Optimal. Leaf size=352 \[ \frac{b^{3/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (5 \sqrt{a} f+3 \sqrt{b} d\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{15 a^{3/4} \sqrt{a+b x^4}}-\frac{2 b^{5/4} d \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 a^{3/4} \sqrt{a+b x^4}}+\frac{2 b^{3/2} d x \sqrt{a+b x^4}}{5 a \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{1}{60} \sqrt{a+b x^4} \left (\frac{10 c}{x^6}+\frac{12 d}{x^5}+\frac{15 e}{x^4}+\frac{20 f}{x^3}\right )-\frac{b c \sqrt{a+b x^4}}{6 a x^2}-\frac{2 b d \sqrt{a+b x^4}}{5 a x}-\frac{b e \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{4 \sqrt{a}} \]
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Rubi [A] time = 0.316338, antiderivative size = 352, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 12, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {14, 1825, 1833, 1252, 807, 266, 63, 208, 1282, 1198, 220, 1196} \[ \frac{b^{3/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (5 \sqrt{a} f+3 \sqrt{b} d\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 a^{3/4} \sqrt{a+b x^4}}-\frac{2 b^{5/4} d \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 a^{3/4} \sqrt{a+b x^4}}+\frac{2 b^{3/2} d x \sqrt{a+b x^4}}{5 a \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{1}{60} \sqrt{a+b x^4} \left (\frac{10 c}{x^6}+\frac{12 d}{x^5}+\frac{15 e}{x^4}+\frac{20 f}{x^3}\right )-\frac{b c \sqrt{a+b x^4}}{6 a x^2}-\frac{2 b d \sqrt{a+b x^4}}{5 a x}-\frac{b e \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{4 \sqrt{a}} \]
Antiderivative was successfully verified.
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Rule 14
Rule 1825
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{\left (c+d x+e x^2+f x^3\right ) \sqrt{a+b x^4}}{x^7} \, dx &=-\frac{1}{60} \left (\frac{10 c}{x^6}+\frac{12 d}{x^5}+\frac{15 e}{x^4}+\frac{20 f}{x^3}\right ) \sqrt{a+b x^4}-(2 b) \int \frac{-\frac{c}{6}-\frac{d x}{5}-\frac{e x^2}{4}-\frac{f x^3}{3}}{x^3 \sqrt{a+b x^4}} \, dx\\ &=-\frac{1}{60} \left (\frac{10 c}{x^6}+\frac{12 d}{x^5}+\frac{15 e}{x^4}+\frac{20 f}{x^3}\right ) \sqrt{a+b x^4}-(2 b) \int \left (\frac{-\frac{c}{6}-\frac{e x^2}{4}}{x^3 \sqrt{a+b x^4}}+\frac{-\frac{d}{5}-\frac{f x^2}{3}}{x^2 \sqrt{a+b x^4}}\right ) \, dx\\ &=-\frac{1}{60} \left (\frac{10 c}{x^6}+\frac{12 d}{x^5}+\frac{15 e}{x^4}+\frac{20 f}{x^3}\right ) \sqrt{a+b x^4}-(2 b) \int \frac{-\frac{c}{6}-\frac{e x^2}{4}}{x^3 \sqrt{a+b x^4}} \, dx-(2 b) \int \frac{-\frac{d}{5}-\frac{f x^2}{3}}{x^2 \sqrt{a+b x^4}} \, dx\\ &=-\frac{1}{60} \left (\frac{10 c}{x^6}+\frac{12 d}{x^5}+\frac{15 e}{x^4}+\frac{20 f}{x^3}\right ) \sqrt{a+b x^4}-\frac{2 b d \sqrt{a+b x^4}}{5 a x}-b \operatorname{Subst}\left (\int \frac{-\frac{c}{6}-\frac{e x}{4}}{x^2 \sqrt{a+b x^2}} \, dx,x,x^2\right )+\frac{(2 b) \int \frac{\frac{a f}{3}+\frac{1}{5} b d x^2}{\sqrt{a+b x^4}} \, dx}{a}\\ &=-\frac{1}{60} \left (\frac{10 c}{x^6}+\frac{12 d}{x^5}+\frac{15 e}{x^4}+\frac{20 f}{x^3}\right ) \sqrt{a+b x^4}-\frac{b c \sqrt{a+b x^4}}{6 a x^2}-\frac{2 b d \sqrt{a+b x^4}}{5 a x}-\frac{\left (2 b^{3/2} d\right ) \int \frac{1-\frac{\sqrt{b} x^2}{\sqrt{a}}}{\sqrt{a+b x^4}} \, dx}{5 \sqrt{a}}+\frac{1}{4} (b e) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x^2}} \, dx,x,x^2\right )+\frac{1}{15} \left (2 b \left (\frac{3 \sqrt{b} d}{\sqrt{a}}+5 f\right )\right ) \int \frac{1}{\sqrt{a+b x^4}} \, dx\\ &=-\frac{1}{60} \left (\frac{10 c}{x^6}+\frac{12 d}{x^5}+\frac{15 e}{x^4}+\frac{20 f}{x^3}\right ) \sqrt{a+b x^4}-\frac{b c \sqrt{a+b x^4}}{6 a x^2}-\frac{2 b d \sqrt{a+b x^4}}{5 a x}+\frac{2 b^{3/2} d x \sqrt{a+b x^4}}{5 a \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{2 b^{5/4} 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 )}{5 a^{3/4} \sqrt{a+b x^4}}+\frac{b^{3/4} \left (3 \sqrt{b} d+5 \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 )}{15 a^{3/4} \sqrt{a+b x^4}}+\frac{1}{8} (b e) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^4\right )\\ &=-\frac{1}{60} \left (\frac{10 c}{x^6}+\frac{12 d}{x^5}+\frac{15 e}{x^4}+\frac{20 f}{x^3}\right ) \sqrt{a+b x^4}-\frac{b c \sqrt{a+b x^4}}{6 a x^2}-\frac{2 b d \sqrt{a+b x^4}}{5 a x}+\frac{2 b^{3/2} d x \sqrt{a+b x^4}}{5 a \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{2 b^{5/4} 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 )}{5 a^{3/4} \sqrt{a+b x^4}}+\frac{b^{3/4} \left (3 \sqrt{b} d+5 \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 )}{15 a^{3/4} \sqrt{a+b x^4}}+\frac{1}{4} e \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^4}\right )\\ &=-\frac{1}{60} \left (\frac{10 c}{x^6}+\frac{12 d}{x^5}+\frac{15 e}{x^4}+\frac{20 f}{x^3}\right ) \sqrt{a+b x^4}-\frac{b c \sqrt{a+b x^4}}{6 a x^2}-\frac{2 b d \sqrt{a+b x^4}}{5 a x}+\frac{2 b^{3/2} d x \sqrt{a+b x^4}}{5 a \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{b e \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{4 \sqrt{a}}-\frac{2 b^{5/4} 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 )}{5 a^{3/4} \sqrt{a+b x^4}}+\frac{b^{3/4} \left (3 \sqrt{b} d+5 \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 )}{15 a^{3/4} \sqrt{a+b x^4}}\\ \end{align*}
Mathematica [C] time = 0.243303, size = 145, normalized size = 0.41 \[ -\frac{\sqrt{a+b x^4} \left (5 \left (\sqrt{\frac{b x^4}{a}+1} \left (2 a c+3 a e x^2+2 b c x^4\right )+3 b e x^6 \tanh ^{-1}\left (\sqrt{\frac{b x^4}{a}+1}\right )+4 a f x^3 \, _2F_1\left (-\frac{3}{4},-\frac{1}{2};\frac{1}{4};-\frac{b x^4}{a}\right )\right )+12 a d x \, _2F_1\left (-\frac{5}{4},-\frac{1}{2};-\frac{1}{4};-\frac{b x^4}{a}\right )\right )}{60 a x^6 \sqrt{\frac{b x^4}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.017, size = 361, normalized size = 1. \begin{align*} -{\frac{f}{3\,{x}^{3}}\sqrt{b{x}^{4}+a}}+{\frac{2\,fb}{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{d}{5\,{x}^{5}}\sqrt{b{x}^{4}+a}}-{\frac{2\,bd}{5\,ax}\sqrt{b{x}^{4}+a}}+{{\frac{2\,i}{5}}d{b}^{{\frac{3}{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 ){\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{{\frac{2\,i}{5}}d{b}^{{\frac{3}{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 ){\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{\frac{e}{4\,a{x}^{4}} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{2}}}}-{\frac{be}{4}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{4}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}+{\frac{be}{4\,a}\sqrt{b{x}^{4}+a}}-{\frac{c}{6\,{x}^{6}a} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{2}}}} \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*} -\frac{{\left (b x^{4} + a\right )}^{\frac{3}{2}} c}{6 \, a x^{6}} + \int \frac{\sqrt{b x^{4} + a}{\left (f x^{2} + e x + d\right )}}{x^{6}}\,{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^{7}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 5.70158, size = 189, normalized size = 0.54 \begin{align*} \frac{\sqrt{a} d \Gamma \left (- \frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, - \frac{1}{2} \\ - \frac{1}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{5} \Gamma \left (- \frac{1}{4}\right )} + \frac{\sqrt{a} f \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 x^{3} \Gamma \left (\frac{1}{4}\right )} - \frac{\sqrt{b} c \sqrt{\frac{a}{b x^{4}} + 1}}{6 x^{4}} - \frac{\sqrt{b} e \sqrt{\frac{a}{b x^{4}} + 1}}{4 x^{2}} - \frac{b^{\frac{3}{2}} c \sqrt{\frac{a}{b x^{4}} + 1}}{6 a} - \frac{b e \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{4 \sqrt{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^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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