Optimal. Leaf size=386 \[ \frac{2 a^{3/4} \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (9 \sqrt{a} f+5 \sqrt{b} d\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{15 \sqrt{a+b x^4}}-\frac{12 a^{5/4} \sqrt [4]{b} f \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 \sqrt{a+b x^4}}-\frac{1}{12} \left (a+b x^4\right )^{3/2} \left (\frac{3 c}{x^4}+\frac{4 d}{x^3}+\frac{6 e}{x^2}+\frac{12 f}{x}\right )+\frac{3}{4} b \sqrt{a+b x^4} \left (c+e x^2\right )-\frac{3}{4} \sqrt{a} b c \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )+\frac{2}{15} b x \sqrt{a+b x^4} \left (5 d+9 f x^2\right )+\frac{3}{4} a \sqrt{b} e \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )+\frac{12 a \sqrt{b} f x \sqrt{a+b x^4}}{5 \left (\sqrt{a}+\sqrt{b} x^2\right )} \]
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Rubi [A] time = 0.346981, antiderivative size = 386, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 15, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {14, 1825, 1833, 1252, 815, 844, 217, 206, 266, 63, 208, 1177, 1198, 220, 1196} \[ \frac{2 a^{3/4} \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (9 \sqrt{a} f+5 \sqrt{b} d\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 \sqrt{a+b x^4}}-\frac{12 a^{5/4} \sqrt [4]{b} f \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 \sqrt{a+b x^4}}-\frac{1}{12} \left (a+b x^4\right )^{3/2} \left (\frac{3 c}{x^4}+\frac{4 d}{x^3}+\frac{6 e}{x^2}+\frac{12 f}{x}\right )+\frac{3}{4} b \sqrt{a+b x^4} \left (c+e x^2\right )-\frac{3}{4} \sqrt{a} b c \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )+\frac{2}{15} b x \sqrt{a+b x^4} \left (5 d+9 f x^2\right )+\frac{3}{4} a \sqrt{b} e \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )+\frac{12 a \sqrt{b} f x \sqrt{a+b x^4}}{5 \left (\sqrt{a}+\sqrt{b} x^2\right )} \]
Antiderivative was successfully verified.
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Rule 14
Rule 1825
Rule 1833
Rule 1252
Rule 815
Rule 844
Rule 217
Rule 206
Rule 266
Rule 63
Rule 208
Rule 1177
Rule 1198
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^5} \, dx &=-\frac{1}{12} \left (\frac{3 c}{x^4}+\frac{4 d}{x^3}+\frac{6 e}{x^2}+\frac{12 f}{x}\right ) \left (a+b x^4\right )^{3/2}-(6 b) \int \frac{\left (-\frac{c}{4}-\frac{d x}{3}-\frac{e x^2}{2}-f x^3\right ) \sqrt{a+b x^4}}{x} \, dx\\ &=-\frac{1}{12} \left (\frac{3 c}{x^4}+\frac{4 d}{x^3}+\frac{6 e}{x^2}+\frac{12 f}{x}\right ) \left (a+b x^4\right )^{3/2}-(6 b) \int \left (\frac{\left (-\frac{c}{4}-\frac{e x^2}{2}\right ) \sqrt{a+b x^4}}{x}+\left (-\frac{d}{3}-f x^2\right ) \sqrt{a+b x^4}\right ) \, dx\\ &=-\frac{1}{12} \left (\frac{3 c}{x^4}+\frac{4 d}{x^3}+\frac{6 e}{x^2}+\frac{12 f}{x}\right ) \left (a+b x^4\right )^{3/2}-(6 b) \int \frac{\left (-\frac{c}{4}-\frac{e x^2}{2}\right ) \sqrt{a+b x^4}}{x} \, dx-(6 b) \int \left (-\frac{d}{3}-f x^2\right ) \sqrt{a+b x^4} \, dx\\ &=\frac{2}{15} b x \left (5 d+9 f x^2\right ) \sqrt{a+b x^4}-\frac{1}{12} \left (\frac{3 c}{x^4}+\frac{4 d}{x^3}+\frac{6 e}{x^2}+\frac{12 f}{x}\right ) \left (a+b x^4\right )^{3/2}-\frac{1}{5} (2 b) \int \frac{-\frac{10 a d}{3}-6 a f x^2}{\sqrt{a+b x^4}} \, dx-(3 b) \operatorname{Subst}\left (\int \frac{\left (-\frac{c}{4}-\frac{e x}{2}\right ) \sqrt{a+b