Integrand size = 16, antiderivative size = 99 \[ \int \sqrt {x} (a-b x)^{3/2} \, dx=-\frac {a^2 \sqrt {x} \sqrt {a-b x}}{8 b}+\frac {1}{4} a x^{3/2} \sqrt {a-b x}+\frac {1}{3} x^{3/2} (a-b x)^{3/2}+\frac {a^3 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{8 b^{3/2}} \]
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
Time = 0.27 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.87 \[ \int \sqrt {x} (a-b x)^{3/2} \, dx=-\frac {\sqrt {x} \sqrt {a-b x} \left (3 a^2-14 a b x+8 b^2 x^2\right )}{24 b}+\frac {a^3 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a-b x}}\right )}{4 b^{3/2}} \]
-1/24*(Sqrt[x]*Sqrt[a - b*x]*(3*a^2 - 14*a*b*x + 8*b^2*x^2))/b + (a^3*ArcT an[(Sqrt[b]*Sqrt[x])/(-Sqrt[a] + Sqrt[a - b*x])])/(4*b^(3/2))
Time = 0.18 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {60, 60, 60, 65, 216}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sqrt {x} (a-b x)^{3/2} \, dx\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{2} a \int \sqrt {x} \sqrt {a-b x}dx+\frac {1}{3} x^{3/2} (a-b x)^{3/2}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{2} a \left (\frac {1}{4} a \int \frac {\sqrt {x}}{\sqrt {a-b x}}dx+\frac {1}{2} x^{3/2} \sqrt {a-b x}\right )+\frac {1}{3} x^{3/2} (a-b x)^{3/2}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{2} a \left (\frac {1}{4} a \left (\frac {a \int \frac {1}{\sqrt {x} \sqrt {a-b x}}dx}{2 b}-\frac {\sqrt {x} \sqrt {a-b x}}{b}\right )+\frac {1}{2} x^{3/2} \sqrt {a-b x}\right )+\frac {1}{3} x^{3/2} (a-b x)^{3/2}\) |
\(\Big \downarrow \) 65 |
\(\displaystyle \frac {1}{2} a \left (\frac {1}{4} a \left (\frac {a \int \frac {1}{\frac {b x}{a-b x}+1}d\frac {\sqrt {x}}{\sqrt {a-b x}}}{b}-\frac {\sqrt {x} \sqrt {a-b x}}{b}\right )+\frac {1}{2} x^{3/2} \sqrt {a-b x}\right )+\frac {1}{3} x^{3/2} (a-b x)^{3/2}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {1}{2} a \left (\frac {1}{4} a \left (\frac {a \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{b^{3/2}}-\frac {\sqrt {x} \sqrt {a-b x}}{b}\right )+\frac {1}{2} x^{3/2} \sqrt {a-b x}\right )+\frac {1}{3} x^{3/2} (a-b x)^{3/2}\) |
(x^(3/2)*(a - b*x)^(3/2))/3 + (a*((x^(3/2)*Sqrt[a - b*x])/2 + (a*(-((Sqrt[ x]*Sqrt[a - b*x])/b) + (a*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a - b*x]])/b^(3/2) ))/4))/2
3.6.29.3.1 Defintions of rubi rules used
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] + Simp[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] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2 Sub st[Int[1/(b - d*x^2), x], x, Sqrt[b*x]/Sqrt[c + d*x]], x] /; FreeQ[{b, c, d }, x] && !GtQ[c, 0]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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 | \(\frac {x^{\frac {3}{2}} \left (-b x +a \right )^{\frac {3}{2}}}{3}+\frac {a \left (\frac {x^{\frac {3}{2}} \sqrt {-b x +a}}{2}+\frac {a \left (-\frac {\sqrt {x}\, \sqrt {-b x +a}}{b}+\frac {a \sqrt {x \left (-b x +a \right )}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-b \,x^{2}+a x}}\right )}{2 b^{\frac {3}{2}} \sqrt {x}\, \sqrt {-b x +a}}\right )}{4}\right )}{2}\) | \(104\) |
-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)
Time = 0.23 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.42 \[ \int \sqrt {x} (a-b x)^{3/2} \, dx=\left [-\frac {3 \, a^{3} \sqrt {-b} \log \left (-2 \, b x + 2 \, \sqrt {-b x + a} \sqrt {-b} \sqrt {x} + a\right ) + 2 \, {\left (8 \, b^{3} x^{2} - 14 \, a b^{2} x + 3 \, a^{2} b\right )} \sqrt {-b x + a} \sqrt {x}}{48 \, b^{2}}, -\frac {3 \, a^{3} \sqrt {b} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) + {\left (8 \, b^{3} x^{2} - 14 \, a b^{2} x + 3 \, a^{2} b\right )} \sqrt {-b x + a} \sqrt {x}}{24 \, b^{2}}\right ] \]
[-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]
Result contains complex when optimal does not.
