Integrand size = 15, antiderivative size = 294 \[ \int \frac {x^7}{\sqrt {a+\frac {b}{x^3}}} \, dx=\frac {91 b^2 \sqrt {a+\frac {b}{x^3}} x^2}{320 a^3}-\frac {13 b \sqrt {a+\frac {b}{x^3}} x^5}{80 a^2}+\frac {\sqrt {a+\frac {b}{x^3}} x^8}{8 a}+\frac {91 \sqrt {2+\sqrt {3}} b^{8/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}+\frac {b^{2/3}}{x^2}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right ),-7-4 \sqrt {3}\right )}{320 \sqrt [4]{3} a^3 \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}} \]
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Time = 0.12 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {342, 331, 224} \[ \int \frac {x^7}{\sqrt {a+\frac {b}{x^3}}} \, dx=\frac {91 b^2 x^2 \sqrt {a+\frac {b}{x^3}}}{320 a^3}-\frac {13 b x^5 \sqrt {a+\frac {b}{x^3}}}{80 a^2}+\frac {91 \sqrt {2+\sqrt {3}} b^{8/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac {b^{2/3}}{x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right ),-7-4 \sqrt {3}\right )}{320 \sqrt [4]{3} a^3 \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}+\frac {x^8 \sqrt {a+\frac {b}{x^3}}}{8 a} \]
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Rule 224
Rule 331
Rule 342
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {1}{x^9 \sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {\sqrt {a+\frac {b}{x^3}} x^8}{8 a}+\frac {(13 b) \text {Subst}\left (\int \frac {1}{x^6 \sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right )}{16 a} \\ & = -\frac {13 b \sqrt {a+\frac {b}{x^3}} x^5}{80 a^2}+\frac {\sqrt {a+\frac {b}{x^3}} x^8}{8 a}-\frac {\left (91 b^2\right ) \text {Subst}\left (\int \frac {1}{x^3 \sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right )}{160 a^2} \\ & = \frac {91 b^2 \sqrt {a+\frac {b}{x^3}} x^2}{320 a^3}-\frac {13 b \sqrt {a+\frac {b}{x^3}} x^5}{80 a^2}+\frac {\sqrt {a+\frac {b}{x^3}} x^8}{8 a}+\frac {\left (91 b^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right )}{640 a^3} \\ & = \frac {91 b^2 \sqrt {a+\frac {b}{x^3}} x^2}{320 a^3}-\frac {13 b \sqrt {a+\frac {b}{x^3}} x^5}{80 a^2}+\frac {\sqrt {a+\frac {b}{x^3}} x^8}{8 a}+\frac {91 \sqrt {2+\sqrt {3}} b^{8/3} \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 )}{320 \sqrt [4]{3} a^3 \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 = 10.05 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.31 \[ \int \frac {x^7}{\sqrt {a+\frac {b}{x^3}}} \, dx=\frac {91 b^3+39 a b^2 x^3-12 a^2 b x^6+40 a^3 x^9-91 b^3 \sqrt {1+\frac {a x^3}{b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},-\frac {a x^3}{b}\right )}{320 a^3 \sqrt {a+\frac {b}{x^3}} x} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 752 vs. \(2 (227 ) = 454\).
Time = 3.56 (sec) , antiderivative size = 753, normalized size of antiderivative = 2.56
method | result | size |
risch | \(\frac {\left (40 a^{2} x^{6}-52 a b \,x^{3}+91 b^{2}\right ) \left (a \,x^{3}+b \right )}{320 a^{3} x \sqrt {\frac {a \,x^{3}+b}{x^{3}}}}-\frac {91 b^{3} \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 ) \sqrt {x \left (a \,x^{3}+b \right )}}{320 a^{2} \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 )}\, x^{2} \sqrt {\frac {a \,x^{3}+b}{x^{3}}}}\) | \(753\) |
default | \(\text {Expression too large to display}\) | \(2233\) |
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\[ \int \frac {x^7}{\sqrt {a+\frac {b}{x^3}}} \, dx=\int { \frac {x^{7}}{\sqrt {a + \frac {b}{x^{3}}}} \,d x } \]
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Time = 0.68 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.16 \[ \int \frac {x^7}{\sqrt {a+\frac {b}{x^3}}} \, dx=- \frac {x^{8} \Gamma \left (- \frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {8}{3}, \frac {1}{2} \\ - \frac {5}{3} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{3}}} \right )}}{3 \sqrt {a} \Gamma \left (- \frac {5}{3}\right )} \]
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\[ \int \frac {x^7}{\sqrt {a+\frac {b}{x^3}}} \, dx=\int { \frac {x^{7}}{\sqrt {a + \frac {b}{x^{3}}}} \,d x } \]
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\[ \int \frac {x^7}{\sqrt {a+\frac {b}{x^3}}} \, dx=\int { \frac {x^{7}}{\sqrt {a + \frac {b}{x^{3}}}} \,d x } \]
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Timed out. \[ \int \frac {x^7}{\sqrt {a+\frac {b}{x^3}}} \, dx=\int \frac {x^7}{\sqrt {a+\frac {b}{x^3}}} \,d x \]
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