Integrand size = 24, antiderivative size = 74 \[ \int \frac {x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\frac {x}{(b c-a d) \sqrt {c+d x^2}}-\frac {\sqrt {a} \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{(b c-a d)^{3/2}} \]
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Time = 0.03 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {482, 12, 385, 211} \[ \int \frac {x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\frac {x}{\sqrt {c+d x^2} (b c-a d)}-\frac {\sqrt {a} \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{(b c-a d)^{3/2}} \]
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Rule 12
Rule 211
Rule 385
Rule 482
Rubi steps \begin{align*} \text {integral}& = \frac {x}{(b c-a d) \sqrt {c+d x^2}}-\frac {\int \frac {a}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{b c-a d} \\ & = \frac {x}{(b c-a d) \sqrt {c+d x^2}}-\frac {a \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{b c-a d} \\ & = \frac {x}{(b c-a d) \sqrt {c+d x^2}}-\frac {a \text {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{b c-a d} \\ & = \frac {x}{(b c-a d) \sqrt {c+d x^2}}-\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{(b c-a d)^{3/2}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(319\) vs. \(2(74)=148\).
Time = 1.38 (sec) , antiderivative size = 319, normalized size of antiderivative = 4.31 \[ \int \frac {x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\frac {\frac {(b c-a d) x}{\sqrt {c+d x^2}}+\frac {\left (-b c+a d+\sqrt {b} \sqrt {c} \sqrt {b c-a d}\right ) \sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} \arctan \left (\frac {\sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (\sqrt {c}-\sqrt {c+d x^2}\right )}\right )}{\sqrt {a} d}+\frac {\sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} \left (b c-a d+\sqrt {b} \sqrt {c} \sqrt {b c-a d}\right ) \arctan \left (\frac {\sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (-\sqrt {c}+\sqrt {c+d x^2}\right )}\right )}{\sqrt {a} d}}{(b c-a d)^2} \]
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Time = 2.92 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.18
method | result | size |
pseudoelliptic | \(\frac {a \,\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right ) \sqrt {d \,x^{2}+c}-x \sqrt {\left (a d -b c \right ) a}}{\left (a d -b c \right ) \sqrt {d \,x^{2}+c}\, \sqrt {\left (a d -b c \right ) a}}\) | \(87\) |
default | \(\frac {x}{b c \sqrt {d \,x^{2}+c}}-\frac {a \left (-\frac {b}{\left (a d -b c \right ) \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}}}+\frac {2 d \sqrt {-a b}\, \left (2 d \left (x -\frac {\sqrt {-a b}}{b}\right )+\frac {2 d \sqrt {-a b}}{b}\right )}{\left (a d -b c \right ) \left (-\frac {4 d \left (a d -b c \right )}{b}+\frac {4 d^{2} a}{b}\right ) \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}}}+\frac {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}}}\right )}{2 \sqrt {-a b}\, b}+\frac {a \left (-\frac {b}{\left (a d -b c \right ) \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}}}-\frac {2 d \sqrt {-a b}\, \left (2 d \left (x +\frac {\sqrt {-a b}}{b}\right )-\frac {2 d \sqrt {-a b}}{b}\right )}{\left (a d -b c \right ) \left (-\frac {4 d \left (a d -b c \right )}{b}+\frac {4 d^{2} a}{b}\right ) \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}}}+\frac {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}}}\right )}{2 \sqrt {-a b}\, b}\) | \(758\) |
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Time = 0.30 (sec) , antiderivative size = 334, normalized size of antiderivative = 4.51 \[ \int \frac {x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\left [-\frac {{\left (d x^{2} + c\right )} \sqrt {-\frac {a}{b c - a d}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} + 4 \, {\left ({\left (b^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{3} - {\left (a b c^{2} - a^{2} c d\right )} x\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {a}{b c - a d}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, \sqrt {d x^{2} + c} x}{4 \, {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x^{2}\right )}}, \frac {{\left (d x^{2} + c\right )} \sqrt {\frac {a}{b c - a d}} \arctan \left (-\frac {{\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c} \sqrt {\frac {a}{b c - a d}}}{2 \, {\left (a d x^{3} + a c x\right )}}\right ) + 2 \, \sqrt {d x^{2} + c} x}{2 \, {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x^{2}\right )}}\right ] \]
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\[ \int \frac {x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {x^{2}}{\left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {x^{2}}{{\left (b x^{2} + a\right )} {\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]
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Time = 0.29 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.39 \[ \int \frac {x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=-\frac {a \sqrt {d} \arctan \left (-\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{\sqrt {a b c d - a^{2} d^{2}} {\left (b c - a d\right )}} + \frac {x}{\sqrt {d x^{2} + c} {\left (b c - a d\right )}} \]
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Timed out. \[ \int \frac {x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {x^2}{\left (b\,x^2+a\right )\,{\left (d\,x^2+c\right )}^{3/2}} \,d x \]
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