\(\int \frac {(2-b x)^{3/2}}{x^{3/2}} \, dx\) [543]

Optimal result

Integrand size = 16, antiderivative size = 60 \[ \int \frac {(2-b x)^{3/2}}{x^{3/2}} \, dx=-3 b \sqrt {x} \sqrt {2-b x}-\frac {2 (2-b x)^{3/2}}{\sqrt {x}}-6 \sqrt {b} \arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right ) \]

[Out]

-6*arcsin(1/2*b^(1/2)*x^(1/2)*2^(1/2))*b^(1/2)-2*(-b*x+2)^(3/2)/x^(1/2)-3*b*x^(1/2)*(-b*x+2)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {49, 52, 56, 222} \[ \int \frac {(2-b x)^{3/2}}{x^{3/2}} \, dx=-6 \sqrt {b} \arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )-\frac {2 (2-b x)^{3/2}}{\sqrt {x}}-3 b \sqrt {x} \sqrt {2-b x} \]

[In]

Int[(2 - b*x)^(3/2)/x^(3/2),x]

[Out]

-3*b*Sqrt[x]*Sqrt[2 - b*x] - (2*(2 - b*x)^(3/2))/Sqrt[x] - 6*Sqrt[b]*ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[2]]

Rule 49

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 52

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

Rule 56

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

Rule 222

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

Rubi steps \begin{align*} \text {integral}& = -\frac {2 (2-b x)^{3/2}}{\sqrt {x}}-(3 b) \int \frac {\sqrt {2-b x}}{\sqrt {x}} \, dx \\ & = -3 b \sqrt {x} \sqrt {2-b x}-\frac {2 (2-b x)^{3/2}}{\sqrt {x}}-(3 b) \int \frac {1}{\sqrt {x} \sqrt {2-b x}} \, dx \\ & = -3 b \sqrt {x} \sqrt {2-b x}-\frac {2 (2-b x)^{3/2}}{\sqrt {x}}-(6 b) \text {Subst}\left (\int \frac {1}{\sqrt {2-b x^2}} \, dx,x,\sqrt {x}\right ) \\ & = -3 b \sqrt {x} \sqrt {2-b x}-\frac {2 (2-b x)^{3/2}}{\sqrt {x}}-6 \sqrt {b} \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.03 \[ \int \frac {(2-b x)^{3/2}}{x^{3/2}} \, dx=\frac {(-4-b x) \sqrt {2-b x}}{\sqrt {x}}+12 \sqrt {b} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}-\sqrt {2-b x}}\right ) \]

[In]

Integrate[(2 - b*x)^(3/2)/x^(3/2),x]

[Out]

((-4 - b*x)*Sqrt[2 - b*x])/Sqrt[x] + 12*Sqrt[b]*ArcTan[(Sqrt[b]*Sqrt[x])/(Sqrt[2] - Sqrt[2 - b*x])]

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.17

method	result	size
meijerg	\(-\frac {3 \left (-b \right )^{\frac {3}{2}} \left (-\frac {8 \sqrt {\pi }\, \sqrt {2}\, \left (\frac {b x}{4}+1\right ) \sqrt {-\frac {b x}{2}+1}}{3 \sqrt {x}\, \sqrt {-b}}-\frac {4 \sqrt {\pi }\, \sqrt {b}\, \arcsin \left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )}{\sqrt {-b}}\right )}{2 \sqrt {\pi }\, b}\)	\(70\)
risch	\(\frac {\left (b^{2} x^{2}+2 b x -8\right ) \sqrt {\left (-b x +2\right ) x}}{\sqrt {-x \left (b x -2\right )}\, \sqrt {x}\, \sqrt {-b x +2}}-\frac {3 \sqrt {b}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {1}{b}\right )}{\sqrt {-b \,x^{2}+2 x}}\right ) \sqrt {\left (-b x +2\right ) x}}{\sqrt {x}\, \sqrt {-b x +2}}\)	\(97\)

[In]

int((-b*x+2)^(3/2)/x^(3/2),x,method=_RETURNVERBOSE)

[Out]

