Integrand size = 30, antiderivative size = 377 \[ \int \frac {c+d x+e x^2+f x^3}{x^6 \sqrt {a+b x^4}} \, dx=-\frac {c \sqrt {a+b x^4}}{5 a x^5}-\frac {d \sqrt {a+b x^4}}{4 a x^4}-\frac {e \sqrt {a+b x^4}}{3 a x^3}-\frac {f \sqrt {a+b x^4}}{2 a x^2}+\frac {3 b c \sqrt {a+b x^4}}{5 a^2 x}-\frac {3 b^{3/2} c x \sqrt {a+b x^4}}{5 a^2 \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {b d \text {arctanh}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{4 a^{3/2}}+\frac {3 b^{5/4} c \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 a^{7/4} \sqrt {a+b x^4}}-\frac {b^{3/4} \left (9 \sqrt {b} c+5 \sqrt {a} e\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{30 a^{7/4} \sqrt {a+b x^4}} \]
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Time = 0.22 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {1847, 1296, 1212, 226, 1210, 1266, 849, 821, 272, 65, 214} \[ \int \frac {c+d x+e x^2+f x^3}{x^6 \sqrt {a+b x^4}} \, dx=-\frac {b^{3/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (5 \sqrt {a} e+9 \sqrt {b} c\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{30 a^{7/4} \sqrt {a+b x^4}}+\frac {3 b^{5/4} c \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 a^{7/4} \sqrt {a+b x^4}}+\frac {b d \text {arctanh}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{4 a^{3/2}}-\frac {3 b^{3/2} c x \sqrt {a+b x^4}}{5 a^2 \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {3 b c \sqrt {a+b x^4}}{5 a^2 x}-\frac {c \sqrt {a+b x^4}}{5 a x^5}-\frac {d \sqrt {a+b x^4}}{4 a x^4}-\frac {e \sqrt {a+b x^4}}{3 a x^3}-\frac {f \sqrt {a+b x^4}}{2 a x^2} \]
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Rule 65
Rule 214
Rule 226
Rule 272
Rule 821
Rule 849
Rule 1210
Rule 1212
Rule 1266
Rule 1296
Rule 1847
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {c+e x^2}{x^6 \sqrt {a+b x^4}}+\frac {d+f x^2}{x^5 \sqrt {a+b x^4}}\right ) \, dx \\ & = \int \frac {c+e x^2}{x^6 \sqrt {a+b x^4}} \, dx+\int \frac {d+f x^2}{x^5 \sqrt {a+b x^4}} \, dx \\ & = -\frac {c \sqrt {a+b x^4}}{5 a x^5}+\frac {1}{2} \text {Subst}\left (\int \frac {d+f x}{x^3 \sqrt {a+b x^2}} \, dx,x,x^2\right )-\frac {\int \frac {-5 a e+3 b c x^2}{x^4 \sqrt {a+b x^4}} \, dx}{5 a} \\ & = -\frac {c \sqrt {a+b x^4}}{5 a x^5}-\frac {d \sqrt {a+b x^4}}{4 a x^4}-\frac {e \sqrt {a+b x^4}}{3 a x^3}+\frac {\int \frac {-9 a b c-5 a b e x^2}{x^2 \sqrt {a+b x^4}} \, dx}{15 a^2}-\frac {\text {Subst}\left (\int \frac {-2 a f+b d x}{x^2 \sqrt {a+b x^2}} \, dx,x,x^2\right )}{4 a} \\ & = -\frac {c \sqrt {a+b x^4}}{5 a x^5}-\frac {d \sqrt {a+b x^4}}{4 a x^4}-\frac {e \sqrt {a+b x^4}}{3 a x^3}-\frac {f \sqrt {a+b x^4}}{2 a x^2}+\frac {3 b c \sqrt {a+b x^4}}{5 a^2 x}-\frac {\int \frac {5 a^2 b e+9 a b^2 c x^2}{\sqrt {a+b x^4}} \, dx}{15 a^3}-\frac {(b d) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x^2}} \, dx,x,x^2\right )}{4 a} \\ & = -\frac {c \sqrt {a+b x^4}}{5 a x^5}-\frac {d \sqrt {a+b x^4}}{4 a x^4}-\frac {e \sqrt {a+b x^4}}{3 a x^3}-\frac {f \sqrt {a+b x^4}}{2 a x^2}+\frac {3 b c \sqrt {a+b x^4}}{5 a^2 x}+\frac {\left (3 b^{3/2} c\right ) \int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a+b x^4}} \, dx}{5 a^{3/2}}-\frac {(b d) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^4\right )}{8 a}-\frac {\left (b \left (9 \sqrt {b} c+5 \sqrt {a} e\right )\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx}{15 a^{3/2}} \\ & = -\frac {c \sqrt {a+b x^4}}{5 a x^5}-\frac {d \sqrt {a+b x^4}}{4 a x^4}-\frac {e \sqrt {a+b x^4}}{3 a x^3}-\frac {f \sqrt {a+b x^4}}{2 a x^2}+\frac {3 b c \sqrt {a+b x^4}}{5 a^2 x}-\frac {3 b^{3/2} c x \sqrt {a+b x^4}}{5 a^2 \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {3 b^{5/4} c \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 )}{5 a^{7/4} \sqrt {a+b x^4}}-\frac {b^{3/4} \left (9 \sqrt {b} c+5 \sqrt {a} e\right ) \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 )}{30 a^{7/4} \sqrt {a+b x^4}}-\frac {d \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^4}\right )}{4 a} \\ & = -\frac {c \sqrt {a+b x^4}}{5 a x^5}-\frac {d \sqrt {a+b x^4}}{4 a x^4}-\frac {e \sqrt {a+b x^4}}{3 a x^3}-\frac {f \sqrt {a+b x^4}}{2 a x^2}+\frac {3 b c \sqrt {a+b x^4}}{5 a^2 x}-\frac {3 b^{3/2} c x \sqrt {a+b x^4}}{5 a^2 \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {b d \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{4 a^{3/2}}+\frac {3 b^{5/4} c \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 )}{5 a^{7/4} \sqrt {a+b x^4}}-\frac {b^{3/4} \left (9 \sqrt {b} c+5 \sqrt {a} e\right ) \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 )}{30 a^{7/4} \sqrt {a+b x^4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.21 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.36 \[ \int \frac {c+d x+e x^2+f x^3}{x^6 \sqrt {a+b x^4}} \, dx=-\frac {\sqrt {a+b x^4} \left (12 a c \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},\frac {1}{2},-\frac {1}{4},-\frac {b x^4}{a}\right )+5 x \left (3 a \left (d+2 f x^2\right ) \sqrt {1+\frac {b x^4}{a}}-3 b d x^4 \text {arctanh}\left (\sqrt {1+\frac {b x^4}{a}}\right )+4 a e x \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {1}{2},\frac {1}{4},-\frac {b x^4}{a}\right )\right )\right )}{60 a^2 x^5 \sqrt {1+\frac {b x^4}{a}}} \]
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Result contains complex when optimal does not.
