Integrand size = 19, antiderivative size = 243 \[ \int \frac {1}{x^2 \sqrt {a x^2+b x^5}} \, dx=-\frac {\sqrt {a x^2+b x^5}}{2 a x^3}-\frac {\sqrt {2+\sqrt {3}} b^{2/3} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{2 \sqrt [4]{3} a \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a x^2+b x^5}} \]
[Out]
Time = 0.09 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2050, 2057, 224} \[ \int \frac {1}{x^2 \sqrt {a x^2+b x^5}} \, dx=-\frac {\sqrt {2+\sqrt {3}} b^{2/3} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{2 \sqrt [4]{3} a \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a x^2+b x^5}}-\frac {\sqrt {a x^2+b x^5}}{2 a x^3} \]
[In]
[Out]
Rule 224
Rule 2050
Rule 2057
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {a x^2+b x^5}}{2 a x^3}-\frac {b \int \frac {x}{\sqrt {a x^2+b x^5}} \, dx}{4 a} \\ & = -\frac {\sqrt {a x^2+b x^5}}{2 a x^3}-\frac {\left (b x \sqrt {a+b x^3}\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx}{4 a \sqrt {a x^2+b x^5}} \\ & = -\frac {\sqrt {a x^2+b x^5}}{2 a x^3}-\frac {\sqrt {2+\sqrt {3}} b^{2/3} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{2 \sqrt [4]{3} a \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a x^2+b x^5}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.02 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.23 \[ \int \frac {1}{x^2 \sqrt {a x^2+b x^5}} \, dx=-\frac {\sqrt {1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{2},\frac {1}{3},-\frac {b x^3}{a}\right )}{2 x \sqrt {x^2 \left (a+b x^3\right )}} \]
[In]
[Out]
Time = 2.02 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.02
method | result | size |
default | \(\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {-\frac {i \left (i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}-2 b x -\left (-a \,b^{2}\right )^{\frac {1}{3}}\right ) \sqrt {3}}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {2 \left (-b x +\left (-a \,b^{2}\right )^{\frac {1}{3}}\right )}{\left (-a \,b^{2}\right )^{\frac {1}{3}} \left (i \sqrt {3}-3\right )}}\, \sqrt {-\frac {i \left (i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}+2 b x +\left (-a \,b^{2}\right )^{\frac {1}{3}}\right ) \sqrt {3}}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, F\left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {-\frac {i \left (i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}-2 b x -\left (-a \,b^{2}\right )^{\frac {1}{3}}\right ) \sqrt {3}}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{6}, \sqrt {2}\, \sqrt {\frac {i \sqrt {3}}{i \sqrt {3}-3}}\right ) x^{2}-6 b \,x^{3}-6 a}{12 x \sqrt {b \,x^{5}+a \,x^{2}}\, a}\) | \(248\) |
risch | \(-\frac {b \,x^{3}+a}{2 a x \sqrt {x^{2} \left (b \,x^{3}+a \right )}}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, F\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right ) x}{6 a \sqrt {x^{2} \left (b \,x^{3}+a \right )}}\) | \(317\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.16 \[ \int \frac {1}{x^2 \sqrt {a x^2+b x^5}} \, dx=-\frac {\sqrt {b} x^{3} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + \sqrt {b x^{5} + a x^{2}}}{2 \, a x^{3}} \]
[In]
[Out]
\[ \int \frac {1}{x^2 \sqrt {a x^2+b x^5}} \, dx=\int \frac {1}{x^{2} \sqrt {x^{2} \left (a + b x^{3}\right )}}\, dx \]
[In]
[Out]
\[ \int \frac {1}{x^2 \sqrt {a x^2+b x^5}} \, dx=\int { \frac {1}{\sqrt {b x^{5} + a x^{2}} x^{2}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{x^2 \sqrt {a x^2+b x^5}} \, dx=\int { \frac {1}{\sqrt {b x^{5} + a x^{2}} x^{2}} \,d x } \]
[In]
[Out]
Time = 9.71 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.18 \[ \int \frac {1}{x^2 \sqrt {a x^2+b x^5}} \, dx=-\frac {2\,\sqrt {\frac {a}{b\,x^3}+1}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{6};\ \frac {13}{6};\ -\frac {a}{b\,x^3}\right )}{7\,x\,\sqrt {b\,x^5+a\,x^2}} \]
[In]
[Out]