Integrand size = 22, antiderivative size = 80 \[ \int \frac {x \sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx=-\frac {\sqrt {c+d x^2}}{2 b \left (a+b x^2\right )}-\frac {d \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 b^{3/2} \sqrt {b c-a d}} \]
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Time = 0.04 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {455, 43, 65, 214} \[ \int \frac {x \sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx=-\frac {d \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 b^{3/2} \sqrt {b c-a d}}-\frac {\sqrt {c+d x^2}}{2 b \left (a+b x^2\right )} \]
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Rule 43
Rule 65
Rule 214
Rule 455
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {c+d x}}{(a+b x)^2} \, dx,x,x^2\right ) \\ & = -\frac {\sqrt {c+d x^2}}{2 b \left (a+b x^2\right )}+\frac {d \text {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^2\right )}{4 b} \\ & = -\frac {\sqrt {c+d x^2}}{2 b \left (a+b x^2\right )}+\frac {\text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{2 b} \\ & = -\frac {\sqrt {c+d x^2}}{2 b \left (a+b x^2\right )}-\frac {d \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 b^{3/2} \sqrt {b c-a d}} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00 \[ \int \frac {x \sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx=-\frac {\sqrt {c+d x^2}}{2 b \left (a+b x^2\right )}+\frac {d \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {-b c+a d}}\right )}{2 b^{3/2} \sqrt {-b c+a d}} \]
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Time = 2.99 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.81
method | result | size |
pseudoelliptic | \(\frac {-\frac {\sqrt {d \,x^{2}+c}}{b \,x^{2}+a}+\frac {d \arctan \left (\frac {b \sqrt {d \,x^{2}+c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}}}{2 b}\) | \(65\) |
default | \(\text {Expression too large to display}\) | \(1320\) |
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Leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (64) = 128\).
Time = 0.27 (sec) , antiderivative size = 356, normalized size of antiderivative = 4.45 \[ \int \frac {x \sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx=\left [\frac {{\left (b d x^{2} + a d\right )} \sqrt {b^{2} c - a b d} \log \left (\frac {b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} - 4 \, {\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {b^{2} c - a b d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, {\left (b^{2} c - a b d\right )} \sqrt {d x^{2} + c}}{8 \, {\left (a b^{3} c - a^{2} b^{2} d + {\left (b^{4} c - a b^{3} d\right )} x^{2}\right )}}, -\frac {{\left (b d x^{2} + a d\right )} \sqrt {-b^{2} c + a b d} \arctan \left (-\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {-b^{2} c + a b d} \sqrt {d x^{2} + c}}{2 \, {\left (b^{2} c^{2} - a b c d + {\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )}}\right ) + 2 \, {\left (b^{2} c - a b d\right )} \sqrt {d x^{2} + c}}{4 \, {\left (a b^{3} c - a^{2} b^{2} d + {\left (b^{4} c - a b^{3} d\right )} x^{2}\right )}}\right ] \]
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\[ \int \frac {x \sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx=\int \frac {x \sqrt {c + d x^{2}}}{\left (a + b x^{2}\right )^{2}}\, dx \]
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Exception generated. \[ \int \frac {x \sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.29 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.99 \[ \int \frac {x \sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx=\frac {d \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{2 \, \sqrt {-b^{2} c + a b d} b} - \frac {\sqrt {d x^{2} + c} d}{2 \, {\left ({\left (d x^{2} + c\right )} b - b c + a d\right )} b} \]
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Time = 5.35 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.88 \[ \int \frac {x \sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx=\frac {d\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d\,x^2+c}}{\sqrt {a\,d-b\,c}}\right )}{2\,b^{3/2}\,\sqrt {a\,d-b\,c}}-\frac {d\,\sqrt {d\,x^2+c}}{2\,\left (d\,b^2\,x^2+a\,d\,b\right )} \]
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