Integrand size = 26, antiderivative size = 307 \[ \int \frac {\sqrt {a+b x^2}}{x^4 \sqrt {c+d x^2}} \, dx=\frac {d (b c-2 a d) x \sqrt {a+b x^2}}{3 a c^2 \sqrt {c+d x^2}}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c x^3}-\frac {(b c-2 a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 a c^2 x}-\frac {\sqrt {d} (b c-2 a d) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a c^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {b \sqrt {d} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 a \sqrt {c} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \]
[Out]
Time = 0.19 (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 = {486, 597, 545, 429, 506, 422} \[ \int \frac {\sqrt {a+b x^2}}{x^4 \sqrt {c+d x^2}} \, dx=-\frac {\sqrt {d} \sqrt {a+b x^2} (b c-2 a d) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a c^{3/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {d} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 a \sqrt {c} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (b c-2 a d)}{3 a c^2 x}+\frac {d x \sqrt {a+b x^2} (b c-2 a d)}{3 a c^2 \sqrt {c+d x^2}}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c x^3} \]
[In]
[Out]
Rule 422
Rule 429
Rule 486
Rule 506
Rule 545
Rule 597
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c x^3}+\frac {\int \frac {b c-2 a d-b d x^2}{x^2 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{3 c} \\ & = -\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c x^3}-\frac {(b c-2 a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 a c^2 x}-\frac {\int \frac {a b c d-b d (b c-2 a d) x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{3 a c^2} \\ & = -\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c x^3}-\frac {(b c-2 a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 a c^2 x}-\frac {(b d) \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{3 c}+\frac {(b d (b c-2 a d)) \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{3 a c^2} \\ & = \frac {d (b c-2 a d) x \sqrt {a+b x^2}}{3 a c^2 \sqrt {c+d x^2}}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c x^3}-\frac {(b c-2 a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 a 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 \sqrt {c} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {(d (b c-2 a d)) \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{3 a c} \\ & = \frac {d (b c-2 a d) x \sqrt {a+b x^2}}{3 a c^2 \sqrt {c+d x^2}}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c x^3}-\frac {(b c-2 a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 a c^2 x}-\frac {\sqrt {d} (b c-2 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 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 \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 = 1.63 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt {a+b x^2}}{x^4 \sqrt {c+d x^2}} \, dx=\frac {-\frac {\left (a+b x^2\right ) \left (c+d x^2\right ) \left (a c+b c x^2-2 a d x^2\right )}{a}+i \sqrt {\frac {b}{a}} c (-b c+2 a d) x^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i \sqrt {\frac {b}{a}} c (b c-a d) x^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{3 c^2 x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}} \]
[In]
[Out]
Time = 5.69 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.02
method | result | size |
risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (-2 a d \,x^{2}+c b \,x^{2}+a c \right )}{3 c^{2} x^{3} a}-\frac {b d \left (\frac {a c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}-\frac {\left (2 a d -b c \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}\, d}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}{3 a \,c^{2} \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(314\) |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (-\frac {\sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}{3 c \,x^{3}}+\frac {\left (2 a d -b c \right ) \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}{3 a \,c^{2} x}-\frac {b d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{3 c \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}+\frac {b \left (2 a d -b c \right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{3 c a \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(341\) |
default | \(\frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (2 \sqrt {-\frac {b}{a}}\, a b \,d^{2} x^{6}-\sqrt {-\frac {b}{a}}\, b^{2} c d \,x^{6}+b d \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) x^{3} a c -\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{2} x^{3}-2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b c d \,x^{3}+\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{2} x^{3}+2 \sqrt {-\frac {b}{a}}\, a^{2} d^{2} x^{4}-\sqrt {-\frac {b}{a}}\, b^{2} c^{2} x^{4}+\sqrt {-\frac {b}{a}}\, a^{2} c d \,x^{2}-2 \sqrt {-\frac {b}{a}}\, a b \,c^{2} x^{2}-\sqrt {-\frac {b}{a}}\, a^{2} c^{2}\right )}{3 \left (b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c \right ) c^{2} x^{3} a \sqrt {-\frac {b}{a}}}\) | \(418\) |
[In]
[Out]
none
Time = 0.08 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.51 \[ \int \frac {\sqrt {a+b x^2}}{x^4 \sqrt {c+d x^2}} \, dx=\frac {{\left (b^{2} c - 2 \, a b d\right )} \sqrt {a c} x^{3} \sqrt {-\frac {b}{a}} E(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left (b^{2} c - {\left (a^{2} + 2 \, a b\right )} d\right )} \sqrt {a c} x^{3} \sqrt {-\frac {b}{a}} F(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left (a^{2} c + {\left (a b c - 2 \, a^{2} d\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{3 \, a^{2} c^{2} x^{3}} \]
[In]
[Out]
\[ \int \frac {\sqrt {a+b x^2}}{x^4 \sqrt {c+d x^2}} \, dx=\int \frac {\sqrt {a + b x^{2}}}{x^{4} \sqrt {c + d x^{2}}}\, dx \]
[In]
[Out]
\[ \int \frac {\sqrt {a+b x^2}}{x^4 \sqrt {c+d x^2}} \, dx=\int { \frac {\sqrt {b x^{2} + a}}{\sqrt {d x^{2} + c} x^{4}} \,d x } \]
[In]
[Out]
\[ \int \frac {\sqrt {a+b x^2}}{x^4 \sqrt {c+d x^2}} \, dx=\int { \frac {\sqrt {b x^{2} + a}}{\sqrt {d x^{2} + c} x^{4}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\sqrt {a+b x^2}}{x^4 \sqrt {c+d x^2}} \, dx=\int \frac {\sqrt {b\,x^2+a}}{x^4\,\sqrt {d\,x^2+c}} \,d x \]
[In]
[Out]