Optimal. Leaf size=476 \[ -\frac{2 a^{11/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (77 \sqrt{a} e+65 \sqrt{b} c\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{5005 b^{7/4} \sqrt{a+b x^4}}-\frac{a^3 d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{32 b^{3/2}}-\frac{4 a^3 e x \sqrt{a+b x^4}}{65 b^{3/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{4 a^{13/4} e \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{65 b^{7/4} \sqrt{a+b x^4}}+\frac{4 a^2 c x \sqrt{a+b x^4}}{77 b}-\frac{a^2 d x^2 \sqrt{a+b x^4}}{32 b}+\frac{4 a^2 e x^3 \sqrt{a+b x^4}}{195 b}-\frac{\left (a+b x^4\right )^{5/2} \left (12 a f-35 b d x^2\right )}{420 b^2}+\frac{1}{143} x^5 \left (a+b x^4\right )^{3/2} \left (13 c+11 e x^2\right )+\frac{2 a x^5 \sqrt{a+b x^4} \left (117 c+77 e x^2\right )}{3003}-\frac{a d x^2 \left (a+b x^4\right )^{3/2}}{48 b}+\frac{f x^4 \left (a+b x^4\right )^{5/2}}{14 b} \]
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Rubi [A] time = 0.439691, antiderivative size = 476, normalized size of antiderivative = 1., number of steps used = 16, 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} \[ -\frac{2 a^{11/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (77 \sqrt{a} e+65 \sqrt{b} c\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5005 b^{7/4} \sqrt{a+b x^4}}-\frac{a^3 d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{32 b^{3/2}}-\frac{4 a^3 e x \sqrt{a+b x^4}}{65 b^{3/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{4 a^{13/4} e \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{65 b^{7/4} \sqrt{a+b x^4}}+\frac{4 a^2 c x \sqrt{a+b x^4}}{77 b}-\frac{a^2 d x^2 \sqrt{a+b x^4}}{32 b}+\frac{4 a^2 e x^3 \sqrt{a+b x^4}}{195 b}-\frac{\left (a+b x^4\right )^{5/2} \left (12 a f-35 b d x^2\right )}{420 b^2}+\frac{1}{143} x^5 \left (a+b x^4\right )^{3/2} \left (13 c+11 e x^2\right )+\frac{2 a x^5 \sqrt{a+b x^4} \left (117 c+77 e x^2\right )}{3003}-\frac{a d x^2 \left (a+b x^4\right )^{3/2}}{48 b}+\frac{f x^4 \left (a+b x^4\right )^{5/2}}{14 b} \]
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 ) \left (a+b x^4\right )^{3/2} \, dx &=\int \left (x^4 \left (c+e x^2\right ) \left (a+b x^4\right )^{3/2}+x^5 \left (d+f x^2\right ) \left (a+b x^4\right )^{3/2}\right ) \, dx\\ &=\int x^4 \left (c+e x^2\right ) \left (a+b x^4\right )^{3/2} \, dx+\int x^5 \left (d+f x^2\right ) \left (a+b x^4\right )^{3/2} \, dx\\ &=\frac{1}{143} x^5 \left (13 c+11 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac{1}{2} \operatorname{Subst}\left (\int x^2 (d+f x) \left (a+b x^2\right )^{3/2} \, dx,x,x^2\right )+\frac{1}{143} (6 a) \int x^4 \left (13 c+11 e x^2\right ) \sqrt{a+b x^4} \, dx\\ &=\frac{2 a x^5 \left (117 c+77 e x^2\right ) \sqrt{a+b x^4}}{3003}+\frac{1}{143} x^5 \left (13 c+11 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac{f x^4 \left (a+b x^4\right )^{5/2}}{14 b}+\frac{\left (4 a^2\right ) \int \frac{x^4 \left (117 c+77 e x^2\right )}{\sqrt{a+b x^4}} \, dx}{3003}+\frac{\operatorname{Subst}\left (\int x (-2 a f+7 b d x) \left (a+b x^2\right )^{3/2} \, dx,x,x^2\right )}{14 b}\\ &=\frac{4 a^2 e x^3 \sqrt{a+b x^4}}{195 b}+\frac{2 a x^5 \left (117 c+77 e x^2\right ) \sqrt{a+b x^4}}{3003}+\frac{1}{143} x^5 \left (13 c+11 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac{f x^4 \left (a+b x^4\right )^{5/2}}{14 b}-\frac{\left (12 a f-35 b d x^2\right ) \left (a+b x^4\right )^{5/2}}{420 b^2}-\frac{\left (4 a^2\right ) \int \frac{x^2 \left (231 a e-585 b c x^2\right )}{\sqrt{a+b x^4}} \, dx}{15015 b}-\frac{(a d) \operatorname{Subst}\left (\int \left (a+b x^2\right )^{3/2} \, dx,x,x^2\right )}{12 b}\\ &=\frac{4 a^2 c x \sqrt{a+b x^4}}{77 b}+\frac{4 a^2 e x^3 \sqrt{a+b x^4}}{195 b}+\frac{2 a x^5 \left (117 c+77 e x^2\right ) \sqrt{a+b x^4}}{3003}-\frac{a d x^2 \left (a+b x^4\right )^{3/2}}{48 b}+\frac{1}{143} x^5 \left (13 c+11 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac{f x^4 \left (a+b x^4\right )^{5/2}}{14 b}-\frac{\left (12 a f-35 b d x^2\right ) \left (a+b x^4\right )^{5/2}}{420 b^2}+\frac{\left (4 a^2\right ) \int \frac{-585 a b c-693 a b e x^2}{\sqrt{a+b x^4}} \, dx}{45045 b^2}-\frac{\left (a^2 d\right ) \operatorname{Subst}\left (\int \sqrt{a+b x^2} \, dx,x,x^2\right )}{16 b}\\ &=\frac{4 a^2 c x \sqrt{a+b x^4}}{77 b}-\frac{a^2 d x^2 \sqrt{a+b x^4}}{32 b}+\frac{4 a^2 e x^3 \sqrt{a+b x^4}}{195 b}+\frac{2 a x^5 \left (117 c+77 e x^2\right ) \sqrt{a+b x^4}}{3003}-\frac{a d x^2 \left (a+b x^4\right )^{3/2}}{48 b}+\frac{1}{143} x^5 \left (13 c+11 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac{f x^4 \left (a+b x^4\right )^{5/2}}{14 b}-\frac{\left (12 a f-35 b d x^2\right ) \left (a+b x^4\right )^{5/2}}{420 b^2}-\frac{\left (a^3 d\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,x^2\right )}{32 b}+\frac{\left (4 a^{7/2} e\right ) \int \frac{1-\frac{\sqrt{b} x^2}{\sqrt{a}}}{\sqrt{a+b x^4}} \, dx}{65 b^{3/2}}-\frac{\left (4 a^3 \left (65 \sqrt{b} c+77 \sqrt{a} e\right )\right ) \int \frac{1}{\sqrt{a+b x^4}} \, dx}{5005 b^{3/2}}\\ &=\frac{4 a^2 c x \sqrt{a+b x^4}}{77 b}-\frac{a^2 d x^2 \sqrt{a+b x^4}}{32 b}+\frac{4 a^2 e x^3 \sqrt{a+b x^4}}{195 b}-\frac{4 a^3 e x \sqrt{a+b x^4}}{65 b^{3/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{2 a x^5 \left (117 c+77 e x^2\right ) \sqrt{a+b x^4}}{3003}-\frac{a d x^2 \left (a+b x^4\right )^{3/2}}{48 b}+\frac{1}{143} x^5 \left (13 c+11 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac{f x^4 \left (a+b x^4\right )^{5/2}}{14 b}-\frac{\left (12 a f-35 b d x^2\right ) \left (a+b x^4\right )^{5/2}}{420 b^2}+\frac{4 a^{13/4} 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 )}{65 b^{7/4} \sqrt{a+b x^4}}-\frac{2 a^{11/4} \left (65 \sqrt{b} c+77 \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 )}{5005 b^{7/4} \sqrt{a+b x^4}}-\frac{\left (a^3 d\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x^2}{\sqrt{a+b x^4}}\right )}{32 b}\\ &=\frac{4 a^2 c x \sqrt{a+b x^4}}{77 b}-\frac{a^2 d x^2 \sqrt{a+b x^4}}{32 b}+\frac{4 a^2 e x^3 \sqrt{a+b x^4}}{195 b}-\frac{4 a^3 e x \sqrt{a+b x^4}}{65 b^{3/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{2 a x^5 \left (117 c+77 e x^2\right ) \sqrt{a+b x^4}}{3003}-\frac{a d x^2 \left (a+b x^4\right )^{3/2}}{48 b}+\frac{1}{143} x^5 \left (13 c+11 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac{f x^4 \left (a+b x^4\right )^{5/2}}{14 b}-\frac{\left (12 a f-35 b d x^2\right ) \left (a+b x^4\right )^{5/2}}{420 b^2}-\frac{a^3 d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{32 b^{3/2}}+\frac{4 a^{13/4} 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 )}{65 b^{7/4} \sqrt{a+b x^4}}-\frac{2 a^{11/4} \left (65 \sqrt{b} c+77 \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 )}{5005 b^{7/4} \sqrt{a+b x^4}}\\ \end{align*}
