Integrand size = 26, antiderivative size = 111 \[ \int \frac {x^3 (1+a x)}{\sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {75 \sqrt {a x} \sqrt {1-a x}}{64 a^4}-\frac {25 (a x)^{3/2} \sqrt {1-a x}}{32 a^4}-\frac {5 (a x)^{5/2} \sqrt {1-a x}}{8 a^4}-\frac {(a x)^{7/2} \sqrt {1-a x}}{4 a^4}-\frac {75 \arcsin (1-2 a x)}{128 a^4} \]
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Time = 0.04 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {16, 81, 52, 55, 633, 222} \[ \int \frac {x^3 (1+a x)}{\sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {75 \arcsin (1-2 a x)}{128 a^4}-\frac {\sqrt {1-a x} (a x)^{7/2}}{4 a^4}-\frac {5 \sqrt {1-a x} (a x)^{5/2}}{8 a^4}-\frac {25 \sqrt {1-a x} (a x)^{3/2}}{32 a^4}-\frac {75 \sqrt {1-a x} \sqrt {a x}}{64 a^4} \]
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Rule 16
Rule 52
Rule 55
Rule 81
Rule 222
Rule 633
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {(a x)^{5/2} (1+a x)}{\sqrt {1-a x}} \, dx}{a^3} \\ & = -\frac {(a x)^{7/2} \sqrt {1-a x}}{4 a^4}+\frac {15 \int \frac {(a x)^{5/2}}{\sqrt {1-a x}} \, dx}{8 a^3} \\ & = -\frac {5 (a x)^{5/2} \sqrt {1-a x}}{8 a^4}-\frac {(a x)^{7/2} \sqrt {1-a x}}{4 a^4}+\frac {25 \int \frac {(a x)^{3/2}}{\sqrt {1-a x}} \, dx}{16 a^3} \\ & = -\frac {25 (a x)^{3/2} \sqrt {1-a x}}{32 a^4}-\frac {5 (a x)^{5/2} \sqrt {1-a x}}{8 a^4}-\frac {(a x)^{7/2} \sqrt {1-a x}}{4 a^4}+\frac {75 \int \frac {\sqrt {a x}}{\sqrt {1-a x}} \, dx}{64 a^3} \\ & = -\frac {75 \sqrt {a x} \sqrt {1-a x}}{64 a^4}-\frac {25 (a x)^{3/2} \sqrt {1-a x}}{32 a^4}-\frac {5 (a x)^{5/2} \sqrt {1-a x}}{8 a^4}-\frac {(a x)^{7/2} \sqrt {1-a x}}{4 a^4}+\frac {75 \int \frac {1}{\sqrt {a x} \sqrt {1-a x}} \, dx}{128 a^3} \\ & = -\frac {75 \sqrt {a x} \sqrt {1-a x}}{64 a^4}-\frac {25 (a x)^{3/2} \sqrt {1-a x}}{32 a^4}-\frac {5 (a x)^{5/2} \sqrt {1-a x}}{8 a^4}-\frac {(a x)^{7/2} \sqrt {1-a x}}{4 a^4}+\frac {75 \int \frac {1}{\sqrt {a x-a^2 x^2}} \, dx}{128 a^3} \\ & = -\frac {75 \sqrt {a x} \sqrt {1-a x}}{64 a^4}-\frac {25 (a x)^{3/2} \sqrt {1-a x}}{32 a^4}-\frac {5 (a x)^{5/2} \sqrt {1-a x}}{8 a^4}-\frac {(a x)^{7/2} \sqrt {1-a x}}{4 a^4}-\frac {75 \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,a-2 a^2 x\right )}{128 a^5} \\ & = -\frac {75 \sqrt {a x} \sqrt {1-a x}}{64 a^4}-\frac {25 (a x)^{3/2} \sqrt {1-a x}}{32 a^4}-\frac {5 (a x)^{5/2} \sqrt {1-a x}}{8 a^4}-\frac {(a x)^{7/2} \sqrt {1-a x}}{4 a^4}-\frac {75 \sin ^{-1}(1-2 a x)}{128 a^4} \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.93 \[ \int \frac {x^3 (1+a x)}{\sqrt {a x} \sqrt {1-a x}} \, dx=\frac {\sqrt {a} x \left (-75+25 a x+10 a^2 x^2+24 a^3 x^3+16 a^4 x^4\right )+150 \sqrt {x} \sqrt {1-a x} \arctan \left (\frac {\sqrt {a} \sqrt {x}}{-1+\sqrt {1-a x}}\right )}{64 a^{7/2} \sqrt {-a x (-1+a x)}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.58 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.19
method | result | size |
