3.2159 \(\int \frac{\left (a+b \sqrt{x}\right )^{10}}{x^4} \, dx\)

Optimal. Leaf size=127 \[ -\frac{a^{10}}{3 x^3}-\frac{4 a^9 b}{x^{5/2}}-\frac{45 a^8 b^2}{2 x^2}-\frac{80 a^7 b^3}{x^{3/2}}-\frac{210 a^6 b^4}{x}-\frac{504 a^5 b^5}{\sqrt{x}}+210 a^4 b^6 \log (x)+240 a^3 b^7 \sqrt{x}+45 a^2 b^8 x+\frac{20}{3} a b^9 x^{3/2}+\frac{b^{10} x^2}{2} \]

[Out]

-a^10/(3*x^3) - (4*a^9*b)/x^(5/2) - (45*a^8*b^2)/(2*x^2) - (80*a^7*b^3)/x^(3/2)
- (210*a^6*b^4)/x - (504*a^5*b^5)/Sqrt[x] + 240*a^3*b^7*Sqrt[x] + 45*a^2*b^8*x +
(20*a*b^9*x^(3/2))/3 + (b^10*x^2)/2 + 210*a^4*b^6*Log[x]

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Rubi [A]  time = 0.180176, antiderivative size = 127, 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 \[ -\frac{a^{10}}{3 x^3}-\frac{4 a^9 b}{x^{5/2}}-\frac{45 a^8 b^2}{2 x^2}-\frac{80 a^7 b^3}{x^{3/2}}-\frac{210 a^6 b^4}{x}-\frac{504 a^5 b^5}{\sqrt{x}}+210 a^4 b^6 \log (x)+240 a^3 b^7 \sqrt{x}+45 a^2 b^8 x+\frac{20}{3} a b^9 x^{3/2}+\frac{b^{10} x^2}{2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^10/x^4,x]

[Out]

-a^10/(3*x^3) - (4*a^9*b)/x^(5/2) - (45*a^8*b^2)/(2*x^2) - (80*a^7*b^3)/x^(3/2)
- (210*a^6*b^4)/x - (504*a^5*b^5)/Sqrt[x] + 240*a^3*b^7*Sqrt[x] + 45*a^2*b^8*x +
(20*a*b^9*x^(3/2))/3 + (b^10*x^2)/2 + 210*a^4*b^6*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{10}}{3 x^{3}} - \frac{4 a^{9} b}{x^{\frac{5}{2}}} - \frac{45 a^{8} b^{2}}{2 x^{2}} - \frac{80 a^{7} b^{3}}{x^{\frac{3}{2}}} - \frac{210 a^{6} b^{4}}{x} - \frac{504 a^{5} b^{5}}{\sqrt{x}} + 420 a^{4} b^{6} \log{\left (\sqrt{x} \right )} + 240 a^{3} b^{7} \sqrt{x} + 90 a^{2} b^{8} \int ^{\sqrt{x}} x\, dx + \frac{20 a b^{9} x^{\frac{3}{2}}}{3} + \frac{b^{10} x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**(1/2))**10/x**4,x)

[Out]

-a**10/(3*x**3) - 4*a**9*b/x**(5/2) - 45*a**8*b**2/(2*x**2) - 80*a**7*b**3/x**(3
/2) - 210*a**6*b**4/x - 504*a**5*b**5/sqrt(x) + 420*a**4*b**6*log(sqrt(x)) + 240
*a**3*b**7*sqrt(x) + 90*a**2*b**8*Integral(x, (x, sqrt(x))) + 20*a*b**9*x**(3/2)
/3 + b**10*x**2/2

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Mathematica [A]  time = 0.0895978, size = 124, normalized size = 0.98 \[ 210 a^4 b^6 \log (x)-\frac{2 a^{10}+24 a^9 b \sqrt{x}+135 a^8 b^2 x+480 a^7 b^3 x^{3/2}+1260 a^6 b^4 x^2+3024 a^5 b^5 x^{5/2}-1440 a^3 b^7 x^{7/2}-270 a^2 b^8 x^4-40 a b^9 x^{9/2}-3 b^{10} x^5}{6 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*Sqrt[x])^10/x^4,x]

[Out]

-(2*a^10 + 24*a^9*b*Sqrt[x] + 135*a^8*b^2*x + 480*a^7*b^3*x^(3/2) + 1260*a^6*b^4
*x^2 + 3024*a^5*b^5*x^(5/2) - 1440*a^3*b^7*x^(7/2) - 270*a^2*b^8*x^4 - 40*a*b^9*
x^(9/2) - 3*b^10*x^5)/(6*x^3) + 210*a^4*b^6*Log[x]

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Maple [A]  time = 0.006, size = 110, normalized size = 0.9 \[ -{\frac{{a}^{10}}{3\,{x}^{3}}}-4\,{\frac{{a}^{9}b}{{x}^{5/2}}}-{\frac{45\,{a}^{8}{b}^{2}}{2\,{x}^{2}}}-80\,{\frac{{a}^{7}{b}^{3}}{{x}^{3/2}}}-210\,{\frac{{a}^{6}{b}^{4}}{x}}+45\,{a}^{2}{b}^{8}x+{\frac{20\,a{b}^{9}}{3}{x}^{{\frac{3}{2}}}}+{\frac{{b}^{10}{x}^{2}}{2}}+210\,{a}^{4}{b}^{6}\ln \left ( x \right ) -504\,{\frac{{a}^{5}{b}^{5}}{\sqrt{x}}}+240\,{a}^{3}{b}^{7}\sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/2))^10/x^4,x)

