Optimal. Leaf size=354 \[ \frac{a^{5/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (21 \sqrt{b} d-5 \sqrt{a} f\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{105 b^{5/4} \sqrt{a+b x^4}}-\frac{2 a^{5/4} d \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt{a+b x^4}}+\frac{1}{4} c x^2 \sqrt{a+b x^4}+\frac{a c \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{4 \sqrt{b}}+\frac{1}{35} x^3 \sqrt{a+b x^4} \left (7 d+5 f x^2\right )+\frac{2 a d x \sqrt{a+b x^4}}{5 \sqrt{b} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{e \left (a+b x^4\right )^{3/2}}{6 b}+\frac{2 a f x \sqrt{a+b x^4}}{21 b} \]
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Rubi [A] time = 0.273206, antiderivative size = 354, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 11, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.393, Rules used = {1833, 1248, 641, 195, 217, 206, 1274, 1280, 1198, 220, 1196} \[ \frac{a^{5/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (21 \sqrt{b} d-5 \sqrt{a} f\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{105 b^{5/4} \sqrt{a+b x^4}}-\frac{2 a^{5/4} d \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt{a+b x^4}}+\frac{1}{4} c x^2 \sqrt{a+b x^4}+\frac{a c \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{4 \sqrt{b}}+\frac{1}{35} x^3 \sqrt{a+b x^4} \left (7 d+5 f x^2\right )+\frac{2 a d x \sqrt{a+b x^4}}{5 \sqrt{b} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{e \left (a+b x^4\right )^{3/2}}{6 b}+\frac{2 a f x \sqrt{a+b x^4}}{21 b} \]
Antiderivative was successfully verified.
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Rule 1833
Rule 1248
Rule 641
Rule 195
Rule 217
Rule 206
Rule 1274
Rule 1280
Rule 1198
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int x \left (c+d x+e x^2+f x^3\right ) \sqrt{a+b x^4} \, dx &=\int \left (x \left (c+e x^2\right ) \sqrt{a+b x^4}+x^2 \left (d+f x^2\right ) \sqrt{a+b x^4}\right ) \, dx\\ &=\int x \left (c+e x^2\right ) \sqrt{a+b x^4} \, dx+\int x^2 \left (d+f x^2\right ) \sqrt{a+b x^4} \, dx\\ &=\frac{1}{35} x^3 \left (7 d+5 f x^2\right ) \sqrt{a+b x^4}+\frac{1}{2} \operatorname{Subst}\left (\int (c+e x) \sqrt{a+b x^2} \, dx,x,x^2\right )+\frac{1}{35} (2 a) \int \frac{x^2 \left (7 d+5 f x^2\right )}{\sqrt{a+b x^4}} \, dx\\ &=\frac{2 a f x \sqrt{a+b x^4}}{21 b}+\frac{1}{35} x^3 \left (7 d+5 f x^2\right ) \sqrt{a+b x^4}+\frac{e \left (a+b x^4\right )^{3/2}}{6 b}-\frac{(2 a) \int \frac{5 a f-21 b d x^2}{\sqrt{a+b x^4}} \, dx}{105 b}+\frac{1}{2} c \operatorname{Subst}\left (\int \sqrt{a+b x^2} \, dx,x,x^2\right )\\ &=\frac{2 a f x \sqrt{a+b x^4}}{21 b}+\frac{1}{4} c x^2 \sqrt{a+b x^4}+\frac{1}{35} x^3 \left (7 d+5 f x^2\right ) \sqrt{a+b x^4}+\frac{e \left (a+b x^4\right )^{3/2}}{6 b}+\frac{1}{4} (a c) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,x^2\right )-\frac{\left (2 a^{3/2} d\right ) \int \frac{1-\frac{\sqrt{b} x^2}{\sqrt{a}}}{\sqrt{a+b x^4}} \, dx}{5 \sqrt{b}}+\frac{\left (2 a^{3/2} \left (21 \sqrt{b} d-5 \sqrt{a} f\right )\right ) \int \frac{1}{\sqrt{a+b x^4}} \, dx}{105 b}\\ &=\frac{2 a f x \sqrt{a+b x^4}}{21 b}+\frac{1}{4} c x^2 \sqrt{a+b x^4}+\frac{2 a d x \sqrt{a+b x^4}}{5 \sqrt{b} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{1}{35} x^3 \left (7 d+5 f x^2\right ) \sqrt{a+b x^4}+\frac{e \left (a+b x^4\right )^{3/2}}{6 b}-\frac{2 a^{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 b^{3/4} \sqrt{a+b x^4}}+\frac{a^{5/4} \left (21 \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 )}{105 b^{5/4} \sqrt{a+b x^4}}+\frac{1}{4} (a c) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x^2}{\sqrt{a+b x^4}}\right )\\ &=\frac{2 a f x \sqrt{a+b x^4}}{21 b}+\frac{1}{4} c x^2 \sqrt{a+b x^4}+\frac{2 a d x \sqrt{a+b x^4}}{5 \sqrt{b} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{1}{35} x^3 \left (7 d+5 f x^2\right ) \sqrt{a+b x^4}+\frac{e \left (a+b x^4\right )^{3/2}}{6 b}+\frac{a c \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{4 \sqrt{b}}-\frac{2 a^{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 b^{3/4} \sqrt{a+b x^4}}+\frac{a^{5/4} \left (21 \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 )}{105 b^{5/4} \sqrt{a+b x^4}}\\ \end{align*}
Mathematica [C] time = 0.184122, size = 211, normalized size = 0.6 \[ \frac{\sqrt{a+b x^4} \left (21 b c x^2 \sqrt{\frac{b x^4}{a}+1}+21 \sqrt{a} \sqrt{b} c \sinh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )+28 b d x^3 \, _2F_1\left (-\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )+14 b e x^4 \sqrt{\frac{b x^4}{a}+1}+14 a e \sqrt{\frac{b x^4}{a}+1}-12 a f x \, _2F_1\left (-\frac{1}{2},\frac{1}{4};\frac{5}{4};-\frac{b x^4}{a}\right )+12 b f x^5 \sqrt{\frac{b x^4}{a}+1}+12 a f x \sqrt{\frac{b x^4}{a}+1}\right )}{84 b \sqrt{\frac{b x^4}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.013, size = 337, normalized size = 1. \begin{align*}{\frac{f{x}^{5}}{7}\sqrt{b{x}^{4}+a}}+{\frac{2\,afx}{21\,b}\sqrt{b{x}^{4}+a}}-{\frac{2\,f{a}^{2}}{21\,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{e}{6\,b} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{2}}}}+{\frac{d{x}^{3}}{5}\sqrt{b{x}^{4}+a}}+{{\frac{2\,i}{5}}d{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 ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}{\frac{1}{\sqrt{b}}}}-{{\frac{2\,i}{5}}d{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 ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}{\frac{1}{\sqrt{b}}}}+{\frac{c{x}^{2}}{4}\sqrt{b{x}^{4}+a}}+{\frac{ac}{4}\ln \left ({x}^{2}\sqrt{b}+\sqrt{b{x}^{4}+a} \right ){\frac{1}{\sqrt{b}}}} \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 (\sqrt{b x^{4} + a}{\left (f x^{4} + e x^{3} + d x^{2} + c x\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.28085, size = 158, normalized size = 0.45 \begin{align*} \frac{\sqrt{a} c x^{2} \sqrt{1 + \frac{b x^{4}}{a}}}{4} + \frac{\sqrt{a} d 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} f 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{a c \operatorname{asinh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{4 \sqrt{b}} + e \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 ) \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 \sqrt{b x^{4} + a}{\left (f x^{3} + e x^{2} + d x + c\right )} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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