\(\int \sqrt {x} (a-b x)^{3/2} \, dx\) [529]

Optimal result

Integrand size = 16, antiderivative size = 99 \[ \int \sqrt {x} (a-b x)^{3/2} \, dx=-\frac {a^2 \sqrt {x} \sqrt {a-b x}}{8 b}+\frac {1}{4} a x^{3/2} \sqrt {a-b x}+\frac {1}{3} x^{3/2} (a-b x)^{3/2}+\frac {a^3 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{8 b^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {52, 65, 223, 209} \[ \int \sqrt {x} (a-b x)^{3/2} \, dx=\frac {a^3 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{8 b^{3/2}}-\frac {a^2 \sqrt {x} \sqrt {a-b x}}{8 b}+\frac {1}{4} a x^{3/2} \sqrt {a-b x}+\frac {1}{3} x^{3/2} (a-b x)^{3/2} \]

[In]

Int[Sqrt[x]*(a - b*x)^(3/2),x]

[Out]

-1/8*(a^2*Sqrt[x]*Sqrt[a - b*x])/b + (a*x^(3/2)*Sqrt[a - b*x])/4 + (x^(3/2)*(a - b*x)^(3/2))/3 + (a^3*ArcTan[(
Sqrt[b]*Sqrt[x])/Sqrt[a - b*x]])/(8*b^(3/2))

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 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}{3} x^{3/2} (a-b x)^{3/2}+\frac {1}{2} a \int \sqrt {x} \sqrt {a-b x} \, dx \\ & = \frac {1}{4} a x^{3/2} \sqrt {a-b x}+\frac {1}{3} x^{3/2} (a-b x)^{3/2}+\frac {1}{8} a^2 \int \frac {\sqrt {x}}{\sqrt {a-b x}} \, dx \\ & = -\frac {a^2 \sqrt {x} \sqrt {a-b x}}{8 b}+\frac {1}{4} a x^{3/2} \sqrt {a-b x}+\frac {1}{3} x^{3/2} (a-b x)^{3/2}+\frac {a^3 \int \frac {1}{\sqrt {x} \sqrt {a-b x}} \, dx}{16 b} \\ & = -\frac {a^2 \sqrt {x} \sqrt {a-b x}}{8 b}+\frac {1}{4} a x^{3/2} \sqrt {a-b x}+\frac {1}{3} x^{3/2} (a-b x)^{3/2}+\frac {a^3 \text {Subst}\left (\int \frac {1}{\sqrt {a-b x^2}} \, dx,x,\sqrt {x}\right )}{8 b} \\ & = -\frac {a^2 \sqrt {x} \sqrt {a-b x}}{8 b}+\frac {1}{4} a x^{3/2} \sqrt {a-b x}+\frac {1}{3} x^{3/2} (a-b x)^{3/2}+\frac {a^3 \text {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a-b x}}\right )}{8 b} \\ & = -\frac {a^2 \sqrt {x} \sqrt {a-b x}}{8 b}+\frac {1}{4} a x^{3/2} \sqrt {a-b x}+\frac {1}{3} x^{3/2} (a-b x)^{3/2}+\frac {a^3 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{8 b^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.87 \[ \int \sqrt {x} (a-b x)^{3/2} \, dx=-\frac {\sqrt {x} \sqrt {a-b x} \left (3 a^2-14 a b x+8 b^2 x^2\right )}{24 b}+\frac {a^3 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a-b x}}\right )}{4 b^{3/2}} \]

[In]

Integrate[Sqrt[x]*(a - b*x)^(3/2),x]

[Out]

-1/24*(Sqrt[x]*Sqrt[a - b*x]*(3*a^2 - 14*a*b*x + 8*b^2*x^2))/b + (a^3*ArcTan[(Sqrt[b]*Sqrt[x])/(-Sqrt[a] + Sqr
t[a - b*x])])/(4*b^(3/2))

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.92

method	result	size
risch	\(-\frac {\left (8 b^{2} x^{2}-14 a b x +3 a^{2}\right ) \sqrt {x}\, \sqrt {-b x +a}}{24 b}+\frac {a^{3} \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-b \,x^{2}+a x}}\right ) \sqrt {x \left (-b x +a \right )}}{16 b^{\frac {3}{2}} \sqrt {x}\, \sqrt {-b x +a}}\)	\(91\)
default	\(\frac {x^{\frac {3}{2}} \left (-b x +a \right )^{\frac {3}{2}}}{3}+\frac {a \left (\frac {x^{\frac {3}{2}} \sqrt {-b x +a}}{2}+\frac {a \left (-\frac {\sqrt {x}\, \sqrt {-b x +a}}{b}+\frac {a \sqrt {x \left (-b x +a \right )}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-b \,x^{2}+a x}}\right )}{2 b^{\frac {3}{2}} \sqrt {x}\, \sqrt {-b x +a}}\right )}{4}\right )}{2}\)	\(104\)

