Optimal. Leaf size=341 \[ \frac{\sqrt [4]{a} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (\sqrt{a} e+3 \sqrt{b} c\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{3 \sqrt [4]{b} \sqrt{a+b x^4}}-\frac{\sqrt{a+b x^4} \left (3 c-e x^2\right )}{3 x}+\frac{2 \sqrt{b} c x \sqrt{a+b x^4}}{\sqrt{a}+\sqrt{b} x^2}-\frac{2 \sqrt [4]{a} \sqrt [4]{b} c \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{\sqrt{a+b x^4}}+\frac{1}{4} \sqrt{a+b x^4} \left (2 d+f x^2\right )-\frac{1}{2} \sqrt{a} d \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )+\frac{a f \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{4 \sqrt{b}} \]
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Rubi [A] time = 0.263187, antiderivative size = 341, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 13, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.433, Rules used = {1833, 1272, 1198, 220, 1196, 1252, 815, 844, 217, 206, 266, 63, 208} \[ -\frac{\sqrt{a+b x^4} \left (3 c-e x^2\right )}{3 x}+\frac{\sqrt [4]{a} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (\sqrt{a} e+3 \sqrt{b} c\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{3 \sqrt [4]{b} \sqrt{a+b x^4}}+\frac{2 \sqrt{b} c x \sqrt{a+b x^4}}{\sqrt{a}+\sqrt{b} x^2}-\frac{2 \sqrt [4]{a} \sqrt [4]{b} c \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{\sqrt{a+b x^4}}+\frac{1}{4} \sqrt{a+b x^4} \left (2 d+f x^2\right )-\frac{1}{2} \sqrt{a} d \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )+\frac{a f \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{4 \sqrt{b}} \]
Antiderivative was successfully verified.
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Rule 1833
Rule 1272
Rule 1198
Rule 220
Rule 1196
Rule 1252
Rule 815
Rule 844
Rule 217
Rule 206
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (c+d x+e x^2+f x^3\right ) \sqrt{a+b x^4}}{x^2} \, dx &=\int \left (\frac{\left (c+e x^2\right ) \sqrt{a+b x^4}}{x^2}+\frac{\left (d+f x^2\right ) \sqrt{a+b x^4}}{x}\right ) \, dx\\ &=\int \frac{\left (c+e x^2\right ) \sqrt{a+b x^4}}{x^2} \, dx+\int \frac{\left (d+f x^2\right ) \sqrt{a+b x^4}}{x} \, dx\\ &=-\frac{\left (3 c-e x^2\right ) \sqrt{a+b x^4}}{3 x}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{(d+f x) \sqrt{a+b x^2}}{x} \, dx,x,x^2\right )-\frac{2}{3} \int \frac{-a e-3 b c x^2}{\sqrt{a+b x^4}} \, dx\\ &=-\frac{\left (3 c-e x^2\right ) \sqrt{a+b x^4}}{3 x}+\frac{1}{4} \left (2 d+f x^2\right ) \sqrt{a+b x^4}+\frac{\operatorname{Subst}\left (\int \frac{2 a b d+a b f x}{x \sqrt{a+b x^2}} \, dx,x,x^2\right )}{4 b}-\left (2 \sqrt{a} \sqrt{b} c\right ) \int \frac{1-\frac{\sqrt{b} x^2}{\sqrt{a}}}{\sqrt{a+b x^4}} \, dx+\frac{1}{3} \left (2 \sqrt{a} \left (3 \sqrt{b} c+\sqrt{a} e\right )\right ) \int \frac{1}{\sqrt{a+b x^4}} \, dx\\ &=\frac{2 \sqrt{b} c x \sqrt{a+b x^4}}{\sqrt{a}+\sqrt{b} x^2}-\frac{\left (3 c-e x^2\right ) \sqrt{a+b x^4}}{3 x}+\frac{1}{4} \left (2 d+f x^2\right ) \sqrt{a+b x^4}-\frac{2 \sqrt [4]{a} \sqrt [4]{b} 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 )}{\sqrt{a+b x^4}}+\frac{\sqrt [4]{a} \left (3 \sqrt{b} c+\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 )}{3 \sqrt [4]{b} \sqrt{a+b x^4}}+\frac{1}{2} (a d) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x^2}} \, dx,x,x^2\right )+\frac{1}{4} (a f) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,x^2\right )\\ &=\frac{2 \sqrt{b} c x \sqrt{a+b x^4}}{\sqrt{a}+\sqrt{b} x^2}-\frac{\left (3 c-e x^2\right ) \sqrt{a+b x^4}}{3 x}+\frac{1}{4} \left (2 d+f x^2\right ) \sqrt{a+b x^4}-\frac{2 \sqrt [4]{a} \sqrt [4]{b} 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 )}{\sqrt{a+b x^4}}+\frac{\sqrt [4]{a} \left (3 \sqrt{b} c+\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 )}{3 \sqrt [4]{b} \sqrt{a+b x^4}}+\frac{1}{4} (a d) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^4\right )+\frac{1}{4} (a f) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x^2}{\sqrt{a+b x^4}}\right )\\ &=\frac{2 \sqrt{b} c x \sqrt{a+b x^4}}{\sqrt{a}+\sqrt{b} x^2}-\frac{\left (3 c-e x^2\right ) \sqrt{a+b x^4}}{3 x}+\frac{1}{4} \left (2 d+f x^2\right ) \sqrt{a+b x^4}+\frac{a f \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{4 \sqrt{b}}-\frac{2 \sqrt [4]{a} \sqrt [4]{b} 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 )}{\sqrt{a+b x^4}}+\frac{\sqrt [4]{a} \left (3 \sqrt{b} c+\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 )}{3 \sqrt [4]{b} \sqrt{a+b x^4}}+\frac{(a d) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^4}\right )}{2 b}\\ &=\frac{2 \sqrt{b} c x \sqrt{a+b x^4}}{\sqrt{a}+\sqrt{b} x^2}-\frac{\left (3 c-e x^2\right ) \sqrt{a+b x^4}}{3 x}+\frac{1}{4} \left (2 d+f x^2\right ) \sqrt{a+b x^4}+\frac{a f \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{4 \sqrt{b}}-\frac{1}{2} \sqrt{a} d \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )-\frac{2 \sqrt [4]{a} \sqrt [4]{b} 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 )}{\sqrt{a+b x^4}}+\frac{\sqrt [4]{a} \left (3 \sqrt{b} c+\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 )}{3 \sqrt [4]{b} \sqrt{a+b x^4}}\\ \end{align*}
Mathematica [C] time = 0.391284, size = 208, normalized size = 0.61 \[ \frac{x \left (\sqrt{b} \sqrt{\frac{b x^4}{a}+1} \left (\sqrt{a+b x^4} \left (2 d+f x^2\right )-2 \sqrt{a} d \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )\right )+4 \sqrt{b} e x \sqrt{a+b x^4} \, _2F_1\left (-\frac{1}{2},\frac{1}{4};\frac{5}{4};-\frac{b x^4}{a}\right )+\sqrt{a} f \sqrt{a+b x^4} \sinh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right )-4 \sqrt{b} c \sqrt{a+b x^4} \, _2F_1\left (-\frac{1}{2},-\frac{1}{4};\frac{3}{4};-\frac{b x^4}{a}\right )}{4 \sqrt{b} x \sqrt{\frac{b x^4}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.012, size = 339, normalized size = 1. \begin{align*}{\frac{f{x}^{2}}{4}\sqrt{b{x}^{4}+a}}+{\frac{af}{4}\ln \left ({x}^{2}\sqrt{b}+\sqrt{b{x}^{4}+a} \right ){\frac{1}{\sqrt{b}}}}+{\frac{ex}{3}\sqrt{b{x}^{4}+a}}+{\frac{2\,ae}{3}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}+{\frac{d}{2}\sqrt{b{x}^{4}+a}}-{\frac{d}{2}\sqrt{a}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{4}+a} \right ) } \right ) }-{\frac{c}{x}\sqrt{b{x}^{4}+a}}+{2\,ic\sqrt{a}\sqrt{b}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{2\,ic\sqrt{a}\sqrt{b}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticE} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{b x^{4} + a}{\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b x^{4} + a}{\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 5.70388, size = 206, normalized size = 0.6 \begin{align*} \frac{\sqrt{a} c \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4} \\ \frac{3}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 x \Gamma \left (\frac{3}{4}\right )} - \frac{\sqrt{a} d \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2} + \frac{\sqrt{a} e x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} + \frac{\sqrt{a} f x^{2} \sqrt{1 + \frac{b x^{4}}{a}}}{4} + \frac{a d}{2 \sqrt{b} x^{2} \sqrt{\frac{a}{b x^{4}} + 1}} + \frac{a f \operatorname{asinh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{4 \sqrt{b}} + \frac{\sqrt{b} d x^{2}}{2 \sqrt{\frac{a}{b x^{4}} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{b x^{4} + a}{\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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