Optimal. Leaf size=404 \[ \frac{2 a^{5/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (5 \sqrt{a} e+21 \sqrt{b} c\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{35 \sqrt [4]{b} \sqrt{a+b x^4}}-\frac{12 a^{5/4} \sqrt [4]{b} c \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 \sqrt{a+b x^4}}-\frac{1}{2} a^{3/2} d \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )+\frac{3 a^2 f \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{16 \sqrt{b}}-\frac{\left (a+b x^4\right )^{3/2} \left (7 c-e x^2\right )}{7 x}+\frac{2}{35} x \sqrt{a+b x^4} \left (5 a e+21 b c x^2\right )+\frac{12 a \sqrt{b} c x \sqrt{a+b x^4}}{5 \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{1}{16} a \sqrt{a+b x^4} \left (8 d+3 f x^2\right )+\frac{1}{24} \left (a+b x^4\right )^{3/2} \left (4 d+3 f x^2\right ) \]
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Rubi [A] time = 0.348052, antiderivative size = 404, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 14, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467, Rules used = {1833, 1272, 1177, 1198, 220, 1196, 1252, 815, 844, 217, 206, 266, 63, 208} \[ \frac{2 a^{5/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (5 \sqrt{a} e+21 \sqrt{b} c\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{35 \sqrt [4]{b} \sqrt{a+b x^4}}-\frac{12 a^{5/4} \sqrt [4]{b} c \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 \sqrt{a+b x^4}}-\frac{1}{2} a^{3/2} d \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )+\frac{3 a^2 f \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{16 \sqrt{b}}-\frac{\left (a+b x^4\right )^{3/2} \left (7 c-e x^2\right )}{7 x}+\frac{2}{35} x \sqrt{a+b x^4} \left (5 a e+21 b c x^2\right )+\frac{12 a \sqrt{b} c x \sqrt{a+b x^4}}{5 \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{1}{16} a \sqrt{a+b x^4} \left (8 d+3 f x^2\right )+\frac{1}{24} \left (a+b x^4\right )^{3/2} \left (4 d+3 f x^2\right ) \]
Antiderivative was successfully verified.
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Rule 1833
Rule 1272
Rule 1177
Rule 1198
Rule 220
Rule 1196
Rule 1252
Rule 815
Rule 844
Rule 217
Rule 206
Rule 266
Rule 63
Rule 208
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^2} \, dx &=\int \left (\frac{\left (c+e x^2\right ) \left (a+b x^4\right )^{3/2}}{x^2}+\frac{\left (d+f x^2\right ) \left (a+b x^4\right )^{3/2}}{x}\right ) \, dx\\ &=\int \frac{\left (c+e x^2\right ) \left (a+b x^4\right )^{3/2}}{x^2} \, dx+\int \frac{\left (d+f x^2\right ) \left (a+b x^4\right )^{3/2}}{x} \, dx\\ &=-\frac{\left (7 c-e x^2\right ) \left (a+b x^4\right )^{3/2}}{7 x}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{(d+f x) \left (a+b x^2\right )^{3/2}}{x} \, dx,x,x^2\right )-\frac{6}{7} \int \left (-a e-7 b c x^2\right ) \sqrt{a+b x^4} \, dx\\ &=\frac{2}{35} x \left (5 a e+21 b c x^2\right ) \sqrt{a+b x^4}-\frac{\left (7 c-e x^2\right ) \left (a+b x^4\right )^{3/2}}{7 x}+\frac{1}{24} \left (4 d+3 f x^2\right ) \left (a+b x^4\right )^{3/2}-\frac{2}{35} \int \frac{-10 a^2 e-42 a b c x^2}{\sqrt{a+b x^4}} \, dx+\frac{\operatorname{Subst}\left (\int \frac{(4 a b d+3 a b f x) \sqrt{a+b x^2}}{x} \, dx,x,x^2\right )}{8 b}\\ &=\frac{2}{35} x \left (5 a e+21 b c x^2\right ) \sqrt{a+b