Integrand size = 21, antiderivative size = 100 \[ \int \frac {1}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\frac {b x \sqrt {c+d x^2}}{2 a (b c-a d) \left (a+b x^2\right )}+\frac {(b c-2 a d) \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{3/2} (b c-a d)^{3/2}} \]
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Time = 0.04 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {390, 385, 211} \[ \int \frac {1}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\frac {(b c-2 a d) \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{3/2} (b c-a d)^{3/2}}+\frac {b x \sqrt {c+d x^2}}{2 a \left (a+b x^2\right ) (b c-a d)} \]
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Rule 211
Rule 385
Rule 390
Rubi steps \begin{align*} \text {integral}& = \frac {b x \sqrt {c+d x^2}}{2 a (b c-a d) \left (a+b x^2\right )}+\frac {(b c-2 a d) \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 a (b c-a d)} \\ & = \frac {b x \sqrt {c+d x^2}}{2 a (b c-a d) \left (a+b x^2\right )}+\frac {(b c-2 a d) \text {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{2 a (b c-a d)} \\ & = \frac {b x \sqrt {c+d x^2}}{2 a (b c-a d) \left (a+b x^2\right )}+\frac {(b c-2 a d) \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{3/2} (b c-a d)^{3/2}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.22 \[ \int \frac {1}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=-\frac {b x \sqrt {c+d x^2}}{2 a (-b c+a d) \left (a+b x^2\right )}+\frac {(-b c+2 a d) \arctan \left (\frac {a \sqrt {d}+b \sqrt {d} x^2-b x \sqrt {c+d x^2}}{\sqrt {a} \sqrt {b c-a d}}\right )}{2 a^{3/2} (b c-a d)^{3/2}} \]
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Time = 3.01 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.88
method | result | size |
pseudoelliptic | \(\frac {-\frac {b \sqrt {d \,x^{2}+c}\, x}{b \,x^{2}+a}+\frac {\left (2 a d -b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{\sqrt {\left (a d -b c \right ) a}}}{2 \left (a d -b c \right ) a}\) | \(88\) |
default | \(-\frac {\frac {b \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{\left (a d -b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}+\frac {d \sqrt {-a b}\, \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}}{4 b a}-\frac {\frac {b \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{\left (a d -b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}-\frac {d \sqrt {-a b}\, \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}}{4 b a}-\frac {\ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{4 a \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}+\frac {\ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{4 a \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}\) | \(822\) |
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Leaf count of result is larger than twice the leaf count of optimal. 209 vs. \(2 (84) = 168\).
Time = 0.34 (sec) , antiderivative size = 459, normalized size of antiderivative = 4.59 \[ \int \frac {1}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\left [\frac {4 \, {\left (a b^{2} c - a^{2} b d\right )} \sqrt {d x^{2} + c} x - {\left (a b c - 2 \, a^{2} d + {\left (b^{2} c - 2 \, a b d\right )} x^{2}\right )} \sqrt {-a b c + a^{2} d} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} - 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{3} - a c x\right )} \sqrt {-a b c + a^{2} d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )}{8 \, {\left (a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2} + {\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )} x^{2}\right )}}, \frac {2 \, {\left (a b^{2} c - a^{2} b d\right )} \sqrt {d x^{2} + c} x + \sqrt {a b c - a^{2} d} {\left (a b c - 2 \, a^{2} d + {\left (b^{2} c - 2 \, a b d\right )} x^{2}\right )} \arctan \left (\frac {\sqrt {a b c - a^{2} d} {\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right )}{4 \, {\left (a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2} + {\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )} x^{2}\right )}}\right ] \]
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\[ \int \frac {1}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\int \frac {1}{\left (a + b x^{2}\right )^{2} \sqrt {c + d x^{2}}}\, dx \]
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\[ \int \frac {1}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{2} \sqrt {d x^{2} + c}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 225 vs. \(2 (84) = 168\).
Time = 0.30 (sec) , antiderivative size = 225, normalized size of antiderivative = 2.25 \[ \int \frac {1}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=-\frac {1}{2} \, d^{\frac {3}{2}} {\left (\frac {{\left (b c - 2 \, a d\right )} \arctan \left (\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{{\left (a b c d - a^{2} d^{2}\right )}^{\frac {3}{2}}} + \frac {2 \, {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d - b c^{2}\right )}}{{\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d + b c^{2}\right )} {\left (a b c d - a^{2} d^{2}\right )}}\right )} \]
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Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^2\,\sqrt {d\,x^2+c}} \,d x \]
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