3.1774 \(\int \frac{\left (a+\frac{b}{x}\right )^{5/2}}{x^{5/2}} \, dx\)

Optimal. Leaf size=126 \[ \frac{5 a^4 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{64 b^{3/2}}-\frac{5 a^3 \sqrt{a+\frac{b}{x}}}{64 b \sqrt{x}}-\frac{5 a^2 \sqrt{a+\frac{b}{x}}}{32 x^{3/2}}-\frac{5 a \left (a+\frac{b}{x}\right )^{3/2}}{24 x^{3/2}}-\frac{\left (a+\frac{b}{x}\right )^{5/2}}{4 x^{3/2}} \]

[Out]

(-5*a^2*Sqrt[a + b/x])/(32*x^(3/2)) - (5*a*(a + b/x)^(3/2))/(24*x^(3/2)) - (a +
b/x)^(5/2)/(4*x^(3/2)) - (5*a^3*Sqrt[a + b/x])/(64*b*Sqrt[x]) + (5*a^4*ArcTanh[S
qrt[b]/(Sqrt[a + b/x]*Sqrt[x])])/(64*b^(3/2))

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Rubi [A]  time = 0.19238, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ \frac{5 a^4 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{64 b^{3/2}}-\frac{5 a^3 \sqrt{a+\frac{b}{x}}}{64 b \sqrt{x}}-\frac{5 a^2 \sqrt{a+\frac{b}{x}}}{32 x^{3/2}}-\frac{5 a \left (a+\frac{b}{x}\right )^{3/2}}{24 x^{3/2}}-\frac{\left (a+\frac{b}{x}\right )^{5/2}}{4 x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^(5/2)/x^(5/2),x]

[Out]

(-5*a^2*Sqrt[a + b/x])/(32*x^(3/2)) - (5*a*(a + b/x)^(3/2))/(24*x^(3/2)) - (a +
b/x)^(5/2)/(4*x^(3/2)) - (5*a^3*Sqrt[a + b/x])/(64*b*Sqrt[x]) + (5*a^4*ArcTanh[S
qrt[b]/(Sqrt[a + b/x]*Sqrt[x])])/(64*b^(3/2))

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Rubi in Sympy [A]  time = 19.5006, size = 107, normalized size = 0.85 \[ \frac{5 a^{4} \operatorname{atanh}{\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a + \frac{b}{x}}} \right )}}{64 b^{\frac{3}{2}}} - \frac{5 a^{3} \sqrt{a + \frac{b}{x}}}{64 b \sqrt{x}} - \frac{5 a^{2} \sqrt{a + \frac{b}{x}}}{32 x^{\frac{3}{2}}} - \frac{5 a \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{24 x^{\frac{3}{2}}} - \frac{\left (a + \frac{b}{x}\right )^{\frac{5}{2}}}{4 x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*a**4*atanh(sqrt(b)/(sqrt(x)*sqrt(a + b/x)))/(64*b**(3/2)) - 5*a**3*sqrt(a + b/
x)/(64*b*sqrt(x)) - 5*a**2*sqrt(a + b/x)/(32*x**(3/2)) - 5*a*(a + b/x)**(3/2)/(2
4*x**(3/2)) - (a + b/x)**(5/2)/(4*x**(3/2))

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Mathematica [A]  time = 0.375347, size = 100, normalized size = 0.79 \[ \frac{30 a^4 \log \left (\sqrt{b} \sqrt{x} \sqrt{a+\frac{b}{x}}+b\right )-15 a^4 \log (x)-\frac{2 \sqrt{b} \sqrt{a+\frac{b}{x}} \left (15 a^3 x^3+118 a^2 b x^2+136 a b^2 x+48 b^3\right )}{x^{7/2}}}{384 b^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^(5/2)/x^(5/2),x]

[Out]

((-2*Sqrt[b]*Sqrt[a + b/x]*(48*b^3 + 136*a*b^2*x + 118*a^2*b*x^2 + 15*a^3*x^3))/
x^(7/2) + 30*a^4*Log[b + Sqrt[b]*Sqrt[a + b/x]*Sqrt[x]] - 15*a^4*Log[x])/(384*b^
(3/2))

