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

Optimal. Leaf size=88 \[ \frac{5 b \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{a^{7/2}}+\frac{5 \sqrt{b x-a}}{a^3 x}+\frac{10}{3 a^2 x \sqrt{b x-a}}-\frac{2}{3 a x (b x-a)^{3/2}} \]

[Out]

-2/(3*a*x*(-a + b*x)^(3/2)) + 10/(3*a^2*x*Sqrt[-a + b*x]) + (5*Sqrt[-a + b*x])/(
a^3*x) + (5*b*ArcTan[Sqrt[-a + b*x]/Sqrt[a]])/a^(7/2)

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Rubi [A]  time = 0.0774951, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{5 b \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{a^{7/2}}+\frac{5 \sqrt{b x-a}}{a^3 x}+\frac{10}{3 a^2 x \sqrt{b x-a}}-\frac{2}{3 a x (b x-a)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(-a + b*x)^(5/2)),x]

[Out]

-2/(3*a*x*(-a + b*x)^(3/2)) + 10/(3*a^2*x*Sqrt[-a + b*x]) + (5*Sqrt[-a + b*x])/(
a^3*x) + (5*b*ArcTan[Sqrt[-a + b*x]/Sqrt[a]])/a^(7/2)

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Rubi in Sympy [A]  time = 10.7394, size = 70, normalized size = 0.8 \[ - \frac{2}{3 a x \left (- a + b x\right )^{\frac{3}{2}}} + \frac{10}{3 a^{2} x \sqrt{- a + b x}} + \frac{5 \sqrt{- a + b x}}{a^{3} x} + \frac{5 b \operatorname{atan}{\left (\frac{\sqrt{- a + b x}}{\sqrt{a}} \right )}}{a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/(3*a*x*(-a + b*x)**(3/2)) + 10/(3*a**2*x*sqrt(-a + b*x)) + 5*sqrt(-a + b*x)/(
a**3*x) + 5*b*atan(sqrt(-a + b*x)/sqrt(a))/a**(7/2)

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Mathematica [A]  time = 0.130028, size = 67, normalized size = 0.76 \[ \frac{5 b \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{a^{7/2}}+\frac{3 a^2-20 a b x+15 b^2 x^2}{3 a^3 x (b x-a)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(3*a^2 - 20*a*b*x + 15*b^2*x^2)/(3*a^3*x*(-a + b*x)^(3/2)) + (5*b*ArcTan[Sqrt[-a
 + b*x]/Sqrt[a]])/a^(7/2)

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Maple [A]  time = 0.021, size = 68, normalized size = 0.8 \[ -{\frac{2\,b}{3\,{a}^{2}} \left ( bx-a \right ) ^{-{\frac{3}{2}}}}+4\,{\frac{b}{{a}^{3}\sqrt{bx-a}}}+{\frac{1}{{a}^{3}x}\sqrt{bx-a}}+5\,{\frac{b}{{a}^{7/2}}\arctan \left ({\frac{\sqrt{bx-a}}{\sqrt{a}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(b*x-a)^(5/2),x)

[Out]

-2/3*b/a^2/(b*x-a)^(3/2)+4*b/a^3/(b*x-a)^(1/2)+(b*x-a)^(1/2)/a^3/x+5*b*arctan((b
*x-a)^(1/2)/a^(1/2))/a^(7/2)

