Integrand size = 24, antiderivative size = 63 \[ \int \frac {x (1+a x)}{\sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {7 \sqrt {a x} \sqrt {1-a x}}{4 a^2}-\frac {(a x)^{3/2} \sqrt {1-a x}}{2 a^2}-\frac {7 \arcsin (1-2 a x)}{8 a^2} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {16, 81, 52, 55, 633, 222} \[ \int \frac {x (1+a x)}{\sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {7 \arcsin (1-2 a x)}{8 a^2}-\frac {\sqrt {1-a x} (a x)^{3/2}}{2 a^2}-\frac {7 \sqrt {1-a x} \sqrt {a x}}{4 a^2} \]
[In]
[Out]
Rule 16
Rule 52
Rule 55
Rule 81
Rule 222
Rule 633
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {\sqrt {a x} (1+a x)}{\sqrt {1-a x}} \, dx}{a} \\ & = -\frac {(a x)^{3/2} \sqrt {1-a x}}{2 a^2}+\frac {7 \int \frac {\sqrt {a x}}{\sqrt {1-a x}} \, dx}{4 a} \\ & = -\frac {7 \sqrt {a x} \sqrt {1-a x}}{4 a^2}-\frac {(a x)^{3/2} \sqrt {1-a x}}{2 a^2}+\frac {7 \int \frac {1}{\sqrt {a x} \sqrt {1-a x}} \, dx}{8 a} \\ & = -\frac {7 \sqrt {a x} \sqrt {1-a x}}{4 a^2}-\frac {(a x)^{3/2} \sqrt {1-a x}}{2 a^2}+\frac {7 \int \frac {1}{\sqrt {a x-a^2 x^2}} \, dx}{8 a} \\ & = -\frac {7 \sqrt {a x} \sqrt {1-a x}}{4 a^2}-\frac {(a x)^{3/2} \sqrt {1-a x}}{2 a^2}-\frac {7 \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,a-2 a^2 x\right )}{8 a^3} \\ & = -\frac {7 \sqrt {a x} \sqrt {1-a x}}{4 a^2}-\frac {(a x)^{3/2} \sqrt {1-a x}}{2 a^2}-\frac {7 \sin ^{-1}(1-2 a x)}{8 a^2} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.38 \[ \int \frac {x (1+a x)}{\sqrt {a x} \sqrt {1-a x}} \, dx=\frac {\sqrt {a} x \left (-7+5 a x+2 a^2 x^2\right )+14 \sqrt {x} \sqrt {1-a x} \arctan \left (\frac {\sqrt {a} \sqrt {x}}{-1+\sqrt {1-a x}}\right )}{4 a^{3/2} \sqrt {-a x (-1+a x)}} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.54 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.43
method | result | size |
default | \(-\frac {\sqrt {-a x +1}\, x \left (4 \,\operatorname {csgn}\left (a \right ) \sqrt {-x \left (a x -1\right ) a}\, a x +14 \,\operatorname {csgn}\left (a \right ) \sqrt {-x \left (a x -1\right ) a}-7 \arctan \left (\frac {\operatorname {csgn}\left (a \right ) \left (2 a x -1\right )}{2 \sqrt {-x \left (a x -1\right ) a}}\right )\right ) \operatorname {csgn}\left (a \right )}{8 a \sqrt {a x}\, \sqrt {-x \left (a x -1\right ) a}}\) | \(90\) |
risch | \(\frac {\left (2 a x +7\right ) x \left (a x -1\right ) \sqrt {a x \left (-a x +1\right )}}{4 a \sqrt {-x \left (a x -1\right ) a}\, \sqrt {a x}\, \sqrt {-a x +1}}+\frac {7 \arctan \left (\frac {\sqrt {a^{2}}\, \left (x -\frac {1}{2 a}\right )}{\sqrt {-a^{2} x^{2}+a x}}\right ) \sqrt {a x \left (-a x +1\right )}}{8 a \sqrt {a^{2}}\, \sqrt {a x}\, \sqrt {-a x +1}}\) | \(116\) |
