3.365 \(\int \frac{1}{x (-a+b x)^{5/2}} \, dx\)
Optimal. Leaf size=60 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{a^{5/2}}+\frac{2}{a^2 \sqrt{b x-a}}-\frac{2}{3 a (b x-a)^{3/2}} \]
[Out]
-2/(3*a*(-a + b*x)^(3/2)) + 2/(a^2*Sqrt[-a + b*x]) + (2*ArcTan[Sqrt[-a + b*x]/Sq
rt[a]])/a^(5/2)
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Rubi [A] time = 0.0558207, antiderivative size = 60, 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
\[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{a^{5/2}}+\frac{2}{a^2 \sqrt{b x-a}}-\frac{2}{3 a (b x-a)^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[1/(x*(-a + b*x)^(5/2)),x]
[Out]
-2/(3*a*(-a + b*x)^(3/2)) + 2/(a^2*Sqrt[-a + b*x]) + (2*ArcTan[Sqrt[-a + b*x]/Sq
rt[a]])/a^(5/2)
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Rubi in Sympy [A] time = 8.22088, size = 48, normalized size = 0.8 \[ - \frac{2}{3 a \left (- a + b x\right )^{\frac{3}{2}}} + \frac{2}{a^{2} \sqrt{- a + b x}} + \frac{2 \operatorname{atan}{\left (\frac{\sqrt{- a + b x}}{\sqrt{a}} \right )}}{a^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x/(b*x-a)**(5/2),x)
[Out]
-2/(3*a*(-a + b*x)**(3/2)) + 2/(a**2*sqrt(-a + b*x)) + 2*atan(sqrt(-a + b*x)/sqr
t(a))/a**(5/2)
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Mathematica [A] time = 0.0844067, size = 52, normalized size = 0.87 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{a^{5/2}}-\frac{8 a-6 b x}{3 a^2 (b x-a)^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x*(-a + b*x)^(5/2)),x]
[Out]
-(8*a - 6*b*x)/(3*a^2*(-a + b*x)^(3/2)) + (2*ArcTan[Sqrt[-a + b*x]/Sqrt[a]])/a^(
5/2)
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Maple [A] time = 0.014, size = 49, normalized size = 0.8 \[ -{\frac{2}{3\,a} \left ( bx-a \right ) ^{-{\frac{3}{2}}}}+2\,{\frac{1}{{a}^{5/2}}\arctan \left ({\frac{\sqrt{bx-a}}{\sqrt{a}}} \right ) }+2\,{\frac{1}{{a}^{2}\sqrt{bx-a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x/(b*x-a)^(5/2),x)
[Out]
-2/3/a/(b*x-a)^(3/2)+2*arctan((b*x-a)^(1/2)/a^(1/2))/a^(5/2)+2/a^2/(b*x-a)^(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)^(5/2)*x),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.230968, size = 1, normalized size = 0.02 \[ \left [\frac{3 \,{\left (b x - a\right )}^{\frac{3}{2}} \log \left (\frac{{\left (b x - 2 \, a\right )} \sqrt{-a} + 2 \, \sqrt{b x - a} a}{x}\right ) + 2 \,{\left (3 \, b x - 4 \, a\right )} \sqrt{-a}}{3 \,{\left (a^{2} b x - a^{3}\right )} \sqrt{b x - a} \sqrt{-a}}, -\frac{2 \,{\left (3 \,{\left (b x - a\right )}^{\frac{3}{2}} \arctan \left (\frac{\sqrt{a}}{\sqrt{b x - a}}\right ) -{\left (3 \, b x - 4 \, a\right )} \sqrt{a}\right )}}{3 \,{\left (a^{2} b x - a^{3}\right )} \sqrt{b x - a} \sqrt{a}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x - a)^(5/2)*x),x, algorithm="fricas")
[Out]
[1/3*(3*(b*x - a)^(3/2)*log(((b*x - 2*a)*sqrt(-a) + 2*sqrt(b*x - a)*a)/x) + 2*(3
*b*x - 4*a)*sqrt(-a))/((a^2*b*x - a^3)*sqrt(b*x - a)*sqrt(-a)), -2/3*(3*(b*x - a
)^(3/2)*arctan(sqrt(a)/sqrt(b*x - a)) - (3*b*x - 4*a)*sqrt(a))/((a^2*b*x - a^3)*
sqrt(b*x - a)*sqrt(a))]
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Sympy [A] time = 10.6633, size = 1950, normalized size = 32.5 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x/(b*x-a)**(5/2),x)
[Out]
Piecewise((8*a**7*sqrt(-1 + b*x/a)/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a**(15/2)
*b**2*x**2 + 3*a**(13/2)*b**3*x**3) + 3*I*a**7*log(b*x/a)/(-3*a**(19/2) + 9*a**(
17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) - 6*I*a**7*log(sqrt(b
)*sqrt(x)/sqrt(a))/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a
