 + 68640*a^3*b^7*
x^(7/2) + 105105*a^4*b^6*x^3 + 112112*a^5*b^5*x^(5/2) + 84084*a^6*b^4*x^2 + 4368
0*a^7*b^3*x^(3/2) + 15015*a^8*b^2*x + 3080*a^9*b*sqrt(x) + 286*a^10)/x^7

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Fricas [A]  time = 0.234056, size = 153, normalized size = 1.59 \[ -\frac{1001 \, b^{10} x^{5} + 30030 \, a^{2} b^{8} x^{4} + 105105 \, a^{4} b^{6} x^{3} + 84084 \, a^{6} b^{4} x^{2} + 15015 \, a^{8} b^{2} x + 286 \, a^{10} + 8 \,{\left (1001 \, a b^{9} x^{4} + 8580 \, a^{3} b^{7} x^{3} + 14014 \, a^{5} b^{5} x^{2} + 5460 \, a^{7} b^{3} x + 385 \, a^{9} b\right )} \sqrt{x}}{2002 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^10/x^8,x, algorithm="fricas")

[Out]

-1/2002*(1001*b^10*x^5 + 30030*a^2*b^8*x^4 + 105105*a^4*b^6*x^3 + 84084*a^6*b^4*
x^2 + 15015*a^8*b^2*x + 286*a^10 + 8*(1001*a*b^9*x^4 + 8580*a^3*b^7*x^3 + 14014*
a^5*b^5*x^2 + 5460*a^7*b^3*x + 385*a^9*b)*sqrt(x))/x^7

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Sympy [A]  time = 18.4966, size = 138, normalized size = 1.44 \[ - \frac{a^{10}}{7 x^{7}} - \frac{20 a^{9} b}{13 x^{\frac{13}{2}}} - \frac{15 a^{8} b^{2}}{2 x^{6}} - \frac{240 a^{7} b^{3}}{11 x^{\frac{11}{2}}} - \frac{42 a^{6} b^{4}}{x^{5}} - \frac{56 a^{5} b^{5}}{x^{\frac{9}{2}}} - \frac{105 a^{4} b^{6}}{2 x^{4}} - \frac{240 a^{3} b^{7}}{7 x^{\frac{7}{2}}} - \frac{15 a^{2} b^{8}}{x^{3}} - \frac{4 a b^{9}}{x^{\frac{5}{2}}} - \frac{b^{10}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**(1/2))**10/x**8,x)

[Out]

-a**10/(7*x**7) - 20*a**9*b/(13*x**(13/2)) - 15*a**8*b**2/(2*x**6) - 240*a**7*b*
*3/(11*x**(11/2)) - 42*a**6*b**4/x**5 - 56*a**5*b**5/x**(9/2) - 105*a**4*b**6/(2
*x**4) - 240*a**3*b**7/(7*x**(7/2)) - 15*a**2*b**8/x**3 - 4*a*b**9/x**(5/2) - b*
*10/(2*x**2)

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GIAC/XCAS [A]  time = 0.216388, size = 151, normalized size = 1.57 \[ -\frac{1001 \, b^{10} x^{5} + 8008 \, a b^{9} x^{\frac{9}{2}} + 30030 \, a^{2} b^{8} x^{4} + 68640 \, a^{3} b^{7} x^{\frac{7}{2}} + 105105 \, a^{4} b^{6} x^{3} + 112112 \, a^{5} b^{5} x^{\frac{5}{2}} + 84084 \, a^{6} b^{4} x^{2} + 43680 \, a^{7} b^{3} x^{\frac{3}{2}} + 15015 \, a^{8} b^{2} x + 3080 \, a^{9} b \sqrt{x} + 286 \, a^{10}}{2002 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^10/x^8,x, algorithm="giac")

[Out]

-1/2002*(1001*b^10*x^5 + 8008*a*b^9*x^(9/2) + 30030*a^2*b^8*x^4 + 68640*a^3*b^7*
x^(7/2) + 105105*a^4*b^6*x^3 + 112112*a^5*b^5*x^(5/2) + 84084*a^6*b^4*x^2 + 4368
0*a^7*b^3*x^(3/2) + 15015*a^8*b^2*x + 3080*a^9*b*sqrt(x) + 286*a^10)/x^7