Optimal. Leaf size=336 \[ -\frac{a^{3/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (9 \sqrt{a} f+5 \sqrt{b} d\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{30 b^{7/4} \sqrt{a+b x^4}}+\frac{3 a^{5/4} f \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 b^{7/4} \sqrt{a+b x^4}}-\frac{a e \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{4 b^{3/2}}-\frac{3 a f x \sqrt{a+b x^4}}{5 b^{3/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{\sqrt{a+b x^4} \left (2 c+e x^2\right )}{4 b}+\frac{d x \sqrt{a+b x^4}}{3 b}+\frac{f x^3 \sqrt{a+b x^4}}{5 b} \]
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Rubi [A] time = 0.265053, antiderivative size = 336, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {1833, 1252, 780, 217, 206, 1280, 1198, 220, 1196} \[ -\frac{a^{3/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (9 \sqrt{a} f+5 \sqrt{b} d\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{30 b^{7/4} \sqrt{a+b x^4}}+\frac{3 a^{5/4} f \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 b^{7/4} \sqrt{a+b x^4}}-\frac{a e \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{4 b^{3/2}}-\frac{3 a f x \sqrt{a+b x^4}}{5 b^{3/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{\sqrt{a+b x^4} \left (2 c+e x^2\right )}{4 b}+\frac{d x \sqrt{a+b x^4}}{3 b}+\frac{f x^3 \sqrt{a+b x^4}}{5 b} \]
Antiderivative was successfully verified.
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Rule 1833
Rule 1252
Rule 780
Rule 217
Rule 206
Rule 1280
Rule 1198
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{x^3 \left (c+d x+e x^2+f x^3\right )}{\sqrt{a+b x^4}} \, dx &=\int \left (\frac{x^3 \left (c+e x^2\right )}{\sqrt{a+b x^4}}+\frac{x^4 \left (d+f x^2\right )}{\sqrt{a+b x^4}}\right ) \, dx\\ &=\int \frac{x^3 \left (c+e x^2\right )}{\sqrt{a+b x^4}} \, dx+\int \frac{x^4 \left (d+f x^2\right )}{\sqrt{a+b x^4}} \, dx\\ &=\frac{f x^3 \sqrt{a+b x^4}}{5 b}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{x (c+e x)}{\sqrt{a+b x^2}} \, dx,x,x^2\right )-\frac{\int \frac{x^2 \left (3 a f-5 b d x^2\right )}{\sqrt{a+b x^4}} \, dx}{5 b}\\ &=\frac{d x \sqrt{a+b x^4}}{3 b}+\frac{f x^3 \sqrt{a+b x^4}}{5 b}+\frac{\left (2 c+e x^2\right ) \sqrt{a+b x^4}}{4 b}+\frac{\int \frac{-5 a b d-9 a b f x^2}{\sqrt{a+b x^4}} \, dx}{15 b^2}-\frac{(a e) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,x^2\right )}{4 b}\\ &=\frac{d x \sqrt{a+b x^4}}{3 b}+\frac{f x^3 \sqrt{a+b x^4}}{5 b}+\frac{\left (2 c+e x^2\right ) \sqrt{a+b x^4}}{4 b}-\frac{(a e) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x^2}{\sqrt{a+b x^4}}\right )}{4 b}+\frac{\left (3 a^{3/2} f\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} d+9 \sqrt{a} f\right )\right ) \int \frac{1}{\sqrt{a+b x^4}} \, dx}{15 b^{3/2}}\\ &=\frac{d x \sqrt{a+b x^4}}{3 b}+\frac{f x^3 \sqrt{a+b x^4}}{5 b}-\frac{3 a f x \sqrt{a+b x^4}}{5 b^{3/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{\left (2 c+e x^2\right ) \sqrt{a+b x^4}}{4 b}-\frac{a e \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{4 b^{3/2}}+\frac{3 a^{5/4} 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 b^{7/4} \sqrt{a+b x^4}}-\frac{a^{3/4} \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 )}{30 b^{7/4} \sqrt{a+b x^4}}\\ \end{align*}
Mathematica [C] time = 0.145449, size = 212, normalized size = 0.63 \[ \frac{30 \sqrt{b} c \left (a+b x^4\right )-20 a \sqrt{b} d x \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^4}{a}\right )+20 \sqrt{b} d x \left (a+b x^4\right )+15 \sqrt{b} e x^2 \left (a+b x^4\right )-15 a e \sqrt{a+b x^4} \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )-12 a \sqrt{b} f x^3 \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )+12 \sqrt{b} f x^3 \left (a+b x^4\right )}{60 b^{3/2} \sqrt{a+b x^4}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.007, size = 325, normalized size = 1. \begin{align*}{\frac{f{x}^{3}}{5\,b}\sqrt{b{x}^{4}+a}}-{{\frac{3\,i}{5}}f{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}}f{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{e{x}^{2}}{4\,b}\sqrt{b{x}^{4}+a}}-{\frac{ae}{4}\ln \left ({x}^{2}\sqrt{b}+\sqrt{b{x}^{4}+a} \right ){b}^{-{\frac{3}{2}}}}+{\frac{dx}{3\,b}\sqrt{b{x}^{4}+a}}-{\frac{ad}{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}}}}+{\frac{c}{2\,b}\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*} \frac{\sqrt{b x^{4} + a} c}{2 \, b} + \int \frac{f x^{6} + e x^{5} + d 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^{6} + e x^{5} + d x^{4} + c x^{3}}{\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 = 5.02286, size = 156, normalized size = 0.46 \begin{align*} \frac{\sqrt{a} e x^{2} \sqrt{1 + \frac{b x^{4}}{a}}}{4 b} - \frac{a e \operatorname{asinh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{4 b^{\frac{3}{2}}} + c \left (\begin{cases} \frac{x^{4}}{4 \sqrt{a}} & \text{for}\: b = 0 \\\frac{\sqrt{a + b x^{4}}}{2 b} & \text{otherwise} \end{cases}\right ) + \frac{d 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{f 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^{3}}{\sqrt{b x^{4} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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