Integrand size = 15, antiderivative size = 243 \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^2} \, dx=-\frac {2 \sqrt {a+\frac {b}{x^3}}}{5 x}-\frac {2\ 3^{3/4} \sqrt {2+\sqrt {3}} a \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}+\frac {b^{2/3}}{x^2}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right ),-7-4 \sqrt {3}\right )}{5 \sqrt [3]{b} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}} \]
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Time = 0.07 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {342, 201, 224} \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^2} \, dx=-\frac {2\ 3^{3/4} \sqrt {2+\sqrt {3}} a \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac {b^{2/3}}{x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right ),-7-4 \sqrt {3}\right )}{5 \sqrt [3]{b} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}-\frac {2 \sqrt {a+\frac {b}{x^3}}}{5 x} \]
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Rule 201
Rule 224
Rule 342
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \sqrt {a+b x^3} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {2 \sqrt {a+\frac {b}{x^3}}}{5 x}-\frac {1}{5} (3 a) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {2 \sqrt {a+\frac {b}{x^3}}}{5 x}-\frac {2\ 3^{3/4} \sqrt {2+\sqrt {3}} a \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}+\frac {b^{2/3}}{x^2}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt {3}\right )}{5 \sqrt [3]{b} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}} \\ \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 = 0.21 \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^2} \, dx=-\frac {2 \sqrt {a+\frac {b}{x^3}} \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},-\frac {1}{2},\frac {1}{6},-\frac {a x^3}{b}\right )}{5 x \sqrt {1+\frac {a x^3}{b}}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 725 vs. \(2 (184 ) = 368\).
Time = 0.39 (sec) , antiderivative size = 726, normalized size of antiderivative = 2.99
method | result | size |
risch | \(-\frac {2 \sqrt {\frac {a \,x^{3}+b}{x^{3}}}}{5 x}+\frac {6 a^{2} \left (\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \sqrt {\frac {\left (-\frac {3 \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) x}{\left (-\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \left (x -\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{a}\right )}}\, {\left (x -\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{a}\right )}^{2} \sqrt {\frac {\left (-a^{2} b \right )^{\frac {1}{3}} \left (x +\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right )}{a \left (-\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \left (x -\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{a}\right )}}\, \sqrt {\frac {\left (-a^{2} b \right )^{\frac {1}{3}} \left (x +\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right )}{a \left (-\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \left (x -\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{a}\right )}}\, F\left (\sqrt {\frac {\left (-\frac {3 \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) x}{\left (-\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \left (x -\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{a}\right )}}, \sqrt {\frac {\left (\frac {3 \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \left (\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right )}{\left (\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \left (\frac {3 \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right )}}\right ) x \sqrt {\frac {a \,x^{3}+b}{x^{3}}}\, \sqrt {x \left (a \,x^{3}+b \right )}}{5 \left (-\frac {3 \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \left (-a^{2} b \right )^{\frac {1}{3}} \sqrt {a x \left (x -\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{a}\right ) \left (x +\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \left (x +\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right )}\, \left (a \,x^{3}+b \right )}\) | \(726\) |
default | \(\text {Expression too large to display}\) | \(1785\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.17 \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^2} \, dx=-\frac {2 \, {\left (3 \, a \sqrt {b} x {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, \frac {1}{x}\right ) + b \sqrt {\frac {a x^{3} + b}{x^{3}}}\right )}}{5 \, b x} \]
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Time = 0.54 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.16 \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^2} \, dx=- \frac {\sqrt {a} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{3}}} \right )}}{3 x \Gamma \left (\frac {4}{3}\right )} \]
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\[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^2} \, dx=\int { \frac {\sqrt {a + \frac {b}{x^{3}}}}{x^{2}} \,d x } \]
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\[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^2} \, dx=\int { \frac {\sqrt {a + \frac {b}{x^{3}}}}{x^{2}} \,d x } \]
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Time = 6.04 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.16 \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^2} \, dx=-\frac {\sqrt {a+\frac {b}{x^3}}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{2},\frac {1}{3};\ \frac {4}{3};\ -\frac {b}{a\,x^3}\right )}{x\,\sqrt {\frac {b}{a\,x^3}+1}} \]
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