Integrand size = 16, antiderivative size = 72 \[ \int \frac {x^{3/2}}{(a-b x)^{5/2}} \, dx=\frac {2 x^{3/2}}{3 b (a-b x)^{3/2}}-\frac {2 \sqrt {x}}{b^2 \sqrt {a-b x}}+\frac {2 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{b^{5/2}} \]
2/3*x^(3/2)/b/(-b*x+a)^(3/2)+2*arctan(b^(1/2)*x^(1/2)/(-b*x+a)^(1/2))/b^(5 /2)-2*x^(1/2)/b^2/(-b*x+a)^(1/2)
Time = 0.20 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.97 \[ \int \frac {x^{3/2}}{(a-b x)^{5/2}} \, dx=\frac {2 \sqrt {x} (-3 a+4 b x)}{3 b^2 (a-b x)^{3/2}}+\frac {4 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a-b x}}\right )}{b^{5/2}} \]
(2*Sqrt[x]*(-3*a + 4*b*x))/(3*b^2*(a - b*x)^(3/2)) + (4*ArcTan[(Sqrt[b]*Sq rt[x])/(-Sqrt[a] + Sqrt[a - b*x])])/b^(5/2)
Time = 0.16 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {57, 57, 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.
\(\displaystyle \int \frac {x^{3/2}}{(a-b x)^{5/2}} \, dx\) |
\(\Big \downarrow \) 57 |
\(\displaystyle \frac {2 x^{3/2}}{3 b (a-b x)^{3/2}}-\frac {\int \frac {\sqrt {x}}{(a-b x)^{3/2}}dx}{b}\) |
\(\Big \downarrow \) 57 |
\(\displaystyle \frac {2 x^{3/2}}{3 b (a-b x)^{3/2}}-\frac {\frac {2 \sqrt {x}}{b \sqrt {a-b x}}-\frac {\int \frac {1}{\sqrt {x} \sqrt {a-b x}}dx}{b}}{b}\) |
\(\Big \downarrow \) 65 |
\(\displaystyle \frac {2 x^{3/2}}{3 b (a-b x)^{3/2}}-\frac {\frac {2 \sqrt {x}}{b \sqrt {a-b x}}-\frac {2 \int \frac {1}{\frac {b x}{a-b x}+1}d\frac {\sqrt {x}}{\sqrt {a-b x}}}{b}}{b}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {2 x^{3/2}}{3 b (a-b x)^{3/2}}-\frac {\frac {2 \sqrt {x}}{b \sqrt {a-b x}}-\frac {2 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{b^{3/2}}}{b}\) |
(2*x^(3/2))/(3*b*(a - b*x)^(3/2)) - ((2*Sqrt[x])/(b*Sqrt[a - b*x]) - (2*Ar cTan[(Sqrt[b]*Sqrt[x])/Sqrt[a - b*x]])/b^(3/2))/b
3.7.3.3.1 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 + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] & & GtQ[n, 0] && LtQ[m, -1] && !(IntegerQ[n] && !IntegerQ[m]) && !(ILeQ[m + n + 2, 0] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[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])
\[\int \frac {x^{\frac {3}{2}}}{\left (-b x +a \right )^{\frac {5}{2}}}d x\]
Time = 0.24 (sec) , antiderivative size = 188, normalized size of antiderivative = 2.61 \[ \int \frac {x^{3/2}}{(a-b x)^{5/2}} \, dx=\left [-\frac {3 \, {\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \sqrt {-b} \log \left (-2 \, b x + 2 \, \sqrt {-b x + a} \sqrt {-b} \sqrt {x} + a\right ) - 2 \, {\left (4 \, b^{2} x - 3 \, a b\right )} \sqrt {-b x + a} \sqrt {x}}{3 \, {\left (b^{5} x^{2} - 2 \, a b^{4} x + a^{2} b^{3}\right )}}, -\frac {2 \, {\left (3 \, {\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \sqrt {b} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) - {\left (4 \, b^{2} x - 3 \, a b\right )} \sqrt {-b x + a} \sqrt {x}\right )}}{3 \, {\left (b^{5} x^{2} - 2 \, a b^{4} x + a^{2} b^{3}\right )}}\right ] \]
[-1/3*(3*(b^2*x^2 - 2*a*b*x + a^2)*sqrt(-b)*log(-2*b*x + 2*sqrt(-b*x + a)* sqrt(-b)*sqrt(x) + a) - 2*(4*b^2*x - 3*a*b)*sqrt(-b*x + a)*sqrt(x))/(b^5*x ^2 - 2*a*b^4*x + a^2*b^3), -2/3*(3*(b^2*x^2 - 2*a*b*x + a^2)*sqrt(b)*arcta n(sqrt(-b*x + a)/(sqrt(b)*sqrt(x))) - (4*b^2*x - 3*a*b)*sqrt(-b*x + a)*sqr t(x))/(b^5*x^2 - 2*a*b^4*x + a^2*b^3)]
Result contains complex when optimal does not.
