Integrand size = 13, antiderivative size = 273 \[ \int \frac {1}{\left (a x+b x^3\right )^{3/2}} \, dx=\frac {1}{a \sqrt {a x+b x^3}}+\frac {3 \sqrt {b} x \left (a+b x^2\right )}{a^2 \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {a x+b x^3}}-\frac {3 \sqrt {a x+b x^3}}{a^2 x}-\frac {3 \sqrt [4]{b} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{a^{7/4} \sqrt {a x+b x^3}}+\frac {3 \sqrt [4]{b} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 a^{7/4} \sqrt {a x+b x^3}} \]
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Time = 0.15 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {2031, 2050, 2057, 335, 311, 226, 1210} \[ \int \frac {1}{\left (a x+b x^3\right )^{3/2}} \, dx=\frac {3 \sqrt [4]{b} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 a^{7/4} \sqrt {a x+b x^3}}-\frac {3 \sqrt [4]{b} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{a^{7/4} \sqrt {a x+b x^3}}-\frac {3 \sqrt {a x+b x^3}}{a^2 x}+\frac {3 \sqrt {b} x \left (a+b x^2\right )}{a^2 \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {a x+b x^3}}+\frac {1}{a \sqrt {a x+b x^3}} \]
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Rule 226
Rule 311
Rule 335
Rule 1210
Rule 2031
Rule 2050
Rule 2057
Rubi steps \begin{align*} \text {integral}& = \frac {1}{a \sqrt {a x+b x^3}}+\frac {3 \int \frac {1}{x \sqrt {a x+b x^3}} \, dx}{2 a} \\ & = \frac {1}{a \sqrt {a x+b x^3}}-\frac {3 \sqrt {a x+b x^3}}{a^2 x}+\frac {(3 b) \int \frac {x}{\sqrt {a x+b x^3}} \, dx}{2 a^2} \\ & = \frac {1}{a \sqrt {a x+b x^3}}-\frac {3 \sqrt {a x+b x^3}}{a^2 x}+\frac {\left (3 b \sqrt {x} \sqrt {a+b x^2}\right ) \int \frac {\sqrt {x}}{\sqrt {a+b x^2}} \, dx}{2 a^2 \sqrt {a x+b x^3}} \\ & = \frac {1}{a \sqrt {a x+b x^3}}-\frac {3 \sqrt {a x+b x^3}}{a^2 x}+\frac {\left (3 b \sqrt {x} \sqrt {a+b x^2}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {a+b x^4}} \, dx,x,\sqrt {x}\right )}{a^2 \sqrt {a x+b x^3}} \\ & = \frac {1}{a \sqrt {a x+b x^3}}-\frac {3 \sqrt {a x+b x^3}}{a^2 x}+\frac {\left (3 \sqrt {b} \sqrt {x} \sqrt {a+b x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^4}} \, dx,x,\sqrt {x}\right )}{a^{3/2} \sqrt {a x+b x^3}}-\frac {\left (3 \sqrt {b} \sqrt {x} \sqrt {a+b x^2}\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a+b x^4}} \, dx,x,\sqrt {x}\right )}{a^{3/2} \sqrt {a x+b x^3}} \\ & = \frac {1}{a \sqrt {a x+b x^3}}+\frac {3 \sqrt {b} x \left (a+b x^2\right )}{a^2 \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {a x+b x^3}}-\frac {3 \sqrt {a x+b x^3}}{a^2 x}-\frac {3 \sqrt [4]{b} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{a^{7/4} \sqrt {a x+b x^3}}+\frac {3 \sqrt [4]{b} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{7/4} \sqrt {a x+b x^3}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.19 \[ \int \frac {1}{\left (a x+b x^3\right )^{3/2}} \, dx=-\frac {2 \sqrt {1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {3}{2},\frac {3}{4},-\frac {b x^2}{a}\right )}{a \sqrt {x \left (a+b x^2\right )}} \]
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Time = 2.79 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.75
method | result | size |
default | \(-\frac {2 \left (b \,x^{2}+a \right )}{a^{2} \sqrt {x \left (b \,x^{2}+a \right )}}-\frac {b \,x^{2}}{a^{2} \sqrt {\left (x^{2}+\frac {a}{b}\right ) b x}}+\frac {3 \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{2 a^{2} \sqrt {b \,x^{3}+a x}}\) | \(206\) |
elliptic | \(-\frac {2 \left (b \,x^{2}+a \right )}{a^{2} \sqrt {x \left (b \,x^{2}+a \right )}}-\frac {b \,x^{2}}{a^{2} \sqrt {\left (x^{2}+\frac {a}{b}\right ) b x}}+\frac {3 \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{2 a^{2} \sqrt {b \,x^{3}+a x}}\) | \(206\) |
risch | \(-\frac {2 \left (b \,x^{2}+a \right )}{a^{2} \sqrt {x \left (b \,x^{2}+a \right )}}+\frac {b^{2} \left (\frac {\sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{b^{2} \sqrt {b \,x^{3}+a x}}-\frac {a \left (\frac {x^{2}}{a \sqrt {\left (x^{2}+\frac {a}{b}\right ) b x}}-\frac {\sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{2 a b \sqrt {b \,x^{3}+a x}}\right )}{b}\right )}{a^{2}}\) | \(379\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.15 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.26 \[ \int \frac {1}{\left (a x+b x^3\right )^{3/2}} \, dx=-\frac {3 \, {\left (b x^{3} + a x\right )} \sqrt {b} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) + \sqrt {b x^{3} + a x} {\left (3 \, b x^{2} + 2 \, a\right )}}{a^{2} b x^{3} + a^{3} x} \]
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\[ \int \frac {1}{\left (a x+b x^3\right )^{3/2}} \, dx=\int \frac {1}{\left (a x + b x^{3}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {1}{\left (a x+b x^3\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{3} + a x\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {1}{\left (a x+b x^3\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{3} + a x\right )}^{\frac {3}{2}}} \,d x } \]
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Time = 9.14 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.15 \[ \int \frac {1}{\left (a x+b x^3\right )^{3/2}} \, dx=-\frac {2\,x\,{\left (\frac {b\,x^2}{a}+1\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {3}{2};\ \frac {3}{4};\ -\frac {b\,x^2}{a}\right )}{{\left (b\,x^3+a\,x\right )}^{3/2}} \]
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