3.1792 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^{3/2} x^{11/2}} \, dx\)

Optimal. Leaf size=130 \[ \frac{35 a^3 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{8 b^{9/2}}-\frac{35 a^2 \sqrt{a+\frac{b}{x}}}{8 b^4 \sqrt{x}}+\frac{35 a \sqrt{a+\frac{b}{x}}}{12 b^3 x^{3/2}}-\frac{7 \sqrt{a+\frac{b}{x}}}{3 b^2 x^{5/2}}+\frac{2}{b x^{7/2} \sqrt{a+\frac{b}{x}}} \]

[Out]

2/(b*Sqrt[a + b/x]*x^(7/2)) - (7*Sqrt[a + b/x])/(3*b^2*x^(5/2)) + (35*a*Sqrt[a +
b/x])/(12*b^3*x^(3/2)) - (35*a^2*Sqrt[a + b/x])/(8*b^4*Sqrt[x]) + (35*a^3*ArcTa
nh[Sqrt[b]/(Sqrt[a + b/x]*Sqrt[x])])/(8*b^(9/2))

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Rubi [A]  time = 0.204955, antiderivative size = 130, 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 \[ \frac{35 a^3 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{8 b^{9/2}}-\frac{35 a^2 \sqrt{a+\frac{b}{x}}}{8 b^4 \sqrt{x}}+\frac{35 a \sqrt{a+\frac{b}{x}}}{12 b^3 x^{3/2}}-\frac{7 \sqrt{a+\frac{b}{x}}}{3 b^2 x^{5/2}}+\frac{2}{b x^{7/2} \sqrt{a+\frac{b}{x}}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^(3/2)*x^(11/2)),x]

[Out]

2/(b*Sqrt[a + b/x]*x^(7/2)) - (7*Sqrt[a + b/x])/(3*b^2*x^(5/2)) + (35*a*Sqrt[a +
b/x])/(12*b^3*x^(3/2)) - (35*a^2*Sqrt[a + b/x])/(8*b^4*Sqrt[x]) + (35*a^3*ArcTa
nh[Sqrt[b]/(Sqrt[a + b/x]*Sqrt[x])])/(8*b^(9/2))

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Rubi in Sympy [A]  time = 21.1822, size = 114, normalized size = 0.88 \[ \frac{35 a^{3} \operatorname{atanh}{\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a + \frac{b}{x}}} \right )}}{8 b^{\frac{9}{2}}} - \frac{35 a^{2} \sqrt{a + \frac{b}{x}}}{8 b^{4} \sqrt{x}} + \frac{35 a \sqrt{a + \frac{b}{x}}}{12 b^{3} x^{\frac{3}{2}}} + \frac{2}{b x^{\frac{7}{2}} \sqrt{a + \frac{b}{x}}} - \frac{7 \sqrt{a + \frac{b}{x}}}{3 b^{2} x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x)**(3/2)/x**(11/2),x)

[Out]

35*a**3*atanh(sqrt(b)/(sqrt(x)*sqrt(a + b/x)))/(8*b**(9/2)) - 35*a**2*sqrt(a + b
/x)/(8*b**4*sqrt(x)) + 35*a*sqrt(a + b/x)/(12*b**3*x**(3/2)) + 2/(b*x**(7/2)*sqr
t(a + b/x)) - 7*sqrt(a + b/x)/(3*b**2*x**(5/2))

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Mathematica [A]  time = 0.424966, size = 107, normalized size = 0.82 \[ \frac{210 a^3 \log \left (\sqrt{b} \sqrt{x} \sqrt{a+\frac{b}{x}}+b\right )-105 a^3 \log (x)-\frac{2 \sqrt{b} \sqrt{a+\frac{b}{x}} \left (105 a^3 x^3+35 a^2 b x^2-14 a b^2 x+8 b^3\right )}{x^{5/2} (a x+b)}}{48 b^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x)^(3/2)*x^(11/2)),x]

[Out]

((-2*Sqrt[b]*Sqrt[a + b/x]*(8*b^3 - 14*a*b^2*x + 35*a^2*b*x^2 + 105*a^3*x^3))/(x
^(5/2)*(b + a*x)) + 210*a^3*Log[b + Sqrt[b]*Sqrt[a + b/x]*Sqrt[x]] - 105*a^3*Log
[x])/(48*b^(9/2))

