Integrand size = 15, antiderivative size = 315 \[ \int \frac {x^7}{\left (a+\frac {b}{x^3}\right )^{3/2}} \, dx=\frac {1729 b^2 \sqrt {a+\frac {b}{x^3}} x^2}{960 a^4}-\frac {247 b \sqrt {a+\frac {b}{x^3}} x^5}{240 a^3}-\frac {2 x^8}{3 a \sqrt {a+\frac {b}{x^3}}}+\frac {19 \sqrt {a+\frac {b}{x^3}} x^8}{24 a^2}+\frac {1729 \sqrt {2+\sqrt {3}} b^{8/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}+\frac {b^{2/3}}{x^2}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right ),-7-4 \sqrt {3}\right )}{960 \sqrt [4]{3} a^4 \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}} \]
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Time = 0.14 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {342, 296, 331, 224} \[ \int \frac {x^7}{\left (a+\frac {b}{x^3}\right )^{3/2}} \, dx=\frac {1729 b^2 x^2 \sqrt {a+\frac {b}{x^3}}}{960 a^4}-\frac {247 b x^5 \sqrt {a+\frac {b}{x^3}}}{240 a^3}+\frac {19 x^8 \sqrt {a+\frac {b}{x^3}}}{24 a^2}+\frac {1729 \sqrt {2+\sqrt {3}} b^{8/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac {b^{2/3}}{x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right ),-7-4 \sqrt {3}\right )}{960 \sqrt [4]{3} a^4 \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}-\frac {2 x^8}{3 a \sqrt {a+\frac {b}{x^3}}} \]
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Rule 224
Rule 296
Rule 331
Rule 342
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {1}{x^9 \left (a+b x^3\right )^{3/2}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {2 x^8}{3 a \sqrt {a+\frac {b}{x^3}}}-\frac {19 \text {Subst}\left (\int \frac {1}{x^9 \sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right )}{3 a} \\ & = -\frac {2 x^8}{3 a \sqrt {a+\frac {b}{x^3}}}+\frac {19 \sqrt {a+\frac {b}{x^3}} x^8}{24 a^2}+\frac {(247 b) \text {Subst}\left (\int \frac {1}{x^6 \sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right )}{48 a^2} \\ & = -\frac {247 b \sqrt {a+\frac {b}{x^3}} x^5}{240 a^3}-\frac {2 x^8}{3 a \sqrt {a+\frac {b}{x^3}}}+\frac {19 \sqrt {a+\frac {b}{x^3}} x^8}{24 a^2}-\frac {\left (1729 b^2\right ) \text {Subst}\left (\int \frac {1}{x^3 \sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right )}{480 a^3} \\ & = \frac {1729 b^2 \sqrt {a+\frac {b}{x^3}} x^2}{960 a^4}-\frac {247 b \sqrt {a+\frac {b}{x^3}} x^5}{240 a^3}-\frac {2 x^8}{3 a \sqrt {a+\frac {b}{x^3}}}+\frac {19 \sqrt {a+\frac {b}{x^3}} x^8}{24 a^2}+\frac {\left (1729 b^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right )}{1920 a^4} \\ & = \frac {1729 b^2 \sqrt {a+\frac {b}{x^3}} x^2}{960 a^4}-\frac {247 b \sqrt {a+\frac {b}{x^3}} x^5}{240 a^3}-\frac {2 x^8}{3 a \sqrt {a+\frac {b}{x^3}}}+\frac {19 \sqrt {a+\frac {b}{x^3}} x^8}{24 a^2}+\frac {1729 \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 )}{960 \sqrt [4]{3} a^4 \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 = 10.04 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.29 \[ \int \frac {x^7}{\left (a+\frac {b}{x^3}\right )^{3/2}} \, dx=\frac {1729 b^3+741 a b^2 x^3-228 a^2 b x^6+120 a^3 x^9-1729 b^3 \sqrt {1+\frac {a x^3}{b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},-\frac {a x^3}{b}\right )}{960 a^4 \sqrt {a+\frac {b}{x^3}} x} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1453 vs. \(2 (244 ) = 488\).
Time = 2.50 (sec) , antiderivative size = 1454, normalized size of antiderivative = 4.62
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1454\) |
default | \(\text {Expression too large to display}\) | \(2540\) |
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\[ \int \frac {x^7}{\left (a+\frac {b}{x^3}\right )^{3/2}} \, dx=\int { \frac {x^{7}}{{\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{2}}} \,d x } \]
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Time = 0.71 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.15 \[ \int \frac {x^7}{\left (a+\frac {b}{x^3}\right )^{3/2}} \, dx=- \frac {x^{8} \Gamma \left (- \frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {8}{3}, \frac {3}{2} \\ - \frac {5}{3} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{3}}} \right )}}{3 a^{\frac {3}{2}} \Gamma \left (- \frac {5}{3}\right )} \]
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\[ \int \frac {x^7}{\left (a+\frac {b}{x^3}\right )^{3/2}} \, dx=\int { \frac {x^{7}}{{\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x^7}{\left (a+\frac {b}{x^3}\right )^{3/2}} \, dx=\int { \frac {x^{7}}{{\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^7}{\left (a+\frac {b}{x^3}\right )^{3/2}} \, dx=\int \frac {x^7}{{\left (a+\frac {b}{x^3}\right )}^{3/2}} \,d x \]
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