Integrand size = 15, antiderivative size = 282 \[ \int \frac {x^{14}}{\left (a+b x^4\right )^{3/2}} \, dx=-\frac {x^{11}}{2 b \sqrt {a+b x^4}}-\frac {77 a x^3 \sqrt {a+b x^4}}{90 b^3}+\frac {11 x^7 \sqrt {a+b x^4}}{18 b^2}+\frac {77 a^2 x \sqrt {a+b x^4}}{30 b^{7/2} \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {77 a^{9/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{30 b^{15/4} \sqrt {a+b x^4}}+\frac {77 a^{9/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{60 b^{15/4} \sqrt {a+b x^4}} \]
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Time = 0.10 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {294, 327, 311, 226, 1210} \[ \int \frac {x^{14}}{\left (a+b x^4\right )^{3/2}} \, dx=\frac {77 a^{9/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{60 b^{15/4} \sqrt {a+b x^4}}-\frac {77 a^{9/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{30 b^{15/4} \sqrt {a+b x^4}}+\frac {77 a^2 x \sqrt {a+b x^4}}{30 b^{7/2} \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {77 a x^3 \sqrt {a+b x^4}}{90 b^3}+\frac {11 x^7 \sqrt {a+b x^4}}{18 b^2}-\frac {x^{11}}{2 b \sqrt {a+b x^4}} \]
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Rule 226
Rule 294
Rule 311
Rule 327
Rule 1210
Rubi steps \begin{align*} \text {integral}& = -\frac {x^{11}}{2 b \sqrt {a+b x^4}}+\frac {11 \int \frac {x^{10}}{\sqrt {a+b x^4}} \, dx}{2 b} \\ & = -\frac {x^{11}}{2 b \sqrt {a+b x^4}}+\frac {11 x^7 \sqrt {a+b x^4}}{18 b^2}-\frac {(77 a) \int \frac {x^6}{\sqrt {a+b x^4}} \, dx}{18 b^2} \\ & = -\frac {x^{11}}{2 b \sqrt {a+b x^4}}-\frac {77 a x^3 \sqrt {a+b x^4}}{90 b^3}+\frac {11 x^7 \sqrt {a+b x^4}}{18 b^2}+\frac {\left (77 a^2\right ) \int \frac {x^2}{\sqrt {a+b x^4}} \, dx}{30 b^3} \\ & = -\frac {x^{11}}{2 b \sqrt {a+b x^4}}-\frac {77 a x^3 \sqrt {a+b x^4}}{90 b^3}+\frac {11 x^7 \sqrt {a+b x^4}}{18 b^2}+\frac {\left (77 a^{5/2}\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx}{30 b^{7/2}}-\frac {\left (77 a^{5/2}\right ) \int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a+b x^4}} \, dx}{30 b^{7/2}} \\ & = -\frac {x^{11}}{2 b \sqrt {a+b x^4}}-\frac {77 a x^3 \sqrt {a+b x^4}}{90 b^3}+\frac {11 x^7 \sqrt {a+b x^4}}{18 b^2}+\frac {77 a^2 x \sqrt {a+b x^4}}{30 b^{7/2} \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {77 a^{9/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{30 b^{15/4} \sqrt {a+b x^4}}+\frac {77 a^{9/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{60 b^{15/4} \sqrt {a+b x^4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.05 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.28 \[ \int \frac {x^{14}}{\left (a+b x^4\right )^{3/2}} \, dx=\frac {x^3 \left (77 a^2-11 a b x^4+5 b^2 x^8-77 a^2 \sqrt {1+\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},-\frac {b x^4}{a}\right )\right )}{45 b^3 \sqrt {a+b x^4}} \]
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Result contains complex when optimal does not.
Time = 5.92 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.56
method | result | size |
default | \(-\frac {x^{3} a^{2}}{2 b^{3} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {x^{7} \sqrt {b \,x^{4}+a}}{9 b^{2}}-\frac {16 a \,x^{3} \sqrt {b \,x^{4}+a}}{45 b^{3}}+\frac {77 i a^{\frac {5}{2}} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{30 b^{\frac {7}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\) | \(157\) |
elliptic | \(-\frac {x^{3} a^{2}}{2 b^{3} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {x^{7} \sqrt {b \,x^{4}+a}}{9 b^{2}}-\frac {16 a \,x^{3} \sqrt {b \,x^{4}+a}}{45 b^{3}}+\frac {77 i a^{\frac {5}{2}} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{30 b^{\frac {7}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\) | \(157\) |
risch | \(-\frac {x^{3} \left (-5 b \,x^{4}+16 a \right ) \sqrt {b \,x^{4}+a}}{45 b^{3}}+\frac {a^{2} \left (31 b \left (-\frac {x^{3}}{2 b \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {3 i \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{2 b^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+16 a \left (\frac {x^{3}}{2 a \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}-\frac {i \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}\right )\right )}{15 b^{3}}\) | \(280\) |
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Time = 0.09 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.51 \[ \int \frac {x^{14}}{\left (a+b x^4\right )^{3/2}} \, dx=\frac {231 \, {\left (a^{2} b x^{5} + a^{3} x\right )} \sqrt {b} \left (-\frac {a}{b}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (-\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - 231 \, {\left (a^{2} b x^{5} + a^{3} x\right )} \sqrt {b} \left (-\frac {a}{b}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + {\left (10 \, b^{3} x^{12} - 22 \, a b^{2} x^{8} + 154 \, a^{2} b x^{4} + 231 \, a^{3}\right )} \sqrt {b x^{4} + a}}{90 \, {\left (b^{5} x^{5} + a b^{4} x\right )}} \]
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Result contains complex when optimal does not.
Time = 0.78 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.13 \[ \int \frac {x^{14}}{\left (a+b x^4\right )^{3/2}} \, dx=\frac {x^{15} \Gamma \left (\frac {15}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {15}{4} \\ \frac {19}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {19}{4}\right )} \]
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\[ \int \frac {x^{14}}{\left (a+b x^4\right )^{3/2}} \, dx=\int { \frac {x^{14}}{{\left (b x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x^{14}}{\left (a+b x^4\right )^{3/2}} \, dx=\int { \frac {x^{14}}{{\left (b x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^{14}}{\left (a+b x^4\right )^{3/2}} \, dx=\int \frac {x^{14}}{{\left (b\,x^4+a\right )}^{3/2}} \,d x \]
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