Time = 4.90 (sec) , antiderivative size = 264, normalized size of antiderivative = 2.67 \[ \int \sqrt {x} (a-b x)^{3/2} \, dx=\begin {cases} \frac {i a^{\frac {5}{2}} \sqrt {x}}{8 b \sqrt {-1 + \frac {b x}{a}}} - \frac {17 i a^{\frac {3}{2}} x^{\frac {3}{2}}}{24 \sqrt {-1 + \frac {b x}{a}}} + \frac {11 i \sqrt {a} b x^{\frac {5}{2}}}{12 \sqrt {-1 + \frac {b x}{a}}} - \frac {i a^{3} \operatorname {acosh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{8 b^{\frac {3}{2}}} - \frac {i b^{2} x^{\frac {7}{2}}}{3 \sqrt {a} \sqrt {-1 + \frac {b x}{a}}} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\- \frac {a^{\frac {5}{2}} \sqrt {x}}{8 b \sqrt {1 - \frac {b x}{a}}} + \frac {17 a^{\frac {3}{2}} x^{\frac {3}{2}}}{24 \sqrt {1 - \frac {b x}{a}}} - \frac {11 \sqrt {a} b x^{\frac {5}{2}}}{12 \sqrt {1 - \frac {b x}{a}}} + \frac {a^{3} \operatorname {asin}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{8 b^{\frac {3}{2}}} + \frac {b^{2} x^{\frac {7}{2}}}{3 \sqrt {a} \sqrt {1 - \frac {b x}{a}}} & \text {otherwise} \end {cases} \]
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*sqrt(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))
Time = 0.33 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.34 \[ \int \sqrt {x} (a-b x)^{3/2} \, dx=-\frac {a^{3} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right )}{8 \, b^{\frac {3}{2}}} + \frac {\frac {3 \, \sqrt {-b x + a} a^{3} b^{2}}{\sqrt {x}} + \frac {8 \, {\left (-b x + a\right )}^{\frac {3}{2}} a^{3} b}{x^{\frac {3}{2}}} - \frac {3 \, {\left (-b x + a\right )}^{\frac {5}{2}} a^{3}}{x^{\frac {5}{2}}}}{24 \, {\left (b^{4} - \frac {3 \, {\left (b x - a\right )} b^{3}}{x} + \frac {3 \, {\left (b x - a\right )}^{2} b^{2}}{x^{2}} - \frac {{\left (b x - a\right )}^{3} b}{x^{3}}\right )}} \]
-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)
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 \[ \int \sqrt {x} (a-b x)^{3/2} \, dx=\frac {{\left (\frac {15 \, a^{3} \log \left ({\left | -\sqrt {-b x + a} \sqrt {-b} + \sqrt {{\left (b x - a\right )} b + a b} \right |}\right )}{\sqrt {-b} b} - \sqrt {{\left (b x - a\right )} b + a b} \sqrt {-b x + a} {\left (2 \, {\left (b x - a\right )} {\left (\frac {4 \, {\left (b x - a\right )}}{b^{2}} + \frac {13 \, a}{b^{2}}\right )} + \frac {33 \, a^{2}}{b^{2}}\right )}\right )} {\left | b \right |} + \frac {24 \, {\left (\frac {a b \log \left ({\left | -\sqrt {-b x + a} \sqrt {-b} + \sqrt {{\left (b x - a\right )} b + a b} \right |}\right )}{\sqrt {-b}} - \sqrt {{\left (b x - a\right )} b + a b} \sqrt {-b x + a}\right )} a^{2} {\left | b \right |}}{b^{2}} - \frac {12 \, {\left (\frac {3 \, a^{2} b \log \left ({\left | -\sqrt {-b x + a} \sqrt {-b} + \sqrt {{\left (b x - a\right )} b + a b} \right |}\right )}{\sqrt {-b}} - \sqrt {{\left (b x - a\right )} b + a b} {\left (2 \, b x + 3 \, a\right )} \sqrt {-b x + a}\right )} a {\left | b \right |}}{b^{2}}}{24 \, b} \]
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)*sq rt(-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
Timed out. \[ \int \sqrt {x} (a-b x)^{3/2} \, dx=\int \sqrt {x}\,{\left (a-b\,x\right )}^{3/2} \,d x \]