Optimal. Leaf size=425 \[ -\frac{b^{7/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (5 \sqrt{a} e+7 \sqrt{b} c\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{105 a^{7/4} \sqrt{a+b x^4}}-\frac{2 b^{5/2} c x \sqrt{a+b x^4}}{15 a^2 \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{2 b^2 c \sqrt{a+b x^4}}{15 a^2 x}+\frac{2 b^{9/4} c \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 a^{7/4} \sqrt{a+b x^4}}+\frac{b^2 d \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{16 a^{3/2}}-\frac{1}{504} \sqrt{a+b x^4} \left (\frac{56 c}{x^9}+\frac{63 d}{x^8}+\frac{72 e}{x^7}+\frac{84 f}{x^6}\right )-\frac{2 b c \sqrt{a+b x^4}}{45 a x^5}-\frac{b d \sqrt{a+b x^4}}{16 a x^4}-\frac{2 b e \sqrt{a+b x^4}}{21 a x^3}-\frac{b f \sqrt{a+b x^4}}{6 a x^2} \]
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Rubi [A] time = 0.433128, antiderivative size = 425, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 13, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.433, Rules used = {14, 1825, 1833, 1282, 1198, 220, 1196, 1252, 835, 807, 266, 63, 208} \[ -\frac{b^{7/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (5 \sqrt{a} e+7 \sqrt{b} c\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{105 a^{7/4} \sqrt{a+b x^4}}-\frac{2 b^{5/2} c x \sqrt{a+b x^4}}{15 a^2 \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{2 b^2 c \sqrt{a+b x^4}}{15 a^2 x}+\frac{2 b^{9/4} c \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 a^{7/4} \sqrt{a+b x^4}}+\frac{b^2 d \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{16 a^{3/2}}-\frac{1}{504} \sqrt{a+b x^4} \left (\frac{56 c}{x^9}+\frac{63 d}{x^8}+\frac{72 e}{x^7}+\frac{84 f}{x^6}\right )-\frac{2 b c \sqrt{a+b x^4}}{45 a x^5}-\frac{b d \sqrt{a+b x^4}}{16 a x^4}-\frac{2 b e \sqrt{a+b x^4}}{21 a x^3}-\frac{b f \sqrt{a+b x^4}}{6 a x^2} \]
Antiderivative was successfully verified.
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Rule 14
Rule 1825
Rule 1833
Rule 1282
Rule 1198
Rule 220
Rule 1196
Rule 1252
Rule 835
Rule 807
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^{10}} \, dx &=-\frac{1}{504} \left (\frac{56 c}{x^9}+\frac{63 d}{x^8}+\frac{72 e}{x^7}+\frac{84 f}{x^6}\right ) \sqrt{a+b x^4}-(2 b) \int \frac{-\frac{c}{9}-\frac{d x}{8}-\frac{e x^2}{7}-\frac{f x^3}{6}}{x^6 \sqrt{a+b x^4}} \, dx\\ &=-\frac{1}{504} \left (\frac{56 c}{x^9}+\frac{63 d}{x^8}+\frac{72 e}{x^7}+\frac{84 f}{x^6}\right ) \sqrt{a+b x^4}-(2 b) \int \left (\frac{-\frac{c}{9}-\frac{e x^2}{7}}{x^6 \sqrt{a+b x^4}}+\frac{-\frac{d}{8}-\frac{f x^2}{6}}{x^5 \sqrt{a+b x^4}}\right ) \, dx\\ &=-\frac{1}{504} \left (\frac{56 c}{x^9}+\frac{63 d}{x^8}+\frac{72 e}{x^7}+\frac{84 f}{x^6}\right ) \sqrt{a+b x^4}-(2 b) \int \frac{-\frac{c}{9}-\frac{e x^2}{7}}{x^6 \sqrt{a+b x^4}} \, dx-(2 b) \int \frac{-\frac{d}{8}-\frac{f x^2}{6}}{x^5 \sqrt{a+b x^4}} \, dx\\ &=-\frac{1}{504} \left (\frac{56 c}{x^9}+\frac{63 d}{x^8}+\frac{72 e}{x^7}+\frac{84 f}{x^6}\right ) \sqrt{a+b x^4}-\frac{2 b c \sqrt{a+b x^4}}{45 a x^5}-b \operatorname{Subst}\left (\int \frac{-\frac{d}{8}-\frac{f x}{6}}{x^3 \sqrt{a+b x^2}} \, dx,x,x^2\right )+\frac{(2 b) \int \frac{\frac{5 a e}{7}-\frac{1}{3} b c x^2}{x^4 \sqrt{a+b x^4}} \, dx}{5 a}\\ &=-\frac{1}{504} \left (\frac{56 c}{x^9}+\frac{63 d}{x^8}+\frac{72 e}{x^7}+\frac{84 f}{x^6}\right ) \sqrt{a+b x^4}-\frac{2 b c \sqrt{a+b x^4}}{45 a x^5}-\frac{b d \sqrt{a+b x^4}}{16 a x^4}-\frac{2 b e \sqrt{a+b x^4}}{21 a x^3}-\frac{(2 b) \int \frac{a b c+\frac{5}{7} a b e x^2}{x^2 \sqrt{a+b x^4}} \, dx}{15 a^2}+\frac{b \operatorname{Subst}\left (\int \frac{\frac{a f}{3}-\frac{b d x}{8}}{x^2 \sqrt{a+b x^2}} \, dx,x,x^2\right )}{2 a}\\ &=-\frac{1}{504} \left (\frac{56 c}{x^9}+\frac{63 d}{x^8}+\frac{72 e}{x^7}+\frac{84 f}{x^6}\right ) \sqrt{a+b x^4}-\frac{2 b c \sqrt{a+b x^4}}{45 a x^5}-\frac{b d \sqrt{a+b x^4}}{16 a x^4}-\frac{2 b e \sqrt{a+b x^4}}{21 a x^3}-\frac{b f \sqrt{a+b x^4}}{6 a x^2}+\frac{2 b^2 c \sqrt{a+b x^4}}{15 a^2 x}+\frac{(2 b) \int \frac{-\frac{5}{7} a^2 b e-a b^2 c x^2}{\sqrt{a+b x^4}} \, dx}{15 a^3}-\frac{\left (b^2 d\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x^2}} \, dx,x,x^2\right )}{16 a}\\ &=-\frac{1}{504} \left (\frac{56 c}{x^9}+\frac{63 d}{x^8}+\frac{72 e}{x^7}+\frac{84 f}{x^6}\right ) \sqrt{a+b x^4}-\frac{2 b c \sqrt{a+b x^4}}{45 a x^5}-\frac{b d \sqrt{a+b x^4}}{16 a x^4}-\frac{2 b e \sqrt{a+b x^4}}{21 a x^3}-\frac{b f \sqrt{a+b x^4}}{6 a x^2}+\frac{2 b^2 c \sqrt{a+b x^4}}{15 a^2 x}+\frac{\left (2 b^{5/2} c\right ) \int \frac{1-\frac{\sqrt{b} x^2}{\sqrt{a}}}{\sqrt{a+b x^4}} \, dx}{15 a^{3/2}}-\frac{\left (b^2 d\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^4\right )}{32 a}-\frac{\left (2 b^2 \left (7 \sqrt{b} c+5 \sqrt{a} e\right )\right ) \int \frac{1}{\sqrt{a+b x^4}} \, dx}{105 a^{3/2}}\\ &=-\frac{1}{504} \left (\frac{56 c}{x^9}+\frac{63 d}{x^8}+\frac{72 e}{x^7}+\frac{84 f}{x^6}\right ) \sqrt{a+b x^4}-\frac{2 b c \sqrt{a+b x^4}}{45 a x^5}-\frac{b d \sqrt{a+b x^4}}{16 a x^4}-\frac{2 b e \sqrt{a+b x^4}}{21 a x^3}-\frac{b f \sqrt{a+b x^4}}{6 a x^2}+\frac{2 b^2 c \sqrt{a+b x^4}}{15 a^2 x}-\frac{2 b^{5/2} c x \sqrt{a+b x^4}}{15 a^2 \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{2 b^{9/4} 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 )}{15 a^{7/4} \sqrt{a+b x^4}}-\frac{b^{7/4} \left (7 \sqrt{b} c+5 \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 )}{105 a^{7/4} \sqrt{a+b x^4}}-\frac{(b d) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^4}\right )}{16 a}\\ &=-\frac{1}{504} \left (\frac{56 c}{x^9}+\frac{63 d}{x^8}+\frac{72 e}{x^7}+\frac{84 f}{x^6}\right ) \sqrt{a+b x^4}-\frac{2 b c \sqrt{a+b x^4}}{45 a x^5}-\frac{b d \sqrt{a+b x^4}}{16 a x^4}-\frac{2 b e \sqrt{a+b x^4}}{21 a x^3}-\frac{b f \sqrt{a+b x^4}}{6 a x^2}+\frac{2 b^2 c \sqrt{a+b x^4}}{15 a^2 x}-\frac{2 b^{5/2} c x \sqrt{a+b x^4}}{15 a^2 \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{b^2 d \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{16 a^{3/2}}+\frac{2 b^{9/4} 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 )}{15 a^{7/4} \sqrt{a+b x^4}}-\frac{b^{7/4} \left (7 \sqrt{b} c+5 \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 )}{105 a^{7/4} \sqrt{a+b x^4}}\\ \end{align*}