[In]

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

[Out]

-1/24*(8*b^2*x^2-14*a*b*x+3*a^2)/b*x^(1/2)*(-b*x+a)^(1/2)+1/16/b^(3/2)*a^3*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 = 141, normalized size of antiderivative = 1.42 \[ \int \sqrt {x} (a-b x)^{3/2} \, dx=\left [-\frac {3 \, a^{3} \sqrt {-b} \log \left (-2 \, b x + 2 \, \sqrt {-b x + a} \sqrt {-b} \sqrt {x} + a\right ) + 2 \, {\left (8 \, b^{3} x^{2} - 14 \, a b^{2} x + 3 \, a^{2} b\right )} \sqrt {-b x + a} \sqrt {x}}{48 \, b^{2}}, -\frac {3 \, a^{3} \sqrt {b} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) + {\left (8 \, b^{3} x^{2} - 14 \, a b^{2} x + 3 \, a^{2} b\right )} \sqrt {-b x + a} \sqrt {x}}{24 \, b^{2}}\right ] \]

[In]

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

[Out]

[-1/48*(3*a^3*sqrt(-b)*log(-2*b*x + 2*sqrt(-b*x + a)*sqrt(-b)*sqrt(x) + a) + 2*(8*b^3*x^2 - 14*a*b^2*x + 3*a^2
*b)*sqrt(-b*x + a)*sqrt(x))/b^2, -1/24*(3*a^3*sqrt(b)*arctan(sqrt(-b*x + a)/(sqrt(b)*sqrt(x))) + (8*b^3*x^2 -
14*a*b^2*x + 3*a^2*b)*sqrt(-b*x + a)*sqrt(x))/b^2]

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 4.90 (sec) , antiderivative size = 264, normalized size of antiderivative = 2.67 \[ \int \sqrt {x} (a-b x)^{3/2} \, dx=\begin {cases} \frac {i a^{\frac {5}{2}} \sqrt {x}}{8 b \sqrt {-1 + \frac {b x}{a}}} - \frac {17 i a^{\frac {3}{2}} x^{\frac {3}{2}}}{24 \sqrt {-1 + \frac {b x}{a}}} + \frac {11 i \sqrt {a} b x^{\frac {5}{2}}}{12 \sqrt {-1 + \frac {b x}{a}}} - \frac {i a^{3} \operatorname {acosh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{8 b^{\frac {3}{2}}} - \frac {i b^{2} x^{\frac {7}{2}}}{3 \sqrt {a} \sqrt {-1 + \frac {b x}{a}}} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\- \frac {a^{\frac {5}{2}} \sqrt {x}}{8 b \sqrt {1 - \frac {b x}{a}}} + \frac {17 a^{\frac {3}{2}} x^{\frac {3}{2}}}{24 \sqrt {1 - \frac {b x}{a}}} - \frac {11 \sqrt {a} b x^{\frac {5}{2}}}{12 \sqrt {1 - \frac {b x}{a}}} + \frac {a^{3} \operatorname {asin}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{8 b^{\frac {3}{2}}} + \frac {b^{2} x^{\frac {7}{2}}}{3 \sqrt {a} \sqrt {1 - \frac {b x}{a}}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((I*a**(5/2)*sqrt(x)/(8*b*sqrt(-1 + b*x/a)) - 17*I*a**(3/2)*x**(3/2)/(24*sqrt(-1 + b*x/a)) + 11*I*sqr
t(a)*b*x**(5/2)/(12*sqrt(-1 + b*x/a)) - I*a**3*acosh(sqrt(b)*sqrt(x)/sqrt(a))/(8*b**(3/2)) - I*b**2*x**(7/2)/(
3*sqrt(a)*sqrt(-1 + b*x/a)), Abs(b*x/a) > 1), (-a**(5/2)*sqrt(x)/(8*b*sqrt(1 - b*x/a)) + 17*a**(3/2)*x**(3/2)/
(24*sqrt(1 - b*x/a)) - 11*sqrt(a)*b*x**(5/2)/(12*sqrt(1 - b*x/a)) + a**3*asin(sqrt(b)*sqrt(x)/sqrt(a))/(8*b**(
3/2)) + b**2*x**(7/2)/(3*sqrt(a)*sqrt(1 - b*x/a)), True))