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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^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.228455, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (b^{2} x^{2} - a b x\right )} \sqrt{b x - a} \log \left (\frac{{\left (b x - 2 \, a\right )} \sqrt{-a} + 2 \, \sqrt{b x - a} a}{x}\right ) + 2 \,{\left (15 \, b^{2} x^{2} - 20 \, a b x + 3 \, a^{2}\right )} \sqrt{-a}}{6 \,{\left (a^{3} b x^{2} - a^{4} x\right )} \sqrt{b x - a} \sqrt{-a}}, -\frac{15 \,{\left (b^{2} x^{2} - a b x\right )} \sqrt{b x - a} \arctan \left (\frac{\sqrt{a}}{\sqrt{b x - a}}\right ) -{\left (15 \, b^{2} x^{2} - 20 \, a b x + 3 \, a^{2}\right )} \sqrt{a}}{3 \,{\left (a^{3} b x^{2} - a^{4} x\right )} \sqrt{b x - a} \sqrt{a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/6*(15*(b^2*x^2 - a*b*x)*sqrt(b*x - a)*log(((b*x - 2*a)*sqrt(-a) + 2*sqrt(b*x
- a)*a)/x) + 2*(15*b^2*x^2 - 20*a*b*x + 3*a^2)*sqrt(-a))/((a^3*b*x^2 - a^4*x)*sq
rt(b*x - a)*sqrt(-a)), -1/3*(15*(b^2*x^2 - a*b*x)*sqrt(b*x - a)*arctan(sqrt(a)/s
qrt(b*x - a)) - (15*b^2*x^2 - 20*a*b*x + 3*a^2)*sqrt(a))/((a^3*b*x^2 - a^4*x)*sq
rt(b*x - a)*sqrt(a))]

