\(\int x^{3/2} (2-b x)^{3/2} \, dx\) [540]

Optimal result

Integrand size = 16, antiderivative size = 109 \[ \int x^{3/2} (2-b x)^{3/2} \, dx=-\frac {3 \sqrt {x} \sqrt {2-b x}}{8 b^2}-\frac {x^{3/2} \sqrt {2-b x}}{8 b}+\frac {1}{4} x^{5/2} \sqrt {2-b x}+\frac {1}{4} x^{5/2} (2-b x)^{3/2}+\frac {3 \arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{4 b^{5/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {52, 56, 222} \[ \int x^{3/2} (2-b x)^{3/2} \, dx=\frac {3 \arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{4 b^{5/2}}-\frac {3 \sqrt {x} \sqrt {2-b x}}{8 b^2}+\frac {1}{4} x^{5/2} (2-b x)^{3/2}+\frac {1}{4} x^{5/2} \sqrt {2-b x}-\frac {x^{3/2} \sqrt {2-b x}}{8 b} \]

[In]

Int[x^(3/2)*(2 - b*x)^(3/2),x]

[Out]

(-3*Sqrt[x]*Sqrt[2 - b*x])/(8*b^2) - (x^(3/2)*Sqrt[2 - b*x])/(8*b) + (x^(5/2)*Sqrt[2 - b*x])/4 + (x^(5/2)*(2 -
b*x)^(3/2))/4 + (3*ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[2]])/(4*b^(5/2))

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

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.78 \[ \int x^{3/2} (2-b x)^{3/2} \, dx=-\frac {\sqrt {x} \sqrt {2-b x} \left (3+b x-6 b^2 x^2+2 b^3 x^3\right )}{8 b^2}-\frac {3 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}-\sqrt {2-b x}}\right )}{2 b^{5/2}} \]

[In]

Integrate[x^(3/2)*(2 - b*x)^(3/2),x]

[Out]

-1/8*(Sqrt[x]*Sqrt[2 - b*x]*(3 + b*x - 6*b^2*x^2 + 2*b^3*x^3))/b^2 - (3*ArcTan[(Sqrt[b]*Sqrt[x])/(Sqrt[2] - Sq
rt[2 - b*x])])/(2*b^(5/2))

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.82

method	result	size
meijerg	\(-\frac {12 \left (-\frac {\sqrt {\pi }\, \sqrt {x}\, \sqrt {2}\, \left (-b \right )^{\frac {5}{2}} \left (10 b^{3} x^{3}-30 b^{2} x^{2}+5 b x +15\right ) \sqrt {-\frac {b x}{2}+1}}{480 b^{2}}+\frac {\sqrt {\pi }\, \left (-b \right )^{\frac {5}{2}} \arcsin \left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )}{16 b^{\frac {5}{2}}}\right )}{\left (-b \right )^{\frac {3}{2}} \sqrt {\pi }\, b}\)	\(89\)
risch	\(\frac {\left (2 b^{3} x^{3}-6 b^{2} x^{2}+b x +3\right ) \sqrt {x}\, \left (b x -2\right ) \sqrt {\left (-b x +2\right ) x}}{8 b^{2} \sqrt {-x \left (b x -2\right )}\, \sqrt {-b x +2}}+\frac {3 \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {1}{b}\right )}{\sqrt {-b \,x^{2}+2 x}}\right ) \sqrt {\left (-b x +2\right ) x}}{8 b^{\frac {5}{2}} \sqrt {x}\, \sqrt {-b x +2}}\)	\(114\)
default	\(-\frac {x^{\frac {3}{2}} \left (-b x +2\right )^{\frac {5}{2}}}{4 b}+\frac {-\frac {\sqrt {x}\, \left (-b x +2\right )^{\frac {5}{2}}}{4 b}+\frac {\frac {\left (-b x +2\right )^{\frac {3}{2}} \sqrt {x}}{2}+\frac {3 \sqrt {x}\, \sqrt {-b x +2}}{2}+\frac {3 \sqrt {\left (-b x +2\right ) x}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {1}{b}\right )}{\sqrt {-b \,x^{2}+2 x}}\right )}{2 \sqrt {-b x +2}\, \sqrt {x}\, \sqrt {b}}}{4 b}}{b}\)	\(122\)

[In]

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

[Out]