Time = 2.01 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.68
method | result | size |
risch | \(-\frac {\sqrt {b \,x^{4}+a}\, \left (-36 b c \,x^{4}+30 a f \,x^{3}+20 a e \,x^{2}+15 a d x +12 a c \right )}{60 a^{2} x^{5}}-\frac {b \left (\frac {10 a e \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {18 i \sqrt {b}\, c \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {15 \sqrt {a}\, d \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{2}\right )}{30 a^{2}}\) | \(255\) |
elliptic | \(-\frac {c \sqrt {b \,x^{4}+a}}{5 a \,x^{5}}-\frac {d \sqrt {b \,x^{4}+a}}{4 a \,x^{4}}-\frac {e \sqrt {b \,x^{4}+a}}{3 a \,x^{3}}-\frac {f \sqrt {b \,x^{4}+a}}{2 a \,x^{2}}+\frac {3 b c \sqrt {b \,x^{4}+a}}{5 a^{2} x}-\frac {b e \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{3 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {3 i b^{\frac {3}{2}} c \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{5 a^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {b d \,\operatorname {arctanh}\left (\frac {\sqrt {a}}{\sqrt {b \,x^{4}+a}}\right )}{4 a^{\frac {3}{2}}}\) | \(286\) |
default | \(e \left (-\frac {\sqrt {b \,x^{4}+a}}{3 a \,x^{3}}-\frac {b \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{3 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )-\frac {f \sqrt {b \,x^{4}+a}}{2 a \,x^{2}}+d \left (-\frac {\sqrt {b \,x^{4}+a}}{4 a \,x^{4}}+\frac {b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{4 a^{\frac {3}{2}}}\right )+c \left (-\frac {\sqrt {b \,x^{4}+a}}{5 a \,x^{5}}+\frac {3 b \sqrt {b \,x^{4}+a}}{5 a^{2} x}-\frac {3 i b^{\frac {3}{2}} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{5 a^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )\) | \(297\) |
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Time = 0.12 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.42 \[ \int \frac {c+d x+e x^2+f x^3}{x^6 \sqrt {a+b x^4}} \, dx=\frac {72 \, \sqrt {a} b c x^{5} \left (-\frac {b}{a}\right )^{\frac {3}{4}} E(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + 15 \, \sqrt {a} b d x^{5} \log \left (-\frac {b x^{4} + 2 \, \sqrt {b x^{4} + a} \sqrt {a} + 2 \, a}{x^{4}}\right ) - 8 \, {\left (9 \, b c - 5 \, a e\right )} \sqrt {a} x^{5} \left (-\frac {b}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + 2 \, {\left (36 \, b c x^{4} - 30 \, a f x^{3} - 20 \, a e x^{2} - 15 \, a d x - 12 \, a c\right )} \sqrt {b x^{4} + a}}{120 \, a^{2} x^{5}} \]
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Result contains complex when optimal does not.
Time = 2.64 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.43 \[ \int \frac {c+d x+e x^2+f x^3}{x^6 \sqrt {a+b x^4}} \, dx=- \frac {\sqrt {b} d \sqrt {\frac {a}{b x^{4}} + 1}}{4 a x^{2}} - \frac {\sqrt {b} f \sqrt {\frac {a}{b x^{4}} + 1}}{2 a} + \frac {c \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, \frac {1}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt {a} x^{5} \Gamma \left (- \frac {1}{4}\right )} + \frac {e \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt {a} x^{3} \Gamma \left (\frac {1}{4}\right )} + \frac {b d \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{4 a^{\frac {3}{2}}} \]
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\[ \int \frac {c+d x+e x^2+f x^3}{x^6 \sqrt {a+b x^4}} \, dx=\int { \frac {f x^{3} + e x^{2} + d x + c}{\sqrt {b x^{4} + a} x^{6}} \,d x } \]
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\[ \int \frac {c+d x+e x^2+f x^3}{x^6 \sqrt {a+b x^4}} \, dx=\int { \frac {f x^{3} + e x^{2} + d x + c}{\sqrt {b x^{4} + a} x^{6}} \,d x } \]
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Timed out. \[ \int \frac {c+d x+e x^2+f x^3}{x^6 \sqrt {a+b x^4}} \, dx=\int \frac {f\,x^3+e\,x^2+d\,x+c}{x^6\,\sqrt {b\,x^4+a}} \,d x \]
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