Optimal. Leaf size=361 \[ -\frac{a^{3/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (9 \sqrt{a} e+5 \sqrt{b} c\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{30 b^{7/4} \sqrt{a+b x^4}}+\frac{3 a^{5/4} e \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 b^{7/4} \sqrt{a+b x^4}}-\frac{\sqrt{a+b x^4} \left (4 a f-3 b d x^2\right )}{12 b^2}-\frac{a d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{4 b^{3/2}}-\frac{3 a e x \sqrt{a+b x^4}}{5 b^{3/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{c x \sqrt{a+b x^4}}{3 b}+\frac{e x^3 \sqrt{a+b x^4}}{5 b}+\frac{f x^4 \sqrt{a+b x^4}}{6 b} \]
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Rubi [A] time = 0.30039, antiderivative size = 361, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {1833, 1280, 1198, 220, 1196, 1252, 833, 780, 217, 206} \[ -\frac{a^{3/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (9 \sqrt{a} e+5 \sqrt{b} c\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{30 b^{7/4} \sqrt{a+b x^4}}+\frac{3 a^{5/4} e \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 b^{7/4} \sqrt{a+b x^4}}-\frac{\sqrt{a+b x^4} \left (4 a f-3 b d x^2\right )}{12 b^2}-\frac{a d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{4 b^{3/2}}-\frac{3 a e x \sqrt{a+b x^4}}{5 b^{3/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{c x \sqrt{a+b x^4}}{3 b}+\frac{e x^3 \sqrt{a+b x^4}}{5 b}+\frac{f x^4 \sqrt{a+b x^4}}{6 b} \]
Antiderivative was successfully verified.
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Rule 1833
Rule 1280
Rule 1198
Rule 220
Rule 1196
Rule 1252
Rule 833
Rule 780
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{x^4 \left (c+d x+e x^2+f x^3\right )}{\sqrt{a+b x^4}} \, dx &=\int \left (\frac{x^4 \left (c+e x^2\right )}{\sqrt{a+b x^4}}+\frac{x^5 \left (d+f x^2\right )}{\sqrt{a+b x^4}}\right ) \, dx\\ &=\int \frac{x^4 \left (c+e x^2\right )}{\sqrt{a+b x^4}} \, dx+\int \frac{x^5 \left (d+f x^2\right )}{\sqrt{a+b x^4}} \, dx\\ &=\frac{e x^3 \sqrt{a+b x^4}}{5 b}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2 (d+f x)}{\sqrt{a+b x^2}} \, dx,x,x^2\right )-\frac{\int \frac{x^2 \left (3 a e-5 b c x^2\right )}{\sqrt{a+b x^4}} \, dx}{5 b}\\ &=\frac{c x \sqrt{a+b x^4}}{3 b}+\frac{e x^3 \sqrt{a+b x^4}}{5 b}+\frac{f x^4 \sqrt{a+b x^4}}{6 b}+\frac{\int \frac{-5 a b c-9 a b e x^2}{\sqrt{a+b x^4}} \, dx}{15 b^2}+\frac{\operatorname{Subst}\left (\int \frac{x (-2 a f+3 b d x)}{\sqrt{a+b x^2}} \, dx,x,x^2\right )}{6 b}\\ &=\frac{c x \sqrt{a+b x^4}}{3 b}+\frac{e x^3 \sqrt{a+b x^4}}{5 b}+\frac{f x^4 \sqrt{a+b x^4}}{6 b}-\frac{\left (4 a f-3 b d x^2\right ) \sqrt{a+b x^4}}{12 b^2}-\frac{(a d) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,x^2\right )}{4 b}+\frac{\left (3 a^{3/2} e\right ) \int \frac{1-\frac{\sqrt{b} x^2}{\sqrt{a}}}{\sqrt{a+b x^4}} \, dx}{5 b^{3/2}}-\frac{\left (a \left (5 \sqrt{b} c+9 \sqrt{a} e\right )\right ) \int \frac{1}{\sqrt{a+b x^4}} \, dx}{15 b^{3/2}}\\ &=\frac{c x \sqrt{a+b x^4}}{3 b}+\frac{e x^3 \sqrt{a+b x^4}}{5 b}+\frac{f x^4 \sqrt{a+b x^4}}{6 b}-\frac{3 a e x \sqrt{a+b x^4}}{5 b^{3/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{\left (4 a f-3 b d x^2\right ) \sqrt{a+b x^4}}{12 