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Maple [A]  time = 0.026, size = 110, normalized size = 0.9 \[ -{\frac{1}{192}\sqrt{{\frac{ax+b}{x}}} \left ( -15\,{\it Artanh} \left ({\frac{\sqrt{ax+b}}{\sqrt{b}}} \right ){a}^{4}{x}^{4}+48\,{b}^{7/2}\sqrt{ax+b}+136\,xa{b}^{5/2}\sqrt{ax+b}+118\,{x}^{2}{a}^{2}{b}^{3/2}\sqrt{ax+b}+15\,{x}^{3}{a}^{3}\sqrt{ax+b}\sqrt{b} \right ){x}^{-{\frac{7}{2}}}{b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{ax+b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(5/2)/x^(5/2),x)

[Out]

-1/192*((a*x+b)/x)^(1/2)*(-15*arctanh((a*x+b)^(1/2)/b^(1/2))*a^4*x^4+48*b^(7/2)*
(a*x+b)^(1/2)+136*x*a*b^(5/2)*(a*x+b)^(1/2)+118*x^2*a^2*b^(3/2)*(a*x+b)^(1/2)+15
*x^3*a^3*(a*x+b)^(1/2)*b^(1/2))/x^(7/2)/b^(3/2)/(a*x+b)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^(5/2)/x^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.24971, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{4} x^{4} \log \left (\frac{2 \, b \sqrt{x} \sqrt{\frac{a x + b}{x}} +{\left (a x + 2 \, b\right )} \sqrt{b}}{x}\right ) - 2 \,{\left (15 \, a^{3} x^{3} + 118 \, a^{2} b x^{2} + 136 \, a b^{2} x + 48 \, b^{3}\right )} \sqrt{b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{384 \, b^{\frac{3}{2}} x^{4}}, -\frac{15 \, a^{4} x^{4} \arctan \left (\frac{b}{\sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}\right ) +{\left (15 \, a^{3} x^{3} + 118 \, a^{2} b x^{2} + 136 \, a b^{2} x + 48 \, b^{3}\right )} \sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{192 \, \sqrt{-b} b x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^(5/2)/x^(5/2),x, algorithm="fricas")

[Out]

[1/384*(15*a^4*x^4*log((2*b*sqrt(x)*sqrt((a*x + b)/x) + (a*x + 2*b)*sqrt(b))/x)
- 2*(15*a^3*x^3 + 118*a^2*b*x^2 + 136*a*b^2*x + 48*b^3)*sqrt(b)*sqrt(x)*sqrt((a*
x + b)/x))/(b^(3/2)*x^4), -1/192*(15*a^4*x^4*arctan(b/(sqrt(-b)*sqrt(x)*sqrt((a*
x + b)/x))) + (15*a^3*x^3 + 118*a^2*b*x^2 + 136*a*b^2*x + 48*b^3)*sqrt(-b)*sqrt(
x)*sqrt((a*x + b)/x))/(sqrt(-b)*b*x^4)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.327285, size = 113, normalized size = 0.9 \[ -\frac{1}{192} \, a^{4}{\left (\frac{15 \, \arctan \left (\frac{\sqrt{a x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b} + \frac{15 \,{\left (a x + b\right )}^{\frac{7}{2}} + 73 \,{\left (a x + b\right )}^{\frac{5}{2}} b - 55 \,{\left (a x + b\right )}^{\frac{3}{2}} b^{2} + 15 \, \sqrt{a x + b} b^{3}}{a^{4} b x^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^(5/2)/x^(5/2),x, algorithm="giac")

[Out]

-1/192*a^4*(15*arctan(sqrt(a*x + b)/sqrt(-b))/(sqrt(-b)*b) + (15*(a*x + b)^(7/2)
+ 73*(a*x + b)^(5/2)*b - 55*(a*x + b)^(3/2)*b^2 + 15*sqrt(a*x + b)*b^3)/(a^4*b*
x^4))