Integrand size = 24, antiderivative size = 327 \[ \int \frac {A+B x}{(e x)^{5/2} \sqrt {a+c x^2}} \, dx=-\frac {2 A \sqrt {a+c x^2}}{3 a e (e x)^{3/2}}-\frac {2 B \sqrt {a+c x^2}}{a e^2 \sqrt {e x}}+\frac {2 B \sqrt {c} x \sqrt {a+c x^2}}{a e^2 \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {2 B \sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{a^{3/4} e^2 \sqrt {e x} \sqrt {a+c x^2}}+\frac {\left (3 \sqrt {a} B-A \sqrt {c}\right ) \sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{3 a^{5/4} e^2 \sqrt {e x} \sqrt {a+c x^2}} \]
[Out]
Time = 0.22 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {849, 856, 854, 1212, 226, 1210} \[ \int \frac {A+B x}{(e x)^{5/2} \sqrt {a+c x^2}} \, dx=\frac {\sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (3 \sqrt {a} B-A \sqrt {c}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{3 a^{5/4} e^2 \sqrt {e x} \sqrt {a+c x^2}}-\frac {2 B \sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{a^{3/4} e^2 \sqrt {e x} \sqrt {a+c x^2}}-\frac {2 A \sqrt {a+c x^2}}{3 a e (e x)^{3/2}}-\frac {2 B \sqrt {a+c x^2}}{a e^2 \sqrt {e x}}+\frac {2 B \sqrt {c} x \sqrt {a+c x^2}}{a e^2 \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )} \]
[In]
[Out]
Rule 226
Rule 849
Rule 854
Rule 856
Rule 1210
Rule 1212
Rubi steps \begin{align*} \text {integral}& = -\frac {2 A \sqrt {a+c x^2}}{3 a e (e x)^{3/2}}-\frac {2 \int \frac {-\frac {3}{2} a B e+\frac {1}{2} A c e x}{(e x)^{3/2} \sqrt {a+c x^2}} \, dx}{3 a e^2} \\ & = -\frac {2 A \sqrt {a+c x^2}}{3 a e (e x)^{3/2}}-\frac {2 B \sqrt {a+c x^2}}{a e^2 \sqrt {e x}}+\frac {4 \int \frac {-\frac {1}{4} a A c e^2+\frac {3}{4} a B c e^2 x}{\sqrt {e x} \sqrt {a+c x^2}} \, dx}{3 a^2 e^4} \\ & = -\frac {2 A \sqrt {a+c x^2}}{3 a e (e x)^{3/2}}-\frac {2 B \sqrt {a+c x^2}}{a e^2 \sqrt {e x}}+\frac {\left (4 \sqrt {x}\right ) \int \frac {-\frac {1}{4} a A c e^2+\frac {3}{4} a B c e^2 x}{\sqrt {x} \sqrt {a+c x^2}} \, dx}{3 a^2 e^4 \sqrt {e x}} \\ & = -\frac {2 A \sqrt {a+c x^2}}{3 a e (e x)^{3/2}}-\frac {2 B \sqrt {a+c x^2}}{a e^2 \sqrt {e x}}+\frac {\left (8 \sqrt {x}\right ) \text {Subst}\left (\int \frac {-\frac {1}{4} a A c e^2+\frac {3}{4} a B c e^2 x^2}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{3 a^2 e^4 \sqrt {e x}} \\ & = -\frac {2 A \sqrt {a+c x^2}}{3 a e (e x)^{3/2}}-\frac {2 B \sqrt {a+c x^2}}{a e^2 \sqrt {e x}}-\frac {\left (2 B \sqrt {c} \sqrt {x}\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {a} e^2 \sqrt {e x}}+\frac {\left (2 \left (3 \sqrt {a} B-A \sqrt {c}\right ) \sqrt {c} \sqrt {x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{3 a e^2 \sqrt {e x}} \\ & = -\frac {2 A \sqrt {a+c x^2}}{3 a e (e x)^{3/2}}-\frac {2 B \sqrt {a+c x^2}}{a e^2 \sqrt {e x}}+\frac {2 B \sqrt {c} x \sqrt {a+c x^2}}{a e^2 \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {2 B \sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{a^{3/4} e^2 \sqrt {e x} \sqrt {a+c x^2}}+\frac {\left (3 \sqrt {a} B-A \sqrt {c}\right ) \sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{3 a^{5/4} e^2 \sqrt {e x} \sqrt {a+c x^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.02 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.25 \[ \int \frac {A+B x}{(e x)^{5/2} \sqrt {a+c x^2}} \, dx=-\frac {2 x \sqrt {1+\frac {c x^2}{a}} \left (A \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {1}{2},\frac {1}{4},-\frac {c x^2}{a}\right )+3 B x \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},-\frac {c x^2}{a}\right )\right )}{3 (e x)^{5/2} \sqrt {a+c x^2}} \]
