3.1790 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^{3/2} x^{7/2}} \, dx\)

Optimal. Leaf size=74 \[ \frac{3 a \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{b^{5/2}}-\frac{3 \sqrt{a+\frac{b}{x}}}{b^2 \sqrt{x}}+\frac{2}{b x^{3/2} \sqrt{a+\frac{b}{x}}} \]

[Out]

2/(b*Sqrt[a + b/x]*x^(3/2)) - (3*Sqrt[a + b/x])/(b^2*Sqrt[x]) + (3*a*ArcTanh[Sqr
t[b]/(Sqrt[a + b/x]*Sqrt[x])])/b^(5/2)

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Rubi [A]  time = 0.116558, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ \frac{3 a \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{b^{5/2}}-\frac{3 \sqrt{a+\frac{b}{x}}}{b^2 \sqrt{x}}+\frac{2}{b x^{3/2} \sqrt{a+\frac{b}{x}}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^(3/2)*x^(7/2)),x]

[Out]

2/(b*Sqrt[a + b/x]*x^(3/2)) - (3*Sqrt[a + b/x])/(b^2*Sqrt[x]) + (3*a*ArcTanh[Sqr
t[b]/(Sqrt[a + b/x]*Sqrt[x])])/b^(5/2)

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Rubi in Sympy [A]  time = 11.9811, size = 63, normalized size = 0.85 \[ \frac{3 a \operatorname{atanh}{\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a + \frac{b}{x}}} \right )}}{b^{\frac{5}{2}}} + \frac{2}{b x^{\frac{3}{2}} \sqrt{a + \frac{b}{x}}} - \frac{3 \sqrt{a + \frac{b}{x}}}{b^{2} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x)**(3/2)/x**(7/2),x)

[Out]

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

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Mathematica [A]  time = 0.395971, size = 79, normalized size = 1.07 \[ \frac{-\frac{2 \sqrt{b} \sqrt{a+\frac{b}{x}} (3 a x+b)}{\sqrt{x} (a x+b)}+6 a \log \left (\sqrt{b} \sqrt{x} \sqrt{a+\frac{b}{x}}+b\right )-3 a \log (x)}{2 b^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x)^(3/2)*x^(7/2)),x]

[Out]

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

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Maple [A]  time = 0.027, size = 61, normalized size = 0.8 \[ -{\frac{1}{ax+b}\sqrt{{\frac{ax+b}{x}}} \left ( -3\,{\it Artanh} \left ({\frac{\sqrt{ax+b}}{\sqrt{b}}} \right ) \sqrt{ax+b}xa+3\,ax\sqrt{b}+{b}^{{\frac{3}{2}}} \right ){\frac{1}{\sqrt{x}}}{b}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^(3/2)/x^(7/2),x)

[Out]

-((a*x+b)/x)^(1/2)*(-3*arctanh((a*x+b)^(1/2)/b^(1/2))*(a*x+b)^(1/2)*x*a+3*a*x*b^
(1/2)+b^(3/2))/x^(1/2)/(a*x+b)/b^(5/2)

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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/((a + b/x)^(3/2)*x^(7/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.249103, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a x^{\frac{3}{2}} \sqrt{\frac{a x + b}{x}} \log \left (\frac{2 \, b \sqrt{x} \sqrt{\frac{a x + b}{x}} +{\left (a x + 2 \, b\right )} \sqrt{b}}{x}\right ) - 2 \,{\left (3 \, a x + b\right )} \sqrt{b}}{2 \, b^{\frac{5}{2}} x^{\frac{3}{2}} \sqrt{\frac{a x + b}{x}}}, -\frac{3 \, a x^{\frac{3}{2}} \sqrt{\frac{a x + b}{x}} \arctan \left (\frac{b}{\sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}\right ) +{\left (3 \, a x + b\right )} \sqrt{-b}}{\sqrt{-b} b^{2} x^{\frac{3}{2}} \sqrt{\frac{a x + b}{x}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^(3/2)*x^(7/2)),x, algorithm="fricas")

[Out]

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

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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)**(3/2)/x**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.25126, size = 78, normalized size = 1.05 \[ -a{\left (\frac{3 \, \arctan \left (\frac{\sqrt{a x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{2}} + \frac{3 \, a x + b}{{\left ({\left (a x + b\right )}^{\frac{3}{2}} - \sqrt{a x + b} b\right )} b^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^(3/2)*x^(7/2)),x, algorithm="giac")

[Out]

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