Integrand size = 16, antiderivative size = 105 \[ \int \frac {x^{5/2}}{\sqrt {a-b x}} \, dx=-\frac {5 a^2 \sqrt {x} \sqrt {a-b x}}{8 b^3}-\frac {5 a x^{3/2} \sqrt {a-b x}}{12 b^2}-\frac {x^{5/2} \sqrt {a-b x}}{3 b}+\frac {5 a^3 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{8 b^{7/2}} \]
5/8*a^3*arctan(b^(1/2)*x^(1/2)/(-b*x+a)^(1/2))/b^(7/2)-5/12*a*x^(3/2)*(-b* x+a)^(1/2)/b^2-1/3*x^(5/2)*(-b*x+a)^(1/2)/b-5/8*a^2*x^(1/2)*(-b*x+a)^(1/2) /b^3
Time = 0.24 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.82 \[ \int \frac {x^{5/2}}{\sqrt {a-b x}} \, dx=-\frac {\sqrt {x} \sqrt {a-b x} \left (15 a^2+10 a b x+8 b^2 x^2\right )}{24 b^3}+\frac {5 a^3 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a-b x}}\right )}{4 b^{7/2}} \]
-1/24*(Sqrt[x]*Sqrt[a - b*x]*(15*a^2 + 10*a*b*x + 8*b^2*x^2))/b^3 + (5*a^3 *ArcTan[(Sqrt[b]*Sqrt[x])/(-Sqrt[a] + Sqrt[a - b*x])])/(4*b^(7/2))
Time = 0.17 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.07, 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.
\(\displaystyle \int \frac {x^{5/2}}{\sqrt {a-b x}} \, dx\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {5 a \int \frac {x^{3/2}}{\sqrt {a-b x}}dx}{6 b}-\frac {x^{5/2} \sqrt {a-b x}}{3 b}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {5 a \left (\frac {3 a \int \frac {\sqrt {x}}{\sqrt {a-b x}}dx}{4 b}-\frac {x^{3/2} \sqrt {a-b x}}{2 b}\right )}{6 b}-\frac {x^{5/2} \sqrt {a-b x}}{3 b}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {5 a \left (\frac {3 a \left (\frac {a \int \frac {1}{\sqrt {x} \sqrt {a-b x}}dx}{2 b}-\frac {\sqrt {x} \sqrt {a-b x}}{b}\right )}{4 b}-\frac {x^{3/2} \sqrt {a-b x}}{2 b}\right )}{6 b}-\frac {x^{5/2} \sqrt {a-b x}}{3 b}\) |
\(\Big \downarrow \) 65 |
\(\displaystyle \frac {5 a \left (\frac {3 a \left (\frac {a \int \frac {1}{\frac {b x}{a-b x}+1}d\frac {\sqrt {x}}{\sqrt {a-b x}}}{b}-\frac {\sqrt {x} \sqrt {a-b x}}{b}\right )}{4 b}-\frac {x^{3/2} \sqrt {a-b x}}{2 b}\right )}{6 b}-\frac {x^{5/2} \sqrt {a-b x}}{3 b}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {5 a \left (\frac {3 a \left (\frac {a \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{b^{3/2}}-\frac {\sqrt {x} \sqrt {a-b x}}{b}\right )}{4 b}-\frac {x^{3/2} \sqrt {a-b x}}{2 b}\right )}{6 b}-\frac {x^{5/2} \sqrt {a-b x}}{3 b}\) |
-1/3*(x^(5/2)*Sqrt[a - b*x])/b + (5*a*(-1/2*(x^(3/2)*Sqrt[a - b*x])/b + (3 *a*(-((Sqrt[x]*Sqrt[a - b*x])/b) + (a*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a - b* x]])/b^(3/2)))/(4*b)))/(6*b)