default | \(-\frac {\sqrt {-a x +1}\, x \left (32 \,\operatorname {csgn}\left (a \right ) a^{3} x^{3} \sqrt {-x \left (a x -1\right ) a}+80 \,\operatorname {csgn}\left (a \right ) x^{2} a^{2} \sqrt {-x \left (a x -1\right ) a}+100 \,\operatorname {csgn}\left (a \right ) \sqrt {-x \left (a x -1\right ) a}\, a x +150 \,\operatorname {csgn}\left (a \right ) \sqrt {-x \left (a x -1\right ) a}-75 \arctan \left (\frac {\operatorname {csgn}\left (a \right ) \left (2 a x -1\right )}{2 \sqrt {-x \left (a x -1\right ) a}}\right )\right ) \operatorname {csgn}\left (a \right )}{128 a^{3} \sqrt {a x}\, \sqrt {-x \left (a x -1\right ) a}}\) | \(132\) |
risch | \(\frac {\left (16 a^{3} x^{3}+40 a^{2} x^{2}+50 a x +75\right ) x \left (a x -1\right ) \sqrt {a x \left (-a x +1\right )}}{64 a^{3} \sqrt {-x \left (a x -1\right ) a}\, \sqrt {a x}\, \sqrt {-a x +1}}+\frac {75 \arctan \left (\frac {\sqrt {a^{2}}\, \left (x -\frac {1}{2 a}\right )}{\sqrt {-a^{2} x^{2}+a x}}\right ) \sqrt {a x \left (-a x +1\right )}}{128 a^{3} \sqrt {a^{2}}\, \sqrt {a x}\, \sqrt {-a x +1}}\) | \(132\) |
meijerg | \(-\frac {\sqrt {x}\, \left (-\frac {\sqrt {\pi }\, \sqrt {x}\, \left (-a \right )^{\frac {9}{2}} \left (144 a^{3} x^{3}+168 a^{2} x^{2}+210 a x +315\right ) \sqrt {-a x +1}}{576 a^{4}}+\frac {35 \sqrt {\pi }\, \left (-a \right )^{\frac {9}{2}} \arcsin \left (\sqrt {a}\, \sqrt {x}\right )}{64 a^{\frac {9}{2}}}\right )}{\left (-a \right )^{\frac {7}{2}} \sqrt {a x}\, \sqrt {\pi }}-\frac {\sqrt {x}\, \left (-\frac {\sqrt {\pi }\, \sqrt {x}\, \left (-a \right )^{\frac {7}{2}} \left (56 a^{2} x^{2}+70 a x +105\right ) \sqrt {-a x +1}}{168 a^{3}}+\frac {5 \sqrt {\pi }\, \left (-a \right )^{\frac {7}{2}} \arcsin \left (\sqrt {a}\, \sqrt {x}\right )}{8 a^{\frac {7}{2}}}\right )}{\left (-a \right )^{\frac {5}{2}} \sqrt {a x}\, \sqrt {\pi }\, a}\) | \(169\) |
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Time = 0.31 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.59 \[ \int \frac {x^3 (1+a x)}{\sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {{\left (16 \, a^{3} x^{3} + 40 \, a^{2} x^{2} + 50 \, a x + 75\right )} \sqrt {a x} \sqrt {-a x + 1} + 75 \, \arctan \left (\frac {\sqrt {a x} \sqrt {-a x + 1}}{a x}\right )}{64 \, a^{4}} \]
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Result contains complex when optimal does not.
Time = 41.97 (sec) , antiderivative size = 484, normalized size of antiderivative = 4.36 \[ \int \frac {x^3 (1+a x)}{\sqrt {a x} \sqrt {1-a x}} \, dx=a \left (\begin {cases} - \frac {35 i \operatorname {acosh}{\left (\sqrt {a} \sqrt {x} \right )}}{64 a^{5}} - \frac {i x^{\frac {9}{2}}}{4 \sqrt {a} \sqrt {a x - 1}} - \frac {i x^{\frac {7}{2}}}{24 a^{\frac {3}{2}} \sqrt {a x - 1}} - \frac {7 i x^{\frac {5}{2}}}{96 a^{\frac {5}{2}} \sqrt {a x - 1}} - \frac {35 i x^{\frac {3}{2}}}{192 a^{\frac {7}{2}} \sqrt {a x - 1}} + \frac {35 i \sqrt {x}}{64 a^{\frac {9}{2}} \sqrt {a x - 1}} & \text {for}\: \left |{a x}\right | > 1 \\\frac {35 \operatorname {asin}{\left (\sqrt {a} \sqrt {x} \right )}}{64 a^{5}} + \frac {x^{\frac {9}{2}}}{4 \sqrt {a} \sqrt {- a x + 1}} + \frac {x^{\frac {7}{2}}}{24 a^{\frac {3}{2}} \sqrt {- a x + 1}} + \frac {7 x^{\frac {5}{2}}}{96 a^{\frac {5}{2}} \sqrt {- a x + 1}} + \frac {35 x^{\frac {3}{2}}}{192 a^{\frac {7}{2}} \sqrt {- a x + 1}} - \frac {35 \sqrt {x}}{64 a^{\frac {9}{2}} \sqrt {- a x + 1}} & \text {otherwise} \end {cases}\right ) + \begin {cases} - \frac {5 i \operatorname {acosh}{\left (\sqrt {a} \sqrt {x} \right )}}{8 a^{4}} - \frac {i x^{\frac {7}{2}}}{3 \sqrt {a} \sqrt {a x - 1}} - \frac {i x^{\frac {5}{2}}}{12 a^{\frac {3}{2}} \sqrt {a x - 1}} - \frac {5 i x^{\frac {3}{2}}}{24 a^{\frac {5}{2}} \sqrt {a x - 1}} + \frac {5 i \sqrt {x}}{8 a^{\frac {7}{2}} \sqrt {a x - 1}} & \text {for}\: \left |{a x}\right | > 1 \\\frac {5 \operatorname {asin}{\left (\sqrt {a} \sqrt {x} \right )}}{8 a^{4}} + \frac {x^{\frac {7}{2}}}{3 \sqrt {a} \sqrt {- a x + 1}} + \frac {x^{\frac {5}{2}}}{12 a^{\frac {3}{2}} \sqrt {- a x + 1}} + \frac {5 x^{\frac {3}{2}}}{24 a^{\frac {5}{2}} \sqrt {- a x + 1}} - \frac {5 \sqrt {x}}{8 a^{\frac {7}{2}} \sqrt {- a x + 1}} & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.95 \[ \int \frac {x^3 (1+a x)}{\sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {\sqrt {-a^{2} x^{2} + a x} x^{3}}{4 \, a} - \frac {5 \, \sqrt {-a^{2} x^{2} + a x} x^{2}}{8 \, a^{2}} - \frac {25 \, \sqrt {-a^{2} x^{2} + a x} x}{32 \, a^{3}} - \frac {75 \, \arcsin \left (-\frac {2 \, a^{2} x - a}{a}\right )}{128 \, a^{4}} - \frac {75 \, \sqrt {-a^{2} x^{2} + a x}}{64 \, a^{4}} \]
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Time = 0.28 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.41 \[ \int \frac {x^3 (1+a x)}{\sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {{\left (2 \, {\left (4 \, {\left (2 \, a x + 5\right )} a x + 25\right )} a x + 75\right )} \sqrt {a x} \sqrt {-a x + 1} - 75 \, \arcsin \left (\sqrt {a x}\right )}{64 \, a^{4}} \]
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Time = 9.03 (sec) , antiderivative size = 345, normalized size of antiderivative = 3.11 \[ \int \frac {x^3 (1+a x)}{\sqrt {a x} \sqrt {1-a x}} \, dx=\frac {75\,\mathrm {atan}\left (\frac {\sqrt {a\,x}}{\sqrt {1-a\,x}-1}\right )}{32\,a^4}-\frac {\frac {5\,\sqrt {a\,x}}{4\,\left (\sqrt {1-a\,x}-1\right )}+\frac {85\,{\left (a\,x\right )}^{3/2}}{12\,{\left (\sqrt {1-a\,x}-1\right )}^3}+\frac {33\,{\left (a\,x\right )}^{5/2}}{2\,{\left (\sqrt {1-a\,x}-1\right )}^5}-\frac {33\,{\left (a\,x\right )}^{7/2}}{2\,{\left (\sqrt {1-a\,x}-1\right )}^7}-\frac {85\,{\left (a\,x\right )}^{9/2}}{12\,{\left (\sqrt {1-a\,x}-1\right )}^9}-\frac {5\,{\left (a\,x\right )}^{11/2}}{4\,{\left (\sqrt {1-a\,x}-1\right )}^{11}}}{a^4\,{\left (\frac {a\,x}{{\left (\sqrt {1-a\,x}-1\right )}^2}+1\right )}^6}-\frac {\frac {35\,\sqrt {a\,x}}{32\,\left (\sqrt {1-a\,x}-1\right )}+\frac {805\,{\left (a\,x\right )}^{3/2}}{96\,{\left (\sqrt {1-a\,x}-1\right )}^3}+\frac {2681\,{\left (a\,x\right )}^{5/2}}{96\,{\left (\sqrt {1-a\,x}-1\right )}^5}+\frac {5053\,{\left (a\,x\right )}^{7/2}}{96\,{\left (\sqrt {1-a\,x}-1\right )}^7}-\frac {5053\,{\left (a\,x\right )}^{9/2}}{96\,{\left (\sqrt {1-a\,x}-1\right )}^9}-\frac {2681\,{\left (a\,x\right )}^{11/2}}{96\,{\left (\sqrt {1-a\,x}-1\right )}^{11}}-\frac {805\,{\left (a\,x\right )}^{13/2}}{96\,{\left (\sqrt {1-a\,x}-1\right )}^{13}}-\frac {35\,{\left (a\,x\right )}^{15/2}}{32\,{\left (\sqrt {1-a\,x}-1\right )}^{15}}}{a^4\,{\left (\frac {a\,x}{{\left (\sqrt {1-a\,x}-1\right )}^2}+1\right )}^8} \]
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