Integrand size = 23, antiderivative size = 237 \[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {x \sqrt {c+d x^2}}{3 a \left (a+b x^2\right )^{3/2}}+\frac {(2 b c-a d) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 a^{3/2} \sqrt {b} (b c-a d) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {c^{3/2} \sqrt {d} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 a^2 (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \]
[Out]
Time = 0.08 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {423, 539, 429, 422} \[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {\sqrt {c+d x^2} (2 b c-a d) E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 a^{3/2} \sqrt {b} \sqrt {a+b x^2} (b c-a d) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {c^{3/2} \sqrt {d} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 a^2 \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {x \sqrt {c+d x^2}}{3 a \left (a+b x^2\right )^{3/2}} \]
[In]
[Out]
Rule 422
Rule 423
Rule 429
Rule 539
Rubi steps \begin{align*} \text {integral}& = \frac {x \sqrt {c+d x^2}}{3 a \left (a+b x^2\right )^{3/2}}-\frac {\int \frac {-2 c-d x^2}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx}{3 a} \\ & = \frac {x \sqrt {c+d x^2}}{3 a \left (a+b x^2\right )^{3/2}}-\frac {(c d) \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{3 a (b c-a d)}+\frac {(2 b c-a d) \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{3/2}} \, dx}{3 a (b c-a d)} \\ & = \frac {x \sqrt {c+d x^2}}{3 a \left (a+b x^2\right )^{3/2}}+\frac {(2 b c-a d) \sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 a^{3/2} \sqrt {b} (b c-a d) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {c^{3/2} \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 (b c-a d) \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 = 2.68 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.03 \[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {\sqrt {\frac {b}{a}} x \left (c+d x^2\right ) \left (2 a^2 d-2 b^2 c x^2+a b \left (-3 c+d x^2\right )\right )+i c (-2 b c+a d) \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-2 i c (-b c+a d) \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{3 a^2 \sqrt {\frac {b}{a}} (-b c+a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \]
[In]
[Out]
Time = 2.37 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.76
method | result | size |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (\frac {x \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}{3 b^{2} a \left (x^{2}+\frac {a}{b}\right )^{2}}+\frac {\left (b d \,x^{2}+b c \right ) x \left (a d -2 b c \right )}{3 b \,a^{2} \left (a d -b c \right ) \sqrt {\left (x^{2}+\frac {a}{b}\right ) \left (b d \,x^{2}+b c \right )}}+\frac {\left (\frac {d}{3 a b}-\frac {a d -2 b c}{3 b \,a^{2}}-\frac {c \left (a d -2 b c \right )}{3 a^{2} \left (a d -b c \right )}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}+\frac {\left (a d -2 b c \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{3 \left (a d -b c \right ) a^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(418\) |
default | \(\frac {\sqrt {-\frac {b}{a}}\, a b \,d^{2} x^{5}-2 \sqrt {-\frac {b}{a}}\, b^{2} c d \,x^{5}+2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b c d \,x^{2}-2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{2} x^{2}-\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b c d \,x^{2}+2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{2} x^{2}+2 \sqrt {-\frac {b}{a}}\, a^{2} d^{2} x^{3}-2 \sqrt {-\frac {b}{a}}\, a b c d \,x^{3}-2 \sqrt {-\frac {b}{a}}\, b^{2} c^{2} x^{3}+2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c d -2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2}-\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c d +2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2}+2 \sqrt {-\frac {b}{a}}\, a^{2} c d x -3 \sqrt {-\frac {b}{a}}\, a b \,c^{2} x}{3 \sqrt {d \,x^{2}+c}\, \sqrt {-\frac {b}{a}}\, \left (a d -b c \right ) a^{2} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}\) | \(617\) |
[In]
[Out]
none
Time = 0.09 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.41 \[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{5/2}} \, dx=-\frac {{\left (2 \, a^{2} b^{2} c - a^{3} b d + {\left (2 \, b^{4} c - a b^{3} d\right )} x^{4} + 2 \, {\left (2 \, a b^{3} c - a^{2} b^{2} d\right )} x^{2}\right )} \sqrt {a c} \sqrt {-\frac {b}{a}} E(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left (2 \, a^{2} b^{2} c + {\left (2 \, b^{4} c + {\left (a^{2} b^{2} - a b^{3}\right )} d\right )} x^{4} + 2 \, {\left (2 \, a b^{3} c + {\left (a^{3} b - a^{2} b^{2}\right )} d\right )} x^{2} + {\left (a^{4} - a^{3} b\right )} d\right )} \sqrt {a c} \sqrt {-\frac {b}{a}} F(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left ({\left (2 \, a b^{3} c - a^{2} b^{2} d\right )} x^{3} + {\left (3 \, a^{2} b^{2} c - 2 \, a^{3} b d\right )} x\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{3 \, {\left (a^{5} b^{2} c - a^{6} b d + {\left (a^{3} b^{4} c - a^{4} b^{3} d\right )} x^{4} + 2 \, {\left (a^{4} b^{3} c - a^{5} b^{2} d\right )} x^{2}\right )}} \]
[In]
[Out]
\[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {c + d x^{2}}}{\left (a + b x^{2}\right )^{\frac {5}{2}}}\, dx \]
[In]
[Out]
\[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{5/2}} \, dx=\int { \frac {\sqrt {d x^{2} + c}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{5/2}} \, dx=\int { \frac {\sqrt {d x^{2} + c}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {d\,x^2+c}}{{\left (b\,x^2+a\right )}^{5/2}} \,d x \]
[In]
[Out]