Integrand size = 21, antiderivative size = 435 \[ \int \frac {1}{x^4 \sqrt {a+\frac {b}{c+d x^2}}} \, dx=\frac {-b-a c-a d x^2}{3 (b+a c) x^3 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}-\frac {(b-a c) d \left (b+a c+a d x^2\right )}{3 c (b+a c)^2 x \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}+\frac {(b-a c) d^2 x \left (b+a c+a d x^2\right )}{3 c (b+a c)^2 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}-\frac {(b-a c) d^{3/2} \left (b+a c+a d x^2\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 \sqrt {c} (b+a c)^2 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}}-\frac {a \sqrt {c} d^{3/2} \left (b+a c+a d x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{3 (b+a c)^2 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}} \]
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Time = 0.30 (sec) , antiderivative size = 431, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {1985, 1986, 486, 597, 545, 429, 506, 422} \[ \int \frac {1}{x^4 \sqrt {a+\frac {b}{c+d x^2}}} \, dx=-\frac {a \sqrt {c} d^{3/2} \left (a c+a d x^2+b\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{3 (a c+b)^2 \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}-\frac {d^{3/2} (b-a c) \left (a c+a d x^2+b\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 \sqrt {c} (a c+b)^2 \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}+\frac {d^2 x (b-a c) \left (a c+a d x^2+b\right )}{3 c (a c+b)^2 \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}-\frac {d (b-a c) \left (a c+a d x^2+b\right )}{3 c x (a c+b)^2 \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}-\frac {a c+a d x^2+b}{3 x^3 (a c+b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}} \]
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Rule 422
Rule 429
Rule 486
Rule 506
Rule 545
Rule 597
Rule 1985
Rule 1986
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x^4 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}} \, dx \\ & = \frac {\sqrt {b+a c+a d x^2} \int \frac {\sqrt {c+d x^2}}{x^4 \sqrt {b+a c+a d x^2}} \, dx}{\sqrt {c+d x^2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}} \\ & = -\frac {b+a c+a d x^2}{3 (b+a c) x^3 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}+\frac {\sqrt {b+a c+a d x^2} \int \frac {(b-a c) d-a d^2 x^2}{x^2 \sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{3 (b+a c) \sqrt {c+d x^2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}} \\ & = -\frac {b+a c+a d x^2}{3 (b+a c) x^3 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}-\frac {(b-a c) d \left (b+a c+a d x^2\right )}{3 c (b+a c)^2 x \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}-\frac {\sqrt {b+a c+a d x^2} \int \frac {a c (b+a c) d^2-a (b-a c) d^3 x^2}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{3 c (b+a c)^2 \sqrt {c+d x^2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}} \\ & = -\frac {b+a c+a d x^2}{3 (b+a c) x^3 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}-\frac {(b-a c) d \left (b+a c+a d x^2\right )}{3 c (b+a c)^2 x \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}-\frac {\left (a d^2 \sqrt {b+a c+a d x^2}\right ) \int \frac {1}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{3 (b+a c) \sqrt {c+d x^2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}+\frac {\left (a (b-a c) d^3 \sqrt {b+a c+a d x^2}\right ) \int \frac {x^2}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{3 c (b+a c)^2 \sqrt {c+d x^2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}} \\ & = -\frac {b+a c+a d x^2}{3 (b+a c) x^3 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}-\frac {(b-a c) d \left (b+a c+a d x^2\right )}{3 c (b+a c)^2 x \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}+\frac {(b-a c) d^2 x \left (b+a c+a d x^2\right )}{3 c (b+a c)^2 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}-\frac {a \sqrt {c} d^{3/2} \left (b+a c+a d x^2\right ) F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 (b+a c)^2 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}}-\frac {\left ((b-a c) d^2 \sqrt {b+a c+a d x^2}\right ) \int \frac {\sqrt {b+a c+a d x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{3 (b+a c)^2 \sqrt {c+d x^2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}} \\ & = -\frac {b+a c+a d x^2}{3 (b+a c) x^3 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}-\frac {(b-a c) d \left (b+a c+a d x^2\right )}{3 c (b+a c)^2 x \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}+\frac {(b-a c) d^2 x \left (b+a c+a d x^2\right )}{3 c (b+a c)^2 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}-\frac {(b-a c) d^{3/2} \left (b+a c+a d x^2\right ) E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 \sqrt {c} (b+a