Mathematica [C] time = 0.16672, size = 148, normalized size = 0.35 \[ -\frac{\sqrt{a+b x^4} \left (3 x^2 \left (7 x \left (a+b x^4\right ) \sqrt{\frac{b x^4}{a}+1} \left (a^2 f+b^2 d x^6 \, _2F_1\left (\frac{3}{2},3;\frac{5}{2};\frac{b x^4}{a}+1\right )\right )+6 a^3 e \, _2F_1\left (-\frac{7}{4},-\frac{1}{2};-\frac{3}{4};-\frac{b x^4}{a}\right )\right )+14 a^3 c \, _2F_1\left (-\frac{9}{4},-\frac{1}{2};-\frac{5}{4};-\frac{b x^4}{a}\right )\right )}{126 a^3 x^9 \sqrt{\frac{b x^4}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.018, size = 429, normalized size = 1. \begin{align*} -{\frac{d}{8\,a{x}^{8}} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{2}}}}+{\frac{bd}{16\,{a}^{2}{x}^{4}} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{2}}}}+{\frac{{b}^{2}d}{16}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{4}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{{b}^{2}d}{16\,{a}^{2}}\sqrt{b{x}^{4}+a}}-{\frac{f}{6\,{x}^{6}a} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{2}}}}-{\frac{e}{7\,{x}^{7}}\sqrt{b{x}^{4}+a}}-{\frac{2\,be}{21\,a{x}^{3}}\sqrt{b{x}^{4}+a}}-{\frac{2\,{b}^{2}e}{21\,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}{9\,{x}^{9}}\sqrt{b{x}^{4}+a}}-{\frac{2\,bc}{45\,a{x}^{5}}\sqrt{b{x}^{4}+a}}+{\frac{2\,{b}^{2}c}{15\,{a}^{2}x}\sqrt{b{x}^{4}+a}}-{{\frac{2\,i}{15}}c{b}^{{\frac{5}{2}}}\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{2\,i}{15}}c{b}^{{\frac{5}{2}}}\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{\sqrt{b x^{4} + a}{\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{10}}\,{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^{10}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 8.46443, size = 246, normalized size = 0.58 \begin{align*} \frac{\sqrt{a} c \Gamma \left (- \frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{9}{4}, - \frac{1}{2} \\ - \frac{5}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{9} \Gamma \left (- \frac{5}{4}\right )} + \frac{\sqrt{a} e \Gamma \left (- \frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{7}{4}, - \frac{1}{2} \\ - \frac{3}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{7} \Gamma \left (- \frac{3}{4}\right )} - \frac{a d}{8 \sqrt{b} x^{10} \sqrt{\frac{a}{b x^{4}} + 1}} - \frac{3 \sqrt{b} d}{16 x^{6} \sqrt{\frac{a}{b x^{4}} + 1}} - \frac{\sqrt{b} f \sqrt{\frac{a}{b x^{4}} + 1}}{6 x^{4}} - \frac{b^{\frac{3}{2}} d}{16 a x^{2} \sqrt{\frac{a}{b x^{4}} + 1}} - \frac{b^{\frac{3}{2}} f \sqrt{\frac{a}{b x^{4}} + 1}}{6 a} + \frac{b^{2} d \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{16 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{\sqrt{b x^{4} + a}{\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{10}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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