Mathematica [C] time = 0.864564, size = 225, normalized size = 0.47 \[ \frac{\sqrt{a+b x^4} \left (5005 b d x^2 \left (3 a^2+14 a b x^4+8 b^2 x^8\right )-\frac{43680 a^2 b c x \, _2F_1\left (-\frac{3}{2},\frac{1}{4};\frac{5}{4};-\frac{b x^4}{a}\right )}{\sqrt{\frac{b x^4}{a}+1}}-\frac{15015 a^{5/2} \sqrt{b} d \sinh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{\sqrt{\frac{b x^4}{a}+1}}-\frac{36960 a^2 b e x^3 \, _2F_1\left (-\frac{3}{2},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )}{\sqrt{\frac{b x^4}{a}+1}}+43680 b c x \left (a+b x^4\right )^2+36960 b e x^3 \left (a+b x^4\right )^2+6864 f \left (a+b x^4\right )^2 \left (5 b x^4-2 a\right )\right )}{480480 b^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.039, size = 462, normalized size = 1. \begin{align*} -{\frac{f \left ( -5\,b{x}^{4}+2\,a \right ) \left ({b}^{2}{x}^{8}+2\,ab{x}^{4}+{a}^{2} \right ) }{70\,{b}^{2}}\sqrt{b{x}^{4}+a}}+{\frac{be{x}^{11}}{13}\sqrt{b{x}^{4}+a}}+{\frac{5\,ae{x}^{7}}{39}\sqrt{b{x}^{4}+a}}+{\frac{4\,{a}^{2}e{x}^{3}}{195\,b}\sqrt{b{x}^{4}+a}}-{{\frac{4\,i}{65}}e{a}^{{\frac{7}{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{4\,i}{65}}e{a}^{{\frac{7}{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{bd{x}^{10}}{12}\sqrt{b{x}^{4}+a}}+{\frac{7\,ad{x}^{6}}{48}\sqrt{b{x}^{4}+a}}+{\frac{{a}^{2}d{x}^{2}}{32\,b}\sqrt{b{x}^{4}+a}}-{\frac{{a}^{3}d}{32}\ln \left ({x}^{2}\sqrt{b}+\sqrt{b{x}^{4}+a} \right ){b}^{-{\frac{3}{2}}}}+{\frac{bc{x}^{9}}{11}\sqrt{b{x}^{4}+a}}+{\frac{13\,ac{x}^{5}}{77}\sqrt{b{x}^{4}+a}}+{\frac{4\,{a}^{2}cx}{77\,b}\sqrt{b{x}^{4}+a}}-{\frac{4\,{a}^{3}c}{77\,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{\left (b x^{4} + a\right )}^{\frac{3}{2}}{\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 (b f x^{11} + b e x^{10} + b d x^{9} + b c x^{8} + a f x^{7} + a e x^{6} + a d x^{5} + a 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 = 18.0418, size = 462, normalized size = 0.97 \begin{align*} \frac{a^{\frac{5}{2}} d x^{2}}{32 b \sqrt{1 + \frac{b x^{4}}{a}}} + \frac{a^{\frac{3}{2}} 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{17 a^{\frac{3}{2}} d x^{6}}{96 \sqrt{1 + \frac{b x^{4}}{a}}} + \frac{a^{\frac{3}{2}} 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{\sqrt{a} b c x^{9} \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{13}{4}\right )} + \frac{11 \sqrt{a} b d x^{10}}{48 \sqrt{1 + \frac{b x^{4}}{a}}} + \frac{\sqrt{a} b e x^{11} \Gamma \left (\frac{11}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{11}{4} \\ \frac{15}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{15}{4}\right )} - \frac{a^{3} d \operatorname{asinh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{32 b^{\frac{3}{2}}} + a 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 ) + b f \left (\begin{cases} \frac{4 a^{3} \sqrt{a + b x^{4}}}{105 b^{3}} - \frac{2 a^{2} x^{4} \sqrt{a + b x^{4}}}{105 b^{2}} + \frac{a x^{8} \sqrt{a + b x^{4}}}{70 b} + \frac{x^{12} \sqrt{a + b x^{4}}}{14} & \text{for}\: b \neq 0 \\\frac{\sqrt{a} x^{12}}{12} & \text{otherwise} \end{cases}\right ) + \frac{b^{2} d x^{14}}{12 \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{\left (b x^{4} + a\right )}^{\frac{3}{2}}{\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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