x^2}}{x} \, dx,x,x^2\right )\\ &=\frac{3}{4} b \left (c+e x^2\right ) \sqrt{a+b x^4}+\frac{2}{15} b x \left (5 d+9 f x^2\right ) \sqrt{a+b x^4}-\frac{1}{12} \left (\frac{3 c}{x^4}+\frac{4 d}{x^3}+\frac{6 e}{x^2}+\frac{12 f}{x}\right ) \left (a+b x^4\right )^{3/2}-\frac{3}{2} \operatorname{Subst}\left (\int \frac{-\frac{1}{2} a b c-\frac{1}{2} a b e x}{x \sqrt{a+b x^2}} \, dx,x,x^2\right )-\frac{1}{5} \left (12 a^{3/2} \sqrt{b} f\right ) \int \frac{1-\frac{\sqrt{b} x^2}{\sqrt{a}}}{\sqrt{a+b x^4}} \, dx+\frac{1}{15} \left (4 a b \left (5 d+\frac{9 \sqrt{a} f}{\sqrt{b}}\right )\right ) \int \frac{1}{\sqrt{a+b x^4}} \, dx\\ &=\frac{12 a \sqrt{b} f x \sqrt{a+b x^4}}{5 \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{3}{4} b \left (c+e x^2\right ) \sqrt{a+b x^4}+\frac{2}{15} b x \left (5 d+9 f x^2\right ) \sqrt{a+b x^4}-\frac{1}{12} \left (\frac{3 c}{x^4}+\frac{4 d}{x^3}+\frac{6 e}{x^2}+\frac{12 f}{x}\right ) \left (a+b x^4\right )^{3/2}-\frac{12 a^{5/4} \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 )}{5 \sqrt{a+b x^4}}+\frac{2 a^{3/4} \sqrt [4]{b} \left (5 \sqrt{b} d+9 \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 \sqrt{a+b x^4}}+\frac{1}{4} (3 a b c) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x^2}} \, dx,x,x^2\right )+\frac{1}{4} (3 a b e) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,x^2\right )\\ &=\frac{12 a \sqrt{b} f x \sqrt{a+b x^4}}{5 \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{3}{4} b \left (c+e x^2\right ) \sqrt{a+b x^4}+\frac{2}{15} b x \left (5 d+9 f x^2\right ) \sqrt{a+b x^4}-\frac{1}{12} \left (\frac{3 c}{x^4}+\frac{4 d}{x^3}+\frac{6 e}{x^2}+\frac{12 f}{x}\right ) \left (a+b x^4\right )^{3/2}-\frac{12 a^{5/4} \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 )}{5 \sqrt{a+b x^4}}+\frac{2 a^{3/4} \sqrt [4]{b} \left (5 \sqrt{b} d+9 \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 \sqrt{a+b x^4}}+\frac{1}{8} (3 a b c) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^4\right )+\frac{1}{4} (3 a b e) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x^2}{\sqrt{a+b x^4}}\right )\\ &=\frac{12 a \sqrt{b} f x \sqrt{a+b x^4}}{5 \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{3}{4} b \left (c+e x^2\right ) \sqrt{a+b x^4}+\frac{2}{15} b x \left (5 d+9 f x^2\right ) \sqrt{a+b x^4}-\frac{1}{12} \left (\frac{3 c}{x^4}+\frac{4 d}{x^3}+\frac{6 e}{x^2}+\frac{12 f}{x}\right ) \left (a+b x^4\right )^{3/2}+\frac{3}{4} a \sqrt{b} e \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )-\frac{12 a^{5/4} \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 )}{5 \sqrt{a+b x^4}}+\frac{2 a^{3/4} \sqrt [4]{b} \left (5 \sqrt{b} d+9 \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 \sqrt{a+b x^4}}+\frac{1}{4} (3 a c) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^4}\right )\\ &=\frac{12 a \sqrt{b} f x \sqrt{a+b x^4}}{5 \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{3}{4} b \left (c+e x^2\right ) \sqrt{a+b x^4}+\frac{2}{15} b x \left (5 d+9 f x^2\right ) \sqrt{a+b x^4}-\frac{1}{12} \left (\frac{3 c}{x^4}+\frac{4 d}{x^3}+\frac{6 e}{x^2}+\frac{12 f}{x}\right ) \left (a+b x^4\right )^{3/2}+\frac{3}{4} a \sqrt{b} e \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )-\frac{3}{4} \sqrt{a} b c \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )-\frac{12 a^{5/4} \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 )}{5 \sqrt{a+b x^4}}+\frac{2 a^{3/4} \sqrt [4]{b} \left (5 \sqrt{b} d+9 \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 \sqrt{a+b x^4}}\\ \end{align*}
