3.474 \(\int \frac{1}{x^{5/2} (-a+b x)} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0460443, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{2 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{5/2}}+\frac{2 b}{a^2 \sqrt{x}}+\frac{2}{3 a x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(5/2)*(-a + b*x)),x]

[Out]

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

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Rubi in Sympy [A]  time = 9.08007, size = 49, normalized size = 0.92 \[ \frac{2}{3 a x^{\frac{3}{2}}} + \frac{2 b}{a^{2} \sqrt{x}} - \frac{2 b^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**(5/2)/(b*x-a),x)

[Out]

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

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Mathematica [A]  time = 0.042775, size = 48, normalized size = 0.91 \[ \frac{2 (a+3 b x)}{3 a^2 x^{3/2}}-\frac{2 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^(5/2)*(-a + b*x)),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(5/2)/(b*x-a),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.219309, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, b x^{\frac{3}{2}} \sqrt{\frac{b}{a}} \log \left (\frac{b x - 2 \, a \sqrt{x} \sqrt{\frac{b}{a}} + a}{b x - a}\right ) + 6 \, b x + 2 \, a}{3 \, a^{2} x^{\frac{3}{2}}}, \frac{2 \,{\left (3 \, b x^{\frac{3}{2}} \sqrt{-\frac{b}{a}} \arctan \left (\frac{a \sqrt{-\frac{b}{a}}}{b \sqrt{x}}\right ) + 3 \, b x + a\right )}}{3 \, a^{2} x^{\frac{3}{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x - a)*x^(5/2)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 6.40469, size = 578, normalized size = 10.91 \[ \begin{cases} \frac{6 a^{\frac{11}{2}} b^{2} x^{2} \operatorname{acoth}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{- 3 a^{8} \sqrt{b} x^{2} + 3 a^{7} b^{\frac{3}{2}} x^{3}} + \frac{3 i \pi a^{\frac{11}{2}} b^{2} x^{2}}{- 3 a^{8} \sqrt{b} x^{2} + 3 a^{7} b^{\frac{3}{2}} x^{3}} - \frac{6 a^{\frac{9}{2}} b^{3} x^{3} \operatorname{acoth}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{- 3 a^{8} \sqrt{b} x^{2} + 3 a^{7} b^{\frac{3}{2}} x^{3}} - \frac{3 i \pi a^{\frac{9}{2}} b^{3} x^{3}}{- 3 a^{8} \sqrt{b} x^{2} + 3 a^{7} b^{\frac{3}{2}} x^{3}} - \frac{2 a^{7} \sqrt{b} \sqrt{x}}{- 3 a^{8} \sqrt{b} x^{2} + 3 a^{7} b^{\frac{3}{2}} x^{3}} - \frac{4 a^{6} b^{\frac{3}{2}} x^{\frac{3}{2}}}{- 3 a^{8} \sqrt{b} x^{2} + 3 a^{7} b^{\frac{3}{2}} x^{3}} + \frac{6 a^{5} b^{\frac{5}{2}} x^{\frac{5}{2}}}{- 3 a^{8} \sqrt{b} x^{2} + 3 a^{7} b^{\frac{3}{2}} x^{3}} & \text{for}\: \left |{\frac{b x}{a}}\right | > 1 \\\frac{6 a^{\frac{11}{2}} b^{2} x^{2} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{- 3 a^{8} \sqrt{b} x^{2} + 3 a^{7} b^{\frac{3}{2}} x^{3}} - \frac{6 a^{\frac{9}{2}} b^{3} x^{3} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{- 3 a^{8} \sqrt{b} x^{2} + 3 a^{7} b^{\frac{3}{2}} x^{3}} - \frac{2 a^{7} \sqrt{b} \sqrt{x}}{- 3 a^{8} \sqrt{b} x^{2} + 3 a^{7} b^{\frac{3}{2}} x^{3}} - \frac{4 a^{6} b^{\frac{3}{2}} x^{\frac{3}{2}}}{- 3 a^{8} \sqrt{b} x^{2} + 3 a^{7} b^{\frac{3}{2}} x^{3}} + \frac{6 a^{5} b^{\frac{5}{2}} x^{\frac{5}{2}}}{- 3 a^{8} \sqrt{b} x^{2} + 3 a^{7} b^{\frac{3}{2}} x^{3}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**(5/2)/(b*x-a),x)

[Out]

Piecewise((6*a**(11/2)*b**2*x**2*acoth(sqrt(b)*sqrt(x)/sqrt(a))/(-3*a**8*sqrt(b)
*x**2 + 3*a**7*b**(3/2)*x**3) + 3*I*pi*a**(11/2)*b**2*x**2/(-3*a**8*sqrt(b)*x**2
+ 3*a**7*b**(3/2)*x**3) - 6*a**(9/2)*b**3*x**3*acoth(sqrt(b)*sqrt(x)/sqrt(a))/(
-3*a**8*sqrt(b)*x**2 + 3*a**7*b**(3/2)*x**3) - 3*I*pi*a**(9/2)*b**3*x**3/(-3*a**
8*sqrt(b)*x**2 + 3*a**7*b**(3/2)*x**3) - 2*a**7*sqrt(b)*sqrt(x)/(-3*a**8*sqrt(b)
*x**2 + 3*a**7*b**(3/2)*x**3) - 4*a**6*b**(3/2)*x**(3/2)/(-3*a**8*sqrt(b)*x**2 +
3*a**7*b**(3/2)*x**3) + 6*a**5*b**(5/2)*x**(5/2)/(-3*a**8*sqrt(b)*x**2 + 3*a**7
*b**(3/2)*x**3), Abs(b*x/a) > 1), (6*a**(11/2)*b**2*x**2*atanh(sqrt(b)*sqrt(x)/s
qrt(a))/(-3*a**8*sqrt(b)*x**2 + 3*a**7*b**(3/2)*x**3) - 6*a**(9/2)*b**3*x**3*ata
nh(sqrt(b)*sqrt(x)/sqrt(a))/(-3*a**8*sqrt(b)*x**2 + 3*a**7*b**(3/2)*x**3) - 2*a*
*7*sqrt(b)*sqrt(x)/(-3*a**8*sqrt(b)*x**2 + 3*a**7*b**(3/2)*x**3) - 4*a**6*b**(3/
2)*x**(3/2)/(-3*a**8*sqrt(b)*x**2 + 3*a**7*b**(3/2)*x**3) + 6*a**5*b**(5/2)*x**(
5/2)/(-3*a**8*sqrt(b)*x**2 + 3*a**7*b**(3/2)*x**3), True))

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GIAC/XCAS [A]  time = 0.203179, size = 55, normalized size = 1.04 \[ \frac{2 \, b^{2} \arctan \left (\frac{b \sqrt{x}}{\sqrt{-a b}}\right )}{\sqrt{-a b} a^{2}} + \frac{2 \,{\left (3 \, b x + a\right )}}{3 \, a^{2} x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x - a)*x^(5/2)),x, algorithm="giac")

[Out]

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