Integrand size = 26, antiderivative size = 307 \[ \int \frac {1}{x^4 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=-\frac {2 d (b c+a d) x \sqrt {a+b x^2}}{3 a^2 c^2 \sqrt {c+d x^2}}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{3 a c x^3}+\frac {2 (b c+a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 a^2 c^2 x}+\frac {2 \sqrt {d} (b c+a d) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a^2 c^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {b \sqrt {d} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 a^2 \sqrt {c} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \]
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Time = 0.18 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {491, 597, 545, 429, 506, 422} \[ \int \frac {1}{x^4 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\frac {2 \sqrt {d} \sqrt {a+b x^2} (a d+b c) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a^2 c^{3/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {d} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 a^2 \sqrt {c} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {2 \sqrt {a+b x^2} \sqrt {c+d x^2} (a d+b c)}{3 a^2 c^2 x}-\frac {2 d x \sqrt {a+b x^2} (a d+b c)}{3 a^2 c^2 \sqrt {c+d x^2}}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{3 a c x^3} \]
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Rule 422
Rule 429
Rule 491
Rule 506
Rule 545
Rule 597
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{3 a c x^3}+\frac {\int \frac {-2 (b c+a d)-b d x^2}{x^2 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{3 a c} \\ & = -\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{3 a c x^3}+\frac {2 (b c+a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 a^2 c^2 x}-\frac {\int \frac {a b c d+2 b d (b c+a d) x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{3 a^2 c^2} \\ & = -\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{3 a c x^3}+\frac {2 (b c+a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 a^2 c^2 x}-\frac {(b d) \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{3 a c}-\frac {(2 b d (b c+a d)) \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{3 a^2 c^2} \\ & = -\frac {2 d (b c+a d) x \sqrt {a+b x^2}}{3 a^2 c^2 \sqrt {c+d x^2}}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{3 a c x^3}+\frac {2 (b c+a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 a^2 c^2 x}-\frac {b \sqrt {d} \sqrt {a+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a^2 \sqrt {c} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {(2 d (b c+a d)) \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{3 a^2 c} \\ & = -\frac {2 d (b c+a d) x \sqrt {a+b x^2}}{3 a^2 c^2 \sqrt {c+d x^2}}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{3 a c x^3}+\frac {2 (b c+a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 a^2 c^2 x}+\frac {2 \sqrt {d} (b c+a d) \sqrt {a+b x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a^2 c^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {b \sqrt {d} \sqrt {a+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a^2 \sqrt {c} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.72 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^4 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\frac {\sqrt {\frac {b}{a}} \left (a+b x^2\right ) \left (c+d x^2\right ) \left (-a c+2 b c x^2+2 a d x^2\right )+2 i b c (b c+a d) x^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i b c (2 b c+a d) x^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{3 a^2 \sqrt {\frac {b}{a}} c^2 x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}} \]
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Time = 6.28 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.03
method | result | size |
risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (-2 a d \,x^{2}-2 c b \,x^{2}+a c \right )}{3 a^{2} c^{2} x^{3}}-\frac {b d \left (\frac {a c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}-\frac {\left (2 a d +2 b c \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}\, d}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}{3 a^{2} c^{2} \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(315\) |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (-\frac {\sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}{3 a c \,x^{3}}+\frac {2 \left (a d +b c \right ) \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}{3 a^{2} c^{2} x}-\frac {b d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{3 a c \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}+\frac {2 b \left (a d +b c \right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{3 a^{2} c \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(343\) |
default | \(\frac {\left (2 \sqrt {-\frac {b}{a}}\, a b \,d^{2} x^{6}+2 \sqrt {-\frac {b}{a}}\, b^{2} c d \,x^{6}+b d \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) x^{3} a c +2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{2} x^{3}-2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b c d \,x^{3}-2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{2} x^{3}+2 \sqrt {-\frac {b}{a}}\, a^{2} d^{2} x^{4}+3 \sqrt {-\frac {b}{a}}\, a b c d \,x^{4}+2 \sqrt {-\frac {b}{a}}\, b^{2} c^{2} x^{4}+\sqrt {-\frac {b}{a}}\, a^{2} c d \,x^{2}+\sqrt {-\frac {b}{a}}\, a b \,c^{2} x^{2}-\sqrt {-\frac {b}{a}}\, a^{2} c^{2}\right ) \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{3 \sqrt {-\frac {b}{a}}\, x^{3} c^{2} a^{2} \left (b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c \right )}\) | \(435\) |
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Time = 0.09 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.51 \[ \int \frac {1}{x^4 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=-\frac {2 \, {\left (b^{2} c + a b d\right )} \sqrt {a c} x^{3} \sqrt {-\frac {b}{a}} E(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left (2 \, b^{2} c + {\left (a^{2} + 2 \, a b\right )} d\right )} \sqrt {a c} x^{3} \sqrt {-\frac {b}{a}} F(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) + {\left (a^{2} c - 2 \, {\left (a b c + a^{2} d\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{3 \, a^{3} c^{2} x^{3}} \]
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\[ \int \frac {1}{x^4 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int \frac {1}{x^{4} \sqrt {a + b x^{2}} \sqrt {c + d x^{2}}}\, dx \]
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\[ \int \frac {1}{x^4 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int { \frac {1}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} x^{4}} \,d x } \]
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\[ \int \frac {1}{x^4 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int { \frac {1}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} x^{4}} \,d x } \]
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Timed out. \[ \int \frac {1}{x^4 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int \frac {1}{x^4\,\sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c}} \,d x \]
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