**(13/2)*b**3*x**3) + 6*a**7*asin(sqrt(a)/(sqrt(b)*sqrt(x)))/(-3*a**(19/2) + 9*a
**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) - 14*a**6*b*x*sqrt
(-1 + b*x/a)/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/
2)*b**3*x**3) - 9*I*a**6*b*x*log(b*x/a)/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a**(
15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) + 18*I*a**6*b*x*log(sqrt(b)*sqrt(x)/sqr
t(a))/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3
*x**3) - 18*a**6*b*x*asin(sqrt(a)/(sqrt(b)*sqrt(x)))/(-3*a**(19/2) + 9*a**(17/2)
*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) + 6*a**5*b**2*x**2*sqrt(-1
+ b*x/a)/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*
b**3*x**3) + 9*I*a**5*b**2*x**2*log(b*x/a)/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a
**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) - 18*I*a**5*b**2*x**2*log(sqrt(b)*sq
rt(x)/sqrt(a))/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(1
3/2)*b**3*x**3) + 18*a**5*b**2*x**2*asin(sqrt(a)/(sqrt(b)*sqrt(x)))/(-3*a**(19/2
) + 9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) - 3*I*a**4*
b**3*x**3*log(b*x/a)/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3
*a**(13/2)*b**3*x**3) + 6*I*a**4*b**3*x**3*log(sqrt(b)*sqrt(x)/sqrt(a))/(-3*a**(
19/2) + 9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) - 6*a**
4*b**3*x**3*asin(sqrt(a)/(sqrt(b)*sqrt(x)))/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*
a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3), Abs(b*x/a) > 1), (8*I*a**7*sqrt(1
- b*x/a)/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b
**3*x**3) + 3*I*a**7*log(b*x/a)/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a**(15/2)*b*
*2*x**2 + 3*a**(13/2)*b**3*x**3) - 6*I*a**7*log(sqrt(1 - b*x/a) + 1)/(-3*a**(19/
2) + 9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) + 3*pi*a**
7/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**
3) - 14*I*a**6*b*x*sqrt(1 - b*x/a)/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a**(15/2)
*b**2*x**2 + 3*a**(13/2)*b**3*x**3) - 9*I*a**6*b*x*log(b*x/a)/(-3*a**(19/2) + 9*
a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) + 18*I*a**6*b*x*l
og(sqrt(1 - b*x/a) + 1)/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2
+ 3*a**(13/2)*b**3*x**3) - 9*pi*a**6*b*x/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a**
(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) + 6*I*a**5*b**2*x**2*sqrt(1 - b*x/a)/(
-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3)
+ 9*I*a**5*b**2*x**2*log(b*x/a)/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a**(15/2)*b*
*2*x**2 + 3*a**(13/2)*b**3*x**3) - 18*I*a**5*b**2*x**2*log(sqrt(1 - b*x/a) + 1)/
(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3)
+ 9*pi*a**5*b**2*x**2/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 +
3*a**(13/2)*b**3*x**3) - 3*I*a**4*b**3*x**3*log(b*x/a)/(-3*a**(19/2) + 9*a**(17
/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) + 6*I*a**4*b**3*x**3*lo
g(sqrt(1 - b*x/a) + 1)/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 +
3*a**(13/2)*b**3*x**3) - 3*pi*a**4*b**3*x**3/(-3*a**(19/2) + 9*a**(17/2)*b*x -
9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3), True))
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GIAC/XCAS [A] time = 0.206596, size = 57, normalized size = 0.95 \[ \frac{2 \, \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right )}{a^{\frac{5}{2}}} + \frac{2 \,{\left (3 \, b x - 4 \, a\right )}}{3 \,{\left (b x - a\right )}^{\frac{3}{2}} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x - a)^(5/2)*x),x, algorithm="giac")
[Out]
2*arctan(sqrt(b*x - a)/sqrt(a))/a^(5/2) + 2/3*(3*b*x - 4*a)/((b*x - a)^(3/2)*a^2
)