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

Optimal. Leaf size=42 \[ -\frac {2 \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {2}{a \sqrt {b x-a}} \]

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Rubi [A]  time = 0.01, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {51, 63, 205} \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {2}{a \sqrt {b x-a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*(-a + b*x)^(3/2)),x]

[Out]

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

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

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Mathematica [C]  time = 0.01, size = 33, normalized size = 0.79 \begin {gather*} -\frac {2 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};1-\frac {b x}{a}\right )}{a \sqrt {b x-a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(-a + b*x)^(3/2)),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.03, size = 42, normalized size = 1.00 \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {2}{a \sqrt {b x-a}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.26, size = 124, normalized size = 2.95 \begin {gather*} \left [-\frac {{\left (b x - a\right )} \sqrt {-a} \log \left (\frac {b x + 2 \, \sqrt {b x - a} \sqrt {-a} - 2 \, a}{x}\right ) + 2 \, \sqrt {b x - a} a}{a^{2} b x - a^{3}}, -\frac {2 \, {\left ({\left (b x - a\right )} \sqrt {a} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right ) + \sqrt {b x - a} a\right )}}{a^{2} b x - a^{3}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.02, size = 34, normalized size = 0.81 \begin {gather*} -\frac {2 \, \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right )}{a^{\frac {3}{2}}} - \frac {2}{\sqrt {b x - a} a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 35, normalized size = 0.83 \begin {gather*} -\frac {2 \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{a^{\frac {3}{2}}}-\frac {2}{\sqrt {b x -a}\, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.97, size = 34, normalized size = 0.81 \begin {gather*} -\frac {2 \, \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right )}{a^{\frac {3}{2}}} - \frac {2}{\sqrt {b x - a} a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.10, size = 34, normalized size = 0.81 \begin {gather*} -\frac {2\,\mathrm {atan}\left (\frac {\sqrt {b\,x-a}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {2}{a\,\sqrt {b\,x-a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(b*x - a)^(3/2)),x)

[Out]

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

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sympy [C]  time = 2.20, size = 478, normalized size = 11.38 \begin {gather*} \begin {cases} \frac {2 i a^{3} \sqrt {-1 + \frac {b x}{a}}}{i a^{\frac {9}{2}} - i a^{\frac {7}{2}} b x} - \frac {a^{3} \log {\left (\frac {b x}{a} \right )}}{i a^{\frac {9}{2}} - i a^{\frac {7}{2}} b x} + \frac {2 a^{3} \log {\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{i a^{\frac {9}{2}} - i a^{\frac {7}{2}} b x} + \frac {2 i a^{3} \operatorname {asin}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{i a^{\frac {9}{2}} - i a^{\frac {7}{2}} b x} + \frac {a^{2} b x \log {\left (\frac {b x}{a} \right )}}{i a^{\frac {9}{2}} - i a^{\frac {7}{2}} b x} - \frac {2 a^{2} b x \log {\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{i a^{\frac {9}{2}} - i a^{\frac {7}{2}} b x} - \frac {2 i a^{2} b x \operatorname {asin}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{i a^{\frac {9}{2}} - i a^{\frac {7}{2}} b x} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\\frac {2 a^{3} \sqrt {1 - \frac {b x}{a}}}{- i a^{\frac {9}{2}} + i a^{\frac {7}{2}} b x} + \frac {a^{3} \log {\left (\frac {b x}{a} \right )}}{- i a^{\frac {9}{2}} + i a^{\frac {7}{2}} b x} - \frac {2 a^{3} \log {\left (\sqrt {1 - \frac {b x}{a}} + 1 \right )}}{- i a^{\frac {9}{2}} + i a^{\frac {7}{2}} b x} - \frac {i \pi a^{3}}{- i a^{\frac {9}{2}} + i a^{\frac {7}{2}} b x} - \frac {a^{2} b x \log {\left (\frac {b x}{a} \right )}}{- i a^{\frac {9}{2}} + i a^{\frac {7}{2}} b x} + \frac {2 a^{2} b x \log {\left (\sqrt {1 - \frac {b x}{a}} + 1 \right )}}{- i a^{\frac {9}{2}} + i a^{\frac {7}{2}} b x} + \frac {i \pi a^{2} b x}{- i a^{\frac {9}{2}} + i a^{\frac {7}{2}} b x} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((2*I*a**3*sqrt(-1 + b*x/a)/(I*a**(9/2) - I*a**(7/2)*b*x) - a**3*log(b*x/a)/(I*a**(9/2) - I*a**(7/2)*
b*x) + 2*a**3*log(sqrt(b)*sqrt(x)/sqrt(a))/(I*a**(9/2) - I*a**(7/2)*b*x) + 2*I*a**3*asin(sqrt(a)/(sqrt(b)*sqrt
(x)))/(I*a**(9/2) - I*a**(7/2)*b*x) + a**2*b*x*log(b*x/a)/(I*a**(9/2) - I*a**(7/2)*b*x) - 2*a**2*b*x*log(sqrt(
b)*sqrt(x)/sqrt(a))/(I*a**(9/2) - I*a**(7/2)*b*x) - 2*I*a**2*b*x*asin(sqrt(a)/(sqrt(b)*sqrt(x)))/(I*a**(9/2) -
I*a**(7/2)*b*x), Abs(b*x/a) > 1), (2*a**3*sqrt(1 - b*x/a)/(-I*a**(9/2) + I*a**(7/2)*b*x) + a**3*log(b*x/a)/(-
I*a**(9/2) + I*a**(7/2)*b*x) - 2*a**3*log(sqrt(1 - b*x/a) + 1)/(-I*a**(9/2) + I*a**(7/2)*b*x) - I*pi*a**3/(-I*
a**(9/2) + I*a**(7/2)*b*x) - a**2*b*x*log(b*x/a)/(-I*a**(9/2) + I*a**(7/2)*b*x) + 2*a**2*b*x*log(sqrt(1 - b*x/
a) + 1)/(-I*a**(9/2) + I*a**(7/2)*b*x) + I*pi*a**2*b*x/(-I*a**(9/2) + I*a**(7/2)*b*x), True))

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