Integrand size = 26, antiderivative size = 261 \[ \int \frac {x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=-\frac {2 (b c+a d) x \sqrt {a+b x^2}}{3 b^2 d \sqrt {c+d x^2}}+\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 b d}+\frac {2 \sqrt {c} (b c+a d) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 b^2 d^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {c^{3/2} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 b d^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \]
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Time = 0.11 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {490, 545, 429, 506, 422} \[ \int \frac {x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\frac {2 \sqrt {c} \sqrt {a+b x^2} (a d+b c) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 b^2 d^{3/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {c^{3/2} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 b d^{3/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {2 x \sqrt {a+b x^2} (a d+b c)}{3 b^2 d \sqrt {c+d x^2}}+\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 b d} \]
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Rule 422
Rule 429
Rule 490
Rule 506
Rule 545
Rubi steps \begin{align*} \text {integral}& = \frac {x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 b d}-\frac {\int \frac {a c+2 (b c+a d) x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{3 b d} \\ & = \frac {x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 b d}-\frac {(a c) \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{3 b d}-\frac {(2 (b c+a d)) \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{3 b d} \\ & = -\frac {2 (b c+a d) x \sqrt {a+b x^2}}{3 b^2 d \sqrt {c+d x^2}}+\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 b d}-\frac {c^{3/2} \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 b d^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {(2 c (b c+a d)) \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{3 b^2 d} \\ & = -\frac {2 (b c+a d) x \sqrt {a+b x^2}}{3 b^2 d \sqrt {c+d x^2}}+\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 b d}+\frac {2 \sqrt {c} (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 b^2 d^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {c^{3/2} \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 b d^{3/2} \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 = 1.47 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.77 \[ \int \frac {x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right )+2 i c (b c+a d) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c (2 b c+a d) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{3 b \sqrt {\frac {b}{a}} d^2 \sqrt {a+b x^2} \sqrt {c+d x^2}} \]
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Time = 5.51 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.12
method | result | size |
risch | \(\frac {x \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}{3 b d}-\frac {\left (\frac {a c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}-\frac {\left (2 a d +2 b c \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}\, d}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}{3 b d \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(293\) |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (\frac {x \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}{3 b d}-\frac {a c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{3 b d \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}+\frac {\left (2 a d +2 b c \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{3 b \,d^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(301\) |
default | \(\frac {\left (\sqrt {-\frac {b}{a}}\, b \,d^{2} x^{5}+\sqrt {-\frac {b}{a}}\, a \,d^{2} x^{3}+\sqrt {-\frac {b}{a}}\, b c d \,x^{3}+a c \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) d +2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b \,c^{2}-2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a c d -2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b \,c^{2}+\sqrt {-\frac {b}{a}}\, a c d x \right ) \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{3 \sqrt {-\frac {b}{a}}\, d^{2} b \left (b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c \right )}\) | \(333\) |
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Time = 0.11 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.60 \[ \int \frac {x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\frac {2 \, {\left (b c^{2} + a c d\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (2 \, b c^{2} + 2 \, a c d + a d^{2}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) + {\left (b d^{2} x^{2} - 2 \, b c d - 2 \, a d^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{3 \, b^{2} d^{3} x} \]
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\[ \int \frac {x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int \frac {x^{4}}{\sqrt {a + b x^{2}} \sqrt {c + d x^{2}}}\, dx \]
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\[ \int \frac {x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int { \frac {x^{4}}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c}} \,d x } \]
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\[ \int \frac {x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int { \frac {x^{4}}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c}} \,d x } \]
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Timed out. \[ \int \frac {x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int \frac {x^4}{\sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c}} \,d x \]
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