3.1682 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^3 \sqrt{x}} \, dx\)

Optimal. Leaf size=82 \[ -\frac{15 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 a^{7/2}}+\frac{15 \sqrt{x}}{4 a^3}-\frac{5 x^{3/2}}{4 a^2 (a x+b)}-\frac{x^{5/2}}{2 a (a x+b)^2} \]

[Out]

(15*Sqrt[x])/(4*a^3) - x^(5/2)/(2*a*(b + a*x)^2) - (5*x^(3/2))/(4*a^2*(b + a*x))
- (15*Sqrt[b]*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]])/(4*a^(7/2))

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Rubi [A]  time = 0.0798338, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{15 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 a^{7/2}}+\frac{15 \sqrt{x}}{4 a^3}-\frac{5 x^{3/2}}{4 a^2 (a x+b)}-\frac{x^{5/2}}{2 a (a x+b)^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^3*Sqrt[x]),x]

[Out]

(15*Sqrt[x])/(4*a^3) - x^(5/2)/(2*a*(b + a*x)^2) - (5*x^(3/2))/(4*a^2*(b + a*x))
- (15*Sqrt[b]*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]])/(4*a^(7/2))

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Rubi in Sympy [A]  time = 14.4742, size = 73, normalized size = 0.89 \[ - \frac{x^{\frac{5}{2}}}{2 a \left (a x + b\right )^{2}} - \frac{5 x^{\frac{3}{2}}}{4 a^{2} \left (a x + b\right )} + \frac{15 \sqrt{x}}{4 a^{3}} - \frac{15 \sqrt{b} \operatorname{atan}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{4 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x**(5/2)/(2*a*(a*x + b)**2) - 5*x**(3/2)/(4*a**2*(a*x + b)) + 15*sqrt(x)/(4*a**
3) - 15*sqrt(b)*atan(sqrt(a)*sqrt(x)/sqrt(b))/(4*a**(7/2))

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Mathematica [A]  time = 0.0668844, size = 70, normalized size = 0.85 \[ \frac{\sqrt{x} \left (8 a^2 x^2+25 a b x+15 b^2\right )}{4 a^3 (a x+b)^2}-\frac{15 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 a^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x]*(15*b^2 + 25*a*b*x + 8*a^2*x^2))/(4*a^3*(b + a*x)^2) - (15*Sqrt[b]*ArcT
an[(Sqrt[a]*Sqrt[x])/Sqrt[b]])/(4*a^(7/2))

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Maple [A]  time = 0.019, size = 66, normalized size = 0.8 \[ 2\,{\frac{\sqrt{x}}{{a}^{3}}}+{\frac{9\,b}{4\,{a}^{2} \left ( ax+b \right ) ^{2}}{x}^{{\frac{3}{2}}}}+{\frac{7\,{b}^{2}}{4\,{a}^{3} \left ( ax+b \right ) ^{2}}\sqrt{x}}-{\frac{15\,b}{4\,{a}^{3}}\arctan \left ({a\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^3/x^(1/2),x)

[Out]

2*x^(1/2)/a^3+9/4/a^2*b/(a*x+b)^2*x^(3/2)+7/4/a^3*b^2/(a*x+b)^2*x^(1/2)-15/4/a^3
*b/(a*b)^(1/2)*arctan(a*x^(1/2)/(a*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/((a + b/x)^3*sqrt(x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.245743, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{a x - 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - b}{a x + b}\right ) + 2 \,{\left (8 \, a^{2} x^{2} + 25 \, a b x + 15 \, b^{2}\right )} \sqrt{x}}{8 \,{\left (a^{5} x^{2} + 2 \, a^{4} b x + a^{3} b^{2}\right )}}, -\frac{15 \,{\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{\sqrt{x}}{\sqrt{\frac{b}{a}}}\right ) -{\left (8 \, a^{2} x^{2} + 25 \, a b x + 15 \, b^{2}\right )} \sqrt{x}}{4 \,{\left (a^{5} x^{2} + 2 \, a^{4} b x + a^{3} b^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(15*(a^2*x^2 + 2*a*b*x + b^2)*sqrt(-b/a)*log((a*x - 2*a*sqrt(x)*sqrt(-b/a)
- b)/(a*x + b)) + 2*(8*a^2*x^2 + 25*a*b*x + 15*b^2)*sqrt(x))/(a^5*x^2 + 2*a^4*b*
x + a^3*b^2), -1/4*(15*(a^2*x^2 + 2*a*b*x + b^2)*sqrt(b/a)*arctan(sqrt(x)/sqrt(b
/a)) - (8*a^2*x^2 + 25*a*b*x + 15*b^2)*sqrt(x))/(a^5*x^2 + 2*a^4*b*x + a^3*b^2)]