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Sympy [A]  time = 18.4644, size = 2234, normalized size = 25.39 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-6*a**17*sqrt(-1 + b*x/a)/(-6*a**(39/2)*x + 18*a**(37/2)*b*x**2 - 18*
a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) + 46*a**16*b*x*sqrt(-1 + b*x/a)/(-6
*a**(39/2)*x + 18*a**(37/2)*b*x**2 - 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x
**4) + 15*I*a**16*b*x*log(b*x/a)/(-6*a**(39/2)*x + 18*a**(37/2)*b*x**2 - 18*a**(
35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) - 30*I*a**16*b*x*log(sqrt(b)*sqrt(x)/sq
rt(a))/(-6*a**(39/2)*x + 18*a**(37/2)*b*x**2 - 18*a**(35/2)*b**2*x**3 + 6*a**(33
/2)*b**3*x**4) + 30*a**16*b*x*asin(sqrt(a)/(sqrt(b)*sqrt(x)))/(-6*a**(39/2)*x +
18*a**(37/2)*b*x**2 - 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) - 70*a**15
*b**2*x**2*sqrt(-1 + b*x/a)/(-6*a**(39/2)*x + 18*a**(37/2)*b*x**2 - 18*a**(35/2)
*b**2*x**3 + 6*a**(33/2)*b**3*x**4) - 45*I*a**15*b**2*x**2*log(b*x/a)/(-6*a**(39
/2)*x + 18*a**(37/2)*b*x**2 - 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) +
90*I*a**15*b**2*x**2*log(sqrt(b)*sqrt(x)/sqrt(a))/(-6*a**(39/2)*x + 18*a**(37/2)
*b*x**2 - 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) - 90*a**15*b**2*x**2*a
sin(sqrt(a)/(sqrt(b)*sqrt(x)))/(-6*a**(39/2)*x + 18*a**(37/2)*b*x**2 - 18*a**(35
/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) + 30*a**14*b**3*x**3*sqrt(-1 + b*x/a)/(-6
*a**(39/2)*x + 18*a**(37/2)*b*x**2 - 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x
**4) + 45*I*a**14*b**3*x**3*log(b*x/a)/(-6*a**(39/2)*x + 18*a**(37/2)*b*x**2 - 1
8*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) - 90*I*a**14*b**3*x**3*log(sqrt(b
)*sqrt(x)/sqrt(a))/(-6*a**(39/2)*x + 18*a**(37/2)*b*x**2 - 18*a**(35/2)*b**2*x**
3 + 6*a**(33/2)*b**3*x**4) + 90*a**14*b**3*x**3*asin(sqrt(a)/(sqrt(b)*sqrt(x)))/
(-6*a**(39/2)*x + 18*a**(37/2)*b*x**2 - 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**
3*x**4) - 15*I*a**13*b**4*x**4*log(b*x/a)/(-6*a**(39/2)*x + 18*a**(37/2)*b*x**2
- 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) + 30*I*a**13*b**4*x**4*log(sqr
t(b)*sqrt(x)/sqrt(a))/(-6*a**(39/2)*x + 18*a**(37/2)*b*x**2 - 18*a**(35/2)*b**2*
x**3 + 6*a**(33/2)*b**3*x**4) - 30*a**13*b**4*x**4*asin(sqrt(a)/(sqrt(b)*sqrt(x)
))/(-6*a**(39/2)*x + 18*a**(37/2)*b*x**2 - 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*
b**3*x**4), Abs(b*x/a) > 1), (-6*I*a**17*sqrt(1 - b*x/a)/(-6*a**(39/2)*x + 18*a*
*(37/2)*b*x**2 - 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) + 46*I*a**16*b*
x*sqrt(1 - b*x/a)/(-6*a**(39/2)*x + 18*a**(37/2)*b*x**2 - 18*a**(35/2)*b**2*x**3
 + 6*a**(33/2)*b**3*x**4) + 15*I*a**16*b*x*log(b*x/a)/(-6*a**(39/2)*x + 18*a**(3
7/2)*b*x**2 - 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) - 30*I*a**16*b*x*l
og(sqrt(1 - b*x/a) + 1)/(-6*a**(39/2)*x + 18*a**(37/2)*b*x**2 - 18*a**(35/2)*b**
2*x**3 + 6*a**(33/2)*b**3*x**4) + 15*pi*a**16*b*x/(-6*a**(39/2)*x + 18*a**(37/2)
*b*x**2 - 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) - 70*I*a**15*b**2*x**2
*sqrt(1 - b*x/a)/(-6*a**(39/2)*x + 18*a**(37/2)*b*x**2 - 18*a**(35/2)*b**2*x**3
+ 6*a**(33/2)*b**3*x**4) - 45*I*a**15*b**2*x**2*log(b*x/a)/(-6*a**(39/2)*x + 18*
a**(37/2)*b*x**2 - 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) + 90*I*a**15*
b**2*x**2*log(sqrt(1 - b*x/a) + 1)/(-6*a**(39/2)*x + 18*a**(37/2)*b*x**2 - 18*a*
*(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) - 45*pi*a**15*b**2*x**2/(-6*a**(39/2)
*x + 18*a**(37/2)*b*x**2 - 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) + 30*
I*a**14*b**3*x**3*sqrt(1 - b*x/a)/(-6*a**(39/2)*x + 18*a**(37/2)*b*x**2 - 18*a**
(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) + 45*I*a**14*b**3*x**3*log(b*x/a)/(-6*
a**(39/2)*x + 18*a**(37/2)*b*x**2 - 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x*
*4) - 90*I*a**14*b**3*x**3*log(sqrt(1 - b*x/a) + 1)/(-6*a**(39/2)*x + 18*a**(37/
2)*b*x**2 - 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) + 45*pi*a**14*b**3*x
**3/(-6*a**(39/2)*x + 18*a**(37/2)*b*x**2 - 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)
*b**3*x**4) - 15*I*a**13*b**4*x**4*log(b*x/a)/(-6*a**(39/2)*x + 18*a**(37/2)*b*x
**2 - 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) + 30*I*a**13*b**4*x**4*log
(sqrt(1 - b*x/a) + 1)/(-6*a**(39/2)*x + 18*a**(37/2)*b*x**2 - 18*a**(35/2)*b**2*
x**3 + 6*a**(33/2)*b**3*x**4) - 15*pi*a**13*b**4*x**4/(-6*a**(39/2)*x + 18*a**(3
7/2)*b*x**2 - 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4), True))

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GIAC/XCAS [A]  time = 0.209397, size = 89, normalized size = 1.01 \[ \frac{5 \, b \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right )}{a^{\frac{7}{2}}} + \frac{2 \,{\left (6 \,{\left (b x - a\right )} b - a b\right )}}{3 \,{\left (b x - a\right )}^{\frac{3}{2}} a^{3}} + \frac{\sqrt{b x - a}}{a^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*b*arctan(sqrt(b*x - a)/sqrt(a))/a^(7/2) + 2/3*(6*(b*x - a)*b - a*b)/((b*x - a)
^(3/2)*a^3) + sqrt(b*x - a)/(a^3*x)

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