Integrand size = 28, antiderivative size = 130 \[ \int \frac {\sqrt {x} \left (A+B x^2\right )}{\sqrt {b x^2+c x^4}} \, dx=\frac {2 B \sqrt {b x^2+c x^4}}{3 c \sqrt {x}}-\frac {(b B-3 A c) x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{3 \sqrt [4]{b} c^{5/4} \sqrt {b x^2+c x^4}} \]
[Out]
Time = 0.13 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2064, 2057, 335, 226} \[ \int \frac {\sqrt {x} \left (A+B x^2\right )}{\sqrt {b x^2+c x^4}} \, dx=\frac {2 B \sqrt {b x^2+c x^4}}{3 c \sqrt {x}}-\frac {x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (b B-3 A c) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{3 \sqrt [4]{b} c^{5/4} \sqrt {b x^2+c x^4}} \]
[In]
[Out]
Rule 226
Rule 335
Rule 2057
Rule 2064
Rubi steps \begin{align*} \text {integral}& = \frac {2 B \sqrt {b x^2+c x^4}}{3 c \sqrt {x}}-\frac {\left (2 \left (\frac {b B}{2}-\frac {3 A c}{2}\right )\right ) \int \frac {\sqrt {x}}{\sqrt {b x^2+c x^4}} \, dx}{3 c} \\ & = \frac {2 B \sqrt {b x^2+c x^4}}{3 c \sqrt {x}}-\frac {\left (2 \left (\frac {b B}{2}-\frac {3 A c}{2}\right ) x \sqrt {b+c x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x^2}} \, dx}{3 c \sqrt {b x^2+c x^4}} \\ & = \frac {2 B \sqrt {b x^2+c x^4}}{3 c \sqrt {x}}-\frac {\left (4 \left (\frac {b B}{2}-\frac {3 A c}{2}\right ) x \sqrt {b+c x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{3 c \sqrt {b x^2+c x^4}} \\ & = \frac {2 B \sqrt {b x^2+c x^4}}{3 c \sqrt {x}}-\frac {(b B-3 A c) x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{3 \sqrt [4]{b} c^{5/4} \sqrt {b x^2+c x^4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.07 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.62 \[ \int \frac {\sqrt {x} \left (A+B x^2\right )}{\sqrt {b x^2+c x^4}} \, dx=\frac {2 x^{3/2} \left (B \left (b+c x^2\right )+(-b B+3 A c) \sqrt {1+\frac {c x^2}{b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {c x^2}{b}\right )\right )}{3 c \sqrt {x^2 \left (b+c x^2\right )}} \]
[In]
[Out]
Time = 2.16 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.35
method | result | size |
risch | \(\frac {2 B \,x^{\frac {3}{2}} \left (c \,x^{2}+b \right )}{3 c \sqrt {x^{2} \left (c \,x^{2}+b \right )}}+\frac {\left (3 A c -B b \right ) \sqrt {-b c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {x}\, \sqrt {x \left (c \,x^{2}+b \right )}}{3 c^{2} \sqrt {c \,x^{3}+b x}\, \sqrt {x^{2} \left (c \,x^{2}+b \right )}}\) | \(175\) |
default | \(\frac {\sqrt {x}\, \left (3 A \sqrt {-b c}\, \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) c -B \sqrt {-b c}\, \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) b +2 B \,c^{2} x^{3}+2 B b c x \right )}{3 \sqrt {x^{4} c +b \,x^{2}}\, c^{2}}\) | \(216\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.39 \[ \int \frac {\sqrt {x} \left (A+B x^2\right )}{\sqrt {b x^2+c x^4}} \, dx=-\frac {2 \, {\left ({\left (B b - 3 \, A c\right )} \sqrt {c} x {\rm weierstrassPInverse}\left (-\frac {4 \, b}{c}, 0, x\right ) - \sqrt {c x^{4} + b x^{2}} B c \sqrt {x}\right )}}{3 \, c^{2} x} \]
[In]
[Out]
\[ \int \frac {\sqrt {x} \left (A+B x^2\right )}{\sqrt {b x^2+c x^4}} \, dx=\int \frac {\sqrt {x} \left (A + B x^{2}\right )}{\sqrt {x^{2} \left (b + c x^{2}\right )}}\, dx \]
[In]
[Out]
\[ \int \frac {\sqrt {x} \left (A+B x^2\right )}{\sqrt {b x^2+c x^4}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} \sqrt {x}}{\sqrt {c x^{4} + b x^{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {\sqrt {x} \left (A+B x^2\right )}{\sqrt {b x^2+c x^4}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} \sqrt {x}}{\sqrt {c x^{4} + b x^{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\sqrt {x} \left (A+B x^2\right )}{\sqrt {b x^2+c x^4}} \, dx=\int \frac {\sqrt {x}\,\left (B\,x^2+A\right )}{\sqrt {c\,x^4+b\,x^2}} \,d x \]
[In]
[Out]