Integrand size = 15, antiderivative size = 114 \[ \int \frac {x}{\left (a x+b x^3\right )^{3/2}} \, dx=\frac {x}{a \sqrt {a x+b x^3}}+\frac {\sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 a^{5/4} \sqrt [4]{b} \sqrt {a x+b x^3}} \]
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Time = 0.05 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {2048, 2036, 335, 226} \[ \int \frac {x}{\left (a x+b x^3\right )^{3/2}} \, dx=\frac {\sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 a^{5/4} \sqrt [4]{b} \sqrt {a x+b x^3}}+\frac {x}{a \sqrt {a x+b x^3}} \]
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Rule 226
Rule 335
Rule 2036
Rule 2048
Rubi steps \begin{align*} \text {integral}& = \frac {x}{a \sqrt {a x+b x^3}}+\frac {\int \frac {1}{\sqrt {a x+b x^3}} \, dx}{2 a} \\ & = \frac {x}{a \sqrt {a x+b x^3}}+\frac {\left (\sqrt {x} \sqrt {a+b x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x^2}} \, dx}{2 a \sqrt {a x+b x^3}} \\ & = \frac {x}{a \sqrt {a x+b x^3}}+\frac {\left (\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 \sqrt {a x+b x^3}} \\ & = \frac {x}{a \sqrt {a x+b x^3}}+\frac {\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^{5/4} \sqrt [4]{b} \sqrt {a x+b x^3}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.03 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.47 \[ \int \frac {x}{\left (a x+b x^3\right )^{3/2}} \, dx=\frac {x+x \sqrt {1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {b x^2}{a}\right )}{a \sqrt {x \left (a+b x^2\right )}} \]
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Time = 2.06 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.16
method | result | size |
default | \(\frac {x}{a \sqrt {\left (x^{2}+\frac {a}{b}\right ) b x}}+\frac {\sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{2 a b \sqrt {b \,x^{3}+a x}}\) | \(132\) |
elliptic | \(\frac {x}{a \sqrt {\left (x^{2}+\frac {a}{b}\right ) b x}}+\frac {\sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{2 a b \sqrt {b \,x^{3}+a x}}\) | \(132\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.45 \[ \int \frac {x}{\left (a x+b x^3\right )^{3/2}} \, dx=\frac {{\left (b x^{2} + a\right )} \sqrt {b} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) + \sqrt {b x^{3} + a x} b}{a b^{2} x^{2} + a^{2} b} \]
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\[ \int \frac {x}{\left (a x+b x^3\right )^{3/2}} \, dx=\int \frac {x}{\left (x \left (a + b x^{2}\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {x}{\left (a x+b x^3\right )^{3/2}} \, dx=\int { \frac {x}{{\left (b x^{3} + a x\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x}{\left (a x+b x^3\right )^{3/2}} \, dx=\int { \frac {x}{{\left (b x^{3} + a x\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x}{\left (a x+b x^3\right )^{3/2}} \, dx=\int \frac {x}{{\left (b\,x^3+a\,x\right )}^{3/2}} \,d x \]
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