3.6.90.3.1 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 = 0.09 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.87
method | result | size |
risch | \(-\frac {\left (8 b^{2} x^{2}+10 a b x +15 a^{2}\right ) \sqrt {x}\, \sqrt {-b x +a}}{24 b^{3}}+\frac {5 a^{3} \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-b \,x^{2}+a x}}\right ) \sqrt {x \left (-b x +a \right )}}{16 b^{\frac {7}{2}} \sqrt {x}\, \sqrt {-b x +a}}\) | \(91\) |
default | \(-\frac {x^{\frac {5}{2}} \sqrt {-b x +a}}{3 b}+\frac {5 a \left (-\frac {x^{\frac {3}{2}} \sqrt {-b x +a}}{2 b}+\frac {3 a \left (-\frac {\sqrt {x}\, \sqrt {-b x +a}}{b}+\frac {a \sqrt {x \left (-b x +a \right )}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-b \,x^{2}+a x}}\right )}{2 b^{\frac {3}{2}} \sqrt {x}\, \sqrt {-b x +a}}\right )}{4 b}\right )}{6 b}\) | \(116\) |
-1/24*(8*b^2*x^2+10*a*b*x+15*a^2)/b^3*x^(1/2)*(-b*x+a)^(1/2)+5/16*a^3/b^(7 /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)
Time = 0.23 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.34 \[ \int \frac {x^{5/2}}{\sqrt {a-b x}} \, dx=\left [-\frac {15 \, a^{3} \sqrt {-b} \log \left (-2 \, b x + 2 \, \sqrt {-b x + a} \sqrt {-b} \sqrt {x} + a\right ) + 2 \, {\left (8 \, b^{3} x^{2} + 10 \, a b^{2} x + 15 \, a^{2} b\right )} \sqrt {-b x + a} \sqrt {x}}{48 \, b^{4}}, -\frac {15 \, a^{3} \sqrt {b} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) + {\left (8 \, b^{3} x^{2} + 10 \, a b^{2} x + 15 \, a^{2} b\right )} \sqrt {-b x + a} \sqrt {x}}{24 \, b^{4}}\right ] \]
[-1/48*(15*a^3*sqrt(-b)*log(-2*b*x + 2*sqrt(-b*x + a)*sqrt(-b)*sqrt(x) + a ) + 2*(8*b^3*x^2 + 10*a*b^2*x + 15*a^2*b)*sqrt(-b*x + a)*sqrt(x))/b^4, -1/ 24*(15*a^3*sqrt(b)*arctan(sqrt(-b*x + a)/(sqrt(b)*sqrt(x))) + (8*b^3*x^2 + 10*a*b^2*x + 15*a^2*b)*sqrt(-b*x + a)*sqrt(x))/b^4]
Result contains complex when optimal does not.
Time = 10.15 (sec) , antiderivative size = 270, normalized size of antiderivative = 2.57 \[ \int \frac {x^{5/2}}{\sqrt {a-b x}} \, dx=\begin {cases} \frac {5 i a^{\frac {5}{2}} \sqrt {x}}{8 b^{3} \sqrt {-1 + \frac {b x}{a}}} - \frac {5 i a^{\frac {3}{2}} x^{\frac {3}{2}}}{24 b^{2} \sqrt {-1 + \frac {b x}{a}}} - \frac {i \sqrt {a} x^{\frac {5}{2}}}{12 b \sqrt {-1 + \frac {b x}{a}}} - \frac {5 i a^{3} \operatorname {acosh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{8 b^{\frac {7}{2}}} - \frac {i x^{\frac {7}{2}}}{3 \sqrt {a} \sqrt {-1 + \frac {b x}{a}}} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\- \frac {5 a^{\frac {5}{2}} \sqrt {x}}{8 