Integrand size = 11, antiderivative size = 122 \[ \int \left (a+c x^4\right )^{3/2} \, dx=\frac {2}{7} a x \sqrt {a+c x^4}+\frac {1}{7} x \left (a+c x^4\right )^{3/2}+\frac {2 a^{7/4} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{7 \sqrt [4]{c} \sqrt {a+c x^4}} \]
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Time = 0.03 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {201, 226} \[ \int \left (a+c x^4\right )^{3/2} \, dx=\frac {2 a^{7/4} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{7 \sqrt [4]{c} \sqrt {a+c x^4}}+\frac {2}{7} a x \sqrt {a+c x^4}+\frac {1}{7} x \left (a+c x^4\right )^{3/2} \]
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Rule 201
Rule 226
Rubi steps \begin{align*} \text {integral}& = \frac {1}{7} x \left (a+c x^4\right )^{3/2}+\frac {1}{7} (6 a) \int \sqrt {a+c x^4} \, dx \\ & = \frac {2}{7} a x \sqrt {a+c x^4}+\frac {1}{7} x \left (a+c x^4\right )^{3/2}+\frac {1}{7} \left (4 a^2\right ) \int \frac {1}{\sqrt {a+c x^4}} \, dx \\ & = \frac {2}{7} a x \sqrt {a+c x^4}+\frac {1}{7} x \left (a+c x^4\right )^{3/2}+\frac {2 a^{7/4} \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 )}{7 \sqrt [4]{c} \sqrt {a+c x^4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 4.38 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.39 \[ \int \left (a+c x^4\right )^{3/2} \, dx=\frac {a x \sqrt {a+c x^4} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4},\frac {5}{4},-\frac {c x^4}{a}\right )}{\sqrt {1+\frac {c x^4}{a}}} \]
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Result contains complex when optimal does not.
Time = 4.16 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.79
method | result | size |
risch | \(\frac {x \left (x^{4} c +3 a \right ) \sqrt {x^{4} c +a}}{7}+\frac {4 a^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{7 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {x^{4} c +a}}\) | \(96\) |
default | \(\frac {x^{5} c \sqrt {x^{4} c +a}}{7}+\frac {3 a x \sqrt {x^{4} c +a}}{7}+\frac {4 a^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{7 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {x^{4} c +a}}\) | \(103\) |
elliptic | \(\frac {x^{5} c \sqrt {x^{4} c +a}}{7}+\frac {3 a x \sqrt {x^{4} c +a}}{7}+\frac {4 a^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{7 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {x^{4} c +a}}\) | \(103\) |
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none
Time = 0.08 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.42 \[ \int \left (a+c x^4\right )^{3/2} \, dx=\frac {4}{7} \, a \sqrt {c} \left (-\frac {a}{c}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + \frac {1}{7} \, {\left (c x^{5} + 3 \, a x\right )} \sqrt {c x^{4} + a} \]
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Result contains complex when optimal does not.
Time = 0.49 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.30 \[ \int \left (a+c x^4\right )^{3/2} \, dx=\frac {a^{\frac {3}{2}} x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} \]
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\[ \int \left (a+c x^4\right )^{3/2} \, dx=\int { {\left (c x^{4} + a\right )}^{\frac {3}{2}} \,d x } \]
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\[ \int \left (a+c x^4\right )^{3/2} \, dx=\int { {\left (c x^{4} + a\right )}^{\frac {3}{2}} \,d x } \]
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Time = 5.42 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.30 \[ \int \left (a+c x^4\right )^{3/2} \, dx=\frac {x\,{\left (c\,x^4+a\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},\frac {1}{4};\ \frac {5}{4};\ -\frac {c\,x^4}{a}\right )}{{\left (\frac {c\,x^4}{a}+1\right )}^{3/2}} \]
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