3.2157 \(\int \frac{\left (a+b \sqrt{x}\right )^{10}}{x^2} \, dx\)

Optimal. Leaf size=123 \[ -\frac{a^{10}}{x}-\frac{20 a^9 b}{\sqrt{x}}+45 a^8 b^2 \log (x)+240 a^7 b^3 \sqrt{x}+210 a^6 b^4 x+168 a^5 b^5 x^{3/2}+105 a^4 b^6 x^2+48 a^3 b^7 x^{5/2}+15 a^2 b^8 x^3+\frac{20}{7} a b^9 x^{7/2}+\frac{b^{10} x^4}{4} \]

[Out]

-(a^10/x) - (20*a^9*b)/Sqrt[x] + 240*a^7*b^3*Sqrt[x] + 210*a^6*b^4*x + 168*a^5*b
^5*x^(3/2) + 105*a^4*b^6*x^2 + 48*a^3*b^7*x^(5/2) + 15*a^2*b^8*x^3 + (20*a*b^9*x
^(7/2))/7 + (b^10*x^4)/4 + 45*a^8*b^2*Log[x]

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Rubi [A]  time = 0.184149, antiderivative size = 123, 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 \[ -\frac{a^{10}}{x}-\frac{20 a^9 b}{\sqrt{x}}+45 a^8 b^2 \log (x)+240 a^7 b^3 \sqrt{x}+210 a^6 b^4 x+168 a^5 b^5 x^{3/2}+105 a^4 b^6 x^2+48 a^3 b^7 x^{5/2}+15 a^2 b^8 x^3+\frac{20}{7} a b^9 x^{7/2}+\frac{b^{10} x^4}{4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^10/x^2,x]

[Out]

-(a^10/x) - (20*a^9*b)/Sqrt[x] + 240*a^7*b^3*Sqrt[x] + 210*a^6*b^4*x + 168*a^5*b
^5*x^(3/2) + 105*a^4*b^6*x^2 + 48*a^3*b^7*x^(5/2) + 15*a^2*b^8*x^3 + (20*a*b^9*x
^(7/2))/7 + (b^10*x^4)/4 + 45*a^8*b^2*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{10}}{x} - \frac{20 a^{9} b}{\sqrt{x}} + 90 a^{8} b^{2} \log{\left (\sqrt{x} \right )} + 240 a^{7} b^{3} \sqrt{x} + 420 a^{6} b^{4} \int ^{\sqrt{x}} x\, dx + 168 a^{5} b^{5} x^{\frac{3}{2}} + 105 a^{4} b^{6} x^{2} + 48 a^{3} b^{7} x^{\frac{5}{2}} + 15 a^{2} b^{8} x^{3} + \frac{20 a b^{9} x^{\frac{7}{2}}}{7} + \frac{b^{10} x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**10/x - 20*a**9*b/sqrt(x) + 90*a**8*b**2*log(sqrt(x)) + 240*a**7*b**3*sqrt(x)
+ 420*a**6*b**4*Integral(x, (x, sqrt(x))) + 168*a**5*b**5*x**(3/2) + 105*a**4*b
**6*x**2 + 48*a**3*b**7*x**(5/2) + 15*a**2*b**8*x**3 + 20*a*b**9*x**(7/2)/7 + b*
*10*x**4/4

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Mathematica [A]  time = 0.0649921, size = 123, normalized size = 1. \[ -\frac{a^{10}}{x}-\frac{20 a^9 b}{\sqrt{x}}+45 a^8 b^2 \log (x)+240 a^7 b^3 \sqrt{x}+210 a^6 b^4 x+168 a^5 b^5 x^{3/2}+105 a^4 b^6 x^2+48 a^3 b^7 x^{5/2}+15 a^2 b^8 x^3+\frac{20}{7} a b^9 x^{7/2}+\frac{b^{10} x^4}{4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*Sqrt[x])^10/x^2,x]

[Out]

-(a^10/x) - (20*a^9*b)/Sqrt[x] + 240*a^7*b^3*Sqrt[x] + 210*a^6*b^4*x + 168*a^5*b
^5*x^(3/2) + 105*a^4*b^6*x^2 + 48*a^3*b^7*x^(5/2) + 15*a^2*b^8*x^3 + (20*a*b^9*x
^(7/2))/7 + (b^10*x^4)/4 + 45*a^8*b^2*Log[x]

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Maple [A]  time = 0.004, size = 110, normalized size = 0.9 \[ -{\frac{{a}^{10}}{x}}+210\,{a}^{6}{b}^{4}x+168\,{a}^{5}{b}^{5}{x}^{3/2}+105\,{a}^{4}{b}^{6}{x}^{2}+48\,{a}^{3}{b}^{7}{x}^{5/2}+15\,{a}^{2}{b}^{8}{x}^{3}+{\frac{20\,a{b}^{9}}{7}{x}^{{\frac{7}{2}}}}+{\frac{{b}^{10}{x}^{4}}{4}}+45\,{a}^{8}{b}^{2}\ln \left ( x \right ) -20\,{\frac{{a}^{9}b}{\sqrt{x}}}+240\,{a}^{7}{b}^{3}\sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/2))^10/x^2,x)

[Out]