b^{3} \sqrt {1 - \frac {b x}{a}}} + \frac {5 a^{\frac {3}{2}} x^{\frac {3}{2}}}{24 b^{2} \sqrt {1 - \frac {b x}{a}}} + \frac {\sqrt {a} x^{\frac {5}{2}}}{12 b \sqrt {1 - \frac {b x}{a}}} + \frac {5 a^{3} \operatorname {asin}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{8 b^{\frac {7}{2}}} + \frac {x^{\frac {7}{2}}}{3 \sqrt {a} \sqrt {1 - \frac {b x}{a}}} & \text {otherwise} \end {cases} \]
Piecewise((5*I*a**(5/2)*sqrt(x)/(8*b**3*sqrt(-1 + b*x/a)) - 5*I*a**(3/2)*x **(3/2)/(24*b**2*sqrt(-1 + b*x/a)) - I*sqrt(a)*x**(5/2)/(12*b*sqrt(-1 + b* x/a)) - 5*I*a**3*acosh(sqrt(b)*sqrt(x)/sqrt(a))/(8*b**(7/2)) - I*x**(7/2)/ (3*sqrt(a)*sqrt(-1 + b*x/a)), Abs(b*x/a) > 1), (-5*a**(5/2)*sqrt(x)/(8*b** 3*sqrt(1 - b*x/a)) + 5*a**(3/2)*x**(3/2)/(24*b**2*sqrt(1 - b*x/a)) + sqrt( a)*x**(5/2)/(12*b*sqrt(1 - b*x/a)) + 5*a**3*asin(sqrt(b)*sqrt(x)/sqrt(a))/ (8*b**(7/2)) + x**(7/2)/(3*sqrt(a)*sqrt(1 - b*x/a)), True))
Time = 0.30 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.29 \[ \int \frac {x^{5/2}}{\sqrt {a-b x}} \, dx=-\frac {5 \, a^{3} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right )}{8 \, b^{\frac {7}{2}}} - \frac {\frac {33 \, \sqrt {-b x + a} a^{3} b^{2}}{\sqrt {x}} + \frac {40 \, {\left (-b x + a\right )}^{\frac {3}{2}} a^{3} b}{x^{\frac {3}{2}}} + \frac {15 \, {\left (-b x + a\right )}^{\frac {5}{2}} a^{3}}{x^{\frac {5}{2}}}}{24 \, {\left (b^{6} - \frac {3 \, {\left (b x - a\right )} b^{5}}{x} + \frac {3 \, {\left (b x - a\right )}^{2} b^{4}}{x^{2}} - \frac {{\left (b x - a\right )}^{3} b^{3}}{x^{3}}\right )}} \]
-5/8*a^3*arctan(sqrt(-b*x + a)/(sqrt(b)*sqrt(x)))/b^(7/2) - 1/24*(33*sqrt( -b*x + a)*a^3*b^2/sqrt(x) + 40*(-b*x + a)^(3/2)*a^3*b/x^(3/2) + 15*(-b*x + a)^(5/2)*a^3/x^(5/2))/(b^6 - 3*(b*x - a)*b^5/x + 3*(b*x - a)^2*b^4/x^2 - (b*x - a)^3*b^3/x^3)
Time = 78.89 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.10 \[ \int \frac {x^{5/2}}{\sqrt {a-b x}} \, dx=\frac {{\left (\frac {15 \, a^{3} \log \left ({\left | -\sqrt {-b x + a} \sqrt {-b} + \sqrt {{\left (b x - a\right )} b + a b} \right |}\right )}{\sqrt {-b} b} - \sqrt {{\left (b x - a\right )} b + a b} \sqrt {-b x + a} {\left (2 \, {\left (b x - a\right )} {\left (\frac {4 \, {\left (b x - a\right )}}{b^{2}} + \frac {13 \, a}{b^{2}}\right )} + \frac {33 \, a^{2}}{b^{2}}\right )}\right )} {\left | b \right |}}{24 \, b^{3}} \]
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)/b^3
Timed out. \[ \int \frac {x^{5/2}}{\sqrt {a-b x}} \, dx=\int \frac {x^{5/2}}{\sqrt {a-b\,x}} \,d x \]