Integrand size = 26, antiderivative size = 266 \[ \int x^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \, dx=-\frac {(2 b c-a d) x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{3 b d}+\frac {x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{3 d}+\frac {\sqrt {c} (2 b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 b d^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {c^{3/2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 d^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]
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Time = 0.15 (sec) , antiderivative size = 266, 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 = {1986, 489, 545, 429, 506, 422} \[ \int x^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \, dx=-\frac {c^{3/2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 d^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {\sqrt {c} (2 b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 b d^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {x \left (c+d x^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{3 d}-\frac {x (2 b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{3 b d} \]
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Rule 422
Rule 429
Rule 489
Rule 506
Rule 545
Rule 1986
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}\right ) \int \frac {x^2 \sqrt {a+b x^2}}{\sqrt {c+d x^2}} \, dx}{\sqrt {a+b x^2}} \\ & = \frac {x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{3 d}-\frac {\left (\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}\right ) \int \frac {a c+(2 b c-a d) x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{3 d \sqrt {a+b x^2}} \\ & = \frac {x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{3 d}-\frac {\left (a c \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}\right ) \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{3 d \sqrt {a+b x^2}}-\frac {\left ((2 b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}\right ) \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{3 d \sqrt {a+b x^2}} \\ & = -\frac {(2 b c-a d) x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{3 b d}+\frac {x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{3 d}-\frac {c^{3/2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 d^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {\left (c (2 b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}\right ) \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{3 b d \sqrt {a+b x^2}} \\ & = -\frac {(2 b c-a d) x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{3 b d}+\frac {x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{3 d}+\frac {\sqrt {c} (2 b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} E\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 )}}}-\frac {c^{3/2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 d^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.22 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.78 \[ \int x^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \, dx=\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right )-i c (-2 b c+a d) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+2 i c (-b c+a d) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{3 \sqrt {\frac {b}{a}} d^2 \left (a+b x^2\right )} \]
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Time = 4.72 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.32
method | result | size |
risch | \(\frac {x \left (d \,x^{2}+c \right ) \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}{3 d}-\frac {\left (\frac {a c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {e d a +e b c}{c b e}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}+\frac {2 \left (a d -2 b c \right ) a c e \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {e d a +e b c}{c b e}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {e d a +e b c}{c b e}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}\, \left (e d a +e b c +e \left (a d -b c \right )\right )}\right ) \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) e \left (b \,x^{2}+a \right )}}{3 d \left (b \,x^{2}+a \right )}\) | \(350\) |
default | \(\frac {\sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right ) \left (\sqrt {-\frac {b}{a}}\, b \,d^{2} x^{5}+\sqrt {-\frac {b}{a}}\, a \,d^{2} x^{3}+\sqrt {-\frac {b}{a}}\, b c d \,x^{3}-2 a c \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) d +2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b \,c^{2}+\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a c d -2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b \,c^{2}+\sqrt {-\frac {b}{a}}\, a c d x \right )}{3 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, d^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}}\) | \(356\) |
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Time = 0.11 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.68 \[ \int x^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \, dx=\frac {{\left (2 \, b c^{2} - a c d\right )} \sqrt {\frac {b e}{d}} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (2 \, b c^{2} - a c d + a d^{2}\right )} \sqrt {\frac {b e}{d}} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) + {\left (b d^{2} x^{4} - 2 \, b c^{2} + a c d - {\left (b c d - a d^{2}\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{3 \, b d^{2} x} \]
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Timed out. \[ \int x^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \, dx=\text {Timed out} \]
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\[ \int x^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \, dx=\int { \sqrt {\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}} x^{2} \,d x } \]
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\[ \int x^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \, dx=\int { \sqrt {\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}} x^{2} \,d x } \]
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Timed out. \[ \int x^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \, dx=\int x^2\,\sqrt {\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}} \,d x \]
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