Optimal. Leaf size=418 \[ -\frac{a^{7/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (7 \sqrt{a} e+5 \sqrt{b} c\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{105 b^{7/4} \sqrt{a+b x^4}}-\frac{a^2 d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{16 b^{3/2}}-\frac{2 a^2 e x \sqrt{a+b x^4}}{15 b^{3/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{2 a^{9/4} e \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 b^{7/4} \sqrt{a+b x^4}}-\frac{\left (a+b x^4\right )^{3/2} \left (8 a f-15 b d x^2\right )}{120 b^2}+\frac{1}{63} x^5 \sqrt{a+b x^4} \left (9 c+7 e x^2\right )+\frac{2 a c x \sqrt{a+b x^4}}{21 b}-\frac{a d x^2 \sqrt{a+b x^4}}{16 b}+\frac{2 a e x^3 \sqrt{a+b x^4}}{45 b}+\frac{f x^4 \left (a+b x^4\right )^{3/2}}{10 b} \]
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Rubi [A] time = 0.381155, antiderivative size = 418, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 12, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {1833, 1274, 1280, 1198, 220, 1196, 1252, 833, 780, 195, 217, 206} \[ -\frac{a^{7/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (7 \sqrt{a} e+5 \sqrt{b} c\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{105 b^{7/4} \sqrt{a+b x^4}}-\frac{a^2 d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{16 b^{3/2}}-\frac{2 a^2 e x \sqrt{a+b x^4}}{15 b^{3/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{2 a^{9/4} e \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 b^{7/4} \sqrt{a+b x^4}}-\frac{\left (a+b x^4\right )^{3/2} \left (8 a f-15 b d x^2\right )}{120 b^2}+\frac{1}{63} x^5 \sqrt{a+b x^4} \left (9 c+7 e x^2\right )+\frac{2 a c x \sqrt{a+b x^4}}{21 b}-\frac{a d x^2 \sqrt{a+b x^4}}{16 b}+\frac{2 a e x^3 \sqrt{a+b x^4}}{45 b}+\frac{f x^4 \left (a+b x^4\right )^{3/2}}{10 b} \]
Antiderivative was successfully verified.
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Rule 1833
Rule 1274
Rule 1280
Rule 1198
Rule 220
Rule 1196
Rule 1252
Rule 833
Rule 780
Rule 195
Rule 217
Rule 206
Rubi steps
\begin{align*} \int x^4 \left (c+d x+e x^2+f x^3\right ) \sqrt{a+b x^4} \, dx &=\int \left (x^4 \left (c+e x^2\right ) \sqrt{a+b x^4}+x^5 \left (d+f x^2\right ) \sqrt{a+b x^4}\right ) \, dx\\ &=\int x^4 \left (c+e x^2\right ) \sqrt{a+b x^4} \, dx+\int x^5 \left (d+f x^2\right ) \sqrt{a+b x^4} \, dx\\ &=\frac{1}{63} x^5 \left (9 c+7 e x^2\right ) \sqrt{a+b x^4}+\frac{1}{2} \operatorname{Subst}\left (\int x^2 (d+f x) \sqrt{a+b x^2} \, dx,x,x^2\right )+\frac{1}{63} (2 a) \int \frac{x^4 \left (9 c+7 e x^2\right )}{\sqrt{a+b x^4}} \, dx\\ &=\frac{2 a e x^3 \sqrt{a+b x^4}}{45 b}+\frac{1}{63} x^5 \left (9 c+7 e x^2\right ) \sqrt{a+b x^4}+\frac{f x^4 \left (a+b x^4\right )^{3/2}}{10 b}+\frac{\operatorname{Subst}\left (\int x (-2 a f+5 b d x) \sqrt{a+b x^2} \, dx,x,x^2\right )}{10 b}-\frac{(2 a) \int \frac{x^2 \left (21 a e-45 b c x^2\right )}{\sqrt{a+b x^4}} \, dx}{315 b}\\ &=\frac{2 a c x \sqrt{a+b x^4}}{21 b}+\frac{2 a e x^3 \sqrt{a+b x^4}}{45 b}+\frac{1}{63} x^5 \left (9 c+7 e x^2\right ) \sqrt{a+b x^4}+\frac{f x^4 \left (a+b x^4\right )^{3/2}}{10 b}-\frac{\left (8 a f-15 b d x^2\right ) \left (a+b x^4\right )^{3/2}}{120 b^2}+\frac{(2 a) \int \frac{-45 a b c-63 a b e x^2}{\sqrt{a+b x^4}} \, dx}{945 b^2}-\frac{(a d) \operatorname{Subst}\left (\int \sqrt{a+b x^2} \, dx,x,x^2\right )}{8 b}\\ &=\frac{2 a c x \sqrt{a+b x^4}}{21 b}-\frac{a d x^2 \sqrt{a+b x^4}}{16 b}+\frac{2 a e x^3 \sqrt{a+b x^4}}{45 b}+\frac{1}{63} x^5 \left (9 c+7 e x^2\right ) \sqrt{a+b x^4}+\frac{f x^4 \left (a+b x^4\right )^{3/2}}{10 b}-\frac{\left (8 a f-15 b d x^2\right ) \left (a+b x^4\right )^{3/2}}{120 b^2}-\frac{\left (a^2 d\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,x^2\right )}{16 b}+\frac{\left (2 a^{5/2} e\right ) \int \frac{1-\frac{\sqrt{b} x^2}{\sqrt{a}}}{\sqrt{a+b x^4}} \, dx}{15 b^{3/2}}-\frac{\left (2 a^2 \left (5 \sqrt{b} c+7 \sqrt{a} e\right )\right ) \int \frac{1}{\sqrt{a+b x^4}} \, dx}{105 b^{3/2}}\\ &=\frac{2 a c x \sqrt{a+b x^4}}{21 b}-\frac{a d x^2 \sqrt{a+b x^4}}{16 b}+\frac{2 a e x^3 \sqrt{a+b x^4}}{45 b}-\frac{2 