3.606 \(\int \frac{1}{x^{3/2} (a-b x)^{5/2}} \, dx\)

Optimal. Leaf size=67 \[ -\frac{16 \sqrt{a-b x}}{3 a^3 \sqrt{x}}+\frac{8}{3 a^2 \sqrt{x} \sqrt{a-b x}}+\frac{2}{3 a \sqrt{x} (a-b x)^{3/2}} \]

[Out]

2/(3*a*Sqrt[x]*(a - b*x)^(3/2)) + 8/(3*a^2*Sqrt[x]*Sqrt[a - b*x]) - (16*Sqrt[a -
b*x])/(3*a^3*Sqrt[x])

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Rubi [A]  time = 0.0467137, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{16 \sqrt{a-b x}}{3 a^3 \sqrt{x}}+\frac{8}{3 a^2 \sqrt{x} \sqrt{a-b x}}+\frac{2}{3 a \sqrt{x} (a-b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(3/2)*(a - b*x)^(5/2)),x]

[Out]

2/(3*a*Sqrt[x]*(a - b*x)^(3/2)) + 8/(3*a^2*Sqrt[x]*Sqrt[a - b*x]) - (16*Sqrt[a -
b*x])/(3*a^3*Sqrt[x])

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Rubi in Sympy [A]  time = 7.5806, size = 58, normalized size = 0.87 \[ \frac{2}{3 a \sqrt{x} \left (a - b x\right )^{\frac{3}{2}}} + \frac{8}{3 a^{2} \sqrt{x} \sqrt{a - b x}} - \frac{16 \sqrt{a - b x}}{3 a^{3} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**(3/2)/(-b*x+a)**(5/2),x)

[Out]

2/(3*a*sqrt(x)*(a - b*x)**(3/2)) + 8/(3*a**2*sqrt(x)*sqrt(a - b*x)) - 16*sqrt(a
- b*x)/(3*a**3*sqrt(x))

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Mathematica [A]  time = 0.0364345, size = 41, normalized size = 0.61 \[ -\frac{2 \left (3 a^2-12 a b x+8 b^2 x^2\right )}{3 a^3 \sqrt{x} (a-b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^(3/2)*(a - b*x)^(5/2)),x]

[Out]

(-2*(3*a^2 - 12*a*b*x + 8*b^2*x^2))/(3*a^3*Sqrt[x]*(a - b*x)^(3/2))

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Maple [A]  time = 0.007, size = 36, normalized size = 0.5 \[ -{\frac{16\,{b}^{2}{x}^{2}-24\,abx+6\,{a}^{2}}{3\,{a}^{3}} \left ( -bx+a \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(3/2)/(-b*x+a)^(5/2),x)

[Out]

-2/3*(8*b^2*x^2-12*a*b*x+3*a^2)/x^(1/2)/(-b*x+a)^(3/2)/a^3

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Maxima [A]  time = 1.34318, size = 68, normalized size = 1.01 \[ \frac{2 \,{\left (b^{2} - \frac{6 \,{\left (b x - a\right )} b}{x}\right )} x^{\frac{3}{2}}}{3 \,{\left (-b x + a\right )}^{\frac{3}{2}} a^{3}} - \frac{2 \, \sqrt{-b x + a}}{a^{3} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x + a)^(5/2)*x^(3/2)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.212388, size = 62, normalized size = 0.93 \[ \frac{2 \,{\left (8 \, b^{2} x^{2} - 12 \, a b x + 3 \, a^{2}\right )}}{3 \,{\left (a^{3} b x - a^{4}\right )} \sqrt{-b x + a} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x + a)^(5/2)*x^(3/2)),x, algorithm="fricas")

[Out]

2/3*(8*b^2*x^2 - 12*a*b*x + 3*a^2)/((a^3*b*x - a^4)*sqrt(-b*x + a)*sqrt(x))

