Integrand size = 15, antiderivative size = 252 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x^6} \, dx=-\frac {6 c \sqrt {a+c x^4}}{5 x}+\frac {12 c^{3/2} x \sqrt {a+c x^4}}{5 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {\left (a+c x^4\right )^{3/2}}{5 x^5}-\frac {12 \sqrt [4]{a} c^{5/4} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 \sqrt {a+c x^4}}+\frac {6 \sqrt [4]{a} c^{5/4} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{5 \sqrt {a+c x^4}} \]
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Time = 0.07 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {283, 311, 226, 1210} \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x^6} \, dx=\frac {6 \sqrt [4]{a} c^{5/4} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{5 \sqrt {a+c x^4}}-\frac {12 \sqrt [4]{a} c^{5/4} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 \sqrt {a+c x^4}}+\frac {12 c^{3/2} x \sqrt {a+c x^4}}{5 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {6 c \sqrt {a+c x^4}}{5 x}-\frac {\left (a+c x^4\right )^{3/2}}{5 x^5} \]
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Rule 226
Rule 283
Rule 311
Rule 1210
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a+c x^4\right )^{3/2}}{5 x^5}+\frac {1}{5} (6 c) \int \frac {\sqrt {a+c x^4}}{x^2} \, dx \\ & = -\frac {6 c \sqrt {a+c x^4}}{5 x}-\frac {\left (a+c x^4\right )^{3/2}}{5 x^5}+\frac {1}{5} \left (12 c^2\right ) \int \frac {x^2}{\sqrt {a+c x^4}} \, dx \\ & = -\frac {6 c \sqrt {a+c x^4}}{5 x}-\frac {\left (a+c x^4\right )^{3/2}}{5 x^5}+\frac {1}{5} \left (12 \sqrt {a} c^{3/2}\right ) \int \frac {1}{\sqrt {a+c x^4}} \, dx-\frac {1}{5} \left (12 \sqrt {a} c^{3/2}\right ) \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+c x^4}} \, dx \\ & = -\frac {6 c \sqrt {a+c x^4}}{5 x}+\frac {12 c^{3/2} x \sqrt {a+c x^4}}{5 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {\left (a+c x^4\right )^{3/2}}{5 x^5}-\frac {12 \sqrt [4]{a} c^{5/4} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 \sqrt {a+c x^4}}+\frac {6 \sqrt [4]{a} c^{5/4} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 \sqrt {a+c x^4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.01 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.21 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x^6} \, dx=-\frac {a \sqrt {a+c x^4} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {5}{4},-\frac {1}{4},-\frac {c x^4}{a}\right )}{5 x^5 \sqrt {1+\frac {c x^4}{a}}} \]
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Result contains complex when optimal does not.
Time = 4.51 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.48
method | result | size |
risch | \(-\frac {\sqrt {x^{4} c +a}\, \left (7 x^{4} c +a \right )}{5 x^{5}}+\frac {12 i c^{\frac {3}{2}} \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {x^{4} c +a}}\) | \(120\) |
default | \(-\frac {a \sqrt {x^{4} c +a}}{5 x^{5}}-\frac {7 c \sqrt {x^{4} c +a}}{5 x}+\frac {12 i c^{\frac {3}{2}} \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {x^{4} c +a}}\) | \(128\) |
elliptic | \(-\frac {a \sqrt {x^{4} c +a}}{5 x^{5}}-\frac {7 c \sqrt {x^{4} c +a}}{5 x}+\frac {12 i c^{\frac {3}{2}} \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {x^{4} c +a}}\) | \(128\) |
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\[ \int \frac {\left (a+c x^4\right )^{3/2}}{x^6} \, dx=\int { \frac {{\left (c x^{4} + a\right )}^{\frac {3}{2}}}{x^{6}} \,d x } \]
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Result contains complex when optimal does not.
Time = 0.58 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.18 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x^6} \, dx=\frac {a^{\frac {3}{2}} \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, - \frac {5}{4} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 x^{5} \Gamma \left (- \frac {1}{4}\right )} \]
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\[ \int \frac {\left (a+c x^4\right )^{3/2}}{x^6} \, dx=\int { \frac {{\left (c x^{4} + a\right )}^{\frac {3}{2}}}{x^{6}} \,d x } \]
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\[ \int \frac {\left (a+c x^4\right )^{3/2}}{x^6} \, dx=\int { \frac {{\left (c x^{4} + a\right )}^{\frac {3}{2}}}{x^{6}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x^6} \, dx=\int \frac {{\left (c\,x^4+a\right )}^{3/2}}{x^6} \,d x \]
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