Integrand size = 24, antiderivative size = 115 \[ \int \frac {x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx=\frac {x}{3 (b c-a d) \left (c+d x^2\right )^{3/2}}+\frac {(2 b c+a d) x}{3 c (b c-a d)^2 \sqrt {c+d x^2}}-\frac {\sqrt {a} b \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{(b c-a d)^{5/2}} \]
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Time = 0.06 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {482, 541, 12, 385, 211} \[ \int \frac {x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx=-\frac {\sqrt {a} b \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{(b c-a d)^{5/2}}+\frac {x (a d+2 b c)}{3 c \sqrt {c+d x^2} (b c-a d)^2}+\frac {x}{3 \left (c+d x^2\right )^{3/2} (b c-a d)} \]
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Rule 12
Rule 211
Rule 385
Rule 482
Rule 541
Rubi steps \begin{align*} \text {integral}& = \frac {x}{3 (b c-a d) \left (c+d x^2\right )^{3/2}}-\frac {\int \frac {a-2 b x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx}{3 (b c-a d)} \\ & = \frac {x}{3 (b c-a d) \left (c+d x^2\right )^{3/2}}+\frac {(2 b c+a d) x}{3 c (b c-a d)^2 \sqrt {c+d x^2}}-\frac {\int \frac {3 a b c}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{3 c (b c-a d)^2} \\ & = \frac {x}{3 (b c-a d) \left (c+d x^2\right )^{3/2}}+\frac {(2 b c+a d) x}{3 c (b c-a d)^2 \sqrt {c+d x^2}}-\frac {(a b) \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{(b c-a d)^2} \\ & = \frac {x}{3 (b c-a d) \left (c+d x^2\right )^{3/2}}+\frac {(2 b c+a d) x}{3 c (b c-a d)^2 \sqrt {c+d x^2}}-\frac {(a b) \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)^2} \\ & = \frac {x}{3 (b c-a d) \left (c+d x^2\right )^{3/2}}+\frac {(2 b c+a d) x}{3 c (b c-a d)^2 \sqrt {c+d x^2}}-\frac {\sqrt {a} b \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{(b c-a d)^{5/2}} \\ \end{align*}
Time = 0.39 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.06 \[ \int \frac {x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx=\frac {a d^2 x^3+b c x \left (3 c+2 d x^2\right )}{3 c (b c-a d)^2 \left (c+d x^2\right )^{3/2}}+\frac {\sqrt {a} b \arctan \left (\frac {a \sqrt {d}+b x \left (\sqrt {d} x-\sqrt {c+d x^2}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{(b c-a d)^{5/2}} \]
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Time = 2.98 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.01
method | result | size |
pseudoelliptic | \(-\frac {a b c \,\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right ) \left (d \,x^{2}+c \right )^{\frac {3}{2}}-\frac {\sqrt {\left (a d -b c \right ) a}\, x \left (a \,d^{2} x^{2}+2 b c d \,x^{2}+3 b \,c^{2}\right )}{3}}{\left (d \,x^{2}+c \right )^{\frac {3}{2}} \sqrt {\left (a d -b c \right ) a}\, \left (a d -b c \right )^{2} c}\) | \(116\) |
default | \(\text {Expression too large to display}\) | \(1439\) |
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Leaf count of result is larger than twice the leaf count of optimal. 229 vs. \(2 (97) = 194\).
Time = 0.43 (sec) , antiderivative size = 550, normalized size of antiderivative = 4.78 \[ \int \frac {x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx=\left [\frac {3 \, {\left (b c d^{2} x^{4} + 2 \, b c^{2} d x^{2} + b c^{3}\right )} \sqrt {-\frac {a}{b c - a d}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} - 4 \, {\left ({\left (b^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{3} - {\left (a b c^{2} - a^{2} c d\right )} x\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {a}{b c - a d}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \, {\left (3 \, b c^{2} x + {\left (2 \, b c d + a d^{2}\right )} x^{3}\right )} \sqrt {d x^{2} + c}}{12 \, {\left (b^{2} c^{5} - 2 \, a b c^{4} d + a^{2} c^{3} d^{2} + {\left (b^{2} c^{3} d^{2} - 2 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x^{4} + 2 \, {\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3}\right )} x^{2}\right )}}, \frac {3 \, {\left (b c d^{2} x^{4} + 2 \, b c^{2} d x^{2} + b c^{3}\right )} \sqrt {\frac {a}{b c - a d}} \arctan \left (-\frac {{\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c} \sqrt {\frac {a}{b c - a d}}}{2 \, {\left (a d x^{3} + a c x\right )}}\right ) + 2 \, {\left (3 \, b c^{2} x + {\left (2 \, b c d + a d^{2}\right )} x^{3}\right )} \sqrt {d x^{2} + c}}{6 \, {\left (b^{2} c^{5} - 2 \, a b c^{4} d + a^{2} c^{3} d^{2} + {\left (b^{2} c^{3} d^{2} - 2 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x^{4} + 2 \, {\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3}\right )} x^{2}\right )}}\right ] \]
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\[ \int \frac {x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx=\int \frac {x^{2}}{\left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {x^{2}}{{\left (b x^{2} + a\right )} {\left (d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 291 vs. \(2 (97) = 194\).
Time = 0.31 (sec) , antiderivative size = 291, normalized size of antiderivative = 2.53 \[ \int \frac {x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx=\frac {a b \sqrt {d} \arctan \left (\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {a b c d - a^{2} d^{2}}} + \frac {{\left (\frac {{\left (2 \, b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{2} d^{3} + a^{3} d^{5}\right )} x^{2}}{b^{4} c^{5} d - 4 \, a b^{3} c^{4} d^{2} + 6 \, a^{2} b^{2} c^{3} d^{3} - 4 \, a^{3} b c^{2} d^{4} + a^{4} c d^{5}} + \frac {3 \, {\left (b^{3} c^{4} d - 2 \, a b^{2} c^{3} d^{2} + a^{2} b c^{2} d^{3}\right )}}{b^{4} c^{5} d - 4 \, a b^{3} c^{4} d^{2} + 6 \, a^{2} b^{2} c^{3} d^{3} - 4 \, a^{3} b c^{2} d^{4} + a^{4} c d^{5}}\right )} x}{3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}}} \]
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Timed out. \[ \int \frac {x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx=\int \frac {x^2}{\left (b\,x^2+a\right )\,{\left (d\,x^2+c\right )}^{5/2}} \,d x \]
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