3.361 \(\int \frac{1}{x^3 \sqrt{-a+b x}} \, dx\)

Optimal. Leaf size=74 \[ \frac{3 b^2 \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{4 a^{5/2}}+\frac{3 b \sqrt{b x-a}}{4 a^2 x}+\frac{\sqrt{b x-a}}{2 a x^2} \]

[Out]

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

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Rubi [A]  time = 0.0170277, antiderivative size = 74, 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, Rules used = {51, 63, 205} \[ \frac{3 b^2 \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{4 a^{5/2}}+\frac{3 b \sqrt{b x-a}}{4 a^2 x}+\frac{\sqrt{b x-a}}{2 a x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*Sqrt[-a + b*x]),x]

[Out]

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

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{1}{x^3 \sqrt{-a+b x}} \, dx &=\frac{\sqrt{-a+b x}}{2 a x^2}+\frac{(3 b) \int \frac{1}{x^2 \sqrt{-a+b x}} \, dx}{4 a}\\ &=\frac{\sqrt{-a+b x}}{2 a x^2}+\frac{3 b \sqrt{-a+b x}}{4 a^2 x}+\frac{\left (3 b^2\right ) \int \frac{1}{x \sqrt{-a+b x}} \, dx}{8 a^2}\\ &=\frac{\sqrt{-a+b x}}{2 a x^2}+\frac{3 b \sqrt{-a+b x}}{4 a^2 x}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{-a+b x}\right )}{4 a^2}\\ &=\frac{\sqrt{-a+b x}}{2 a x^2}+\frac{3 b \sqrt{-a+b x}}{4 a^2 x}+\frac{3 b^2 \tan ^{-1}\left (\frac{\sqrt{-a+b x}}{\sqrt{a}}\right )}{4 a^{5/2}}\\ \end{align*}

Mathematica [C]  time = 0.0061519, size = 36, normalized size = 0.49 \[ \frac{2 b^2 \sqrt{b x-a} \, _2F_1\left (\frac{1}{2},3;\frac{3}{2};1-\frac{b x}{a}\right )}{a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*Sqrt[-a + b*x]),x]

[Out]

(2*b^2*Sqrt[-a + b*x]*Hypergeometric2F1[1/2, 3, 3/2, 1 - (b*x)/a])/a^3

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Maple [A]  time = 0.006, size = 59, normalized size = 0.8 \begin{align*}{\frac{3\,{b}^{2}}{4}\arctan \left ({\sqrt{bx-a}{\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{5}{2}}}}+{\frac{1}{2\,a{x}^{2}}\sqrt{bx-a}}+{\frac{3\,b}{4\,{a}^{2}x}\sqrt{bx-a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(b*x-a)^(1/2),x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.54129, size = 300, normalized size = 4.05 \begin{align*} \left [-\frac{3 \, \sqrt{-a} b^{2} x^{2} \log \left (\frac{b x - 2 \, \sqrt{b x - a} \sqrt{-a} - 2 \, a}{x}\right ) - 2 \,{\left (3 \, a b x + 2 \, a^{2}\right )} \sqrt{b x - a}}{8 \, a^{3} x^{2}}, \frac{3 \, \sqrt{a} b^{2} x^{2} \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right ) +{\left (3 \, a b x + 2 \, a^{2}\right )} \sqrt{b x - a}}{4 \, a^{3} x^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 5.75695, size = 219, normalized size = 2.96 \begin{align*} \begin{cases} \frac{i}{2 \sqrt{b} x^{\frac{5}{2}} \sqrt{\frac{a}{b x} - 1}} + \frac{i \sqrt{b}}{4 a x^{\frac{3}{2}} \sqrt{\frac{a}{b x} - 1}} - \frac{3 i b^{\frac{3}{2}}}{4 a^{2} \sqrt{x} \sqrt{\frac{a}{b x} - 1}} + \frac{3 i b^{2} \operatorname{acosh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{4 a^{\frac{5}{2}}} & \text{for}\: \frac{\left |{a}\right |}{\left |{b}\right | \left |{x}\right |} > 1 \\- \frac{1}{2 \sqrt{b} x^{\frac{5}{2}} \sqrt{- \frac{a}{b x} + 1}} - \frac{\sqrt{b}}{4 a x^{\frac{3}{2}} \sqrt{- \frac{a}{b x} + 1}} + \frac{3 b^{\frac{3}{2}}}{4 a^{2} \sqrt{x} \sqrt{- \frac{a}{b x} + 1}} - \frac{3 b^{2} \operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{4 a^{\frac{5}{2}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.18143, size = 92, normalized size = 1.24 \begin{align*} \frac{\frac{3 \, b^{3} \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right )}{a^{\frac{5}{2}}} + \frac{3 \,{\left (b x - a\right )}^{\frac{3}{2}} b^{3} + 5 \, \sqrt{b x - a} a b^{3}}{a^{2} b^{2} x^{2}}}{4 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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