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

Optimal. Leaf size=93 \[ -\frac{15 b^2 \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{4 a^{7/2}}-\frac{15 b \sqrt{b x-a}}{4 a^3 x}-\frac{5 \sqrt{b x-a}}{2 a^2 x^2}-\frac{2}{a x^2 \sqrt{b x-a}} \]

[Out]

-2/(a*x^2*Sqrt[-a + b*x]) - (5*Sqrt[-a + b*x])/(2*a^2*x^2) - (15*b*Sqrt[-a + b*x
])/(4*a^3*x) - (15*b^2*ArcTan[Sqrt[-a + b*x]/Sqrt[a]])/(4*a^(7/2))

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Rubi [A]  time = 0.0797001, antiderivative size = 93, 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 \[ -\frac{15 b^2 \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{4 a^{7/2}}-\frac{15 b \sqrt{b x-a}}{4 a^3 x}-\frac{5 \sqrt{b x-a}}{2 a^2 x^2}-\frac{2}{a x^2 \sqrt{b x-a}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(-a + b*x)^(3/2)),x]

[Out]

-2/(a*x^2*Sqrt[-a + b*x]) - (5*Sqrt[-a + b*x])/(2*a^2*x^2) - (15*b*Sqrt[-a + b*x
])/(4*a^3*x) - (15*b^2*ArcTan[Sqrt[-a + b*x]/Sqrt[a]])/(4*a^(7/2))

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Rubi in Sympy [A]  time = 11.0034, size = 80, normalized size = 0.86 \[ - \frac{2}{a x^{2} \sqrt{- a + b x}} - \frac{5 \sqrt{- a + b x}}{2 a^{2} x^{2}} - \frac{15 b \sqrt{- a + b x}}{4 a^{3} x} - \frac{15 b^{2} \operatorname{atan}{\left (\frac{\sqrt{- a + b x}}{\sqrt{a}} \right )}}{4 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/(a*x**2*sqrt(-a + b*x)) - 5*sqrt(-a + b*x)/(2*a**2*x**2) - 15*b*sqrt(-a + b*x
)/(4*a**3*x) - 15*b**2*atan(sqrt(-a + b*x)/sqrt(a))/(4*a**(7/2))

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Mathematica [A]  time = 0.0987135, size = 71, normalized size = 0.76 \[ \frac{2 a^2+5 a b x-15 b^2 x^2}{4 a^3 x^2 \sqrt{b x-a}}-\frac{15 b^2 \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{4 a^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^2 + 5*a*b*x - 15*b^2*x^2)/(4*a^3*x^2*Sqrt[-a + b*x]) - (15*b^2*ArcTan[Sqrt[
-a + b*x]/Sqrt[a]])/(4*a^(7/2))

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Maple [A]  time = 0.017, size = 75, normalized size = 0.8 \[ -2\,{\frac{{b}^{2}}{{a}^{3}\sqrt{bx-a}}}-{\frac{7}{4\,{a}^{3}{x}^{2}} \left ( bx-a \right ) ^{{\frac{3}{2}}}}-{\frac{9}{4\,{a}^{2}{x}^{2}}\sqrt{bx-a}}-{\frac{15\,{b}^{2}}{4}\arctan \left ({1\sqrt{bx-a}{\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(b*x-a)^(3/2),x)

[Out]