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Sympy [A]  time = 102.587, size = 314, normalized size = 4.69 \[ \begin{cases} - \frac{6 a^{2} b^{\frac{9}{2}} \sqrt{\frac{a}{b x} - 1}}{3 a^{5} b^{4} - 6 a^{4} b^{5} x + 3 a^{3} b^{6} x^{2}} + \frac{24 a b^{\frac{11}{2}} x \sqrt{\frac{a}{b x} - 1}}{3 a^{5} b^{4} - 6 a^{4} b^{5} x + 3 a^{3} b^{6} x^{2}} - \frac{16 b^{\frac{13}{2}} x^{2} \sqrt{\frac{a}{b x} - 1}}{3 a^{5} b^{4} - 6 a^{4} b^{5} x + 3 a^{3} b^{6} x^{2}} & \text{for}\: \left |{\frac{a}{b x}}\right | > 1 \\- \frac{6 i a^{2} b^{\frac{9}{2}} \sqrt{- \frac{a}{b x} + 1}}{3 a^{5} b^{4} - 6 a^{4} b^{5} x + 3 a^{3} b^{6} x^{2}} + \frac{24 i a b^{\frac{11}{2}} x \sqrt{- \frac{a}{b x} + 1}}{3 a^{5} b^{4} - 6 a^{4} b^{5} x + 3 a^{3} b^{6} x^{2}} - \frac{16 i b^{\frac{13}{2}} x^{2} \sqrt{- \frac{a}{b x} + 1}}{3 a^{5} b^{4} - 6 a^{4} b^{5} x + 3 a^{3} b^{6} x^{2}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**(3/2)/(-b*x+a)**(5/2),x)

[Out]

Piecewise((-6*a**2*b**(9/2)*sqrt(a/(b*x) - 1)/(3*a**5*b**4 - 6*a**4*b**5*x + 3*a
**3*b**6*x**2) + 24*a*b**(11/2)*x*sqrt(a/(b*x) - 1)/(3*a**5*b**4 - 6*a**4*b**5*x
+ 3*a**3*b**6*x**2) - 16*b**(13/2)*x**2*sqrt(a/(b*x) - 1)/(3*a**5*b**4 - 6*a**4
*b**5*x + 3*a**3*b**6*x**2), Abs(a/(b*x)) > 1), (-6*I*a**2*b**(9/2)*sqrt(-a/(b*x
) + 1)/(3*a**5*b**4 - 6*a**4*b**5*x + 3*a**3*b**6*x**2) + 24*I*a*b**(11/2)*x*sqr
t(-a/(b*x) + 1)/(3*a**5*b**4 - 6*a**4*b**5*x + 3*a**3*b**6*x**2) - 16*I*b**(13/2
)*x**2*sqrt(-a/(b*x) + 1)/(3*a**5*b**4 - 6*a**4*b**5*x + 3*a**3*b**6*x**2), True
))

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GIAC/XCAS [A]  time = 0.231043, size = 255, normalized size = 3.81 \[ -\frac{2 \, \sqrt{-b x + a} b^{2}}{\sqrt{{\left (b x - a\right )} b + a b} a^{3}{\left | b \right |}} - \frac{4 \,{\left (3 \,{\left (\sqrt{-b x + a} \sqrt{-b} - \sqrt{{\left (b x - a\right )} b + a b}\right )}^{4} \sqrt{-b} b^{2} - 12 \, a{\left (\sqrt{-b x + a} \sqrt{-b} - \sqrt{{\left (b x - a\right )} b + a b}\right )}^{2} \sqrt{-b} b^{3} + 5 \, a^{2} \sqrt{-b} b^{4}\right )}}{3 \,{\left ({\left (\sqrt{-b x + a} \sqrt{-b} - \sqrt{{\left (b x - a\right )} b + a b}\right )}^{2} - a b\right )}^{3} a^{2}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x + a)^(5/2)*x^(3/2)),x, algorithm="giac")

[Out]

-2*sqrt(-b*x + a)*b^2/(sqrt((b*x - a)*b + a*b)*a^3*abs(b)) - 4/3*(3*(sqrt(-b*x +
a)*sqrt(-b) - sqrt((b*x - a)*b + a*b))^4*sqrt(-b)*b^2 - 12*a*(sqrt(-b*x + a)*sq
rt(-b) - sqrt((b*x - a)*b + a*b))^2*sqrt(-b)*b^3 + 5*a^2*sqrt(-b)*b^4)/(((sqrt(-
b*x + a)*sqrt(-b) - sqrt((b*x - a)*b + a*b))^2 - a*b)^3*a^2*abs(b))