Integrand size = 26, antiderivative size = 307 \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^2} \, dx=-\frac {(b c-a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{c d x}-\frac {(b c-2 a d) e x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{c^2}+\frac {(b c-2 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{c^2 d x}+\frac {(b c-2 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{c^{3/2} \sqrt {d} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {b e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]
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Time = 0.23 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {1986, 479, 597, 545, 429, 506, 422} \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^2} \, dx=\frac {e (b c-2 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{c^{3/2} \sqrt {d} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {b e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {e \left (c+d x^2\right ) (b c-2 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{c^2 d x}-\frac {e x (b c-2 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{c^2}-\frac {e (b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{c d x} \]
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Rule 422
Rule 429
Rule 479
Rule 506
Rule 545
Rule 597
Rule 1986
Rubi steps \begin{align*} \text {integral}& = \frac {\left (e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}\right ) \int \frac {\left (a+b x^2\right )^{3/2}}{x^2 \left (c+d x^2\right )^{3/2}} \, dx}{\sqrt {a+b x^2}} \\ & = -\frac {(b c-a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{c d x}-\frac {\left (e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}\right ) \int \frac {a (b c-2 a d)-a b d x^2}{x^2 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{c d \sqrt {a+b x^2}} \\ & = -\frac {(b c-a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{c d x}+\frac {(b c-2 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{c^2 d x}+\frac {\left (e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}\right ) \int \frac {a^2 b c d-a b d (b c-2 a d) x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{a c^2 d \sqrt {a+b x^2}} \\ & = -\frac {(b c-a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{c d x}+\frac {(b c-2 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{c^2 d x}+\frac {\left (a b e \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}{c \sqrt {a+b x^2}}-\frac {\left (b (b c-2 a d) e \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}{c^2 \sqrt {a+b x^2}} \\ & = -\frac {(b c-a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{c d x}-\frac {(b c-2 a d) e x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{c^2}+\frac {(b c-2 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{c^2 d x}+\frac {b e \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 )}{\sqrt {c} \sqrt {d} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {\left ((b c-2 a d) e \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}{c \sqrt {a+b x^2}} \\ & = -\frac {(b c-a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{c d x}-\frac {(b c-2 a d) e x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{c^2}+\frac {(b c-2 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{c^2 d x}+\frac {(b c-2 a d) e \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 )}{c^{3/2} \sqrt {d} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {b e \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 )}{\sqrt {c} \sqrt {d} \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 = 5.25 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.74 \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^2} \, dx=-\frac {e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\sqrt {\frac {b}{a}} d \left (a+b x^2\right ) \left (a c-b c x^2+2 a d x^2\right )+i b c (-b c+2 a d) x \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i b c (-b c+a d) x \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{\sqrt {\frac {b}{a}} c^2 d x \left (a+b x^2\right )} \]
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Time = 8.05 (sec) , antiderivative size = 670, normalized size of antiderivative = 2.18
method | result | size |
default | \(-\frac {{\left (\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}\right )}^{\frac {3}{2}} \left (d \,x^{2}+c \right ) \left (\sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, a b \,d^{2} x^{4}+\sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {-\frac {b}{a}}\, a b \,d^{2} x^{4}-\sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {-\frac {b}{a}}\, b^{2} c d \,x^{4}+\sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b c d x -\sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{2} x -2 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b c d x +\sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{2} x +\sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, a^{2} d^{2} x^{2}+\sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, a b c d \,x^{2}+\sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {-\frac {b}{a}}\, a^{2} d^{2} x^{2}-\sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {-\frac {b}{a}}\, a b c d \,x^{2}+\sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, a^{2} c d \right )}{\left (b \,x^{2}+a \right )^{2} c^{2} x \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, d}\) | \(670\) |
risch | \(-\frac {a \left (d \,x^{2}+c \right ) e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}{c^{2} x}+\frac {\left (\frac {b^{2} c^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {e d a +e b c}{c b e}}\right )}{d \sqrt {-\frac {b}{a}}\, \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}-\frac {2 d \,a^{2} b c e \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {e d a +e b c}{c b e}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {e d a +e b c}{c b e}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}\, \left (e d a +e b c +e \left (a d -b c \right )\right )}-\frac {c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {\left (b d e \,x^{2}+e d a \right ) x}{c \left (a d -b c \right ) e \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (b d e \,x^{2}+e d a \right )}}+\frac {\left (\frac {1}{c}-\frac {d a}{c \left (a d -b c \right )}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {e d a +e b c}{c b e}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}+\frac {2 b d a e \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {e d a +e b c}{c b e}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {e d a +e b c}{c b e}}\right )\right )}{\left (a d -b c \right ) \sqrt {-\frac {b}{a}}\, \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}\, \left (e d a +e b c +e \left (a d -b c \right )\right )}\right )}{d}\right ) e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) e \left (b \,x^{2}+a \right )}}{c^{2} \left (b \,x^{2}+a \right )}\) | \(723\) |
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Time = 0.12 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.55 \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^2} \, dx=-\frac {{\left (b^{2} c - 2 \, a b d\right )} \sqrt {\frac {a c e}{d^{2}}} e x \sqrt {-\frac {b}{a}} E(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left (b^{2} c - {\left (a^{2} + 2 \, a b\right )} d\right )} \sqrt {\frac {a c e}{d^{2}}} e x \sqrt {-\frac {b}{a}} F(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) + {\left (a^{2} c e - {\left (a b c - 2 \, a^{2} d\right )} e x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{a c^{2} x} \]
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Timed out. \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^2} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^2} \, dx=\int { \frac {\left (\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}\right )^{\frac {3}{2}}}{x^{2}} \,d x } \]
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\[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^2} \, dx=\int { \frac {\left (\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}\right )^{\frac {3}{2}}}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^2} \, dx=\int \frac {{\left (\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}\right )}^{3/2}}{x^2} \,d x \]
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