Optimal. Leaf size=314 \[ \frac{\left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (3 \sqrt{a} e+\sqrt{b} c\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{4 \sqrt [4]{a} b^{7/4} \sqrt{a+b x^4}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{2 b^{3/2}}+\frac{3 e x \sqrt{a+b x^4}}{2 b^{3/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{3 \sqrt [4]{a} e \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 b^{7/4} \sqrt{a+b x^4}}+\frac{f \sqrt{a+b x^4}}{b^2}-\frac{x \left (c+d x+e x^2+f x^3\right )}{2 b \sqrt{a+b x^4}} \]
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Rubi [A] time = 0.26789, antiderivative size = 314, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {1828, 1885, 1198, 220, 1196, 1248, 641, 217, 206} \[ \frac{\left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (3 \sqrt{a} e+\sqrt{b} c\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{a} b^{7/4} \sqrt{a+b x^4}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{2 b^{3/2}}+\frac{3 e x \sqrt{a+b x^4}}{2 b^{3/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{3 \sqrt [4]{a} e \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 b^{7/4} \sqrt{a+b x^4}}+\frac{f \sqrt{a+b x^4}}{b^2}-\frac{x \left (c+d x+e x^2+f x^3\right )}{2 b \sqrt{a+b x^4}} \]
Antiderivative was successfully verified.
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Rule 1828
Rule 1885
Rule 1198
Rule 220
Rule 1196
Rule 1248
Rule 641
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{x^4 \left (c+d x+e x^2+f x^3\right )}{\left (a+b x^4\right )^{3/2}} \, dx &=-\frac{x \left (c+d x+e x^2+f x^3\right )}{2 b \sqrt{a+b x^4}}-\frac{\int \frac{-a b c-2 a b d x-3 a b e x^2-4 a b f x^3}{\sqrt{a+b x^4}} \, dx}{2 a b^2}\\ &=-\frac{x \left (c+d x+e x^2+f x^3\right )}{2 b \sqrt{a+b x^4}}-\frac{\int \left (\frac{-a b c-3 a b e x^2}{\sqrt{a+b x^4}}+\frac{x \left (-2 a b d-4 a b f x^2\right )}{\sqrt{a+b x^4}}\right ) \, dx}{2 a b^2}\\ &=-\frac{x \left (c+d x+e x^2+f x^3\right )}{2 b \sqrt{a+b x^4}}-\frac{\int \frac{-a b c-3 a b e x^2}{\sqrt{a+b x^4}} \, dx}{2 a b^2}-\frac{\int \frac{x \left (-2 a b d-4 a b f x^2\right )}{\sqrt{a+b x^4}} \, dx}{2 a b^2}\\ &=-\frac{x \left (c+d x+e x^2+f x^3\right )}{2 b \sqrt{a+b x^4}}-\frac{\operatorname{Subst}\left (\int \frac{-2 a b d-4 a b f x}{\sqrt{a+b x^2}} \, dx,x,x^2\right )}{4 a b^2}-\frac{\left (3 \sqrt{a} e\right ) \int \frac{1-\frac{\sqrt{b} x^2}{\sqrt{a}}}{\sqrt{a+b x^4}} \, dx}{2 b^{3/2}}+\frac{\left (\sqrt{b} c+3 \sqrt{a} e\right ) \int \frac{1}{\sqrt{a+b x^4}} \, dx}{2 b^{3/2}}\\ &=-\frac{x \left (c+d x+e x^2+f x^3\right )}{2 b \sqrt{a+b x^4}}+\frac{f \sqrt{a+b x^4}}{b^2}+\frac{3 e x \sqrt{a+b x^4}}{2 b^{3/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{3 \sqrt [4]{a} 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 )}{2 b^{7/4} \sqrt{a+b x^4}}+\frac{\left (\sqrt{b} c+3 \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 )}{4 \sqrt [4]{a} b^{7/4} \sqrt{a+b x^4}}+\frac{d \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,x^2\right )}{2 b}\\ &=-\frac{x \left (c+d x+e x^2+f x^3\right )}{2 b \sqrt{a+b x^4}}+\frac{f \sqrt{a+b x^4}}{b^2}+\frac{3 e x \sqrt{a+b x^4}}{2 b^{3/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{3 \sqrt [4]{a} 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 )}{2 b^{7/4} \sqrt{a+b x^4}}+\frac{\left (\sqrt{b} c+3 \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 )}{4 \sqrt [4]{a} b^{7/4} \sqrt{a+b x^4}}+\frac{d \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x^2}{\sqrt{a+b x^4}}\right )}{2 b}\\ &=-\frac{x \left (c+d x+e x^2+f x^3\right )}{2 b \sqrt{a+b x^4}}+\frac{f \sqrt{a+b x^4}}{b^2}+\frac{3 e x \sqrt{a+b x^4}}{2 b^{3/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{2 b^{3/2}}-\frac{3 \sqrt [4]{a} 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 )}{2 b^{7/4} \sqrt{a+b x^4}}+\frac{\left (\sqrt{b} c+3 \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 )}{4 \sqrt [4]{a} b^{7/4} \sqrt{a+b x^4}}\\ \end{align*}
Mathematica [C] time = 0.145698, size = 166, normalized size = 0.53 \[ \frac{b c x \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^4}{a}\right )+\sqrt{a} \sqrt{b} d \sqrt{\frac{b x^4}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )-2 b e x^3 \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{3}{4},\frac{3}{2};\frac{7}{4};-\frac{b x^4}{a}\right )+2 a f-b c x-b d x^2+2 b e x^3+b f x^4}{2 b^2 \sqrt{a+b x^4}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.008, size = 340, normalized size = 1.1 \begin{align*}{\frac{f \left ( b{x}^{4}+2\,a \right ) }{2\,{b}^{2}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{\frac{e{x}^{3}}{2\,b}{\frac{1}{\sqrt{ \left ({x}^{4}+{\frac{a}{b}} \right ) b}}}}+{{\frac{3\,i}{2}}e\sqrt{a}\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}{2}}e\sqrt{a}\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}}{2\,b}{\frac{1}{\sqrt{b{x}^{4}+a}}}}+{\frac{d}{2}\ln \left ({x}^{2}\sqrt{b}+\sqrt{b{x}^{4}+a} \right ){b}^{-{\frac{3}{2}}}}-{\frac{cx}{2\,b}{\frac{1}{\sqrt{ \left ({x}^{4}+{\frac{a}{b}} \right ) b}}}}+{\frac{c}{2\,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}}{{\left (b x^{4} + a\right )}^{\frac{3}{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 (f x^{7} + e x^{6} + d x^{5} + c x^{4}\right )} \sqrt{b x^{4} + a}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 19.9864, size = 172, normalized size = 0.55 \begin{align*} d \left (\frac{\operatorname{asinh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2 b^{\frac{3}{2}}} - \frac{x^{2}}{2 \sqrt{a} b \sqrt{1 + \frac{b x^{4}}{a}}}\right ) + f \left (\begin{cases} \frac{a}{b^{2} \sqrt{a + b x^{4}}} + \frac{x^{4}}{2 b \sqrt{a + b x^{4}}} & \text{for}\: b \neq 0 \\\frac{x^{8}}{8 a^{\frac{3}{2}}} & \text{otherwise} \end{cases}\right ) + \frac{c x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{5}{4}, \frac{3}{2} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{3}{2}} \Gamma \left (\frac{9}{4}\right )} + \frac{e x^{7} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{3}{2}} \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}}{{\left (b x^{4} + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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