[In]
[Out]
Time = 0.65 (sec) , antiderivative size = 306, normalized size of antiderivative = 0.94
method | result | size |
default | \(-\frac {A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a c}\, x +3 B \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a x -6 B \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, E\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a x +6 B c \,x^{3}+2 A c \,x^{2}+6 a B x +2 a A}{3 x \sqrt {c \,x^{2}+a}\, a \,e^{2} \sqrt {e x}}\) | \(306\) |
risch | \(-\frac {2 \sqrt {c \,x^{2}+a}\, \left (3 B x +A \right )}{3 a x \,e^{2} \sqrt {e x}}-\frac {c \left (\frac {A \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c \sqrt {c e \,x^{3}+a e x}}-\frac {3 B \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{c \sqrt {c e \,x^{3}+a e x}}\right ) \sqrt {\left (c \,x^{2}+a \right ) e x}}{3 a \,e^{2} \sqrt {e x}\, \sqrt {c \,x^{2}+a}}\) | \(340\) |
elliptic | \(\frac {\sqrt {\left (c \,x^{2}+a \right ) e x}\, \left (-\frac {2 A \sqrt {c e \,x^{3}+a e x}}{3 e^{3} a \,x^{2}}-\frac {2 \left (c e \,x^{2}+a e \right ) B}{e^{3} a \sqrt {x \left (c e \,x^{2}+a e \right )}}-\frac {A \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{3 a \,e^{2} \sqrt {c e \,x^{3}+a e x}}+\frac {B \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{a \,e^{2} \sqrt {c e \,x^{3}+a e x}}\right )}{\sqrt {e x}\, \sqrt {c \,x^{2}+a}}\) | \(364\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.24 \[ \int \frac {A+B x}{(e x)^{5/2} \sqrt {a+c x^2}} \, dx=-\frac {2 \, {\left (\sqrt {c e} A x^{2} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) + 3 \, \sqrt {c e} B x^{2} {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) + \sqrt {c x^{2} + a} {\left (3 \, B x + A\right )} \sqrt {e x}\right )}}{3 \, a e^{3} x^{2}} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 4.60 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.31 \[ \int \frac {A+B x}{(e x)^{5/2} \sqrt {a+c x^2}} \, dx=\frac {A \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} e^{\frac {5}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {1}{4}\right )} + \frac {B \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} e^{\frac {5}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} \]
[In]
[Out]
\[ \int \frac {A+B x}{(e x)^{5/2} \sqrt {a+c x^2}} \, dx=\int { \frac {B x + A}{\sqrt {c x^{2} + a} \left (e x\right )^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {A+B x}{(e x)^{5/2} \sqrt {a+c x^2}} \, dx=\int { \frac {B x + A}{\sqrt {c x^{2} + a} \left (e x\right )^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {A+B x}{(e x)^{5/2} \sqrt {a+c x^2}} \, dx=\int \frac {A+B\,x}{{\left (e\,x\right )}^{5/2}\,\sqrt {c\,x^2+a}} \,d x \]
[In]
[Out]