Integrand size = 20, antiderivative size = 361 \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^2} \, dx=\frac {\left (b^2+12 a c\right ) x \sqrt {a+b x^2+c x^4}}{5 \sqrt {c} \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {1}{5} x \left (7 b+6 c x^2\right ) \sqrt {a+b x^2+c x^4}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{x}-\frac {\sqrt [4]{a} \left (b^2+12 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{5 c^{3/4} \sqrt {a+b x^2+c x^4}}+\frac {\sqrt [4]{a} \left (b^2+8 \sqrt {a} b \sqrt {c}+12 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{10 c^{3/4} \sqrt {a+b x^2+c x^4}} \]
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Time = 0.12 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1131, 1190, 1211, 1117, 1209} \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^2} \, dx=\frac {\sqrt [4]{a} \left (8 \sqrt {a} b \sqrt {c}+12 a c+b^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{10 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\sqrt [4]{a} \left (12 a c+b^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{5 c^{3/4} \sqrt {a+b x^2+c x^4}}+\frac {x \left (12 a c+b^2\right ) \sqrt {a+b x^2+c x^4}}{5 \sqrt {c} \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{x}+\frac {1}{5} x \left (7 b+6 c x^2\right ) \sqrt {a+b x^2+c x^4} \]
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Rule 1117
Rule 1131
Rule 1190
Rule 1209
Rule 1211
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a+b x^2+c x^4\right )^{3/2}}{x}+3 \int \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4} \, dx \\ & = \frac {1}{5} x \left (7 b+6 c x^2\right ) \sqrt {a+b x^2+c x^4}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{x}+\frac {\int \frac {8 a b c+c \left (b^2+12 a c\right ) x^2}{\sqrt {a+b x^2+c x^4}} \, dx}{5 c} \\ & = \frac {1}{5} x \left (7 b+6 c x^2\right ) \sqrt {a+b x^2+c x^4}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{x}-\frac {\left (\sqrt {a} \left (b^2+12 a c\right )\right ) \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+b x^2+c x^4}} \, dx}{5 \sqrt {c}}+\frac {\left (\sqrt {a} \left (b^2+8 \sqrt {a} b \sqrt {c}+12 a c\right )\right ) \int \frac {1}{\sqrt {a+b x^2+c x^4}} \, dx}{5 \sqrt {c}} \\ & = \frac {\left (b^2+12 a c\right ) x \sqrt {a+b x^2+c x^4}}{5 \sqrt {c} \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {1}{5} x \left (7 b+6 c x^2\right ) \sqrt {a+b x^2+c x^4}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{x}-\frac {\sqrt [4]{a} \left (b^2+12 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{5 c^{3/4} \sqrt {a+b x^2+c x^4}}+\frac {\sqrt [4]{a} \left (b^2+8 \sqrt {a} b \sqrt {c}+12 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{10 c^{3/4} \sqrt {a+b x^2+c x^4}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 10.80 (sec) , antiderivative size = 505, normalized size of antiderivative = 1.40 \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^2} \, dx=\frac {4 c \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \left (-5 a^2-3 a b x^2+2 b^2 x^4-4 a c x^4+3 b c x^6+c^2 x^8\right )+i \left (b^2+12 a c\right ) \left (-b+\sqrt {b^2-4 a c}\right ) x \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-i \left (-b^3+4 a b c+b^2 \sqrt {b^2-4 a c}+12 a c \sqrt {b^2-4 a c}\right ) x \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{20 c \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x \sqrt {a+b x^2+c x^4}} \]
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Time = 1.28 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.12
method | result | size |
risch | \(-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (-c \,x^{4}-2 b \,x^{2}+5 a \right )}{5 x}+\frac {2 a b \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{5 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {\left (12 a c +b^{2}\right ) a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-E\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{10 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\) | \(405\) |
default | \(-\frac {a \sqrt {c \,x^{4}+b \,x^{2}+a}}{x}+\frac {c \,x^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}{5}+\frac {2 b x \sqrt {c \,x^{4}+b \,x^{2}+a}}{5}+\frac {2 a b \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{5 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {\left (\frac {12 a c}{5}+\frac {b^{2}}{5}\right ) a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-E\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\) | \(430\) |
elliptic | \(-\frac {a \sqrt {c \,x^{4}+b \,x^{2}+a}}{x}+\frac {c \,x^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}{5}+\frac {2 b x \sqrt {c \,x^{4}+b \,x^{2}+a}}{5}+\frac {2 a b \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{5 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {\left (\frac {12 a c}{5}+\frac {b^{2}}{5}\right ) a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-E\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\) | \(430\) |
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\[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^2} \, dx=\int { \frac {{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}}}{x^{2}} \,d x } \]
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\[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^2} \, dx=\int \frac {\left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}{x^{2}}\, dx \]
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\[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^2} \, dx=\int { \frac {{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}}}{x^{2}} \,d x } \]
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\[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^2} \, dx=\int { \frac {{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}}}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^2} \, dx=\int \frac {{\left (c\,x^4+b\,x^2+a\right )}^{3/2}}{x^2} \,d x \]
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