Maxima [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.34 \[ \int \sqrt {x} (a-b x)^{3/2} \, dx=-\frac {a^{3} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right )}{8 \, b^{\frac {3}{2}}} + \frac {\frac {3 \, \sqrt {-b x + a} a^{3} b^{2}}{\sqrt {x}} + \frac {8 \, {\left (-b x + a\right )}^{\frac {3}{2}} a^{3} b}{x^{\frac {3}{2}}} - \frac {3 \, {\left (-b x + a\right )}^{\frac {5}{2}} a^{3}}{x^{\frac {5}{2}}}}{24 \, {\left (b^{4} - \frac {3 \, {\left (b x - a\right )} b^{3}}{x} + \frac {3 \, {\left (b x - a\right )}^{2} b^{2}}{x^{2}} - \frac {{\left (b x - a\right )}^{3} b}{x^{3}}\right )}} \]

[In]

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

[Out]

-1/8*a^3*arctan(sqrt(-b*x + a)/(sqrt(b)*sqrt(x)))/b^(3/2) + 1/24*(3*sqrt(-b*x + a)*a^3*b^2/sqrt(x) + 8*(-b*x +
a)^(3/2)*a^3*b/x^(3/2) - 3*(-b*x + a)^(5/2)*a^3/x^(5/2))/(b^4 - 3*(b*x - a)*b^3/x + 3*(b*x - a)^2*b^2/x^2 - (
b*x - a)^3*b/x^3)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 281 vs. \(2 (71) = 142\).

Time = 228.30 (sec) , antiderivative size = 281, normalized size of antiderivative = 2.84 \[ \int \sqrt {x} (a-b x)^{3/2} \, dx=\frac {{\left (\frac {15 \, a^{3} \log \left ({\left | -\sqrt {-b x + a} \sqrt {-b} + \sqrt {{\left (b x - a\right )} b + a b} \right |}\right )}{\sqrt {-b} b} - \sqrt {{\left (b x - a\right )} b + a b} \sqrt {-b x + a} {\left (2 \, {\left (b x - a\right )} {\left (\frac {4 \, {\left (b x - a\right )}}{b^{2}} + \frac {13 \, a}{b^{2}}\right )} + \frac {33 \, a^{2}}{b^{2}}\right )}\right )} {\left | b \right |} + \frac {24 \, {\left (\frac {a b \log \left ({\left | -\sqrt {-b x + a} \sqrt {-b} + \sqrt {{\left (b x - a\right )} b + a b} \right |}\right )}{\sqrt {-b}} - \sqrt {{\left (b x - a\right )} b + a b} \sqrt {-b x + a}\right )} a^{2} {\left | b \right |}}{b^{2}} - \frac {12 \, {\left (\frac {3 \, a^{2} b \log \left ({\left | -\sqrt {-b x + a} \sqrt {-b} + \sqrt {{\left (b x - a\right )} b + a b} \right |}\right )}{\sqrt {-b}} - \sqrt {{\left (b x - a\right )} b + a b} {\left (2 \, b x + 3 \, a\right )} \sqrt {-b x + a}\right )} a {\left | b \right |}}{b^{2}}}{24 \, b} \]

[In]

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

[Out]

1/24*((15*a^3*log(abs(-sqrt(-b*x + a)*sqrt(-b) + sqrt((b*x - a)*b + a*b)))/(sqrt(-b)*b) - sqrt((b*x - a)*b + a
*b)*sqrt(-b*x + a)*(2*(b*x - a)*(4*(b*x - a)/b^2 + 13*a/b^2) + 33*a^2/b^2))*abs(b) + 24*(a*b*log(abs(-sqrt(-b*
x + a)*sqrt(-b) + sqrt((b*x - a)*b + a*b)))/sqrt(-b) - sqrt((b*x - a)*b + a*b)*sqrt(-b*x + a))*a^2*abs(b)/b^2
- 12*(3*a^2*b*log(abs(-sqrt(-b*x + a)*sqrt(-b) + sqrt((b*x - a)*b + a*b)))/sqrt(-b) - sqrt((b*x - a)*b + a*b)*
(2*b*x + 3*a)*sqrt(-b*x + a))*a*abs(b)/b^2)/b

Mupad [F(-1)]

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

[In]

int(x^(1/2)*(a - b*x)^(3/2),x)

[Out]

int(x^(1/2)*(a - b*x)^(3/2), x)