3.2160 \(\int \frac{\left (a+b \sqrt{x}\right )^{10}}{x^5} \, dx\)

Optimal. Leaf size=122 \[ -\frac{a^{10}}{4 x^4}-\frac{20 a^9 b}{7 x^{7/2}}-\frac{15 a^8 b^2}{x^3}-\frac{48 a^7 b^3}{x^{5/2}}-\frac{105 a^6 b^4}{x^2}-\frac{168 a^5 b^5}{x^{3/2}}-\frac{210 a^4 b^6}{x}-\frac{240 a^3 b^7}{\sqrt{x}}+45 a^2 b^8 \log (x)+20 a b^9 \sqrt{x}+b^{10} x \]

[Out]

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

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Rubi [A]  time = 0.177279, antiderivative size = 122, 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 \[ -\frac{a^{10}}{4 x^4}-\frac{20 a^9 b}{7 x^{7/2}}-\frac{15 a^8 b^2}{x^3}-\frac{48 a^7 b^3}{x^{5/2}}-\frac{105 a^6 b^4}{x^2}-\frac{168 a^5 b^5}{x^{3/2}}-\frac{210 a^4 b^6}{x}-\frac{240 a^3 b^7}{\sqrt{x}}+45 a^2 b^8 \log (x)+20 a b^9 \sqrt{x}+b^{10} x \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^10/x^5,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{10}}{4 x^{4}} - \frac{20 a^{9} b}{7 x^{\frac{7}{2}}} - \frac{15 a^{8} b^{2}}{x^{3}} - \frac{48 a^{7} b^{3}}{x^{\frac{5}{2}}} - \frac{105 a^{6} b^{4}}{x^{2}} - \frac{168 a^{5} b^{5}}{x^{\frac{3}{2}}} - \frac{210 a^{4} b^{6}}{x} - \frac{240 a^{3} b^{7}}{\sqrt{x}} + 90 a^{2} b^{8} \log{\left (\sqrt{x} \right )} + 20 a b^{9} \sqrt{x} + 2 b^{10} \int ^{\sqrt{x}} x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**(1/2))**10/x**5,x)

[Out]

-a**10/(4*x**4) - 20*a**9*b/(7*x**(7/2)) - 15*a**8*b**2/x**3 - 48*a**7*b**3/x**(
5/2) - 105*a**6*b**4/x**2 - 168*a**5*b**5/x**(3/2) - 210*a**4*b**6/x - 240*a**3*
b**7/sqrt(x) + 90*a**2*b**8*log(sqrt(x)) + 20*a*b**9*sqrt(x) + 2*b**10*Integral(
x, (x, sqrt(x)))

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Mathematica [A]  time = 0.131011, size = 126, normalized size = 1.03 \[ -\frac{a^{10}}{4 x^4}-\frac{20 a^9 b}{7 x^{7/2}}-\frac{15 a^8 b^2}{x^3}-\frac{48 a^7 b^3}{x^{5/2}}-\frac{105 a^6 b^4}{x^2}-\frac{168 a^5 b^5}{x^{3/2}}-\frac{210 a^4 b^6}{x}-\frac{240 a^3 b^7}{\sqrt{x}}+90 a^2 b^8 \log \left (\sqrt{x}\right )+20 a b^9 \sqrt{x}+b^{10} x \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*Sqrt[x])^10/x^5,x]

[Out]

-a^10/(4*x^4) - (20*a^9*b)/(7*x^(7/2)) - (15*a^8*b^2)/x^3 - (48*a^7*b^3)/x^(5/2)
- (105*a^6*b^4)/x^2 - (168*a^5*b^5)/x^(3/2) - (210*a^4*b^6)/x - (240*a^3*b^7)/S
qrt[x] + 20*a*b^9*Sqrt[x] + b^10*x + 90*a^2*b^8*Log[Sqrt[x]]

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Maple [A]  time = 0.006, size = 109, normalized size = 0.9 \[ -{\frac{{a}^{10}}{4\,{x}^{4}}}-{\frac{20\,{a}^{9}b}{7}{x}^{-{\frac{7}{2}}}}-15\,{\frac{{a}^{8}{b}^{2}}{{x}^{3}}}-48\,{\frac{{a}^{7}{b}^{3}}{{x}^{5/2}}}-105\,{\frac{{a}^{6}{b}^{4}}{{x}^{2}}}-168\,{\frac{{a}^{5}{b}^{5}}{{x}^{3/2}}}-210\,{\frac{{a}^{4}{b}^{6}}{x}}+{b}^{10}x+45\,{a}^{2}{b}^{8}\ln \left ( x \right ) -240\,{\frac{{a}^{3}{b}^{7}}{\sqrt{x}}}+20\,a{b}^{9}\sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/2))^10/x^5,x)