Time = 2.49 (sec) , antiderivative size = 833, normalized size of antiderivative = 11.57 \[ \int \frac {x^{3/2}}{(a-b x)^{5/2}} \, dx=\begin {cases} - \frac {6 i a^{\frac {39}{2}} b^{11} x^{\frac {27}{2}} \sqrt {-1 + \frac {b x}{a}} \operatorname {acosh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {-1 + \frac {b x}{a}} - 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{\frac {29}{2}} \sqrt {-1 + \frac {b x}{a}}} + \frac {3 \pi a^{\frac {39}{2}} b^{11} x^{\frac {27}{2}} \sqrt {-1 + \frac {b x}{a}}}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {-1 + \frac {b x}{a}} - 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{\frac {29}{2}} \sqrt {-1 + \frac {b x}{a}}} + \frac {6 i a^{\frac {37}{2}} b^{12} x^{\frac {29}{2}} \sqrt {-1 + \frac {b x}{a}} \operatorname {acosh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {-1 + \frac {b x}{a}} - 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{\frac {29}{2}} \sqrt {-1 + \frac {b x}{a}}} - \frac {3 \pi a^{\frac {37}{2}} b^{12} x^{\frac {29}{2}} \sqrt {-1 + \frac {b x}{a}}}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {-1 + \frac {b x}{a}} - 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{\frac {29}{2}} \sqrt {-1 + \frac {b x}{a}}} + \frac {6 i a^{19} b^{\frac {23}{2}} x^{14}}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {-1 + \frac {b x}{a}} - 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{\frac {29}{2}} \sqrt {-1 + \frac {b x}{a}}} - \frac {8 i a^{18} b^{\frac {25}{2}} x^{15}}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {-1 + \frac {b x}{a}} - 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{\frac {29}{2}} \sqrt {-1 + \frac {b x}{a}}} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\\frac {6 a^{\frac {39}{2}} b^{11} x^{\frac {27}{2}} \sqrt {1 - \frac {b x}{a}} \operatorname {asin}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {1 - \frac {b x}{a}} - 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{\frac {29}{2}} \sqrt {1 - \frac {b x}{a}}} - \frac {6 a^{\frac {37}{2}} b^{12} x^{\frac {29}{2}} \sqrt {1 - \frac {b x}{a}} \operatorname {asin}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {1 - \frac {b x}{a}} - 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{\frac {29}{2}} \sqrt {1 - \frac {b x}{a}}} - \frac {6 a^{19} b^{\frac {23}{2}} x^{14}}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {1 - \frac {b x}{a}} - 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{\frac {29}{2}} \sqrt {1 - \frac {b x}{a}}} + \frac {8 a^{18} b^{\frac {25}{2}} x^{15}}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {1 - \frac {b x}{a}} - 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{\frac {29}{2}} \sqrt {1 - \frac {b x}{a}}} & \text {otherwise} \end {cases} \]
