3.598 \(\int \frac {\sqrt {x}}{(a-b x)^{3/2}} \, dx\)

Optimal. Leaf size=50 \[ \frac {2 \sqrt {x}}{b \sqrt {a-b x}}-\frac {2 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{b^{3/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {47, 63, 217, 203} \[ \frac {2 \sqrt {x}}{b \sqrt {a-b x}}-\frac {2 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{b^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/(a - b*x)^(3/2),x]

[Out]

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

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && 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 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.08, size = 66, normalized size = 1.32 \[ \frac {2 \sqrt {b} \sqrt {x}-2 \sqrt {a} \sqrt {1-\frac {b x}{a}} \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{3/2} \sqrt {a-b x}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]/(a - b*x)^(3/2),x]

[Out]

(2*Sqrt[b]*Sqrt[x] - 2*Sqrt[a]*Sqrt[1 - (b*x)/a]*ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(b^(3/2)*Sqrt[a - b*x])

________________________________________________________________________________________

fricas [A]  time = 0.46, size = 128, normalized size = 2.56 \[ \left [-\frac {{\left (b x - a\right )} \sqrt {-b} \log \left (-2 \, b x - 2 \, \sqrt {-b x + a} \sqrt {-b} \sqrt {x} + a\right ) + 2 \, \sqrt {-b x + a} b \sqrt {x}}{b^{3} x - a b^{2}}, \frac {2 \, {\left ({\left (b x - a\right )} \sqrt {b} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) - \sqrt {-b x + a} b \sqrt {x}\right )}}{b^{3} x - a b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [B]  time = 113.04, size = 98, normalized size = 1.96 \[ -\frac {{\left (\frac {4 \, a \sqrt {-b}}{{\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{2} - a b} + \frac {\log \left ({\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{2}\right )}{\sqrt {-b}}\right )} {\left | b \right |}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(4*a*sqrt(-b)/((sqrt(-b*x + a)*sqrt(-b) - sqrt((b*x - a)*b + a*b))^2 - a*b) + log((sqrt(-b*x + a)*sqrt(-b) -
sqrt((b*x - a)*b + a*b))^2)/sqrt(-b))*abs(b)/b^2

________________________________________________________________________________________

maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {x}}{\left (-b x +a \right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 3.01, size = 38, normalized size = 0.76 \[ \frac {2 \, \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right )}{b^{\frac {3}{2}}} + \frac {2 \, \sqrt {x}}{\sqrt {-b x + a} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {x}}{{\left (a-b\,x\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)/(a - b*x)^(3/2),x)

[Out]

int(x^(1/2)/(a - b*x)^(3/2), x)

________________________________________________________________________________________

sympy [A]  time = 1.89, size = 102, normalized size = 2.04 \[ \begin {cases} \frac {2 i \operatorname {acosh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{b^{\frac {3}{2}}} - \frac {2 i \sqrt {x}}{\sqrt {a} b \sqrt {-1 + \frac {b x}{a}}} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\- \frac {2 \operatorname {asin}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{b^{\frac {3}{2}}} + \frac {2 \sqrt {x}}{\sqrt {a} b \sqrt {1 - \frac {b x}{a}}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________