-12/(-b)^(3/2)/Pi^(1/2)/b*(-1/480*Pi^(1/2)*x^(1/2)*2^(1/2)*(-b)^(5/2)*(10*b^3*x^3-30*b^2*x^2+5*b*x+15)/b^2*(-1
/2*b*x+1)^(1/2)+1/16*Pi^(1/2)*(-b)^(5/2)/b^(5/2)*arcsin(1/2*b^(1/2)*x^(1/2)*2^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.28 \[ \int x^{3/2} (2-b x)^{3/2} \, dx=\left [-\frac {{\left (2 \, b^{4} x^{3} - 6 \, b^{3} x^{2} + b^{2} x + 3 \, b\right )} \sqrt {-b x + 2} \sqrt {x} + 3 \, \sqrt {-b} \log \left (-b x + \sqrt {-b x + 2} \sqrt {-b} \sqrt {x} + 1\right )}{8 \, b^{3}}, -\frac {{\left (2 \, b^{4} x^{3} - 6 \, b^{3} x^{2} + b^{2} x + 3 \, b\right )} \sqrt {-b x + 2} \sqrt {x} + 6 \, \sqrt {b} \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right )}{8 \, b^{3}}\right ] \]

[In]

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

[Out]

[-1/8*((2*b^4*x^3 - 6*b^3*x^2 + b^2*x + 3*b)*sqrt(-b*x + 2)*sqrt(x) + 3*sqrt(-b)*log(-b*x + sqrt(-b*x + 2)*sqr
t(-b)*sqrt(x) + 1))/b^3, -1/8*((2*b^4*x^3 - 6*b^3*x^2 + b^2*x + 3*b)*sqrt(-b*x + 2)*sqrt(x) + 6*sqrt(b)*arctan
(sqrt(-b*x + 2)/(sqrt(b)*sqrt(x))))/b^3]

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 18.74 (sec) , antiderivative size = 250, normalized size of antiderivative = 2.29 \[ \int x^{3/2} (2-b x)^{3/2} \, dx=\begin {cases} - \frac {i b^{2} x^{\frac {9}{2}}}{4 \sqrt {b x - 2}} + \frac {5 i b x^{\frac {7}{2}}}{4 \sqrt {b x - 2}} - \frac {13 i x^{\frac {5}{2}}}{8 \sqrt {b x - 2}} - \frac {i x^{\frac {3}{2}}}{8 b \sqrt {b x - 2}} + \frac {3 i \sqrt {x}}{4 b^{2} \sqrt {b x - 2}} - \frac {3 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{4 b^{\frac {5}{2}}} & \text {for}\: \left |{b x}\right | > 2 \\\frac {b^{2} x^{\frac {9}{2}}}{4 \sqrt {- b x + 2}} - \frac {5 b x^{\frac {7}{2}}}{4 \sqrt {- b x + 2}} + \frac {13 x^{\frac {5}{2}}}{8 \sqrt {- b x + 2}} + \frac {x^{\frac {3}{2}}}{8 b \sqrt {- b x + 2}} - \frac {3 \sqrt {x}}{4 b^{2} \sqrt {- b x + 2}} + \frac {3 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{4 b^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((-I*b**2*x**(9/2)/(4*sqrt(b*x - 2)) + 5*I*b*x**(7/2)/(4*sqrt(b*x - 2)) - 13*I*x**(5/2)/(8*sqrt(b*x -
2)) - I*x**(3/2)/(8*b*sqrt(b*x - 2)) + 3*I*sqrt(x)/(4*b**2*sqrt(b*x - 2)) - 3*I*acosh(sqrt(2)*sqrt(b)*sqrt(x)
/2)/(4*b**(5/2)), Abs(b*x) > 2), (b**2*x**(9/2)/(4*sqrt(-b*x + 2)) - 5*b*x**(7/2)/(4*sqrt(-b*x + 2)) + 13*x**(
5/2)/(8*sqrt(-b*x + 2)) + x**(3/2)/(8*b*sqrt(-b*x + 2)) - 3*sqrt(x)/(4*b**2*sqrt(-b*x + 2)) + 3*asin(sqrt(2)*s
qrt(b)*sqrt(x)/2)/(4*b**(5/2)), True))

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.35 \[ \int x^{3/2} (2-b x)^{3/2} \, dx=\frac {\frac {3 \, \sqrt {-b x + 2} b^{3}}{\sqrt {x}} + \frac {11 \, {\left (-b x + 2\right )}^{\frac {3}{2}} b^{2}}{x^{\frac {3}{2}}} - \frac {11 \, {\left (-b x + 2\right )}^{\frac {5}{2}} b}{x^{\frac {5}{2}}} - \frac {3 \, {\left (-b x + 2\right )}^{\frac {7}{2}}}{x^{\frac {7}{2}}}}{4 \, {\left (b^{6} - \frac {4 \, {\left (b x - 2\right )} b^{5}}{x} + \frac {6 \, {\left (b x - 2\right )}^{2} b^{4}}{x^{2}} - \frac {4 \, {\left (b x - 2\right )}^{3} b^{3}}{x^{3}} + \frac {{\left (b x - 2\right )}^{4} b^{2}}{x^{4}}\right )}} - \frac {3 \, \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right )}{4 \, b^{\frac {5}{2}}} \]

