Integrand size = 24, antiderivative size = 128 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^2 \left (a+b x^2\right )^2} \, dx=-\frac {(3 b c-a d) \sqrt {c+d x^2}}{2 a^2 b x}+\frac {(b c-a d) \sqrt {c+d x^2}}{2 a b x \left (a+b x^2\right )}-\frac {3 c \sqrt {b c-a d} \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{5/2}} \]
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Time = 0.09 (sec) , antiderivative size = 128, 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 = {479, 597, 12, 385, 211} \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^2 \left (a+b x^2\right )^2} \, dx=-\frac {3 c \sqrt {b c-a d} \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{5/2}}-\frac {\sqrt {c+d x^2} (3 b c-a d)}{2 a^2 b x}+\frac {\sqrt {c+d x^2} (b c-a d)}{2 a b x \left (a+b x^2\right )} \]
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Rule 12
Rule 211
Rule 385
Rule 479
Rule 597
Rubi steps \begin{align*} \text {integral}& = \frac {(b c-a d) \sqrt {c+d x^2}}{2 a b x \left (a+b x^2\right )}-\frac {\int \frac {-c (3 b c-a d)-2 b c d x^2}{x^2 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 a b} \\ & = -\frac {(3 b c-a d) \sqrt {c+d x^2}}{2 a^2 b x}+\frac {(b c-a d) \sqrt {c+d x^2}}{2 a b x \left (a+b x^2\right )}+\frac {\int -\frac {3 b c^2 (b c-a d)}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 a^2 b c} \\ & = -\frac {(3 b c-a d) \sqrt {c+d x^2}}{2 a^2 b x}+\frac {(b c-a d) \sqrt {c+d x^2}}{2 a b x \left (a+b x^2\right )}-\frac {(3 c (b c-a d)) \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 a^2} \\ & = -\frac {(3 b c-a d) \sqrt {c+d x^2}}{2 a^2 b x}+\frac {(b c-a d) \sqrt {c+d x^2}}{2 a b x \left (a+b x^2\right )}-\frac {(3 c (b c-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^2} \\ & = -\frac {(3 b c-a d) \sqrt {c+d x^2}}{2 a^2 b x}+\frac {(b c-a d) \sqrt {c+d x^2}}{2 a b x \left (a+b x^2\right )}-\frac {3 c \sqrt {b c-a d} \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{5/2}} \\ \end{align*}
Time = 0.41 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.95 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^2 \left (a+b x^2\right )^2} \, dx=\frac {\sqrt {c+d x^2} \left (-2 a c-3 b c x^2+a d x^2\right )}{2 a^2 x \left (a+b x^2\right )}+\frac {3 c \sqrt {b c-a d} \arctan \left (\frac {a \sqrt {d}+b x \left (\sqrt {d} x-\sqrt {c+d x^2}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{2 a^{5/2}} \]
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Time = 3.05 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.76
method | result | size |
pseudoelliptic | \(-\frac {c \left (\frac {2 \sqrt {d \,x^{2}+c}}{x}+\left (a d -b c \right ) \left (-\frac {\sqrt {d \,x^{2}+c}\, x}{c \left (b \,x^{2}+a \right )}-\frac {3 \,\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{\sqrt {\left (a d -b c \right ) a}}\right )\right )}{2 a^{2}}\) | \(97\) |
risch | \(-\frac {c \sqrt {d \,x^{2}+c}}{a^{2} x}+\frac {\frac {\left (\frac {1}{4} a^{2} d^{2}-\frac {1}{2} a b c d +\frac {1}{4} b^{2} c^{2}\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 )}{b^{2}}+\frac {\left (\frac {1}{4} a^{2} d^{2}-\frac {1}{2} a b c d +\frac {1}{4} b^{2} c^{2}\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 )}{b^{2}}-\frac {\left (a^{2} d^{2}+2 a b c d -3 b^{2} c^{2}\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}\, b \sqrt {-\frac {a d -b c}{b}}}-\frac {\left (-a^{2} d^{2}-2 a b c d +3 b^{2} c^{2}\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}\, b \sqrt {-\frac {a d -b c}{b}}}}{a^{2}}\) | \(928\) |
default | \(\text {Expression too large to display}\) | \(3462\) |
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Time = 0.28 (sec) , antiderivative size = 351, normalized size of antiderivative = 2.74 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^2 \left (a+b x^2\right )^2} \, dx=\left [\frac {3 \, {\left (b c x^{3} + a c x\right )} \sqrt {-\frac {b c - a d}{a}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} + 4 \, {\left (a^{2} c x - {\left (a b c - 2 \, a^{2} d\right )} x^{3}\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b c - a d}{a}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, {\left ({\left (3 \, b c - a d\right )} x^{2} + 2 \, a c\right )} \sqrt {d x^{2} + c}}{8 \, {\left (a^{2} b x^{3} + a^{3} x\right )}}, -\frac {3 \, {\left (b c x^{3} + a c x\right )} \sqrt {\frac {b c - a d}{a}} \arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b c - a d}{a}}}{2 \, {\left ({\left (b c d - a d^{2}\right )} x^{3} + {\left (b c^{2} - a c d\right )} x\right )}}\right ) + 2 \, {\left ({\left (3 \, b c - a d\right )} x^{2} + 2 \, a c\right )} \sqrt {d x^{2} + c}}{4 \, {\left (a^{2} b x^{3} + a^{3} x\right )}}\right ] \]
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\[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^2 \left (a+b x^2\right )^2} \, dx=\int \frac {\left (c + d x^{2}\right )^{\frac {3}{2}}}{x^{2} \left (a + b x^{2}\right )^{2}}\, dx \]
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\[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^2 \left (a+b x^2\right )^2} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}}}{{\left (b x^{2} + a\right )}^{2} x^{2}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 412 vs. \(2 (108) = 216\).
Time = 0.84 (sec) , antiderivative size = 412, normalized size of antiderivative = 3.22 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^2 \left (a+b x^2\right )^2} \, dx=\frac {3 \, {\left (b c^{2} \sqrt {d} - a c d^{\frac {3}{2}}\right )} \arctan \left (\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{2 \, \sqrt {a b c d - a^{2} d^{2}} a^{2}} + \frac {3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b^{2} c^{2} \sqrt {d} - 3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a b c d^{\frac {3}{2}} + 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a^{2} d^{\frac {5}{2}} - 6 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b^{2} c^{3} \sqrt {d} + 12 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b c^{2} d^{\frac {3}{2}} - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{2} c d^{\frac {5}{2}} + 3 \, b^{2} c^{4} \sqrt {d} - a b c^{3} d^{\frac {3}{2}}}{{\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} b - 3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b c + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a d + 3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c^{2} - 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a c d - b c^{3}\right )} a^{2} b} \]
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Timed out. \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^2 \left (a+b x^2\right )^2} \, dx=\int \frac {{\left (d\,x^2+c\right )}^{3/2}}{x^2\,{\left (b\,x^2+a\right )}^2} \,d x \]
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