x^4}+\frac{1}{16} a \left (8 d+3 f x^2\right ) \sqrt{a+b x^4}-\frac{\left (7 c-e x^2\right ) \left (a+b x^4\right )^{3/2}}{7 x}+\frac{1}{24} \left (4 d+3 f x^2\right ) \left (a+b x^4\right )^{3/2}+\frac{\operatorname{Subst}\left (\int \frac{8 a^2 b^2 d+3 a^2 b^2 f x}{x \sqrt{a+b x^2}} \, dx,x,x^2\right )}{16 b^2}-\frac{1}{5} \left (12 a^{3/2} \sqrt{b} c\right ) \int \frac{1-\frac{\sqrt{b} x^2}{\sqrt{a}}}{\sqrt{a+b x^4}} \, dx+\frac{1}{35} \left (4 a^{3/2} \left (21 \sqrt{b} c+5 \sqrt{a} e\right )\right ) \int \frac{1}{\sqrt{a+b x^4}} \, dx\\ &=\frac{12 a \sqrt{b} c x \sqrt{a+b x^4}}{5 \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{2}{35} x \left (5 a e+21 b c x^2\right ) \sqrt{a+b x^4}+\frac{1}{16} a \left (8 d+3 f x^2\right ) \sqrt{a+b x^4}-\frac{\left (7 c-e x^2\right ) \left (a+b x^4\right )^{3/2}}{7 x}+\frac{1}{24} \left (4 d+3 f x^2\right ) \left (a+b x^4\right )^{3/2}-\frac{12 a^{5/4} \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 )}{5 \sqrt{a+b x^4}}+\frac{2 a^{5/4} \left (21 \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 )}{35 \sqrt [4]{b} \sqrt{a+b x^4}}+\frac{1}{2} \left (a^2 d\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x^2}} \, dx,x,x^2\right )+\frac{1}{16} \left (3 a^2 f\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,x^2\right )\\ &=\frac{12 a \sqrt{b} c x \sqrt{a+b x^4}}{5 \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{2}{35} x \left (5 a e+21 b c x^2\right ) \sqrt{a+b x^4}+\frac{1}{16} a \left (8 d+3 f x^2\right ) \sqrt{a+b x^4}-\frac{\left (7 c-e x^2\right ) \left (a+b x^4\right )^{3/2}}{7 x}+\frac{1}{24} \left (4 d+3 f x^2\right ) \left (a+b x^4\right )^{3/2}-\frac{12 a^{5/4} \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 )}{5 \sqrt{a+b x^4}}+\frac{2 a^{5/4} \left (21 \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 )}{35 \sqrt [4]{b} \sqrt{a+b x^4}}+\frac{1}{4} \left (a^2 d\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^4\right )+\frac{1}{16} \left (3 a^2 f\right ) \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} c x \sqrt{a+b x^4}}{5 \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{2}{35} x \left (5 a e+21 b c x^2\right ) \sqrt{a+b x^4}+\frac{1}{16} a \left (8 d+3 f x^2\right ) \sqrt{a+b x^4}-\frac{\left (7 c-e x^2\right ) \left (a+b x^4\right )^{3/2}}{7 x}+\frac{1}{24} \left (4 d+3 f x^2\right ) \left (a+b x^4\right )^{3/2}+\frac{3 a^2 f \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{16 \sqrt{b}}-\frac{12 a^{5/4} \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 )}{5 \sqrt{a+b x^4}}+\frac{2 a^{5/4} \left (21 \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 )}{35 \sqrt [4]{b} \sqrt{a+b x^4}}+\frac{\left (a^2 d\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^4}\right )}{2 b}\\ &=\frac{12 a \sqrt{b} c x \sqrt{a+b x^4}}{5 \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{2}{35} x \left (5 a e+21 b c x^2\right ) \sqrt{a+b x^4}+\frac{1}{16} a \left (8 d+3 f x^2\right ) \sqrt{a+b x^4}-\frac{\left (7 c-e x^2\right ) \left (a+b x^4\right )^{3/2}}{7 x}+\frac{1}{24} \left (4 d+3 f x^2\right ) \left (a+b x^4\right )^{3/2}+\frac{3 a^2 f \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{16 \sqrt{b}}-\frac{1}{2} a^{3/2} d \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )-\frac{12 a^{5/4} \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 )}{5 \sqrt{a+b x^4}}+\frac{2 a^{5/4} \left (21 \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 )}{35 \sqrt [4]{b} \sqrt{a+b x^4}}\\ \end{align*}
