3.1669 \(\int \frac{1}{\left (a+\frac{b}{x}\right ) x^{7/2}} \, dx\)

Optimal. Leaf size=53 \[ \frac{2 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{b^{5/2}}+\frac{2 a}{b^2 \sqrt{x}}-\frac{2}{3 b x^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0649546, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{2 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{b^{5/2}}+\frac{2 a}{b^2 \sqrt{x}}-\frac{2}{3 b x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)*x^(7/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 10.7323, size = 49, normalized size = 0.92 \[ \frac{2 a^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{b^{\frac{5}{2}}} + \frac{2 a}{b^{2} \sqrt{x}} - \frac{2}{3 b x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.044517, size = 50, normalized size = 0.94 \[ \frac{2 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{b^{5/2}}+\frac{2 (3 a x-b)}{3 b^2 x^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 43, normalized size = 0.8 \[ -{\frac{2}{3\,b}{x}^{-{\frac{3}{2}}}}+2\,{\frac{a}{{b}^{2}\sqrt{x}}}+2\,{\frac{{a}^{2}}{{b}^{2}\sqrt{ab}}\arctan \left ({\frac{a\sqrt{x}}{\sqrt{ab}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)/x^(7/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.245342, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, a x^{\frac{3}{2}} \sqrt{-\frac{a}{b}} \log \left (\frac{a x + 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - b}{a x + b}\right ) + 6 \, a x - 2 \, b}{3 \, b^{2} x^{\frac{3}{2}}}, -\frac{2 \,{\left (3 \, a x^{\frac{3}{2}} \sqrt{\frac{a}{b}} \arctan \left (\frac{b \sqrt{\frac{a}{b}}}{a \sqrt{x}}\right ) - 3 \, a x + b\right )}}{3 \, b^{2} x^{\frac{3}{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 116.555, size = 121, normalized size = 2.28 \[ \begin{cases} \frac{\tilde{\infty }}{x^{\frac{3}{2}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{2}{5 a x^{\frac{5}{2}}} & \text{for}\: b = 0 \\- \frac{2}{3 b x^{\frac{3}{2}}} & \text{for}\: a = 0 \\\frac{2 a}{b^{2} \sqrt{x}} - \frac{i a \log{\left (- i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{b^{\frac{5}{2}} \sqrt{\frac{1}{a}}} + \frac{i a \log{\left (i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{b^{\frac{5}{2}} \sqrt{\frac{1}{a}}} - \frac{2}{3 b x^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x)/x**(7/2),x)

[Out]

Piecewise((zoo/x**(3/2), Eq(a, 0) & Eq(b, 0)), (-2/(5*a*x**(5/2)), Eq(b, 0)), (-
2/(3*b*x**(3/2)), Eq(a, 0)), (2*a/(b**2*sqrt(x)) - I*a*log(-I*sqrt(b)*sqrt(1/a)
+ sqrt(x))/(b**(5/2)*sqrt(1/a)) + I*a*log(I*sqrt(b)*sqrt(1/a) + sqrt(x))/(b**(5/
2)*sqrt(1/a)) - 2/(3*b*x**(3/2)), True))

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GIAC/XCAS [A]  time = 0.219167, size = 55, normalized size = 1.04 \[ \frac{2 \, a^{2} \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{2}} + \frac{2 \,{\left (3 \, a x - b\right )}}{3 \, b^{2} x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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