Integrand size = 25, antiderivative size = 47 \[ \int \frac {x}{\sqrt {a-b x^2} \sqrt {c+d x^2}} \, dx=-\frac {\arctan \left (\frac {\sqrt {d} \sqrt {a-b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{\sqrt {b} \sqrt {d}} \]
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Time = 0.04 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {455, 65, 223, 209} \[ \int \frac {x}{\sqrt {a-b x^2} \sqrt {c+d x^2}} \, dx=-\frac {\arctan \left (\frac {\sqrt {d} \sqrt {a-b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{\sqrt {b} \sqrt {d}} \]
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Rule 65
Rule 209
Rule 223
Rule 455
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {a-b x} \sqrt {c+d x}} \, dx,x,x^2\right ) \\ & = -\frac {\text {Subst}\left (\int \frac {1}{\sqrt {c+\frac {a d}{b}-\frac {d x^2}{b}}} \, dx,x,\sqrt {a-b x^2}\right )}{b} \\ & = -\frac {\text {Subst}\left (\int \frac {1}{1+\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a-b x^2}}{\sqrt {c+d x^2}}\right )}{b} \\ & = -\frac {\tan ^{-1}\left (\frac {\sqrt {d} \sqrt {a-b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{\sqrt {b} \sqrt {d}} \\ \end{align*}
Time = 0.35 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.98 \[ \int \frac {x}{\sqrt {a-b x^2} \sqrt {c+d x^2}} \, dx=\frac {\arctan \left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {d} \sqrt {a-b x^2}}\right )}{\sqrt {b} \sqrt {d}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(91\) vs. \(2(35)=70\).
Time = 3.09 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.96
method | result | size |
default | \(-\frac {\arctan \left (\frac {\sqrt {b d}\, \left (-2 b d \,x^{2}+a d -b c \right )}{2 b d \sqrt {\left (-b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}\right ) \sqrt {-b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}{2 \sqrt {b d}\, \sqrt {\left (-b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}\) | \(92\) |
elliptic | \(\frac {\sqrt {\left (-b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \arctan \left (\frac {\sqrt {b d}\, \left (x^{2}-\frac {a d -b c}{2 b d}\right )}{\sqrt {-b d \,x^{4}+\left (a d -b c \right ) x^{2}+a c}}\right )}{2 \sqrt {-b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \sqrt {b d}}\) | \(97\) |
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Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (35) = 70\).
Time = 0.30 (sec) , antiderivative size = 201, normalized size of antiderivative = 4.28 \[ \int \frac {x}{\sqrt {a-b x^2} \sqrt {c+d x^2}} \, dx=\left [-\frac {\sqrt {-b d} \log \left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} - 6 \, a b c d + a^{2} d^{2} + 8 \, {\left (b^{2} c d - a b d^{2}\right )} x^{2} - 4 \, {\left (2 \, b d x^{2} + b c - a d\right )} \sqrt {-b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {-b d}\right )}{4 \, b d}, -\frac {\sqrt {b d} \arctan \left (\frac {{\left (2 \, b d x^{2} + b c - a d\right )} \sqrt {-b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {b d}}{2 \, {\left (b^{2} d^{2} x^{4} - a b c d + {\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )}}\right )}{2 \, b d}\right ] \]
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\[ \int \frac {x}{\sqrt {a-b x^2} \sqrt {c+d x^2}} \, dx=\int \frac {x}{\sqrt {a - b x^{2}} \sqrt {c + d x^{2}}}\, dx \]
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Exception generated. \[ \int \frac {x}{\sqrt {a-b x^2} \sqrt {c+d x^2}} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.31 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.21 \[ \int \frac {x}{\sqrt {a-b x^2} \sqrt {c+d x^2}} \, dx=\frac {b \log \left ({\left | -\sqrt {-b x^{2} + a} \sqrt {-b d} + \sqrt {b^{2} c + {\left (b x^{2} - a\right )} b d + a b d} \right |}\right )}{\sqrt {-b d} {\left | b \right |}} \]
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Time = 5.68 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.02 \[ \int \frac {x}{\sqrt {a-b x^2} \sqrt {c+d x^2}} \, dx=-\frac {2\,\mathrm {atan}\left (\frac {d\,\left (\sqrt {a-b\,x^2}-\sqrt {a}\right )}{\sqrt {b\,d}\,\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}\right )}{\sqrt {b\,d}} \]
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