Piecewise((-6*I*a**(39/2)*b**11*x**(27/2)*sqrt(-1 + b*x/a)*acosh(sqrt(b)*s qrt(x)/sqrt(a))/(3*a**(39/2)*b**(27/2)*x**(27/2)*sqrt(-1 + b*x/a) - 3*a**( 37/2)*b**(29/2)*x**(29/2)*sqrt(-1 + b*x/a)) + 3*pi*a**(39/2)*b**11*x**(27/ 2)*sqrt(-1 + b*x/a)/(3*a**(39/2)*b**(27/2)*x**(27/2)*sqrt(-1 + b*x/a) - 3* a**(37/2)*b**(29/2)*x**(29/2)*sqrt(-1 + b*x/a)) + 6*I*a**(37/2)*b**12*x**( 29/2)*sqrt(-1 + b*x/a)*acosh(sqrt(b)*sqrt(x)/sqrt(a))/(3*a**(39/2)*b**(27/ 2)*x**(27/2)*sqrt(-1 + b*x/a) - 3*a**(37/2)*b**(29/2)*x**(29/2)*sqrt(-1 + b*x/a)) - 3*pi*a**(37/2)*b**12*x**(29/2)*sqrt(-1 + b*x/a)/(3*a**(39/2)*b** (27/2)*x**(27/2)*sqrt(-1 + b*x/a) - 3*a**(37/2)*b**(29/2)*x**(29/2)*sqrt(- 1 + b*x/a)) + 6*I*a**19*b**(23/2)*x**14/(3*a**(39/2)*b**(27/2)*x**(27/2)*s qrt(-1 + b*x/a) - 3*a**(37/2)*b**(29/2)*x**(29/2)*sqrt(-1 + b*x/a)) - 8*I* a**18*b**(25/2)*x**15/(3*a**(39/2)*b**(27/2)*x**(27/2)*sqrt(-1 + b*x/a) - 3*a**(37/2)*b**(29/2)*x**(29/2)*sqrt(-1 + b*x/a)), Abs(b*x/a) > 1), (6*a** (39/2)*b**11*x**(27/2)*sqrt(1 - b*x/a)*asin(sqrt(b)*sqrt(x)/sqrt(a))/(3*a* *(39/2)*b**(27/2)*x**(27/2)*sqrt(1 - b*x/a) - 3*a**(37/2)*b**(29/2)*x**(29 /2)*sqrt(1 - b*x/a)) - 6*a**(37/2)*b**12*x**(29/2)*sqrt(1 - b*x/a)*asin(sq rt(b)*sqrt(x)/sqrt(a))/(3*a**(39/2)*b**(27/2)*x**(27/2)*sqrt(1 - b*x/a) - 3*a**(37/2)*b**(29/2)*x**(29/2)*sqrt(1 - b*x/a)) - 6*a**19*b**(23/2)*x**14 /(3*a**(39/2)*b**(27/2)*x**(27/2)*sqrt(1 - b*x/a) - 3*a**(37/2)*b**(29/2)* x**(29/2)*sqrt(1 - b*x/a)) + 8*a**18*b**(25/2)*x**15/(3*a**(39/2)*b**(2...
Time = 0.30 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.72 \[ \int \frac {x^{3/2}}{(a-b x)^{5/2}} \, dx=\frac {2 \, {\left (b + \frac {3 \, {\left (b x - a\right )}}{x}\right )} x^{\frac {3}{2}}}{3 \, {\left (-b x + a\right )}^{\frac {3}{2}} b^{2}} - \frac {2 \, \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right )}{b^{\frac {5}{2}}} \]
2/3*(b + 3*(b*x - a)/x)*x^(3/2)/((-b*x + a)^(3/2)*b^2) - 2*arctan(sqrt(-b* x + a)/(sqrt(b)*sqrt(x)))/b^(5/2)
Leaf count of result is larger than twice the leaf count of optimal. 194 vs. \(2 (54) = 108\).
Time = 16.02 (sec) , antiderivative size = 194, normalized size of antiderivative = 2.69 \[ \int \frac {x^{3/2}}{(a-b x)^{5/2}} \, dx=\frac {{\left (\frac {3 \, \log \left ({\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{2}\right )}{\sqrt {-b}} + \frac {8 \, {\left (3 \, a {\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{4} \sqrt {-b} - 3 \, a^{2} {\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{2} \sqrt {-b} b + 2 \, a^{3} \sqrt {-b} b^{2}\right )}}{{\left ({\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{2} - a b\right )}^{3}}\right )} {\left | b \right |}}{3 \, b^{3}} \]
1/3*(3*log((sqrt(-b*x + a)*sqrt(-b) - sqrt((b*x - a)*b + a*b))^2)/sqrt(-b) + 8*(3*a*(sqrt(-b*x + a)*sqrt(-b) - sqrt((b*x - a)*b + a*b))^4*sqrt(-b) - 3*a^2*(sqrt(-b*x + a)*sqrt(-b) - sqrt((b*x - a)*b + a*b))^2*sqrt(-b)*b + 2*a^3*sqrt(-b)*b^2)/((sqrt(-b*x + a)*sqrt(-b) - sqrt((b*x - a)*b + a*b))^2 - a*b)^3)*abs(b)/b^3
Timed out. \[ \int \frac {x^{3/2}}{(a-b x)^{5/2}} \, dx=\int \frac {x^{3/2}}{{\left (a-b\,x\right )}^{5/2}} \,d x \]