meijerg | \(-\frac {\sqrt {x}\, \left (-\frac {\sqrt {\pi }\, \sqrt {x}\, \left (-a \right )^{\frac {5}{2}} \left (10 a x +15\right ) \sqrt {-a x +1}}{20 a^{2}}+\frac {3 \sqrt {\pi }\, \left (-a \right )^{\frac {5}{2}} \arcsin \left (\sqrt {a}\, \sqrt {x}\right )}{4 a^{\frac {5}{2}}}\right )}{\left (-a \right )^{\frac {3}{2}} \sqrt {a x}\, \sqrt {\pi }}-\frac {\sqrt {x}\, \left (-\frac {\sqrt {\pi }\, \sqrt {x}\, \left (-a \right )^{\frac {3}{2}} \sqrt {-a x +1}}{a}+\frac {\sqrt {\pi }\, \left (-a \right )^{\frac {3}{2}} \arcsin \left (\sqrt {a}\, \sqrt {x}\right )}{a^{\frac {3}{2}}}\right )}{\sqrt {-a}\, \sqrt {a x}\, \sqrt {\pi }\, a}\) | \(138\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.78 \[ \int \frac {x (1+a x)}{\sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {{\left (2 \, a x + 7\right )} \sqrt {a x} \sqrt {-a x + 1} + 7 \, \arctan \left (\frac {\sqrt {a x} \sqrt {-a x + 1}}{a x}\right )}{4 \, a^{2}} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 4.94 (sec) , antiderivative size = 269, normalized size of antiderivative = 4.27 \[ \int \frac {x (1+a x)}{\sqrt {a x} \sqrt {1-a x}} \, dx=a \left (\begin {cases} - \frac {3 i \operatorname {acosh}{\left (\sqrt {a} \sqrt {x} \right )}}{4 a^{3}} - \frac {i x^{\frac {5}{2}}}{2 \sqrt {a} \sqrt {a x - 1}} - \frac {i x^{\frac {3}{2}}}{4 a^{\frac {3}{2}} \sqrt {a x - 1}} + \frac {3 i \sqrt {x}}{4 a^{\frac {5}{2}} \sqrt {a x - 1}} & \text {for}\: \left |{a x}\right | > 1 \\\frac {3 \operatorname {asin}{\left (\sqrt {a} \sqrt {x} \right )}}{4 a^{3}} + \frac {x^{\frac {5}{2}}}{2 \sqrt {a} \sqrt {- a x + 1}} + \frac {x^{\frac {3}{2}}}{4 a^{\frac {3}{2}} \sqrt {- a x + 1}} - \frac {3 \sqrt {x}}{4 a^{\frac {5}{2}} \sqrt {- a x + 1}} & \text {otherwise} \end {cases}\right ) + \begin {cases} - \frac {i \operatorname {acosh}{\left (\sqrt {a} \sqrt {x} \right )}}{a^{2}} - \frac {i \sqrt {x} \sqrt {a x - 1}}{a^{\frac {3}{2}}} & \text {for}\: \left |{a x}\right | > 1 \\\frac {\operatorname {asin}{\left (\sqrt {a} \sqrt {x} \right )}}{a^{2}} + \frac {x^{\frac {3}{2}}}{\sqrt {a} \sqrt {- a x + 1}} - \frac {\sqrt {x}}{a^{\frac {3}{2}} \sqrt {- a x + 1}} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.97 \[ \int \frac {x (1+a x)}{\sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {\sqrt {-a^{2} x^{2} + a x} x}{2 \, a} - \frac {7 \, \arcsin \left (-\frac {2 \, a^{2} x - a}{a}\right )}{8 \, a^{2}} - \frac {7 \, \sqrt {-a^{2} x^{2} + a x}}{4 \, a^{2}} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.54 \[ \int \frac {x (1+a x)}{\sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {{\left (2 \, a x + 7\right )} \sqrt {a x} \sqrt {-a x + 1} - 7 \, \arcsin \left (\sqrt {a x}\right )}{4 \, a^{2}} \]
[In]
[Out]
Time = 5.14 (sec) , antiderivative size = 191, normalized size of antiderivative = 3.03 \[ \int \frac {x (1+a x)}{\sqrt {a x} \sqrt {1-a x}} \, dx=\frac {7\,\mathrm {atan}\left (\frac {\sqrt {a\,x}}{\sqrt {1-a\,x}-1}\right )}{2\,a^2}-\frac {\frac {2\,\sqrt {a\,x}}{\sqrt {1-a\,x}-1}-\frac {2\,{\left (a\,x\right )}^{3/2}}{{\left (\sqrt {1-a\,x}-1\right )}^3}}{a^2\,{\left (\frac {a\,x}{{\left (\sqrt {1-a\,x}-1\right )}^2}+1\right )}^2}-\frac {\frac {3\,\sqrt {a\,x}}{2\,\left (\sqrt {1-a\,x}-1\right )}+\frac {11\,{\left (a\,x\right )}^{3/2}}{2\,{\left (\sqrt {1-a\,x}-1\right )}^3}-\frac {11\,{\left (a\,x\right )}^{5/2}}{2\,{\left (\sqrt {1-a\,x}-1\right )}^5}-\frac {3\,{\left (a\,x\right )}^{7/2}}{2\,{\left (\sqrt {1-a\,x}-1\right )}^7}}{a^2\,{\left (\frac {a\,x}{{\left (\sqrt {1-a\,x}-1\right )}^2}+1\right )}^4} \]
[In]
[Out]