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Sympy [A]  time = 144.385, size = 816, normalized size = 9.95 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((zoo*x**(7/2), Eq(a, 0) & Eq(b, 0)), (2*sqrt(x)/a**3, Eq(b, 0)), (2*x*
*(7/2)/(7*b**3), Eq(a, 0)), (16*I*a**3*sqrt(b)*x**(5/2)*sqrt(1/a)/(8*I*a**6*sqrt
(b)*x**2*sqrt(1/a) + 16*I*a**5*b**(3/2)*x*sqrt(1/a) + 8*I*a**4*b**(5/2)*sqrt(1/a
)) + 50*I*a**2*b**(3/2)*x**(3/2)*sqrt(1/a)/(8*I*a**6*sqrt(b)*x**2*sqrt(1/a) + 16
*I*a**5*b**(3/2)*x*sqrt(1/a) + 8*I*a**4*b**(5/2)*sqrt(1/a)) - 15*a**2*b*x**2*log
(-I*sqrt(b)*sqrt(1/a) + sqrt(x))/(8*I*a**6*sqrt(b)*x**2*sqrt(1/a) + 16*I*a**5*b*
*(3/2)*x*sqrt(1/a) + 8*I*a**4*b**(5/2)*sqrt(1/a)) + 15*a**2*b*x**2*log(I*sqrt(b)
*sqrt(1/a) + sqrt(x))/(8*I*a**6*sqrt(b)*x**2*sqrt(1/a) + 16*I*a**5*b**(3/2)*x*sq
rt(1/a) + 8*I*a**4*b**(5/2)*sqrt(1/a)) + 30*I*a*b**(5/2)*sqrt(x)*sqrt(1/a)/(8*I*
a**6*sqrt(b)*x**2*sqrt(1/a) + 16*I*a**5*b**(3/2)*x*sqrt(1/a) + 8*I*a**4*b**(5/2)
*sqrt(1/a)) - 30*a*b**2*x*log(-I*sqrt(b)*sqrt(1/a) + sqrt(x))/(8*I*a**6*sqrt(b)*
x**2*sqrt(1/a) + 16*I*a**5*b**(3/2)*x*sqrt(1/a) + 8*I*a**4*b**(5/2)*sqrt(1/a)) +
30*a*b**2*x*log(I*sqrt(b)*sqrt(1/a) + sqrt(x))/(8*I*a**6*sqrt(b)*x**2*sqrt(1/a)
+ 16*I*a**5*b**(3/2)*x*sqrt(1/a) + 8*I*a**4*b**(5/2)*sqrt(1/a)) - 15*b**3*log(-
I*sqrt(b)*sqrt(1/a) + sqrt(x))/(8*I*a**6*sqrt(b)*x**2*sqrt(1/a) + 16*I*a**5*b**(
3/2)*x*sqrt(1/a) + 8*I*a**4*b**(5/2)*sqrt(1/a)) + 15*b**3*log(I*sqrt(b)*sqrt(1/a
) + sqrt(x))/(8*I*a**6*sqrt(b)*x**2*sqrt(1/a) + 16*I*a**5*b**(3/2)*x*sqrt(1/a) +
8*I*a**4*b**(5/2)*sqrt(1/a)), True))

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GIAC/XCAS [A]  time = 0.219049, size = 80, normalized size = 0.98 \[ -\frac{15 \, b \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} a^{3}} + \frac{2 \, \sqrt{x}}{a^{3}} + \frac{9 \, a b x^{\frac{3}{2}} + 7 \, b^{2} \sqrt{x}}{4 \,{\left (a x + b\right )}^{2} a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-15/4*b*arctan(a*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^3) + 2*sqrt(x)/a^3 + 1/4*(9*a*b
*x^(3/2) + 7*b^2*sqrt(x))/((a*x + b)^2*a^3)