Mathematica [C] time = 0.523102, size = 222, normalized size = 0.55 \[ \frac{1}{6} d \left (\sqrt{a+b x^4} \left (4 a+b x^4\right )-3 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )\right )+\frac{1}{16} f \sqrt{a+b x^4} \left (\frac{3 a^{3/2} \sinh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{\sqrt{b} \sqrt{\frac{b x^4}{a}+1}}+5 a x^2+2 b x^6\right )-\frac{a c \sqrt{a+b x^4} \, _2F_1\left (-\frac{3}{2},-\frac{1}{4};\frac{3}{4};-\frac{b x^4}{a}\right )}{x \sqrt{\frac{b x^4}{a}+1}}+\frac{a e x \sqrt{a+b x^4} \, _2F_1\left (-\frac{3}{2},\frac{1}{4};\frac{5}{4};-\frac{b x^4}{a}\right )}{\sqrt{\frac{b x^4}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.013, size = 411, normalized size = 1. \begin{align*}{\frac{bf{x}^{6}}{8}\sqrt{b{x}^{4}+a}}+{\frac{5\,af{x}^{2}}{16}\sqrt{b{x}^{4}+a}}+{\frac{3\,f{a}^{2}}{16}\ln \left ({x}^{2}\sqrt{b}+\sqrt{b{x}^{4}+a} \right ){\frac{1}{\sqrt{b}}}}+{\frac{be{x}^{5}}{7}\sqrt{b{x}^{4}+a}}+{\frac{3\,aex}{7}\sqrt{b{x}^{4}+a}}+{\frac{4\,{a}^{2}e}{7}\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{bd{x}^{4}}{6}\sqrt{b{x}^{4}+a}}+{\frac{2\,ad}{3}\sqrt{b{x}^{4}+a}}-{\frac{d}{2}{a}^{{\frac{3}{2}}}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{4}+a} \right ) } \right ) }-{\frac{ac}{x}\sqrt{b{x}^{4}+a}}+{\frac{bc{x}^{3}}{5}\sqrt{b{x}^{4}+a}}+{{\frac{12\,i}{5}}c{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}}c{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] 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^{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{{\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^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 11.26, size = 406, normalized size = 1. \begin{align*} \frac{a^{\frac{3}{2}} c \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{a^{\frac{3}{2}} d \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2} + \frac{a^{\frac{3}{2}} e 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{a^{\frac{3}{2}} f x^{2} \sqrt{1 + \frac{b x^{4}}{a}}}{4} + \frac{a^{\frac{3}{2}} f x^{2}}{16 \sqrt{1 + \frac{b x^{4}}{a}}} + \frac{\sqrt{a} b c 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{\sqrt{a} b e x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{9}{4}\right )} + \frac{3 \sqrt{a} b f x^{6}}{16 \sqrt{1 + \frac{b x^{4}}{a}}} + \frac{a^{2} d}{2 \sqrt{b} x^{2} \sqrt{\frac{a}{b x^{4}} + 1}} + \frac{3 a^{2} f \operatorname{asinh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{16 \sqrt{b}} + \frac{a \sqrt{b} d x^{2}}{2 \sqrt{\frac{a}{b x^{4}} + 1}} + b d \left (\begin{cases} \frac{\sqrt{a} x^{4}}{4} & \text{for}\: b = 0 \\\frac{\left (a + b x^{4}\right )^{\frac{3}{2}}}{6 b} & \text{otherwise} \end{cases}\right ) + \frac{b^{2} f x^{10}}{8 \sqrt{a} \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{{\left (b x^{4} + a\right )}^{\frac{3}{2}}{\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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