-a^10/x+210*a^6*b^4*x+168*a^5*b^5*x^(3/2)+105*a^4*b^6*x^2+48*a^3*b^7*x^(5/2)+15*
a^2*b^8*x^3+20/7*a*b^9*x^(7/2)+1/4*b^10*x^4+45*a^8*b^2*ln(x)-20*a^9*b/x^(1/2)+24
0*a^7*b^3*x^(1/2)

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Maxima [A]  time = 1.43775, size = 149, normalized size = 1.21 \[ \frac{1}{4} \, b^{10} x^{4} + \frac{20}{7} \, a b^{9} x^{\frac{7}{2}} + 15 \, a^{2} b^{8} x^{3} + 48 \, a^{3} b^{7} x^{\frac{5}{2}} + 105 \, a^{4} b^{6} x^{2} + 168 \, a^{5} b^{5} x^{\frac{3}{2}} + 210 \, a^{6} b^{4} x + 45 \, a^{8} b^{2} \log \left (x\right ) + 240 \, a^{7} b^{3} \sqrt{x} - \frac{20 \, a^{9} b \sqrt{x} + a^{10}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^10/x^2,x, algorithm="maxima")

[Out]

1/4*b^10*x^4 + 20/7*a*b^9*x^(7/2) + 15*a^2*b^8*x^3 + 48*a^3*b^7*x^(5/2) + 105*a^
4*b^6*x^2 + 168*a^5*b^5*x^(3/2) + 210*a^6*b^4*x + 45*a^8*b^2*log(x) + 240*a^7*b^
3*sqrt(x) - (20*a^9*b*sqrt(x) + a^10)/x

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Fricas [A]  time = 0.232829, size = 158, normalized size = 1.28 \[ \frac{7 \, b^{10} x^{5} + 420 \, a^{2} b^{8} x^{4} + 2940 \, a^{4} b^{6} x^{3} + 5880 \, a^{6} b^{4} x^{2} + 2520 \, a^{8} b^{2} x \log \left (\sqrt{x}\right ) - 28 \, a^{10} + 16 \,{\left (5 \, a b^{9} x^{4} + 84 \, a^{3} b^{7} x^{3} + 294 \, a^{5} b^{5} x^{2} + 420 \, a^{7} b^{3} x - 35 \, a^{9} b\right )} \sqrt{x}}{28 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^10/x^2,x, algorithm="fricas")

[Out]

1/28*(7*b^10*x^5 + 420*a^2*b^8*x^4 + 2940*a^4*b^6*x^3 + 5880*a^6*b^4*x^2 + 2520*
a^8*b^2*x*log(sqrt(x)) - 28*a^10 + 16*(5*a*b^9*x^4 + 84*a^3*b^7*x^3 + 294*a^5*b^
5*x^2 + 420*a^7*b^3*x - 35*a^9*b)*sqrt(x))/x

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Sympy [A]  time = 6.55711, size = 124, normalized size = 1.01 \[ - \frac{a^{10}}{x} - \frac{20 a^{9} b}{\sqrt{x}} + 45 a^{8} b^{2} \log{\left (x \right )} + 240 a^{7} b^{3} \sqrt{x} + 210 a^{6} b^{4} x + 168 a^{5} b^{5} x^{\frac{3}{2}} + 105 a^{4} b^{6} x^{2} + 48 a^{3} b^{7} x^{\frac{5}{2}} + 15 a^{2} b^{8} x^{3} + \frac{20 a b^{9} x^{\frac{7}{2}}}{7} + \frac{b^{10} x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**10/x - 20*a**9*b/sqrt(x) + 45*a**8*b**2*log(x) + 240*a**7*b**3*sqrt(x) + 210
*a**6*b**4*x + 168*a**5*b**5*x**(3/2) + 105*a**4*b**6*x**2 + 48*a**3*b**7*x**(5/
2) + 15*a**2*b**8*x**3 + 20*a*b**9*x**(7/2)/7 + b**10*x**4/4

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GIAC/XCAS [A]  time = 0.21679, size = 150, normalized size = 1.22 \[ \frac{1}{4} \, b^{10} x^{4} + \frac{20}{7} \, a b^{9} x^{\frac{7}{2}} + 15 \, a^{2} b^{8} x^{3} + 48 \, a^{3} b^{7} x^{\frac{5}{2}} + 105 \, a^{4} b^{6} x^{2} + 168 \, a^{5} b^{5} x^{\frac{3}{2}} + 210 \, a^{6} b^{4} x + 45 \, a^{8} b^{2}{\rm ln}\left ({\left | x \right |}\right ) + 240 \, a^{7} b^{3} \sqrt{x} - \frac{20 \, a^{9} b \sqrt{x} + a^{10}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^10/x^2,x, algorithm="giac")

[Out]

1/4*b^10*x^4 + 20/7*a*b^9*x^(7/2) + 15*a^2*b^8*x^3 + 48*a^3*b^7*x^(5/2) + 105*a^
4*b^6*x^2 + 168*a^5*b^5*x^(3/2) + 210*a^6*b^4*x + 45*a^8*b^2*ln(abs(x)) + 240*a^
7*b^3*sqrt(x) - (20*a^9*b*sqrt(x) + a^10)/x