Integrand size = 21, antiderivative size = 203 \[ \int \frac {\sqrt {x}}{\sqrt {a x^2+b x^5}} \, dx=\frac {x^{3/2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt [4]{3} \sqrt [3]{a} \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a x^2+b x^5}} \]
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Time = 0.12 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2057, 335, 231} \[ \int \frac {\sqrt {x}}{\sqrt {a x^2+b x^5}} \, dx=\frac {x^{3/2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt [4]{3} \sqrt [3]{a} \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a x^2+b x^5}} \]
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Rule 231
Rule 335
Rule 2057
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x \sqrt {a+b x^3}\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x^3}} \, dx}{\sqrt {a x^2+b x^5}} \\ & = \frac {\left (2 x \sqrt {a+b x^3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {a x^2+b x^5}} \\ & = \frac {x^{3/2} \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 (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} F\left (\cos ^{-1}\left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt [4]{3} \sqrt [3]{a} \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \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 = 0.27 \[ \int \frac {\sqrt {x}}{\sqrt {a x^2+b x^5}} \, dx=\frac {2 x^{3/2} \sqrt {1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},-\frac {b x^3}{a}\right )}{\sqrt {x^2 \left (a+b x^3\right )}} \]
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Result contains complex when optimal does not.
Time = 1.87 (sec) , antiderivative size = 437, normalized size of antiderivative = 2.15
method | result | size |
default | \(-\frac {4 x^{\frac {3}{2}} \left (b \,x^{3}+a \right ) \sqrt {-\frac {\left (i \sqrt {3}-3\right ) x b}{\left (i \sqrt {3}-1\right ) \left (-b x +\left (-a \,b^{2}\right )^{\frac {1}{3}}\right )}}\, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}+2 b x +\left (-a \,b^{2}\right )^{\frac {1}{3}}}{\left (1+i \sqrt {3}\right ) \left (-b x +\left (-a \,b^{2}\right )^{\frac {1}{3}}\right )}}\, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}-2 b x -\left (-a \,b^{2}\right )^{\frac {1}{3}}}{\left (i \sqrt {3}-1\right ) \left (-b x +\left (-a \,b^{2}\right )^{\frac {1}{3}}\right )}}\, F\left (\sqrt {-\frac {\left (i \sqrt {3}-3\right ) x b}{\left (i \sqrt {3}-1\right ) \left (-b x +\left (-a \,b^{2}\right )^{\frac {1}{3}}\right )}}, \sqrt {\frac {\left (i \sqrt {3}+3\right ) \left (i \sqrt {3}-1\right )}{\left (1+i \sqrt {3}\right ) \left (i \sqrt {3}-3\right )}}\right ) \left (i \sqrt {3}\, b^{2} x^{2}-2 i \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {3}\, b x +i \left (-a \,b^{2}\right )^{\frac {2}{3}} \sqrt {3}-b^{2} x^{2}+2 \left (-a \,b^{2}\right )^{\frac {1}{3}} b x -\left (-a \,b^{2}\right )^{\frac {2}{3}}\right )}{\sqrt {b \,x^{5}+a \,x^{2}}\, b \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {x \left (b \,x^{3}+a \right )}\, \left (i \sqrt {3}-3\right ) \sqrt {\frac {x \left (-b x +\left (-a \,b^{2}\right )^{\frac {1}{3}}\right ) \left (i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}+2 b x +\left (-a \,b^{2}\right )^{\frac {1}{3}}\right ) \left (i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}-2 b x -\left (-a \,b^{2}\right )^{\frac {1}{3}}\right )}{b^{2}}}}\) | \(437\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.08 \[ \int \frac {\sqrt {x}}{\sqrt {a x^2+b x^5}} \, dx=-\frac {2 \, {\rm weierstrassPInverse}\left (0, -\frac {4 \, b}{a}, \frac {1}{x}\right )}{\sqrt {a}} \]
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\[ \int \frac {\sqrt {x}}{\sqrt {a x^2+b x^5}} \, dx=\int \frac {\sqrt {x}}{\sqrt {x^{2} \left (a + b x^{3}\right )}}\, dx \]
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\[ \int \frac {\sqrt {x}}{\sqrt {a x^2+b x^5}} \, dx=\int { \frac {\sqrt {x}}{\sqrt {b x^{5} + a x^{2}}} \,d x } \]
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\[ \int \frac {\sqrt {x}}{\sqrt {a x^2+b x^5}} \, dx=\int { \frac {\sqrt {x}}{\sqrt {b x^{5} + a x^{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {x}}{\sqrt {a x^2+b x^5}} \, dx=\int \frac {\sqrt {x}}{\sqrt {b\,x^5+a\,x^2}} \,d x \]
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