Mathematica [C] time = 0.218816, size = 163, normalized size = 0.42 \[ \frac{\sqrt{a+b x^4} \left (3 x \left (-5 a^3 e \, _2F_1\left (-\frac{3}{2},-\frac{1}{2};\frac{1}{2};-\frac{b x^4}{a}\right )-10 a^3 f x \, _2F_1\left (-\frac{3}{2},-\frac{1}{4};\frac{3}{4};-\frac{b x^4}{a}\right )+b c x^2 \left (a+b x^4\right )^2 \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left (2,\frac{5}{2};\frac{7}{2};\frac{b x^4}{a}+1\right )\right )-10 a^3 d \, _2F_1\left (-\frac{3}{2},-\frac{3}{4};\frac{1}{4};-\frac{b x^4}{a}\right )\right )}{30 a^2 x^3 \sqrt{\frac{b x^4}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.021, size = 409, normalized size = 1.1 \begin{align*} -{\frac{ad}{3\,{x}^{3}}\sqrt{b{x}^{4}+a}}+{\frac{bdx}{3}\sqrt{b{x}^{4}+a}}+{\frac{4\,bda}{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{bc}{2}\sqrt{b{x}^{4}+a}}-{\frac{3\,bc}{4}\sqrt{a}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{4}+a} \right ) } \right ) }-{\frac{ac}{4\,{x}^{4}}\sqrt{b{x}^{4}+a}}+{\frac{be{x}^{2}}{4}\sqrt{b{x}^{4}+a}}+{\frac{3\,ae}{4}\sqrt{b}\ln \left ({x}^{2}\sqrt{b}+\sqrt{b{x}^{4}+a} \right ) }-{\frac{ae}{2\,{x}^{2}}\sqrt{b{x}^{4}+a}}-{\frac{af}{x}\sqrt{b{x}^{4}+a}}+{\frac{bf{x}^{3}}{5}\sqrt{b{x}^{4}+a}}+{{\frac{12\,i}{5}}f{a}^{{\frac{3}{2}}}\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{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{{\frac{12\,i}{5}}f{a}^{{\frac{3}{2}}}\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{{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{{\left (b f x^{7} + b e x^{6} + b d x^{5} + b c x^{4} + a f x^{3} + a e x^{2} + a d x + a c\right )} \sqrt{b x^{4} + a}}{x^{5}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 10.7074, size = 379, normalized size = 0.98 \begin{align*} \frac{a^{\frac{3}{2}} 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 x^{3} \Gamma \left (\frac{1}{4}\right )} - \frac{a^{\frac{3}{2}} e}{2 x^{2} \sqrt{1 + \frac{b x^{4}}{a}}} + \frac{a^{\frac{3}{2}} f \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4} \\ \frac{3}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 x \Gamma \left (\frac{3}{4}\right )} - \frac{3 \sqrt{a} b c \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{4} + \frac{\sqrt{a} b d x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} + \frac{\sqrt{a} b e x^{2} \sqrt{1 + \frac{b x^{4}}{a}}}{4} - \frac{\sqrt{a} b e x^{2}}{2 \sqrt{1 + \frac{b x^{4}}{a}}} + \frac{\sqrt{a} b f x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{7}{4}\right )} - \frac{a \sqrt{b} c \sqrt{\frac{a}{b x^{4}} + 1}}{4 x^{2}} + \frac{a \sqrt{b} c}{2 x^{2} \sqrt{\frac{a}{b x^{4}} + 1}} + \frac{3 a \sqrt{b} e \operatorname{asinh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{4} + \frac{b^{\frac{3}{2}} c x^{2}}{2 \sqrt{\frac{a}{b x^{4}} + 1}} \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{{\left (b x^{4} + a\right )}^{\frac{3}{2}}{\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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