3.1782 \(\int \frac{1}{\sqrt{a+\frac{b}{x}} x^{7/2}} \, dx\)

Optimal. Leaf size=83 \[ -\frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{4 b^{5/2}}+\frac{3 a \sqrt{a+\frac{b}{x}}}{4 b^2 \sqrt{x}}-\frac{\sqrt{a+\frac{b}{x}}}{2 b x^{3/2}} \]

[Out]

-Sqrt[a + b/x]/(2*b*x^(3/2)) + (3*a*Sqrt[a + b/x])/(4*b^2*Sqrt[x]) - (3*a^2*ArcT
anh[Sqrt[b]/(Sqrt[a + b/x]*Sqrt[x])])/(4*b^(5/2))

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Rubi [A]  time = 0.117942, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ -\frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{4 b^{5/2}}+\frac{3 a \sqrt{a+\frac{b}{x}}}{4 b^2 \sqrt{x}}-\frac{\sqrt{a+\frac{b}{x}}}{2 b x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[a + b/x]*x^(7/2)),x]

[Out]

-Sqrt[a + b/x]/(2*b*x^(3/2)) + (3*a*Sqrt[a + b/x])/(4*b^2*Sqrt[x]) - (3*a^2*ArcT
anh[Sqrt[b]/(Sqrt[a + b/x]*Sqrt[x])])/(4*b^(5/2))

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Rubi in Sympy [A]  time = 11.9218, size = 70, normalized size = 0.84 \[ - \frac{3 a^{2} \operatorname{atanh}{\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a + \frac{b}{x}}} \right )}}{4 b^{\frac{5}{2}}} + \frac{3 a \sqrt{a + \frac{b}{x}}}{4 b^{2} \sqrt{x}} - \frac{\sqrt{a + \frac{b}{x}}}{2 b x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x)**(1/2)/x**(7/2),x)

[Out]

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

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Mathematica [A]  time = 0.242186, size = 78, normalized size = 0.94 \[ \frac{-6 a^2 \log \left (\sqrt{b} \sqrt{x} \sqrt{a+\frac{b}{x}}+b\right )+3 a^2 \log (x)+\frac{2 \sqrt{b} \sqrt{a+\frac{b}{x}} (3 a x-2 b)}{x^{3/2}}}{8 b^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[a + b/x]*x^(7/2)),x]

[Out]

((2*Sqrt[b]*Sqrt[a + b/x]*(-2*b + 3*a*x))/x^(3/2) - 6*a^2*Log[b + Sqrt[b]*Sqrt[a
+ b/x]*Sqrt[x]] + 3*a^2*Log[x])/(8*b^(5/2))

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Maple [A]  time = 0.019, size = 74, normalized size = 0.9 \[ -{\frac{1}{4}\sqrt{{\frac{ax+b}{x}}} \left ( 3\,{\it Artanh} \left ({\frac{\sqrt{ax+b}}{\sqrt{b}}} \right ){a}^{2}{x}^{2}-3\,xa\sqrt{ax+b}\sqrt{b}+2\,{b}^{3/2}\sqrt{ax+b} \right ){x}^{-{\frac{3}{2}}}{b}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{ax+b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^(1/2)/x^(7/2),x)

[Out]

-1/4*((a*x+b)/x)^(1/2)/x^(3/2)/b^(5/2)*(3*arctanh((a*x+b)^(1/2)/b^(1/2))*a^2*x^2
-3*x*a*(a*x+b)^(1/2)*b^(1/2)+2*b^(3/2)*(a*x+b)^(1/2))/(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(1/(sqrt(a + b/x)*x^(7/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.253822, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a^{2} x^{2} \log \left (-\frac{2 \, b \sqrt{x} \sqrt{\frac{a x + b}{x}} -{\left (a x + 2 \, b\right )} \sqrt{b}}{x}\right ) + 2 \,{\left (3 \, a x - 2 \, b\right )} \sqrt{b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{8 \, b^{\frac{5}{2}} x^{2}}, \frac{3 \, a^{2} x^{2} \arctan \left (\frac{b}{\sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}\right ) +{\left (3 \, a x - 2 \, b\right )} \sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{4 \, \sqrt{-b} b^{2} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a + b/x)*x^(7/2)),x, algorithm="fricas")

[Out]

[1/8*(3*a^2*x^2*log(-(2*b*sqrt(x)*sqrt((a*x + b)/x) - (a*x + 2*b)*sqrt(b))/x) +
2*(3*a*x - 2*b)*sqrt(b)*sqrt(x)*sqrt((a*x + b)/x))/(b^(5/2)*x^2), 1/4*(3*a^2*x^2
*arctan(b/(sqrt(-b)*sqrt(x)*sqrt((a*x + b)/x))) + (3*a*x - 2*b)*sqrt(-b)*sqrt(x)
*sqrt((a*x + b)/x))/(sqrt(-b)*b^2*x^2)]

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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(1/(a+b/x)**(1/2)/x**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.251149, size = 81, normalized size = 0.98 \[ \frac{1}{4} \, a^{2}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{a x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{2}} + \frac{3 \,{\left (a x + b\right )}^{\frac{3}{2}} - 5 \, \sqrt{a x + b} b}{a^{2} b^{2} x^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a + b/x)*x^(7/2)),x, algorithm="giac")

[Out]

1/4*a^2*(3*arctan(sqrt(a*x + b)/sqrt(-b))/(sqrt(-b)*b^2) + (3*(a*x + b)^(3/2) -
5*sqrt(a*x + b)*b)/(a^2*b^2*x^2))