[Out]

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

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Maxima [A]  time = 1.44092, size = 147, normalized size = 1.2 \[ b^{10} x + 45 \, a^{2} b^{8} \log \left (x\right ) + 20 \, a b^{9} \sqrt{x} - \frac{6720 \, a^{3} b^{7} x^{\frac{7}{2}} + 5880 \, a^{4} b^{6} x^{3} + 4704 \, a^{5} b^{5} x^{\frac{5}{2}} + 2940 \, a^{6} b^{4} x^{2} + 1344 \, a^{7} b^{3} x^{\frac{3}{2}} + 420 \, a^{8} b^{2} x + 80 \, a^{9} b \sqrt{x} + 7 \, a^{10}}{28 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^10/x^5,x, algorithm="maxima")

[Out]

b^10*x + 45*a^2*b^8*log(x) + 20*a*b^9*sqrt(x) - 1/28*(6720*a^3*b^7*x^(7/2) + 588
0*a^4*b^6*x^3 + 4704*a^5*b^5*x^(5/2) + 2940*a^6*b^4*x^2 + 1344*a^7*b^3*x^(3/2) +
420*a^8*b^2*x + 80*a^9*b*sqrt(x) + 7*a^10)/x^4

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Fricas [A]  time = 0.235328, size = 158, normalized size = 1.3 \[ \frac{28 \, b^{10} x^{5} + 2520 \, a^{2} b^{8} x^{4} \log \left (\sqrt{x}\right ) - 5880 \, a^{4} b^{6} x^{3} - 2940 \, a^{6} b^{4} x^{2} - 420 \, a^{8} b^{2} x - 7 \, a^{10} + 16 \,{\left (35 \, a b^{9} x^{4} - 420 \, a^{3} b^{7} x^{3} - 294 \, a^{5} b^{5} x^{2} - 84 \, a^{7} b^{3} x - 5 \, a^{9} b\right )} \sqrt{x}}{28 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^10/x^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 5.84552, size = 124, normalized size = 1.02 \[ - \frac{a^{10}}{4 x^{4}} - \frac{20 a^{9} b}{7 x^{\frac{7}{2}}} - \frac{15 a^{8} b^{2}}{x^{3}} - \frac{48 a^{7} b^{3}}{x^{\frac{5}{2}}} - \frac{105 a^{6} b^{4}}{x^{2}} - \frac{168 a^{5} b^{5}}{x^{\frac{3}{2}}} - \frac{210 a^{4} b^{6}}{x} - \frac{240 a^{3} b^{7}}{\sqrt{x}} + 45 a^{2} b^{8} \log{\left (x \right )} + 20 a b^{9} \sqrt{x} + b^{10} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**(1/2))**10/x**5,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.221342, size = 149, normalized size = 1.22 \[ b^{10} x + 45 \, a^{2} b^{8}{\rm ln}\left ({\left | x \right |}\right ) + 20 \, a b^{9} \sqrt{x} - \frac{6720 \, a^{3} b^{7} x^{\frac{7}{2}} + 5880 \, a^{4} b^{6} x^{3} + 4704 \, a^{5} b^{5} x^{\frac{5}{2}} + 2940 \, a^{6} b^{4} x^{2} + 1344 \, a^{7} b^{3} x^{\frac{3}{2}} + 420 \, a^{8} b^{2} x + 80 \, a^{9} b \sqrt{x} + 7 \, a^{10}}{28 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^10/x^5,x, algorithm="giac")

[Out]

b^10*x + 45*a^2*b^8*ln(abs(x)) + 20*a*b^9*sqrt(x) - 1/28*(6720*a^3*b^7*x^(7/2) +
5880*a^4*b^6*x^3 + 4704*a^5*b^5*x^(5/2) + 2940*a^6*b^4*x^2 + 1344*a^7*b^3*x^(3/
2) + 420*a^8*b^2*x + 80*a^9*b*sqrt(x) + 7*a^10)/x^4