Integrand size = 24, antiderivative size = 147 \[ \int \frac {\sqrt {c+d x^2}}{x^4 \left (a+b x^2\right )^2} \, dx=-\frac {5 \sqrt {c+d x^2}}{6 a^2 x^3}+\frac {(15 b c-2 a d) \sqrt {c+d x^2}}{6 a^3 c x}+\frac {\sqrt {c+d x^2}}{2 a x^3 \left (a+b x^2\right )}+\frac {b (5 b c-4 a d) \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{7/2} \sqrt {b c-a d}} \]
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Time = 0.15 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {480, 597, 12, 385, 211} \[ \int \frac {\sqrt {c+d x^2}}{x^4 \left (a+b x^2\right )^2} \, dx=\frac {b (5 b c-4 a d) \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{7/2} \sqrt {b c-a d}}+\frac {\sqrt {c+d x^2} (15 b c-2 a d)}{6 a^3 c x}-\frac {5 \sqrt {c+d x^2}}{6 a^2 x^3}+\frac {\sqrt {c+d x^2}}{2 a x^3 \left (a+b x^2\right )} \]
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Rule 12
Rule 211
Rule 385
Rule 480
Rule 597
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c+d x^2}}{2 a x^3 \left (a+b x^2\right )}-\frac {\int \frac {-5 c-4 d x^2}{x^4 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 a} \\ & = -\frac {5 \sqrt {c+d x^2}}{6 a^2 x^3}+\frac {\sqrt {c+d x^2}}{2 a x^3 \left (a+b x^2\right )}+\frac {\int \frac {-c (15 b c-2 a d)-10 b c d x^2}{x^2 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{6 a^2 c} \\ & = -\frac {5 \sqrt {c+d x^2}}{6 a^2 x^3}+\frac {(15 b c-2 a d) \sqrt {c+d x^2}}{6 a^3 c x}+\frac {\sqrt {c+d x^2}}{2 a x^3 \left (a+b x^2\right )}-\frac {\int -\frac {3 b c^2 (5 b c-4 a d)}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{6 a^3 c^2} \\ & = -\frac {5 \sqrt {c+d x^2}}{6 a^2 x^3}+\frac {(15 b c-2 a d) \sqrt {c+d x^2}}{6 a^3 c x}+\frac {\sqrt {c+d x^2}}{2 a x^3 \left (a+b x^2\right )}+\frac {(b (5 b c-4 a d)) \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 a^3} \\ & = -\frac {5 \sqrt {c+d x^2}}{6 a^2 x^3}+\frac {(15 b c-2 a d) \sqrt {c+d x^2}}{6 a^3 c x}+\frac {\sqrt {c+d x^2}}{2 a x^3 \left (a+b x^2\right )}+\frac {(b (5 b c-4 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^3} \\ & = -\frac {5 \sqrt {c+d x^2}}{6 a^2 x^3}+\frac {(15 b c-2 a d) \sqrt {c+d x^2}}{6 a^3 c x}+\frac {\sqrt {c+d x^2}}{2 a x^3 \left (a+b x^2\right )}+\frac {b (5 b c-4 a d) \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{7/2} \sqrt {b c-a d}} \\ \end{align*}
Time = 0.87 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.05 \[ \int \frac {\sqrt {c+d x^2}}{x^4 \left (a+b x^2\right )^2} \, dx=\frac {\sqrt {c+d x^2} \left (15 b^2 c x^4-2 a b x^2 \left (-5 c+d x^2\right )-2 a^2 \left (c+d x^2\right )\right )}{6 a^3 c x^3 \left (a+b x^2\right )}-\frac {b (5 b c-4 a d) \arctan \left (\frac {a \sqrt {d}+b x \left (\sqrt {d} x-\sqrt {c+d x^2}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{2 a^{7/2} \sqrt {b c-a d}} \]
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Time = 3.10 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.79
method | result | size |
pseudoelliptic | \(\frac {-\frac {\sqrt {d \,x^{2}+c}\, \left (a d \,x^{2}-6 c b \,x^{2}+a c \right )}{3 x^{3}}+\frac {b c \left (\frac {b \sqrt {d \,x^{2}+c}\, x}{b \,x^{2}+a}-\frac {\left (4 a d -5 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{\sqrt {\left (a d -b c \right ) a}}\right )}{2}}{a^{3} c}\) | \(116\) |
risch | \(-\frac {\sqrt {d \,x^{2}+c}\, \left (a d \,x^{2}-6 c b \,x^{2}+a c \right )}{3 c \,a^{3} x^{3}}-\frac {b \left (-\frac {\left (a d -b c \right ) \left (\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}}}\right )}{4 b}-\frac {\left (a d -b c \right ) \left (\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}}}\right )}{4 b}+\frac {\left (3 a d -5 b c \right ) \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}\, \sqrt {-\frac {a d -b c}{b}}}-\frac {\left (3 a d -5 b c \right ) \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}\, \sqrt {-\frac {a d -b c}{b}}}\right )}{a^{3}}\) | \(888\) |
default | \(\text {Expression too large to display}\) | \(2052\) |
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Leaf count of result is larger than twice the leaf count of optimal. 281 vs. \(2 (123) = 246\).