-2*b^2/a^3/(b*x-a)^(1/2)-7/4/a^3/x^2*(b*x-a)^(3/2)-9/4*(b*x-a)^(1/2)/a^2/x^2-15/
4*b^2*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)^(3/2)*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.243475, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, \sqrt{b x - a} b^{2} x^{2} \log \left (\frac{{\left (b x - 2 \, a\right )} \sqrt{-a} - 2 \, \sqrt{b x - a} a}{x}\right ) - 2 \,{\left (15 \, b^{2} x^{2} - 5 \, a b x - 2 \, a^{2}\right )} \sqrt{-a}}{8 \, \sqrt{b x - a} \sqrt{-a} a^{3} x^{2}}, \frac{15 \, \sqrt{b x - a} b^{2} x^{2} \arctan \left (\frac{\sqrt{a}}{\sqrt{b x - a}}\right ) -{\left (15 \, b^{2} x^{2} - 5 \, a b x - 2 \, a^{2}\right )} \sqrt{a}}{4 \, \sqrt{b x - a} a^{\frac{7}{2}} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(15*sqrt(b*x - a)*b^2*x^2*log(((b*x - 2*a)*sqrt(-a) - 2*sqrt(b*x - a)*a)/x)
- 2*(15*b^2*x^2 - 5*a*b*x - 2*a^2)*sqrt(-a))/(sqrt(b*x - a)*sqrt(-a)*a^3*x^2),
1/4*(15*sqrt(b*x - a)*b^2*x^2*arctan(sqrt(a)/sqrt(b*x - a)) - (15*b^2*x^2 - 5*a*
b*x - 2*a^2)*sqrt(a))/(sqrt(b*x - a)*a^(7/2)*x^2)]

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Sympy [A]  time = 18.5893, size = 226, normalized size = 2.43 \[ \begin{cases} - \frac{i}{2 a \sqrt{b} x^{\frac{5}{2}} \sqrt{\frac{a}{b x} - 1}} - \frac{5 i \sqrt{b}}{4 a^{2} x^{\frac{3}{2}} \sqrt{\frac{a}{b x} - 1}} + \frac{15 i b^{\frac{3}{2}}}{4 a^{3} \sqrt{x} \sqrt{\frac{a}{b x} - 1}} - \frac{15 i b^{2} \operatorname{acosh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{4 a^{\frac{7}{2}}} & \text{for}\: \left |{\frac{a}{b x}}\right | > 1 \\\frac{1}{2 a \sqrt{b} x^{\frac{5}{2}} \sqrt{- \frac{a}{b x} + 1}} + \frac{5 \sqrt{b}}{4 a^{2} x^{\frac{3}{2}} \sqrt{- \frac{a}{b x} + 1}} - \frac{15 b^{\frac{3}{2}}}{4 a^{3} \sqrt{x} \sqrt{- \frac{a}{b x} + 1}} + \frac{15 b^{2} \operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{4 a^{\frac{7}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-I/(2*a*sqrt(b)*x**(5/2)*sqrt(a/(b*x) - 1)) - 5*I*sqrt(b)/(4*a**2*x**
(3/2)*sqrt(a/(b*x) - 1)) + 15*I*b**(3/2)/(4*a**3*sqrt(x)*sqrt(a/(b*x) - 1)) - 15
*I*b**2*acosh(sqrt(a)/(sqrt(b)*sqrt(x)))/(4*a**(7/2)), Abs(a/(b*x)) > 1), (1/(2*
a*sqrt(b)*x**(5/2)*sqrt(-a/(b*x) + 1)) + 5*sqrt(b)/(4*a**2*x**(3/2)*sqrt(-a/(b*x
) + 1)) - 15*b**(3/2)/(4*a**3*sqrt(x)*sqrt(-a/(b*x) + 1)) + 15*b**2*asin(sqrt(a)
/(sqrt(b)*sqrt(x)))/(4*a**(7/2)), True))

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GIAC/XCAS [A]  time = 0.205313, size = 109, normalized size = 1.17 \[ -\frac{15 \, b^{2} \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right )}{4 \, a^{\frac{7}{2}}} - \frac{2 \, b^{2}}{\sqrt{b x - a} a^{3}} - \frac{7 \,{\left (b x - a\right )}^{\frac{3}{2}} b^{2} + 9 \, \sqrt{b x - a} a b^{2}}{4 \, a^{3} b^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-15/4*b^2*arctan(sqrt(b*x - a)/sqrt(a))/a^(7/2) - 2*b^2/(sqrt(b*x - a)*a^3) - 1/
4*(7*(b*x - a)^(3/2)*b^2 + 9*sqrt(b*x - a)*a*b^2)/(a^3*b^2*x^2)