Optimal. Leaf size=346 \[ -\frac{\sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (\sqrt{b} d-3 \sqrt{a} f\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{6 a^{5/4} \sqrt{a+b x^4}}+\frac{b c \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{4 a^{3/2}}-\frac{\sqrt [4]{b} f \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{a^{3/4} \sqrt{a+b x^4}}-\frac{c \sqrt{a+b x^4}}{4 a x^4}-\frac{d \sqrt{a+b x^4}}{3 a x^3}-\frac{e \sqrt{a+b x^4}}{2 a x^2}-\frac{f \sqrt{a+b x^4}}{a x}+\frac{\sqrt{b} f x \sqrt{a+b x^4}}{a \left (\sqrt{a}+\sqrt{b} x^2\right )} \]
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Rubi [A] time = 0.2813, antiderivative size = 346, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 11, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.367, Rules used = {1833, 1252, 835, 807, 266, 63, 208, 1282, 1198, 220, 1196} \[ \frac{b c \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{4 a^{3/2}}-\frac{\sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (\sqrt{b} d-3 \sqrt{a} f\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{6 a^{5/4} \sqrt{a+b x^4}}-\frac{\sqrt [4]{b} f \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{a^{3/4} \sqrt{a+b x^4}}-\frac{c \sqrt{a+b x^4}}{4 a x^4}-\frac{d \sqrt{a+b x^4}}{3 a x^3}-\frac{e \sqrt{a+b x^4}}{2 a x^2}-\frac{f \sqrt{a+b x^4}}{a x}+\frac{\sqrt{b} f x \sqrt{a+b x^4}}{a \left (\sqrt{a}+\sqrt{b} x^2\right )} \]
Antiderivative was successfully verified.
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Rule 1833
Rule 1252
Rule 835
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^5 \sqrt{a+b x^4}} \, dx &=\int \left (\frac{c+e x^2}{x^5 \sqrt{a+b x^4}}+\frac{d+f x^2}{x^4 \sqrt{a+b x^4}}\right ) \, dx\\ &=\int \frac{c+e x^2}{x^5 \sqrt{a+b x^4}} \, dx+\int \frac{d+f x^2}{x^4 \sqrt{a+b x^4}} \, dx\\ &=-\frac{d \sqrt{a+b x^4}}{3 a x^3}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{c+e x}{x^3 \sqrt{a+b x^2}} \, dx,x,x^2\right )-\frac{\int \frac{-3 a f+b d x^2}{x^2 \sqrt{a+b x^4}} \, dx}{3 a}\\ &=-\frac{c \sqrt{a+b x^4}}{4 a x^4}-\frac{d \sqrt{a+b x^4}}{3 a x^3}-\frac{f \sqrt{a+b x^4}}{a x}+\frac{\int \frac{-a b d+3 a b f x^2}{\sqrt{a+b x^4}} \, dx}{3 a^2}-\frac{\operatorname{Subst}\left (\int \frac{-2 a e+b c x}{x^2 \sqrt{a+b x^2}} \, dx,x,x^2\right )}{4 a}\\ &=-\frac{c \sqrt{a+b x^4}}{4 a x^4}-\frac{d \sqrt{a+b x^4}}{3 a x^3}-\frac{e \sqrt{a+b x^4}}{2 a x^2}-\frac{f \sqrt{a+b x^4}}{a x}-\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x^2}} \, dx,x,x^2\right )}{4 a}-\frac{\left (\sqrt{b} f\right ) \int \frac{1-\frac{\sqrt{b} x^2}{\sqrt{a}}}{\sqrt{a+b x^4}} \, dx}{\sqrt{a}}-\frac{\left (\sqrt{b} \left (\sqrt{b} d-3 \sqrt{a} f\right )\right ) \int \frac{1}{\sqrt{a+b x^4}} \, dx}{3 a}\\ &=-\frac{c \sqrt{a+b x^4}}{4 a x^4}-\frac{d \sqrt{a+b x^4}}{3 a x^3}-\frac{e \sqrt{a+b x^4}}{2 a x^2}-\frac{f \sqrt{a+b x^4}}{a x}+\frac{\sqrt{b} f x \sqrt{a+b x^4}}{a \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{\sqrt [4]{b} f \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{\sqrt [4]{b} \left (\sqrt{b} d-3 \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 )}{6 a^{5/4} \sqrt{a+b x^4}}-\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^4\right )}{8 a}\\ &=-\frac{c \sqrt{a+b x^4}}{4 a x^4}-\frac{d \sqrt{a+b x^4}}{3 a x^3}-\frac{e \sqrt{a+b x^4}}{2 a x^2}-\frac{f \sqrt{a+b x^4}}{a x}+\frac{\sqrt{b} f x \sqrt{a+b x^4}}{a \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{\sqrt [4]{b} f \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{\sqrt [4]{b} \left (\sqrt{b} d-3 \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 )}{6 a^{5/4} \sqrt{a+b x^4}}-\frac{c \operatorname{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}}{4 a x^4}-\frac{d \sqrt{a+b x^4}}{3 a x^3}-\frac{e \sqrt{a+b x^4}}{2 a x^2}-\frac{f \sqrt{a+b x^4}}{a x}+\frac{\sqrt{b} f x \sqrt{a+b x^4}}{a \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{b c \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{4 a^{3/2}}-\frac{\sqrt [4]{b} f \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{\sqrt [4]{b} \left (\sqrt{b} d-3 \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 )}{6 a^{5/4} \sqrt{a+b x^4}}\\ \end{align*}
Mathematica [C] time = 0.147496, size = 147, normalized size = 0.42 \[ -\frac{\sqrt{a+b x^4} \left (3 a c \sqrt{\frac{b x^4}{a}+1}-3 b c x^4 \tanh ^{-1}\left (\sqrt{\frac{b x^4}{a}+1}\right )+4 a d x \, _2F_1\left (-\frac{3}{4},\frac{1}{2};\frac{1}{4};-\frac{b x^4}{a}\right )+6 a e x^2 \sqrt{\frac{b x^4}{a}+1}+12 a f x^3 \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{b x^4}{a}\right )\right )}{12 a^2 x^4 \sqrt{\frac{b x^4}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.017, size = 335, normalized size = 1. \begin{align*} -{\frac{d}{3\,a{x}^{3}}\sqrt{b{x}^{4}+a}}-{\frac{bd}{3\,a}\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{c}{4\,a{x}^{4}}\sqrt{b{x}^{4}+a}}+{\frac{bc}{4}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{4}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{e}{2\,a{x}^{2}}\sqrt{b{x}^{4}+a}}-{\frac{f}{ax}\sqrt{b{x}^{4}+a}}+{if\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}}}}-{if\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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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^{9} + a x^{5}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 4.53686, size = 158, normalized size = 0.46 \begin{align*} - \frac{\sqrt{b} c \sqrt{\frac{a}{b x^{4}} + 1}}{4 a x^{2}} - \frac{\sqrt{b} e \sqrt{\frac{a}{b x^{4}} + 1}}{2 a} + \frac{d \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{f \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{b c \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{4 a^{\frac{3}{2}}} \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^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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