[Out]

-1/3*a^10/x^3-4*a^9*b/x^(5/2)-45/2*a^8*b^2/x^2-80*a^7*b^3/x^(3/2)-210*a^6*b^4/x+
45*a^2*b^8*x+20/3*a*b^9*x^(3/2)+1/2*b^10*x^2+210*a^4*b^6*ln(x)-504*a^5*b^5/x^(1/
2)+240*a^3*b^7*x^(1/2)

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Maxima [A]  time = 1.44064, size = 149, normalized size = 1.17 \[ \frac{1}{2} \, b^{10} x^{2} + \frac{20}{3} \, a b^{9} x^{\frac{3}{2}} + 45 \, a^{2} b^{8} x + 210 \, a^{4} b^{6} \log \left (x\right ) + 240 \, a^{3} b^{7} \sqrt{x} - \frac{3024 \, a^{5} b^{5} x^{\frac{5}{2}} + 1260 \, a^{6} b^{4} x^{2} + 480 \, a^{7} b^{3} x^{\frac{3}{2}} + 135 \, a^{8} b^{2} x + 24 \, a^{9} b \sqrt{x} + 2 \, a^{10}}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^10/x^4,x, algorithm="maxima")

[Out]

1/2*b^10*x^2 + 20/3*a*b^9*x^(3/2) + 45*a^2*b^8*x + 210*a^4*b^6*log(x) + 240*a^3*
b^7*sqrt(x) - 1/6*(3024*a^5*b^5*x^(5/2) + 1260*a^6*b^4*x^2 + 480*a^7*b^3*x^(3/2)
+ 135*a^8*b^2*x + 24*a^9*b*sqrt(x) + 2*a^10)/x^3

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Fricas [A]  time = 0.240245, size = 158, normalized size = 1.24 \[ \frac{3 \, b^{10} x^{5} + 270 \, a^{2} b^{8} x^{4} + 2520 \, a^{4} b^{6} x^{3} \log \left (\sqrt{x}\right ) - 1260 \, a^{6} b^{4} x^{2} - 135 \, a^{8} b^{2} x - 2 \, a^{10} + 8 \,{\left (5 \, a b^{9} x^{4} + 180 \, a^{3} b^{7} x^{3} - 378 \, a^{5} b^{5} x^{2} - 60 \, a^{7} b^{3} x - 3 \, a^{9} b\right )} \sqrt{x}}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^10/x^4,x, algorithm="fricas")

[Out]

1/6*(3*b^10*x^5 + 270*a^2*b^8*x^4 + 2520*a^4*b^6*x^3*log(sqrt(x)) - 1260*a^6*b^4
*x^2 - 135*a^8*b^2*x - 2*a^10 + 8*(5*a*b^9*x^4 + 180*a^3*b^7*x^3 - 378*a^5*b^5*x
^2 - 60*a^7*b^3*x - 3*a^9*b)*sqrt(x))/x^3

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Sympy [A]  time = 6.13967, size = 128, normalized size = 1.01 \[ - \frac{a^{10}}{3 x^{3}} - \frac{4 a^{9} b}{x^{\frac{5}{2}}} - \frac{45 a^{8} b^{2}}{2 x^{2}} - \frac{80 a^{7} b^{3}}{x^{\frac{3}{2}}} - \frac{210 a^{6} b^{4}}{x} - \frac{504 a^{5} b^{5}}{\sqrt{x}} + 210 a^{4} b^{6} \log{\left (x \right )} + 240 a^{3} b^{7} \sqrt{x} + 45 a^{2} b^{8} x + \frac{20 a b^{9} x^{\frac{3}{2}}}{3} + \frac{b^{10} x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**(1/2))**10/x**4,x)

[Out]

-a**10/(3*x**3) - 4*a**9*b/x**(5/2) - 45*a**8*b**2/(2*x**2) - 80*a**7*b**3/x**(3
/2) - 210*a**6*b**4/x - 504*a**5*b**5/sqrt(x) + 210*a**4*b**6*log(x) + 240*a**3*
b**7*sqrt(x) + 45*a**2*b**8*x + 20*a*b**9*x**(3/2)/3 + b**10*x**2/2

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GIAC/XCAS [A]  time = 0.220573, size = 150, normalized size = 1.18 \[ \frac{1}{2} \, b^{10} x^{2} + \frac{20}{3} \, a b^{9} x^{\frac{3}{2}} + 45 \, a^{2} b^{8} x + 210 \, a^{4} b^{6}{\rm ln}\left ({\left | x \right |}\right ) + 240 \, a^{3} b^{7} \sqrt{x} - \frac{3024 \, a^{5} b^{5} x^{\frac{5}{2}} + 1260 \, a^{6} b^{4} x^{2} + 480 \, a^{7} b^{3} x^{\frac{3}{2}} + 135 \, a^{8} b^{2} x + 24 \, a^{9} b \sqrt{x} + 2 \, a^{10}}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^10/x^4,x, algorithm="giac")

[Out]

1/2*b^10*x^2 + 20/3*a*b^9*x^(3/2) + 45*a^2*b^8*x + 210*a^4*b^6*ln(abs(x)) + 240*
a^3*b^7*sqrt(x) - 1/6*(3024*a^5*b^5*x^(5/2) + 1260*a^6*b^4*x^2 + 480*a^7*b^3*x^(
3/2) + 135*a^8*b^2*x + 24*a^9*b*sqrt(x) + 2*a^10)/x^3