Integrand size = 31, antiderivative size = 80 \[ \int \frac {\sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right )}{x} \, dx=a \sqrt {-c+d x} \sqrt {c+d x}+\frac {b (-c+d x)^{3/2} (c+d x)^{3/2}}{3 d^2}-a c \arctan \left (\frac {\sqrt {-c+d x} \sqrt {c+d x}}{c}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {471, 103, 12, 94, 211} \[ \int \frac {\sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right )}{x} \, dx=-a c \arctan \left (\frac {\sqrt {d x-c} \sqrt {c+d x}}{c}\right )+a \sqrt {d x-c} \sqrt {c+d x}+\frac {b (d x-c)^{3/2} (c+d x)^{3/2}}{3 d^2} \]
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Rule 12
Rule 94
Rule 103
Rule 211
Rule 471
Rubi steps \begin{align*} \text {integral}& = \frac {b (-c+d x)^{3/2} (c+d x)^{3/2}}{3 d^2}+a \int \frac {\sqrt {-c+d x} \sqrt {c+d x}}{x} \, dx \\ & = a \sqrt {-c+d x} \sqrt {c+d x}+\frac {b (-c+d x)^{3/2} (c+d x)^{3/2}}{3 d^2}-a \int \frac {c^2}{x \sqrt {-c+d x} \sqrt {c+d x}} \, dx \\ & = a \sqrt {-c+d x} \sqrt {c+d x}+\frac {b (-c+d x)^{3/2} (c+d x)^{3/2}}{3 d^2}-\left (a c^2\right ) \int \frac {1}{x \sqrt {-c+d x} \sqrt {c+d x}} \, dx \\ & = a \sqrt {-c+d x} \sqrt {c+d x}+\frac {b (-c+d x)^{3/2} (c+d x)^{3/2}}{3 d^2}-\left (a c^2 d\right ) \text {Subst}\left (\int \frac {1}{c^2 d+d x^2} \, dx,x,\sqrt {-c+d x} \sqrt {c+d x}\right ) \\ & = a \sqrt {-c+d x} \sqrt {c+d x}+\frac {b (-c+d x)^{3/2} (c+d x)^{3/2}}{3 d^2}-a c \tan ^{-1}\left (\frac {\sqrt {-c+d x} \sqrt {c+d x}}{c}\right ) \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right )}{x} \, dx=\frac {\sqrt {-c+d x} \sqrt {c+d x} \left (-b c^2+3 a d^2+b d^2 x^2\right )}{3 d^2}-2 a c \arctan \left (\frac {\sqrt {-c+d x}}{\sqrt {c+d x}}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(173\) vs. \(2(66)=132\).
Time = 4.02 (sec) , antiderivative size = 174, normalized size of antiderivative = 2.18
method | result | size |
default | \(\frac {\sqrt {d x -c}\, \sqrt {d x +c}\, \left (b \,d^{2} x^{2} \sqrt {-c^{2}}\, \sqrt {d^{2} x^{2}-c^{2}}+3 \ln \left (-\frac {2 \left (c^{2}-\sqrt {-c^{2}}\, \sqrt {d^{2} x^{2}-c^{2}}\right )}{x}\right ) a \,c^{2} d^{2}+3 \sqrt {-c^{2}}\, \sqrt {d^{2} x^{2}-c^{2}}\, a \,d^{2}-b \,c^{2} \sqrt {-c^{2}}\, \sqrt {d^{2} x^{2}-c^{2}}\right )}{3 \sqrt {d^{2} x^{2}-c^{2}}\, d^{2} \sqrt {-c^{2}}}\) | \(174\) |
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Time = 0.26 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right )}{x} \, dx=-\frac {6 \, a c d^{2} \arctan \left (-\frac {d x - \sqrt {d x + c} \sqrt {d x - c}}{c}\right ) - {\left (b d^{2} x^{2} - b c^{2} + 3 \, a d^{2}\right )} \sqrt {d x + c} \sqrt {d x - c}}{3 \, d^{2}} \]
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\[ \int \frac {\sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right )}{x} \, dx=\int \frac {\left (a + b x^{2}\right ) \sqrt {- c + d x} \sqrt {c + d x}}{x}\, dx \]
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Time = 0.34 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.65 \[ \int \frac {\sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right )}{x} \, dx=a c \arcsin \left (\frac {c}{d {\left | x \right |}}\right ) + \sqrt {d^{2} x^{2} - c^{2}} a + \frac {{\left (d^{2} x^{2} - c^{2}\right )}^{\frac {3}{2}} b}{3 \, d^{2}} \]
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Time = 0.33 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.98 \[ \int \frac {\sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right )}{x} \, dx=2 \, a c \arctan \left (\frac {{\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{2}}{2 \, c}\right ) + \frac {1}{3} \, \sqrt {d x + c} \sqrt {d x - c} {\left ({\left (d x + c\right )} {\left (\frac {{\left (d x + c\right )} b}{d^{2}} - \frac {2 \, b c}{d^{2}}\right )} + 3 \, a\right )} \]
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Time = 7.77 (sec) , antiderivative size = 248, normalized size of antiderivative = 3.10 \[ \int \frac {\sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right )}{x} \, dx=a\,\sqrt {-c}\,\sqrt {c}\,\ln \left (\frac {{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^2}+1\right )-a\,\sqrt {-c}\,\sqrt {c}\,\ln \left (\frac {\sqrt {c+d\,x}-\sqrt {c}}{\sqrt {-c}-\sqrt {d\,x-c}}\right )-\frac {b\,\left (c^2-d^2\,x^2\right )\,\sqrt {c+d\,x}\,\sqrt {d\,x-c}}{3\,d^2}-\frac {8\,a\,\sqrt {-c}\,\sqrt {c}\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^2\,\left (\frac {{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^4}-\frac {2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^2}+1\right )} \]
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