\(\int \frac {(a-b x)^{3/2}}{\sqrt {x}} \, dx\) [530]

Optimal result

Integrand size = 16, antiderivative size = 74 \[ \int \frac {(a-b x)^{3/2}}{\sqrt {x}} \, dx=\frac {3}{4} a \sqrt {x} \sqrt {a-b x}+\frac {1}{2} \sqrt {x} (a-b x)^{3/2}+\frac {3 a^2 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{4 \sqrt {b}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {52, 65, 223, 209} \[ \int \frac {(a-b x)^{3/2}}{\sqrt {x}} \, dx=\frac {3 a^2 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{4 \sqrt {b}}+\frac {3}{4} a \sqrt {x} \sqrt {a-b x}+\frac {1}{2} \sqrt {x} (a-b x)^{3/2} \]

[In]

Int[(a - b*x)^(3/2)/Sqrt[x],x]

[Out]

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

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 65

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 209

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

Rule 223

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

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.97 \[ \int \frac {(a-b x)^{3/2}}{\sqrt {x}} \, dx=-\frac {1}{4} \sqrt {x} \sqrt {a-b x} (-5 a+2 b x)+\frac {3 a^2 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a-b x}}\right )}{2 \sqrt {b}} \]

[In]

Integrate[(a - b*x)^(3/2)/Sqrt[x],x]

[Out]

-1/4*(Sqrt[x]*Sqrt[a - b*x]*(-5*a + 2*b*x)) + (3*a^2*ArcTan[(Sqrt[b]*Sqrt[x])/(-Sqrt[a] + Sqrt[a - b*x])])/(2*
Sqrt[b])

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.04

method	result	size
risch	\(\frac {\left (-2 b x +5 a \right ) \sqrt {x}\, \sqrt {-b x +a}}{4}+\frac {3 a^{2} \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-b \,x^{2}+a x}}\right ) \sqrt {x \left (-b x +a \right )}}{8 \sqrt {b}\, \sqrt {x}\, \sqrt {-b x +a}}\)	\(77\)
default	\(\frac {\left (-b x +a \right )^{\frac {3}{2}} \sqrt {x}}{2}+\frac {3 a \left (\sqrt {x}\, \sqrt {-b x +a}+\frac {a \sqrt {x \left (-b x +a \right )}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-b \,x^{2}+a x}}\right )}{2 \sqrt {-b x +a}\, \sqrt {x}\, \sqrt {b}}\right )}{4}\)	\(83\)

[In]

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

[Out]

1/4*(-2*b*x+5*a)*x^(1/2)*(-b*x+a)^(1/2)+3/8*a^2/b^(1/2)*arctan(b^(1/2)*(x-1/2*a/b)/(-b*x^2+a*x)^(1/2))*(x*(-b*
x+a))^(1/2)/x^(1/2)/(-b*x+a)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.61 \[ \int \frac {(a-b x)^{3/2}}{\sqrt {x}} \, dx=\left [-\frac {3 \, a^{2} \sqrt {-b} \log \left (-2 \, b x + 2 \, \sqrt {-b x + a} \sqrt {-b} \sqrt {x} + a\right ) + 2 \, {\left (2 \, b^{2} x - 5 \, a b\right )} \sqrt {-b x + a} \sqrt {x}}{8 \, b}, -\frac {3 \, a^{2} \sqrt {b} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) + {\left (2 \, b^{2} x - 5 \, a b\right )} \sqrt {-b x + a} \sqrt {x}}{4 \, b}\right ] \]

[In]

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

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.90 (sec) , antiderivative size = 190, normalized size of antiderivative = 2.57 \[ \int \frac {(a-b x)^{3/2}}{\sqrt {x}} \, dx=\begin {cases} - \frac {5 i a^{\frac {3}{2}} \sqrt {x}}{4 \sqrt {-1 + \frac {b x}{a}}} + \frac {7 i \sqrt {a} b x^{\frac {3}{2}}}{4 \sqrt {-1 + \frac {b x}{a}}} - \frac {3 i a^{2} \operatorname {acosh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{4 \sqrt {b}} - \frac {i b^{2} x^{\frac {5}{2}}}{2 \sqrt {a} \sqrt {-1 + \frac {b x}{a}}} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\\frac {5 a^{\frac {3}{2}} \sqrt {x} \sqrt {1 - \frac {b x}{a}}}{4} - \frac {\sqrt {a} b x^{\frac {3}{2}} \sqrt {1 - \frac {b x}{a}}}{2} + \frac {3 a^{2} \operatorname {asin}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{4 \sqrt {b}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.26 \[ \int \frac {(a-b x)^{3/2}}{\sqrt {x}} \, dx=-\frac {3 \, a^{2} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right )}{4 \, \sqrt {b}} + \frac {\frac {3 \, \sqrt {-b x + a} a^{2} b}{\sqrt {x}} + \frac {5 \, {\left (-b x + a\right )}^{\frac {3}{2}} a^{2}}{x^{\frac {3}{2}}}}{4 \, {\left (b^{2} - \frac {2 \, {\left (b x - a\right )} b}{x} + \frac {{\left (b x - a\right )}^{2}}{x^{2}}\right )}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 76.75 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.28 \[ \int \frac {(a-b x)^{3/2}}{\sqrt {x}} \, dx=\frac {{\left (\frac {3 \, a^{2} \log \left ({\left | -\sqrt {-b x + a} \sqrt {-b} + \sqrt {{\left (b x - a\right )} b + a b} \right |}\right )}{\sqrt {-b}} - \sqrt {{\left (b x - a\right )} b + a b} \sqrt {-b x + a} {\left (\frac {2 \, {\left (b x - a\right )}}{b} - \frac {3 \, a}{b}\right )}\right )} b}{4 \, {\left | b \right |}} \]

[In]

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

[Out]

1/4*(3*a^2*log(abs(-sqrt(-b*x + a)*sqrt(-b) + sqrt((b*x - a)*b + a*b)))/sqrt(-b) - sqrt((b*x - a)*b + a*b)*sqr
t(-b*x + a)*(2*(b*x - a)/b - 3*a/b))*b/abs(b)

Mupad [F(-1)]

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

[In]

int((a - b*x)^(3/2)/x^(1/2),x)

[Out]

int((a - b*x)^(3/2)/x^(1/2), x)