a^2 e x \sqrt{a+b x^4}}{15 b^{3/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{1}{63} x^5 \left (9 c+7 e x^2\right ) \sqrt{a+b x^4}+\frac{f x^4 \left (a+b x^4\right )^{3/2}}{10 b}-\frac{\left (8 a f-15 b d x^2\right ) \left (a+b x^4\right )^{3/2}}{120 b^2}+\frac{2 a^{9/4} 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 )}{15 b^{7/4} \sqrt{a+b x^4}}-\frac{a^{7/4} \left (5 \sqrt{b} c+7 \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 )}{105 b^{7/4} \sqrt{a+b x^4}}-\frac{\left (a^2 d\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x^2}{\sqrt{a+b x^4}}\right )}{16 b}\\ &=\frac{2 a c x \sqrt{a+b x^4}}{21 b}-\frac{a d x^2 \sqrt{a+b x^4}}{16 b}+\frac{2 a e x^3 \sqrt{a+b x^4}}{45 b}-\frac{2 a^2 e x \sqrt{a+b x^4}}{15 b^{3/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{1}{63} x^5 \left (9 c+7 e x^2\right ) \sqrt{a+b x^4}+\frac{f x^4 \left (a+b x^4\right )^{3/2}}{10 b}-\frac{\left (8 a f-15 b d x^2\right ) \left (a+b x^4\right )^{3/2}}{120 b^2}-\frac{a^2 d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{16 b^{3/2}}+\frac{2 a^{9/4} 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 )}{15 b^{7/4} \sqrt{a+b x^4}}-\frac{a^{7/4} \left (5 \sqrt{b} c+7 \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 )}{105 b^{7/4} \sqrt{a+b x^4}}\\ \end{align*}
Mathematica [C] time = 0.63576, size = 202, normalized size = 0.48 \[ \frac{\sqrt{a+b x^4} \left (-\frac{315 a^{3/2} \sqrt{b} d \sinh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{\sqrt{\frac{b x^4}{a}+1}}-\frac{720 a b c x \, _2F_1\left (-\frac{1}{2},\frac{1}{4};\frac{5}{4};-\frac{b x^4}{a}\right )}{\sqrt{\frac{b x^4}{a}+1}}+720 b c x \left (a+b x^4\right )+315 b d x^2 \left (a+2 b x^4\right )-\frac{560 a b e x^3 \, _2F_1\left (-\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )}{\sqrt{\frac{b x^4}{a}+1}}+560 b e x^3 \left (a+b x^4\right )+168 f \left (a+b x^4\right ) \left (3 b x^4-2 a\right )\right )}{5040 b^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.032, size = 390, normalized size = 0.9 \begin{align*} -{\frac{f \left ( -3\,b{x}^{4}+2\,a \right ) }{30\,{b}^{2}} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{2}}}}+{\frac{e{x}^{7}}{9}\sqrt{b{x}^{4}+a}}+{\frac{2\,ae{x}^{3}}{45\,b}\sqrt{b{x}^{4}+a}}-{{\frac{2\,i}{15}}e{a}^{{\frac{5}{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{2\,i}{15}}e{a}^{{\frac{5}{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}}{8\,b} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{2}}}}-{\frac{ad{x}^{2}}{16\,b}\sqrt{b{x}^{4}+a}}-{\frac{{a}^{2}d}{16}\ln \left ({x}^{2}\sqrt{b}+\sqrt{b{x}^{4}+a} \right ){b}^{-{\frac{3}{2}}}}+{\frac{c{x}^{5}}{7}\sqrt{b{x}^{4}+a}}+{\frac{2\,acx}{21\,b}\sqrt{b{x}^{4}+a}}-{\frac{2\,{a}^{2}c}{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}}}} \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 \sqrt{b x^{4} + a}{\left (f x^{3} + e x^{2} + d x + c\right )} x^{4}\,{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 ({\left (f x^{7} + e x^{6} + d x^{5} + c x^{4}\right )} \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 = 7.09639, size = 252, normalized size = 0.6 \begin{align*} \frac{a^{\frac{3}{2}} d x^{2}}{16 b \sqrt{1 + \frac{b x^{4}}{a}}} + \frac{\sqrt{a} 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 \Gamma \left (\frac{9}{4}\right )} + \frac{3 \sqrt{a} d x^{6}}{16 \sqrt{1 + \frac{b x^{4}}{a}}} + \frac{\sqrt{a} 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 \Gamma \left (\frac{11}{4}\right )} - \frac{a^{2} d \operatorname{asinh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{16 b^{\frac{3}{2}}} + f \left (\begin{cases} - \frac{a^{2} \sqrt{a + b x^{4}}}{15 b^{2}} + \frac{a x^{4} \sqrt{a + b x^{4}}}{30 b} + \frac{x^{8} \sqrt{a + b x^{4}}}{10} & \text{for}\: b \neq 0 \\\frac{\sqrt{a} x^{8}}{8} & \text{otherwise} \end{cases}\right ) + \frac{b d 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 \sqrt{b x^{4} + a}{\left (f x^{3} + e x^{2} + d x + c\right )} x^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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