-3/2*(-b)^(3/2)/Pi^(1/2)/b*(-8/3*Pi^(1/2)/x^(1/2)*2^(1/2)/(-b)^(1/2)*(1/4*b*x+1)*(-1/2*b*x+1)^(1/2)-4*Pi^(1/2)
/(-b)^(1/2)*b^(1/2)*arcsin(1/2*b^(1/2)*x^(1/2)*2^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.68 \[ \int \frac {(2-b x)^{3/2}}{x^{3/2}} \, dx=\left [\frac {3 \, \sqrt {-b} x \log \left (-b x + \sqrt {-b x + 2} \sqrt {-b} \sqrt {x} + 1\right ) - {\left (b x + 4\right )} \sqrt {-b x + 2} \sqrt {x}}{x}, \frac {6 \, \sqrt {b} x \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right ) - {\left (b x + 4\right )} \sqrt {-b x + 2} \sqrt {x}}{x}\right ] \]

[In]

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

[Out]

[(3*sqrt(-b)*x*log(-b*x + sqrt(-b*x + 2)*sqrt(-b)*sqrt(x) + 1) - (b*x + 4)*sqrt(-b*x + 2)*sqrt(x))/x, (6*sqrt(
b)*x*arctan(sqrt(-b*x + 2)/(sqrt(b)*sqrt(x))) - (b*x + 4)*sqrt(-b*x + 2)*sqrt(x))/x]

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.58 (sec) , antiderivative size = 158, normalized size of antiderivative = 2.63 \[ \int \frac {(2-b x)^{3/2}}{x^{3/2}} \, dx=\begin {cases} 6 i \sqrt {b} \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )} - \frac {i b^{2} x^{\frac {3}{2}}}{\sqrt {b x - 2}} - \frac {2 i b \sqrt {x}}{\sqrt {b x - 2}} + \frac {8 i}{\sqrt {x} \sqrt {b x - 2}} & \text {for}\: \left |{b x}\right | > 2 \\- 6 \sqrt {b} \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )} + \frac {b^{2} x^{\frac {3}{2}}}{\sqrt {- b x + 2}} + \frac {2 b \sqrt {x}}{\sqrt {- b x + 2}} - \frac {8}{\sqrt {x} \sqrt {- b x + 2}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((6*I*sqrt(b)*acosh(sqrt(2)*sqrt(b)*sqrt(x)/2) - I*b**2*x**(3/2)/sqrt(b*x - 2) - 2*I*b*sqrt(x)/sqrt(b
*x - 2) + 8*I/(sqrt(x)*sqrt(b*x - 2)), Abs(b*x) > 2), (-6*sqrt(b)*asin(sqrt(2)*sqrt(b)*sqrt(x)/2) + b**2*x**(3
/2)/sqrt(-b*x + 2) + 2*b*sqrt(x)/sqrt(-b*x + 2) - 8/(sqrt(x)*sqrt(-b*x + 2)), True))

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.05 \[ \int \frac {(2-b x)^{3/2}}{x^{3/2}} \, dx=6 \, \sqrt {b} \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right ) - \frac {4 \, \sqrt {-b x + 2}}{\sqrt {x}} - \frac {2 \, \sqrt {-b x + 2} b}{{\left (b - \frac {b x - 2}{x}\right )} \sqrt {x}} \]

[In]

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

[Out]

6*sqrt(b)*arctan(sqrt(-b*x + 2)/(sqrt(b)*sqrt(x))) - 4*sqrt(-b*x + 2)/sqrt(x) - 2*sqrt(-b*x + 2)*b/((b - (b*x
- 2)/x)*sqrt(x))

Giac [A] (verification not implemented)

none

Time = 6.15 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.25 \[ \int \frac {(2-b x)^{3/2}}{x^{3/2}} \, dx=-\frac {{\left (\frac {{\left (b x + 4\right )} \sqrt {-b x + 2}}{\sqrt {{\left (b x - 2\right )} b + 2 \, b}} + \frac {6 \, \log \left ({\left | -\sqrt {-b x + 2} \sqrt {-b} + \sqrt {{\left (b x - 2\right )} b + 2 \, b} \right |}\right )}{\sqrt {-b}}\right )} b^{2}}{{\left | b \right |}} \]

[In]

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

[Out]

-((b*x + 4)*sqrt(-b*x + 2)/sqrt((b*x - 2)*b + 2*b) + 6*log(abs(-sqrt(-b*x + 2)*sqrt(-b) + sqrt((b*x - 2)*b + 2
*b)))/sqrt(-b))*b^2/abs(b)

Mupad [F(-1)]

Timed out. \[ \int \frac {(2-b x)^{3/2}}{x^{3/2}} \, dx=\int \frac {{\left (2-b\,x\right )}^{3/2}}{x^{3/2}} \,d x \]

[In]

int((2 - b*x)^(3/2)/x^(3/2),x)

[Out]

int((2 - b*x)^(3/2)/x^(3/2), x)