[In]

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

[Out]

1/4*(3*sqrt(-b*x + 2)*b^3/sqrt(x) + 11*(-b*x + 2)^(3/2)*b^2/x^(3/2) - 11*(-b*x + 2)^(5/2)*b/x^(5/2) - 3*(-b*x
+ 2)^(7/2)/x^(7/2))/(b^6 - 4*(b*x - 2)*b^5/x + 6*(b*x - 2)^2*b^4/x^2 - 4*(b*x - 2)^3*b^3/x^3 + (b*x - 2)^4*b^2
/x^4) - 3/4*arctan(sqrt(-b*x + 2)/(sqrt(b)*sqrt(x)))/b^(5/2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 287 vs. \(2 (76) = 152\).

Time = 17.73 (sec) , antiderivative size = 287, normalized size of antiderivative = 2.63 \[ \int x^{3/2} (2-b x)^{3/2} \, dx=-\frac {{\left ({\left ({\left (b x - 2\right )} {\left (2 \, {\left (b x - 2\right )} {\left (\frac {3 \, {\left (b x - 2\right )}}{b^{3}} + \frac {25}{b^{3}}\right )} + \frac {163}{b^{3}}\right )} + \frac {279}{b^{3}}\right )} \sqrt {{\left (b x - 2\right )} b + 2 \, b} \sqrt {-b x + 2} - \frac {210 \, \log \left ({\left | -\sqrt {-b x + 2} \sqrt {-b} + \sqrt {{\left (b x - 2\right )} b + 2 \, b} \right |}\right )}{\sqrt {-b} b^{2}}\right )} {\left | b \right |} - \frac {16 \, {\left (\sqrt {{\left (b x - 2\right )} b + 2 \, b} \sqrt {-b x + 2} {\left ({\left (b x - 2\right )} {\left (\frac {2 \, {\left (b x - 2\right )}}{b^{2}} + \frac {13}{b^{2}}\right )} + \frac {33}{b^{2}}\right )} - \frac {30 \, \log \left ({\left | -\sqrt {-b x + 2} \sqrt {-b} + \sqrt {{\left (b x - 2\right )} b + 2 \, b} \right |}\right )}{\sqrt {-b} b}\right )} {\left | b \right |}}{b} + \frac {48 \, {\left (\sqrt {{\left (b x - 2\right )} b + 2 \, b} {\left (b x + 3\right )} \sqrt {-b x + 2} - \frac {6 \, b \log \left ({\left | -\sqrt {-b x + 2} \sqrt {-b} + \sqrt {{\left (b x - 2\right )} b + 2 \, b} \right |}\right )}{\sqrt {-b}}\right )} {\left | b \right |}}{b^{3}}}{24 \, b} \]

[In]

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

[Out]

-1/24*((((b*x - 2)*(2*(b*x - 2)*(3*(b*x - 2)/b^3 + 25/b^3) + 163/b^3) + 279/b^3)*sqrt((b*x - 2)*b + 2*b)*sqrt(
-b*x + 2) - 210*log(abs(-sqrt(-b*x + 2)*sqrt(-b) + sqrt((b*x - 2)*b + 2*b)))/(sqrt(-b)*b^2))*abs(b) - 16*(sqrt
((b*x - 2)*b + 2*b)*sqrt(-b*x + 2)*((b*x - 2)*(2*(b*x - 2)/b^2 + 13/b^2) + 33/b^2) - 30*log(abs(-sqrt(-b*x + 2
)*sqrt(-b) + sqrt((b*x - 2)*b + 2*b)))/(sqrt(-b)*b))*abs(b)/b + 48*(sqrt((b*x - 2)*b + 2*b)*(b*x + 3)*sqrt(-b*
x + 2) - 6*b*log(abs(-sqrt(-b*x + 2)*sqrt(-b) + sqrt((b*x - 2)*b + 2*b)))/sqrt(-b))*abs(b)/b^3)/b

Mupad [F(-1)]

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

[In]

int(x^(3/2)*(2 - b*x)^(3/2),x)

[Out]

int(x^(3/2)*(2 - b*x)^(3/2), x)