Optimal. Leaf size=300 \[ \frac{\left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (\sqrt{a} f+\sqrt{b} d\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{2 a^{3/4} \sqrt [4]{b} \sqrt{a+b x^4}}-\frac{\sqrt [4]{b} d \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{a^{3/4} \sqrt{a+b x^4}}-\frac{c \sqrt{a+b x^4}}{2 a x^2}-\frac{d \sqrt{a+b x^4}}{a x}+\frac{\sqrt{b} d x \sqrt{a+b x^4}}{a \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{e \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{2 \sqrt{a}} \]
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Rubi [A] time = 0.215868, antiderivative size = 300, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {1833, 1252, 807, 266, 63, 208, 1282, 1198, 220, 1196} \[ \frac{\left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (\sqrt{a} f+\sqrt{b} d\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 a^{3/4} \sqrt [4]{b} \sqrt{a+b x^4}}-\frac{\sqrt [4]{b} d \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{a^{3/4} \sqrt{a+b x^4}}-\frac{c \sqrt{a+b x^4}}{2 a x^2}-\frac{d \sqrt{a+b x^4}}{a x}+\frac{\sqrt{b} d x \sqrt{a+b x^4}}{a \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{e \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{2 \sqrt{a}} \]
Antiderivative was successfully verified.
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Rule 1833
Rule 1252
Rule 807
Rule 266
Rule 63
Rule 208
Rule 1282
Rule 1198
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{c+d x+e x^2+f x^3}{x^3 \sqrt{a+b x^4}} \, dx &=\int \left (\frac{c+e x^2}{x^3 \sqrt{a+b x^4}}+\frac{d+f x^2}{x^2 \sqrt{a+b x^4}}\right ) \, dx\\ &=\int \frac{c+e x^2}{x^3 \sqrt{a+b x^4}} \, dx+\int \frac{d+f x^2}{x^2 \sqrt{a+b x^4}} \, dx\\ &=-\frac{d \sqrt{a+b x^4}}{a x}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{c+e x}{x^2 \sqrt{a+b x^2}} \, dx,x,x^2\right )-\frac{\int \frac{-a f-b d x^2}{\sqrt{a+b x^4}} \, dx}{a}\\ &=-\frac{c \sqrt{a+b x^4}}{2 a x^2}-\frac{d \sqrt{a+b x^4}}{a x}-\frac{\left (\sqrt{b} d\right ) \int \frac{1-\frac{\sqrt{b} x^2}{\sqrt{a}}}{\sqrt{a+b x^4}} \, dx}{\sqrt{a}}+\frac{1}{2} e \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x^2}} \, dx,x,x^2\right )+\left (\frac{\sqrt{b} d}{\sqrt{a}}+f\right ) \int \frac{1}{\sqrt{a+b x^4}} \, dx\\ &=-\frac{c \sqrt{a+b x^4}}{2 a x^2}-\frac{d \sqrt{a+b x^4}}{a x}+\frac{\sqrt{b} d x \sqrt{a+b x^4}}{a \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{\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 )}{a^{3/4} \sqrt{a+b x^4}}+\frac{\left (\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 )}{2 a^{3/4} \sqrt [4]{b} \sqrt{a+b x^4}}+\frac{1}{4} e \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^4\right )\\ &=-\frac{c \sqrt{a+b x^4}}{2 a x^2}-\frac{d \sqrt{a+b x^4}}{a x}+\frac{\sqrt{b} d x \sqrt{a+b x^4}}{a \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{\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 )}{a^{3/4} \sqrt{a+b x^4}}+\frac{\left (\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 )}{2 a^{3/4} \sqrt [4]{b} \sqrt{a+b x^4}}+\frac{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{c \sqrt{a+b x^4}}{2 a x^2}-\frac{d \sqrt{a+b x^4}}{a x}+\frac{\sqrt{b} d x \sqrt{a+b x^4}}{a \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{e \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{2 \sqrt{a}}-\frac{\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 )}{a^{3/4} \sqrt{a+b x^4}}+\frac{\left (\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 )}{2 a^{3/4} \sqrt [4]{b} \sqrt{a+b x^4}}\\ \end{align*}
Mathematica [C] time = 0.139661, size = 148, normalized size = 0.49 \[ -\frac{c \sqrt{a+b x^4}}{2 a x^2}-\frac{d \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{b x^4}{a}\right )}{x \sqrt{a+b x^4}}-\frac{e \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{2 \sqrt{a}}+\frac{f x \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^4}{a}\right )}{\sqrt{a+b x^4}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.015, size = 293, normalized size = 1. \begin{align*}{f\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}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{4}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}-{\frac{c}{2\,a{x}^{2}}\sqrt{b{x}^{4}+a}}-{\frac{d}{ax}\sqrt{b{x}^{4}+a}}+{id\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{a}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{id\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{a}}}{\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{f x^{3} + e x^{2} + d x + c}{\sqrt{b x^{4} + a} 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 )}}{b x^{7} + a x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 3.09551, size = 126, normalized size = 0.42 \begin{align*} - \frac{\sqrt{b} c \sqrt{\frac{a}{b x^{4}} + 1}}{2 a} + \frac{d \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{2} \\ \frac{3}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt{a} x \Gamma \left (\frac{3}{4}\right )} - \frac{e \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2 \sqrt{a}} + \frac{f x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{5}{4}\right )} \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{f x^{3} + e x^{2} + d x + c}{\sqrt{b x^{4} + a} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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