b^2}+\frac{3 a^{5/4} e \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 b^{7/4} \sqrt{a+b x^4}}-\frac{a^{3/4} \left (5 \sqrt{b} c+9 \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 )}{30 b^{7/4} \sqrt{a+b x^4}}-\frac{(a d) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x^2}{\sqrt{a+b x^4}}\right )}{4 b}\\ &=\frac{c x \sqrt{a+b x^4}}{3 b}+\frac{e x^3 \sqrt{a+b x^4}}{5 b}+\frac{f x^4 \sqrt{a+b x^4}}{6 b}-\frac{3 a e x \sqrt{a+b x^4}}{5 b^{3/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{\left (4 a f-3 b d x^2\right ) \sqrt{a+b x^4}}{12 b^2}-\frac{a d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{4 b^{3/2}}+\frac{3 a^{5/4} e \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 b^{7/4} \sqrt{a+b x^4}}-\frac{a^{3/4} \left (5 \sqrt{b} c+9 \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 )}{30 b^{7/4} \sqrt{a+b x^4}}\\ \end{align*}
Mathematica [C] time = 0.130861, size = 212, normalized size = 0.59 \[ \frac{-20 a^2 f-20 a b c x \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^4}{a}\right )+20 a b c x+15 a b d x^2-15 a \sqrt{b} d \sqrt{a+b x^4} \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )-12 a b e x^3 \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )+12 a b e x^3-10 a b f x^4+20 b^2 c x^5+15 b^2 d x^6+12 b^2 e x^7+10 b^2 f x^8}{60 b^2 \sqrt{a+b x^4}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.018, size = 335, normalized size = 0.9 \begin{align*} -{\frac{f \left ( -b{x}^{4}+2\,a \right ) }{6\,{b}^{2}}\sqrt{b{x}^{4}+a}}+{\frac{e{x}^{3}}{5\,b}\sqrt{b{x}^{4}+a}}-{{\frac{3\,i}{5}}e{a}^{{\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 ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}+{{\frac{3\,i}{5}}e{a}^{{\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 ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}+{\frac{d{x}^{2}}{4\,b}\sqrt{b{x}^{4}+a}}-{\frac{ad}{4}\ln \left ({x}^{2}\sqrt{b}+\sqrt{b{x}^{4}+a} \right ){b}^{-{\frac{3}{2}}}}+{\frac{cx}{3\,b}\sqrt{b{x}^{4}+a}}-{\frac{ac}{3\,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}}}} \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 (f x^{3} + e x^{2} + d x + c\right )} x^{4}}{\sqrt{b x^{4} + a}}\,{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{f x^{7} + e x^{6} + d x^{5} + c x^{4}}{\sqrt{b x^{4} + a}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.13838, size = 177, normalized size = 0.49 \begin{align*} \frac{\sqrt{a} d x^{2} \sqrt{1 + \frac{b x^{4}}{a}}}{4 b} - \frac{a d \operatorname{asinh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{4 b^{\frac{3}{2}}} + f \left (\begin{cases} - \frac{a \sqrt{a + b x^{4}}}{3 b^{2}} + \frac{x^{4} \sqrt{a + b x^{4}}}{6 b} & \text{for}\: b \neq 0 \\\frac{x^{8}}{8 \sqrt{a}} & \text{otherwise} \end{cases}\right ) + \frac{c 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 \sqrt{a} \Gamma \left (\frac{9}{4}\right )} + \frac{e x^{7} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{11}{4}\right )} \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 (f x^{3} + e x^{2} + d x + c\right )} x^{4}}{\sqrt{b x^{4} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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