Integrand size = 13, antiderivative size = 255 \[ \int \sqrt {a x+b x^3} \, dx=\frac {4 a x \left (a+b x^2\right )}{5 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {a x+b x^3}}+\frac {2}{5} x \sqrt {a x+b x^3}-\frac {4 a^{5/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 b^{3/4} \sqrt {a x+b x^3}}+\frac {2 a^{5/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{5 b^{3/4} \sqrt {a x+b x^3}} \]
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Time = 0.12 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2029, 2057, 335, 311, 226, 1210} \[ \int \sqrt {a x+b x^3} \, dx=\frac {2 a^{5/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{5 b^{3/4} \sqrt {a x+b x^3}}-\frac {4 a^{5/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 b^{3/4} \sqrt {a x+b x^3}}+\frac {2}{5} x \sqrt {a x+b x^3}+\frac {4 a x \left (a+b x^2\right )}{5 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {a x+b x^3}} \]
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Rule 226
Rule 311
Rule 335
Rule 1210
Rule 2029
Rule 2057
Rubi steps \begin{align*} \text {integral}& = \frac {2}{5} x \sqrt {a x+b x^3}+\frac {1}{5} (2 a) \int \frac {x}{\sqrt {a x+b x^3}} \, dx \\ & = \frac {2}{5} x \sqrt {a x+b x^3}+\frac {\left (2 a \sqrt {x} \sqrt {a+b x^2}\right ) \int \frac {\sqrt {x}}{\sqrt {a+b x^2}} \, dx}{5 \sqrt {a x+b x^3}} \\ & = \frac {2}{5} x \sqrt {a x+b x^3}+\frac {\left (4 a \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 )}{5 \sqrt {a x+b x^3}} \\ & = \frac {2}{5} x \sqrt {a x+b x^3}+\frac {\left (4 a^{3/2} \sqrt {x} \sqrt {a+b x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^4}} \, dx,x,\sqrt {x}\right )}{5 \sqrt {b} \sqrt {a x+b x^3}}-\frac {\left (4 a^{3/2} \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 )}{5 \sqrt {b} \sqrt {a x+b x^3}} \\ & = \frac {4 a x \left (a+b x^2\right )}{5 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {a x+b x^3}}+\frac {2}{5} x \sqrt {a x+b x^3}-\frac {4 a^{5/4} \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 )}{5 b^{3/4} \sqrt {a x+b x^3}}+\frac {2 a^{5/4} \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 )}{5 b^{3/4} \sqrt {a x+b x^3}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.20 \[ \int \sqrt {a x+b x^3} \, dx=\frac {2 x \sqrt {x \left (a+b x^2\right )} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3}{4},\frac {7}{4},-\frac {b x^2}{a}\right )}{3 \sqrt {1+\frac {b x^2}{a}}} \]
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Time = 2.22 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.69
method | result | size |
default | \(\frac {2 x \sqrt {b \,x^{3}+a x}}{5}+\frac {2 a \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{5 b \sqrt {b \,x^{3}+a x}}\) | \(175\) |
elliptic | \(\frac {2 x \sqrt {b \,x^{3}+a x}}{5}+\frac {2 a \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{5 b \sqrt {b \,x^{3}+a x}}\) | \(175\) |
risch | \(\frac {2 x^{2} \left (b \,x^{2}+a \right )}{5 \sqrt {x \left (b \,x^{2}+a \right )}}+\frac {2 a \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{5 b \sqrt {b \,x^{3}+a x}}\) | \(184\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.17 \[ \int \sqrt {a x+b x^3} \, dx=\frac {2 \, {\left (\sqrt {b x^{3} + a x} b x - 2 \, a \sqrt {b} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right )\right )}}{5 \, b} \]
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\[ \int \sqrt {a x+b x^3} \, dx=\int \sqrt {a x + b x^{3}}\, dx \]
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\[ \int \sqrt {a x+b x^3} \, dx=\int { \sqrt {b x^{3} + a x} \,d x } \]
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\[ \int \sqrt {a x+b x^3} \, dx=\int { \sqrt {b x^{3} + a x} \,d x } \]
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Time = 11.14 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.16 \[ \int \sqrt {a x+b x^3} \, dx=\frac {2\,x\,\sqrt {b\,x^3+a\,x}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{2},\frac {3}{4};\ \frac {7}{4};\ -\frac {b\,x^2}{a}\right )}{3\,\sqrt {\frac {b\,x^2}{a}+1}} \]
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