Integrand size = 24, antiderivative size = 89 \[ \int \frac {x^2}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=-\frac {x \sqrt {c+d x^2}}{2 (b c-a d) \left (a+b x^2\right )}+\frac {c \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 \sqrt {a} (b c-a d)^{3/2}} \]
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Time = 0.03 (sec) , antiderivative size = 89, 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} \[ \int \frac {x^2}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\frac {c \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 \sqrt {a} (b c-a d)^{3/2}}-\frac {x \sqrt {c+d x^2}}{2 \left (a+b x^2\right ) (b c-a d)} \]
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Rule 12
Rule 211
Rule 385
Rule 482
Rubi steps \begin{align*} \text {integral}& = -\frac {x \sqrt {c+d x^2}}{2 (b c-a d) \left (a+b x^2\right )}+\frac {\int \frac {c}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 (b c-a d)} \\ & = -\frac {x \sqrt {c+d x^2}}{2 (b c-a d) \left (a+b x^2\right )}+\frac {c \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 (b c-a d)} \\ & = -\frac {x \sqrt {c+d x^2}}{2 (b c-a d) \left (a+b x^2\right )}+\frac {c \text {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{2 (b c-a d)} \\ & = -\frac {x \sqrt {c+d x^2}}{2 (b c-a d) \left (a+b x^2\right )}+\frac {c \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 \sqrt {a} (b c-a d)^{3/2}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(608\) vs. \(2(89)=178\).
Time = 3.00 (sec) , antiderivative size = 608, normalized size of antiderivative = 6.83 \[ \int \frac {x^2}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\frac {1}{2} \left (\frac {2 c^{3/2} x+2 \sqrt {c} d x^3-2 c x \sqrt {c+d x^2}-d x^3 \sqrt {c+d x^2}}{(b c-a d) \left (a+b x^2\right ) \left (2 c+d x^2-2 \sqrt {c} \sqrt {c+d x^2}\right )}+\frac {\sqrt {b} c^{3/2} \arctan \left (\frac {\sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (\sqrt {c}-\sqrt {c+d x^2}\right )}\right )}{\sqrt {a} (b c-a d)^{3/2} \sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}}}+\frac {c \arctan \left (\frac {\sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (-\sqrt {c}+\sqrt {c+d x^2}\right )}\right )}{\sqrt {a} (b c-a d) \sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}}}+\frac {\sqrt {b} c^{3/2} \arctan \left (\frac {\sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (-\sqrt {c}+\sqrt {c+d x^2}\right )}\right )}{\sqrt {a} (b c-a d)^{3/2} \sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}}}+\frac {c \arctan \left (\frac {\sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (-\sqrt {c}+\sqrt {c+d x^2}\right )}\right )}{\sqrt {a} (b c-a d) \sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}}}\right ) \]
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Time = 2.99 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.89
method | result | size |
pseudoelliptic | \(-\frac {c \left (-\frac {\sqrt {d \,x^{2}+c}\, x}{c \left (b \,x^{2}+a \right )}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{\sqrt {\left (a d -b c \right ) a}}\right )}{2 \left (a d -b c \right )}\) | \(79\) |
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^{2}}+\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^{2}}-\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 \sqrt {-a b}\, 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 \sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}\) | \(816\) |
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Leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (73) = 146\).
Time = 0.32 (sec) , antiderivative size = 418, normalized size of antiderivative = 4.70 \[ \int \frac {x^2}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\left [-\frac {4 \, {\left (a b c - a^{2} d\right )} \sqrt {d x^{2} + c} x - {\left (b c x^{2} + a c\right )} \sqrt {-a b c + a^{2} d} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} + 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{3} - a c x\right )} \sqrt {-a b c + a^{2} d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )}{8 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} + {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{2}\right )}}, -\frac {2 \, {\left (a b c - a^{2} d\right )} \sqrt {d x^{2} + c} x - {\left (b c x^{2} + a c\right )} \sqrt {a b c - a^{2} d} \arctan \left (\frac {\sqrt {a b c - a^{2} d} {\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right )}{4 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} + {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{2}\right )}}\right ] \]
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\[ \int \frac {x^2}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\int \frac {x^{2}}{\left (a + b x^{2}\right )^{2} \sqrt {c + d x^{2}}}\, dx \]
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\[ \int \frac {x^2}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\int { \frac {x^{2}}{{\left (b x^{2} + a\right )}^{2} \sqrt {d x^{2} + c}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 231 vs. \(2 (73) = 146\).
Time = 0.84 (sec) , antiderivative size = 231, normalized size of antiderivative = 2.60 \[ \int \frac {x^2}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\frac {c \sqrt {d} \arctan \left (-\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{2 \, \sqrt {a b c d - a^{2} d^{2}} {\left (b c - a d\right )}} + \frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c \sqrt {d} - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d^{\frac {3}{2}} - b c^{2} \sqrt {d}}{{\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d + b c^{2}\right )} {\left (b^{2} c - a b d\right )}} \]
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Timed out. \[ \int \frac {x^2}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\int \frac {x^2}{{\left (b\,x^2+a\right )}^2\,\sqrt {d\,x^2+c}} \,d x \]
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