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Maple [A]  time = 0.028, size = 89, normalized size = 0.7 \[ -{\frac{1}{24\,ax+24\,b}\sqrt{{\frac{ax+b}{x}}} \left ( -105\,{\it Artanh} \left ({\frac{\sqrt{ax+b}}{\sqrt{b}}} \right ) \sqrt{ax+b}{x}^{3}{a}^{3}-14\,{b}^{5/2}xa+35\,{b}^{3/2}{x}^{2}{a}^{2}+105\,{x}^{3}{a}^{3}\sqrt{b}+8\,{b}^{7/2} \right ){x}^{-{\frac{5}{2}}}{b}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^(3/2)/x^(11/2),x)

[Out]

-1/24*((a*x+b)/x)^(1/2)*(-105*arctanh((a*x+b)^(1/2)/b^(1/2))*(a*x+b)^(1/2)*x^3*a
^3-14*b^(5/2)*x*a+35*b^(3/2)*x^2*a^2+105*x^3*a^3*b^(1/2)+8*b^(7/2))/x^(5/2)/(a*x
+b)/b^(9/2)

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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^(11/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.24994, size = 1, normalized size = 0.01 \[ \left [\frac{105 \, a^{3} x^{\frac{7}{2}} \sqrt{\frac{a x + b}{x}} \log \left (\frac{2 \, b \sqrt{x} \sqrt{\frac{a x + b}{x}} +{\left (a x + 2 \, b\right )} \sqrt{b}}{x}\right ) - 2 \,{\left (105 \, a^{3} x^{3} + 35 \, a^{2} b x^{2} - 14 \, a b^{2} x + 8 \, b^{3}\right )} \sqrt{b}}{48 \, b^{\frac{9}{2}} x^{\frac{7}{2}} \sqrt{\frac{a x + b}{x}}}, -\frac{105 \, a^{3} x^{\frac{7}{2}} \sqrt{\frac{a x + b}{x}} \arctan \left (\frac{b}{\sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}\right ) +{\left (105 \, a^{3} x^{3} + 35 \, a^{2} b x^{2} - 14 \, a b^{2} x + 8 \, b^{3}\right )} \sqrt{-b}}{24 \, \sqrt{-b} b^{4} x^{\frac{7}{2}} \sqrt{\frac{a x + b}{x}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^(3/2)*x^(11/2)),x, algorithm="fricas")

[Out]

[1/48*(105*a^3*x^(7/2)*sqrt((a*x + b)/x)*log((2*b*sqrt(x)*sqrt((a*x + b)/x) + (a
*x + 2*b)*sqrt(b))/x) - 2*(105*a^3*x^3 + 35*a^2*b*x^2 - 14*a*b^2*x + 8*b^3)*sqrt
(b))/(b^(9/2)*x^(7/2)*sqrt((a*x + b)/x)), -1/24*(105*a^3*x^(7/2)*sqrt((a*x + b)/
x)*arctan(b/(sqrt(-b)*sqrt(x)*sqrt((a*x + b)/x))) + (105*a^3*x^3 + 35*a^2*b*x^2
- 14*a*b^2*x + 8*b^3)*sqrt(-b))/(sqrt(-b)*b^4*x^(7/2)*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**(11/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.365273, size = 113, normalized size = 0.87 \[ -\frac{1}{24} \, a^{3}{\left (\frac{105 \, \arctan \left (\frac{\sqrt{a x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{4}} + \frac{48}{\sqrt{a x + b} b^{4}} + \frac{57 \,{\left (a x + b\right )}^{\frac{5}{2}} - 136 \,{\left (a x + b\right )}^{\frac{3}{2}} b + 87 \, \sqrt{a x + b} b^{2}}{a^{3} b^{4} x^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^(3/2)*x^(11/2)),x, algorithm="giac")

[Out]

-1/24*a^3*(105*arctan(sqrt(a*x + b)/sqrt(-b))/(sqrt(-b)*b^4) + 48/(sqrt(a*x + b)
*b^4) + (57*(a*x + b)^(5/2) - 136*(a*x + b)^(3/2)*b + 87*sqrt(a*x + b)*b^2)/(a^3
*b^4*x^3))