Optimal. Leaf size=342 \[ \frac{\sqrt [4]{a} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (\sqrt{a} f+3 \sqrt{b} d\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{3 \sqrt [4]{b} \sqrt{a+b x^4}}-\frac{\sqrt{a+b x^4} \left (c-e x^2\right )}{2 x^2}+\frac{1}{2} \sqrt{b} c \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )-\frac{\sqrt{a+b x^4} \left (3 d-f x^2\right )}{3 x}+\frac{2 \sqrt{b} d x \sqrt{a+b x^4}}{\sqrt{a}+\sqrt{b} x^2}-\frac{2 \sqrt [4]{a} \sqrt [4]{b} d \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{\sqrt{a+b x^4}}-\frac{1}{2} \sqrt{a} e \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right ) \]
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Rubi [A] time = 0.262424, antiderivative size = 342, 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, 1252, 813, 844, 217, 206, 266, 63, 208, 1272, 1198, 220, 1196} \[ -\frac{\sqrt{a+b x^4} \left (c-e x^2\right )}{2 x^2}+\frac{1}{2} \sqrt{b} c \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )-\frac{\sqrt{a+b x^4} \left (3 d-f x^2\right )}{3 x}+\frac{\sqrt [4]{a} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (\sqrt{a} f+3 \sqrt{b} d\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{3 \sqrt [4]{b} \sqrt{a+b x^4}}+\frac{2 \sqrt{b} d x \sqrt{a+b x^4}}{\sqrt{a}+\sqrt{b} x^2}-\frac{2 \sqrt [4]{a} \sqrt [4]{b} d \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{\sqrt{a+b x^4}}-\frac{1}{2} \sqrt{a} e \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right ) \]
Antiderivative was successfully verified.
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Rule 1833
Rule 1252
Rule 813
Rule 844
Rule 217
Rule 206
Rule 266
Rule 63
Rule 208
Rule 1272
Rule 1198
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{\left (c+d x+e x^2+f x^3\right ) \sqrt{a+b x^4}}{x^3} \, dx &=\int \left (\frac{\left (c+e x^2\right ) \sqrt{a+b x^4}}{x^3}+\frac{\left (d+f x^2\right ) \sqrt{a+b x^4}}{x^2}\right ) \, dx\\ &=\int \frac{\left (c+e x^2\right ) \sqrt{a+b x^4}}{x^3} \, dx+\int \frac{\left (d+f x^2\right ) \sqrt{a+b x^4}}{x^2} \, dx\\ &=-\frac{\left (3 d-f x^2\right ) \sqrt{a+b x^4}}{3 x}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{(c+e x) \sqrt{a+b x^2}}{x^2} \, dx,x,x^2\right )-\frac{2}{3} \int \frac{-a f-3 b d x^2}{\sqrt{a+b x^4}} \, dx\\ &=-\frac{\left (c-e x^2\right ) \sqrt{a+b x^4}}{2 x^2}-\frac{\left (3 d-f x^2\right ) \sqrt{a+b x^4}}{3 x}-\frac{1}{4} \operatorname{Subst}\left (\int \frac{-2 a e-2 b c x}{x \sqrt{a+b x^2}} \, dx,x,x^2\right )-\left (2 \sqrt{a} \sqrt{b} d\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} d+\sqrt{a} f\right )\right ) \int \frac{1}{\sqrt{a+b x^4}} \, dx\\ &=\frac{2 \sqrt{b} d x \sqrt{a+b x^4}}{\sqrt{a}+\sqrt{b} x^2}-\frac{\left (c-e x^2\right ) \sqrt{a+b x^4}}{2 x^2}-\frac{\left (3 d-f x^2\right ) \sqrt{a+b x^4}}{3 x}-\frac{2 \sqrt [4]{a} \sqrt [4]{b} d \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} d+\sqrt{a} f\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} (b c) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,x^2\right )+\frac{1}{2} (a e) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x^2}} \, dx,x,x^2\right )\\ &=\frac{2 \sqrt{b} d x \sqrt{a+b x^4}}{\sqrt{a}+\sqrt{b} x^2}-\frac{\left (c-e x^2\right ) \sqrt{a+b x^4}}{2 x^2}-\frac{\left (3 d-f x^2\right ) \sqrt{a+b x^4}}{3 x}-\frac{2 \sqrt [4]{a} \sqrt [4]{b} d \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} d+\sqrt{a} f\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} (b c) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x^2}{\sqrt{a+b x^4}}\right )+\frac{1}{4} (a e) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^4\right )\\ &=\frac{2 \sqrt{b} d x \sqrt{a+b x^4}}{\sqrt{a}+\sqrt{b} x^2}-\frac{\left (c-e x^2\right ) \sqrt{a+b x^4}}{2 x^2}-\frac{\left (3 d-f x^2\right ) \sqrt{a+b x^4}}{3 x}+\frac{1}{2} \sqrt{b} c \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )-\frac{2 \sqrt [4]{a} \sqrt [4]{b} d \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} d+\sqrt{a} f\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 e) \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} d x \sqrt{a+b x^4}}{\sqrt{a}+\sqrt{b} x^2}-\frac{\left (c-e x^2\right ) \sqrt{a+b x^4}}{2 x^2}-\frac{\left (3 d-f x^2\right ) \sqrt{a+b x^4}}{3 x}+\frac{1}{2} \sqrt{b} c \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )-\frac{1}{2} \sqrt{a} e \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )-\frac{2 \sqrt [4]{a} \sqrt [4]{b} d \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} d+\sqrt{a} f\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 = 0.233381, size = 204, normalized size = 0.6 \[ \frac{\sqrt{a} \sqrt{b} c x^2 \sqrt{\frac{b x^4}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )-2 a d x \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left (-\frac{1}{2},-\frac{1}{4};\frac{3}{4};-\frac{b x^4}{a}\right )-\sqrt{a} e x^2 \sqrt{a+b x^4} \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )+2 a f x^3 \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left (-\frac{1}{2},\frac{1}{4};\frac{5}{4};-\frac{b x^4}{a}\right )-a c+a e x^2-b c x^4+b e x^6}{2 x^2 \sqrt{a+b x^4}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.016, size = 360, normalized size = 1.1 \begin{align*}{\frac{fx}{3}\sqrt{b{x}^{4}+a}}+{\frac{2\,af}{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{e}{2}\sqrt{b{x}^{4}+a}}-{\frac{e}{2}\sqrt{a}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{4}+a} \right ) } \right ) }-{\frac{c}{2\,a{x}^{2}} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{2}}}}+{\frac{bc{x}^{2}}{2\,a}\sqrt{b{x}^{4}+a}}+{\frac{c}{2}\sqrt{b}\ln \left ({x}^{2}\sqrt{b}+\sqrt{b{x}^{4}+a} \right ) }-{\frac{d}{x}\sqrt{b{x}^{4}+a}}+{2\,id\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\,id\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^{3}}\,{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^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 5.27143, size = 230, normalized size = 0.67 \begin{align*} - \frac{\sqrt{a} c}{2 x^{2} \sqrt{1 + \frac{b x^{4}}{a}}} + \frac{\sqrt{a} d \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} e \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2} + \frac{\sqrt{a} f 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{a e}{2 \sqrt{b} x^{2} \sqrt{\frac{a}{b x^{4}} + 1}} + \frac{\sqrt{b} c \operatorname{asinh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2} + \frac{\sqrt{b} e x^{2}}{2 \sqrt{\frac{a}{b x^{4}} + 1}} - \frac{b c x^{2}}{2 \sqrt{a} \sqrt{1 + \frac{b x^{4}}{a}}} \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^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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