Optimal. Leaf size=344 \[ \frac{\left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (\sqrt{a} e+3 \sqrt{b} c\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{4 a^{7/4} \sqrt [4]{b} \sqrt{a+b x^4}}+\frac{x \left (a e+a f x-b c x^2-b d x^3\right )}{2 a^2 \sqrt{a+b x^4}}-\frac{c \sqrt{a+b x^4}}{a^2 x}+\frac{3 \sqrt{b} c x \sqrt{a+b x^4}}{2 a^2 \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{3 \sqrt [4]{b} c \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 a^{7/4} \sqrt{a+b x^4}}+\frac{d \sqrt{a+b x^4}}{2 a^2}-\frac{d \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{2 a^{3/2}} \]
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Rubi [A] time = 0.382575, antiderivative size = 344, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 12, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {1829, 1833, 1835, 1584, 1198, 220, 1196, 21, 266, 50, 63, 208} \[ \frac{x \left (a e+a f x-b c x^2-b d x^3\right )}{2 a^2 \sqrt{a+b x^4}}+\frac{\left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (\sqrt{a} e+3 \sqrt{b} c\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 a^{7/4} \sqrt [4]{b} \sqrt{a+b x^4}}-\frac{c \sqrt{a+b x^4}}{a^2 x}+\frac{3 \sqrt{b} c x \sqrt{a+b x^4}}{2 a^2 \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{3 \sqrt [4]{b} c \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 a^{7/4} \sqrt{a+b x^4}}+\frac{d \sqrt{a+b x^4}}{2 a^2}-\frac{d \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{2 a^{3/2}} \]
Antiderivative was successfully verified.
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Rule 1829
Rule 1833
Rule 1835
Rule 1584
Rule 1198
Rule 220
Rule 1196
Rule 21
Rule 266
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{c+d x+e x^2+f x^3}{x^2 \left (a+b x^4\right )^{3/2}} \, dx &=\frac{x \left (a e+a f x-b c x^2-b d x^3\right )}{2 a^2 \sqrt{a+b x^4}}-\frac{\int \frac{-2 b c-2 b d x-b e x^2-\frac{b^2 c x^4}{a}-\frac{2 b^2 d x^5}{a}}{x^2 \sqrt{a+b x^4}} \, dx}{2 a b}\\ &=\frac{x \left (a e+a f x-b c x^2-b d x^3\right )}{2 a^2 \sqrt{a+b x^4}}-\frac{\int \left (\frac{-2 b c-b e x^2-\frac{b^2 c x^4}{a}}{x^2 \sqrt{a+b x^4}}+\frac{-2 b d-\frac{2 b^2 d x^4}{a}}{x \sqrt{a+b x^4}}\right ) \, dx}{2 a b}\\ &=\frac{x \left (a e+a f x-b c x^2-b d x^3\right )}{2 a^2 \sqrt{a+b x^4}}-\frac{\int \frac{-2 b c-b e x^2-\frac{b^2 c x^4}{a}}{x^2 \sqrt{a+b x^4}} \, dx}{2 a b}-\frac{\int \frac{-2 b d-\frac{2 b^2 d x^4}{a}}{x \sqrt{a+b x^4}} \, dx}{2 a b}\\ &=\frac{x \left (a e+a f x-b c x^2-b d x^3\right )}{2 a^2 \sqrt{a+b x^4}}-\frac{c \sqrt{a+b x^4}}{a^2 x}+\frac{\int \frac{2 a b e x+6 b^2 c x^3}{x \sqrt{a+b x^4}} \, dx}{4 a^2 b}+\frac{d \int \frac{\sqrt{a+b x^4}}{x} \, dx}{a^2}\\ &=\frac{x \left (a e+a f x-b c x^2-b d x^3\right )}{2 a^2 \sqrt{a+b x^4}}-\frac{c \sqrt{a+b x^4}}{a^2 x}+\frac{\int \frac{2 a b e+6 b^2 c x^2}{\sqrt{a+b x^4}} \, dx}{4 a^2 b}+\frac{d \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,x^4\right )}{4 a^2}\\ &=\frac{x \left (a e+a f x-b c x^2-b d x^3\right )}{2 a^2 \sqrt{a+b x^4}}+\frac{d \sqrt{a+b x^4}}{2 a^2}-\frac{c \sqrt{a+b x^4}}{a^2 x}-\frac{\left (3 \sqrt{b} c\right ) \int \frac{1-\frac{\sqrt{b} x^2}{\sqrt{a}}}{\sqrt{a+b x^4}} \, dx}{2 a^{3/2}}+\frac{d \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^4\right )}{4 a}+\frac{\left (3 \sqrt{b} c+\sqrt{a} e\right ) \int \frac{1}{\sqrt{a+b x^4}} \, dx}{2 a^{3/2}}\\ &=\frac{x \left (a e+a f x-b c x^2-b d x^3\right )}{2 a^2 \sqrt{a+b x^4}}+\frac{d \sqrt{a+b x^4}}{2 a^2}-\frac{c \sqrt{a+b x^4}}{a^2 x}+\frac{3 \sqrt{b} c x \sqrt{a+b x^4}}{2 a^2 \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{3 \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 )}{2 a^{7/4} \sqrt{a+b x^4}}+\frac{\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 )}{4 a^{7/4} \sqrt [4]{b} \sqrt{a+b x^4}}+\frac{d \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^4}\right )}{2 a b}\\ &=\frac{x \left (a e+a f x-b c x^2-b d x^3\right )}{2 a^2 \sqrt{a+b x^4}}+\frac{d \sqrt{a+b x^4}}{2 a^2}-\frac{c \sqrt{a+b x^4}}{a^2 x}+\frac{3 \sqrt{b} c x \sqrt{a+b x^4}}{2 a^2 \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{d \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{2 a^{3/2}}-\frac{3 \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 )}{2 a^{7/4} \sqrt{a+b x^4}}+\frac{\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 )}{4 a^{7/4} \sqrt [4]{b} \sqrt{a+b x^4}}\\ \end{align*}
Mathematica [C] time = 0.114255, size = 123, normalized size = 0.36 \[ \frac{-2 c \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left (-\frac{1}{4},\frac{3}{2};\frac{3}{4};-\frac{b x^4}{a}\right )+d x \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{b x^4}{a}+1\right )+x^2 \left (e \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^4}{a}\right )+e+f x\right )}{2 a x \sqrt{a+b x^4}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.01, size = 355, normalized size = 1. \begin{align*}{\frac{f{x}^{2}}{2\,a}{\frac{1}{\sqrt{b{x}^{4}+a}}}}+{\frac{ex}{2\,a}{\frac{1}{\sqrt{ \left ({x}^{4}+{\frac{a}{b}} \right ) b}}}}+{\frac{e}{2\,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{d}{2\,a}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{\frac{d}{2}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{4}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{bc{x}^{3}}{2\,{a}^{2}}{\frac{1}{\sqrt{ \left ({x}^{4}+{\frac{a}{b}} \right ) b}}}}-{\frac{c}{{a}^{2}x}\sqrt{b{x}^{4}+a}}+{{\frac{3\,i}{2}}c\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 ){a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{{\frac{3\,i}{2}}c\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 ){a}^{-{\frac{3}{2}}}{\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}{{\left (b x^{4} + a\right )}^{\frac{3}{2}} 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 )}}{b^{2} x^{10} + 2 \, a b x^{6} + a^{2} x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 34.1444, size = 291, normalized size = 0.85 \begin{align*} d \left (\frac{2 a^{3} \sqrt{1 + \frac{b x^{4}}{a}}}{4 a^{\frac{9}{2}} + 4 a^{\frac{7}{2}} b x^{4}} + \frac{a^{3} \log{\left (\frac{b x^{4}}{a} \right )}}{4 a^{\frac{9}{2}} + 4 a^{\frac{7}{2}} b x^{4}} - \frac{2 a^{3} \log{\left (\sqrt{1 + \frac{b x^{4}}{a}} + 1 \right )}}{4 a^{\frac{9}{2}} + 4 a^{\frac{7}{2}} b x^{4}} + \frac{a^{2} b x^{4} \log{\left (\frac{b x^{4}}{a} \right )}}{4 a^{\frac{9}{2}} + 4 a^{\frac{7}{2}} b x^{4}} - \frac{2 a^{2} b x^{4} \log{\left (\sqrt{1 + \frac{b x^{4}}{a}} + 1 \right )}}{4 a^{\frac{9}{2}} + 4 a^{\frac{7}{2}} b x^{4}}\right ) + \frac{c \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{3}{2} \\ \frac{3}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{3}{2}} x \Gamma \left (\frac{3}{4}\right )} + \frac{e x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{3}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{3}{2}} \Gamma \left (\frac{5}{4}\right )} + \frac{f x^{2}}{2 a^{\frac{3}{2}} \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{f x^{3} + e x^{2} + d x + c}{{\left (b x^{4} + a\right )}^{\frac{3}{2}} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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