c)^2 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}}-\frac {a \sqrt {c} d^{3/2} \left (b+a c+a d x^2\right ) F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 (b+a c)^2 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 9.85 (sec) , antiderivative size = 299, normalized size of antiderivative = 0.69 \[ \int \frac {1}{x^4 \sqrt {a+\frac {b}{c+d x^2}}} \, dx=-\frac {\sqrt {\frac {d}{c}} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (\sqrt {\frac {d}{c}} \left (c+d x^2\right ) \left (b^2 \left (c+d x^2\right )+a^2 c \left (c^2-d^2 x^4\right )+a b \left (2 c^2+c d x^2+d^2 x^4\right )\right )+i \left (b^2-a^2 c^2\right ) d^2 x^3 \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right )|\frac {a c}{b+a c}\right )-i b (b+a c) d^2 x^3 \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {a c}{b+a c}\right )\right )}{3 (b+a c)^2 d x^3 \left (b+a \left (c+d x^2\right )\right )} \]
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Time = 7.00 (sec) , antiderivative size = 568, normalized size of antiderivative = 1.31
method | result | size |
risch | \(-\frac {\left (a d \,x^{2}+a c +b \right ) \left (-a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c \right )}{3 \left (a c +b \right )^{2} x^{3} c \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}-\frac {d^{2} a \left (\frac {a \,c^{2} \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}}+\frac {b c \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}}-\frac {2 \left (a c d -b d \right ) \left (a \,c^{2}+b c \right ) \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )-E\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \left (2 a c d +2 b d \right )}\right ) \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}}{3 \left (a c +b \right )^{2} c \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right )}\) | \(568\) |
default | \(-\frac {\left (-\sqrt {-\frac {a d}{a c +b}}\, a^{2} c \,d^{3} x^{6}+\sqrt {-\frac {a d}{a c +b}}\, a b \,d^{3} x^{6}+\sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) a^{2} c^{2} d^{2} x^{3}-\sqrt {-\frac {a d}{a c +b}}\, a^{2} c^{2} d^{2} x^{4}+2 \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) a b c \,d^{2} x^{3}-\sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) a b c \,d^{2} x^{3}+2 \sqrt {-\frac {a d}{a c +b}}\, a b c \,d^{2} x^{4}+\sqrt {-\frac {a d}{a c +b}}\, a^{2} c^{3} d \,x^{2}+\sqrt {-\frac {a d}{a c +b}}\, b^{2} d^{2} x^{4}+3 \sqrt {-\frac {a d}{a c +b}}\, a b \,c^{2} d \,x^{2}+\sqrt {-\frac {a d}{a c +b}}\, a^{2} c^{4}+2 \sqrt {-\frac {a d}{a c +b}}\, b^{2} c d \,x^{2}+2 \sqrt {-\frac {a d}{a c +b}}\, a b \,c^{3}+\sqrt {-\frac {a d}{a c +b}}\, b^{2} c^{2}\right ) \left (d \,x^{2}+c \right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{3 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {-\frac {a d}{a c +b}}\, c \,x^{3} \left (a c +b \right )^{2} \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}}\) | \(593\) |
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none
Time = 0.11 (sec) , antiderivative size = 297, normalized size of antiderivative = 0.68 \[ \int \frac {1}{x^4 \sqrt {a+\frac {b}{c+d x^2}}} \, dx=-\frac {{\left (a^{2} c - a b\right )} \sqrt {-\frac {a d}{a c + b}} d^{3} x^{3} \sqrt {\frac {a c^{2} + b c}{d^{2}}} E(\arcsin \left (\sqrt {-\frac {a d}{a c + b}} x\right )\,|\,\frac {a c + b}{a c}) - {\left ({\left (a^{2} c - a b\right )} d^{3} + {\left (a^{2} c^{2} + 2 \, a b c + b^{2}\right )} d^{2}\right )} \sqrt {-\frac {a d}{a c + b}} x^{3} \sqrt {\frac {a c^{2} + b c}{d^{2}}} F(\arcsin \left (\sqrt {-\frac {a d}{a c + b}} x\right )\,|\,\frac {a c + b}{a c}) - {\left ({\left (a^{2} c^{2} - b^{2}\right )} d^{2} x^{4} - a^{2} c^{4} - 2 \, a b c^{3} - b^{2} c^{2} - 2 \, {\left (a b c^{2} + b^{2} c\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{3 \, {\left (a^{3} c^{4} + 3 \, a^{2} b c^{3} + 3 \, a b^{2} c^{2} + b^{3} c\right )} x^{3}} \]
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\[ \int \frac {1}{x^4 \sqrt {a+\frac {b}{c+d x^2}}} \, dx=\int \frac {1}{x^{4} \sqrt {\frac {a c + a d x^{2} + b}{c + d x^{2}}}}\, dx \]
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\[ \int \frac {1}{x^4 \sqrt {a+\frac {b}{c+d x^2}}} \, dx=\int { \frac {1}{\sqrt {a + \frac {b}{d x^{2} + c}} x^{4}} \,d x } \]
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\[ \int \frac {1}{x^4 \sqrt {a+\frac {b}{c+d x^2}}} \, dx=\int { \frac {1}{\sqrt {a + \frac {b}{d x^{2} + c}} x^{4}} \,d x } \]
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Timed out. \[ \int \frac {1}{x^4 \sqrt {a+\frac {b}{c+d x^2}}} \, dx=\int \frac {1}{x^4\,\sqrt {a+\frac {b}{d\,x^2+c}}} \,d x \]
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