Time = 0.35 (sec) , antiderivative size = 602, normalized size of antiderivative = 4.10 \[ \int \frac {\sqrt {c+d x^2}}{x^4 \left (a+b x^2\right )^2} \, dx=\left [\frac {3 \, {\left ({\left (5 \, b^{3} c^{2} - 4 \, a b^{2} c d\right )} x^{5} + {\left (5 \, a b^{2} c^{2} - 4 \, a^{2} b c d\right )} x^{3}\right )} \sqrt {-a b c + a^{2} d} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} + 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{3} - a c x\right )} \sqrt {-a b c + a^{2} d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, {\left (2 \, a^{3} b c^{2} - 2 \, a^{4} c d - {\left (15 \, a b^{3} c^{2} - 17 \, a^{2} b^{2} c d + 2 \, a^{3} b d^{2}\right )} x^{4} - 2 \, {\left (5 \, a^{2} b^{2} c^{2} - 6 \, a^{3} b c d + a^{4} d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{24 \, {\left ({\left (a^{4} b^{2} c^{2} - a^{5} b c d\right )} x^{5} + {\left (a^{5} b c^{2} - a^{6} c d\right )} x^{3}\right )}}, \frac {3 \, {\left ({\left (5 \, b^{3} c^{2} - 4 \, a b^{2} c d\right )} x^{5} + {\left (5 \, a b^{2} c^{2} - 4 \, a^{2} b c d\right )} x^{3}\right )} \sqrt {a b c - a^{2} d} \arctan \left (\frac {\sqrt {a b c - a^{2} d} {\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left (2 \, a^{3} b c^{2} - 2 \, a^{4} c d - {\left (15 \, a b^{3} c^{2} - 17 \, a^{2} b^{2} c d + 2 \, a^{3} b d^{2}\right )} x^{4} - 2 \, {\left (5 \, a^{2} b^{2} c^{2} - 6 \, a^{3} b c d + a^{4} d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{12 \, {\left ({\left (a^{4} b^{2} c^{2} - a^{5} b c d\right )} x^{5} + {\left (a^{5} b c^{2} - a^{6} c d\right )} x^{3}\right )}}\right ] \]
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\[ \int \frac {\sqrt {c+d x^2}}{x^4 \left (a+b x^2\right )^2} \, dx=\int \frac {\sqrt {c + d x^{2}}}{x^{4} \left (a + b x^{2}\right )^{2}}\, dx \]
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\[ \int \frac {\sqrt {c+d x^2}}{x^4 \left (a+b x^2\right )^2} \, dx=\int { \frac {\sqrt {d x^{2} + c}}{{\left (b x^{2} + a\right )}^{2} x^{4}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 361 vs. \(2 (123) = 246\).
Time = 1.08 (sec) , antiderivative size = 361, normalized size of antiderivative = 2.46 \[ \int \frac {\sqrt {c+d x^2}}{x^4 \left (a+b x^2\right )^2} \, dx=-\frac {{\left (5 \, b^{2} c \sqrt {d} - 4 \, a b d^{\frac {3}{2}}\right )} \arctan \left (\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{2 \, \sqrt {a b c d - a^{2} d^{2}} a^{3}} - \frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b^{2} c \sqrt {d} - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b d^{\frac {3}{2}} - b^{2} c^{2} \sqrt {d}}{{\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d + b c^{2}\right )} a^{3}} - \frac {2 \, {\left (6 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b c \sqrt {d} - 3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a d^{\frac {3}{2}} - 12 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c^{2} \sqrt {d} + 6 \, b c^{3} \sqrt {d} - a c^{2} d^{\frac {3}{2}}\right )}}{3 \, {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c\right )}^{3} a^{3}} \]
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Timed out. \[ \int \frac {\sqrt {c+d x^2}}{x^4 \left (a+b x^2\right )^2} \, dx=\int \frac {\sqrt {d\,x^2+c}}{x^4\,{\left (b\,x^2+a\right )}^2} \,d x \]
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