Optimal. Leaf size=476 \[ -\frac{2 a^{11/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (77 \sqrt{a} e+65 \sqrt{b} c\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5005 b^{7/4} \sqrt{a+b x^4}}+\frac{4 a^{13/4} e \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{65 b^{7/4} \sqrt{a+b x^4}}-\frac{a^3 d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{32 b^{3/2}}-\frac{4 a^3 e x \sqrt{a+b x^4}}{65 b^{3/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{4 a^2 c x \sqrt{a+b x^4}}{77 b}-\frac{a^2 d x^2 \sqrt{a+b x^4}}{32 b}+\frac{4 a^2 e x^3 \sqrt{a+b x^4}}{195 b}-\frac{\left (a+b x^4\right )^{5/2} \left (12 a f-35 b d x^2\right )}{420 b^2}+\frac{1}{143} x^5 \left (a+b x^4\right )^{3/2} \left (13 c+11 e x^2\right )+\frac{2 a x^5 \sqrt{a+b x^4} \left (117 c+77 e x^2\right )}{3003}-\frac{a d x^2 \left (a+b x^4\right )^{3/2}}{48 b}+\frac{f x^4 \left (a+b x^4\right )^{5/2}}{14 b} \]
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Rubi [A] time = 1.31135, 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 \[ -\frac{2 a^{11/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (77 \sqrt{a} e+65 \sqrt{b} c\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5005 b^{7/4} \sqrt{a+b x^4}}+\frac{4 a^{13/4} e \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{65 b^{7/4} \sqrt{a+b x^4}}-\frac{a^3 d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{32 b^{3/2}}-\frac{4 a^3 e x \sqrt{a+b x^4}}{65 b^{3/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{4 a^2 c x \sqrt{a+b x^4}}{77 b}-\frac{a^2 d x^2 \sqrt{a+b x^4}}{32 b}+\frac{4 a^2 e x^3 \sqrt{a+b x^4}}{195 b}-\frac{\left (a+b x^4\right )^{5/2} \left (12 a f-35 b d x^2\right )}{420 b^2}+\frac{1}{143} x^5 \left (a+b x^4\right )^{3/2} \left (13 c+11 e x^2\right )+\frac{2 a x^5 \sqrt{a+b x^4} \left (117 c+77 e x^2\right )}{3003}-\frac{a d x^2 \left (a+b x^4\right )^{3/2}}{48 b}+\frac{f x^4 \left (a+b x^4\right )^{5/2}}{14 b} \]
Antiderivative was successfully verified.
[In] Int[x^4*(c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 116.903, size = 442, normalized size = 0.93 \[ \frac{4 a^{\frac{13}{4}} e \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{65 b^{\frac{7}{4}} \sqrt{a + b x^{4}}} - \frac{2 a^{\frac{11}{4}} \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) \left (77 \sqrt{a} e + 65 \sqrt{b} c\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{5005 b^{\frac{7}{4}} \sqrt{a + b x^{4}}} - \frac{a^{3} d \operatorname{atanh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a + b x^{4}}} \right )}}{32 b^{\frac{3}{2}}} - \frac{4 a^{3} e x \sqrt{a + b x^{4}}}{65 b^{\frac{3}{2}} \left (\sqrt{a} + \sqrt{b} x^{2}\right )} + \frac{4 a^{2} c x \sqrt{a + b x^{4}}}{77 b} - \frac{a^{2} d x^{2} \sqrt{a + b x^{4}}}{32 b} + \frac{4 a^{2} e x^{3} \sqrt{a + b x^{4}}}{195 b} + \frac{2 a x^{5} \sqrt{a + b x^{4}} \left (117 c + 77 e x^{2}\right )}{3003} - \frac{a d x^{2} \left (a + b x^{4}\right )^{\frac{3}{2}}}{48 b} + \frac{x^{5} \left (a + b x^{4}\right )^{\frac{3}{2}} \left (13 c + 11 e x^{2}\right )}{143} + \frac{f x^{4} \left (a + b x^{4}\right )^{\frac{5}{2}}}{14 b} - \frac{\left (a + b x^{4}\right )^{\frac{5}{2}} \left (12 a f - 35 b d x^{2}\right )}{420 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4*(f*x**3+e*x**2+d*x+c)*(b*x**4+a)**(3/2),x)
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Mathematica [C] time = 1.09568, size = 327, normalized size = 0.69 \[ \frac{-29568 a^{7/2} \sqrt{b} e \sqrt{\frac{b x^4}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )+384 a^3 \sqrt{b} \sqrt{\frac{b x^4}{a}+1} \left (77 \sqrt{a} e+65 i \sqrt{b} c\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )+\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \left (-15015 a^3 \sqrt{b} d \sqrt{a+b x^4} \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )-\left (a+b x^4\right ) \left (13728 a^3 f-a^2 b x \left (24960 c+11 x \left (1365 d+896 e x+624 f x^2\right )\right )-2 a b^2 x^5 \left (40560 c+11 x \left (3185 d+2800 e x+2496 f x^2\right )\right )-40 b^3 x^9 \left (1092 c+11 x \left (91 d+84 e x+78 f x^2\right )\right )\right )\right )}{480480 b^2 \sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \sqrt{a+b x^4}} \]
Antiderivative was successfully verified.
[In] Integrate[x^4*(c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^(3/2),x]
[Out]
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Maple [C] time = 0.047, size = 462, normalized size = 1. \[{\frac{bc{x}^{9}}{11}\sqrt{b{x}^{4}+a}}+{\frac{13\,ac{x}^{5}}{77}\sqrt{b{x}^{4}+a}}+{\frac{4\,{a}^{2}cx}{77\,b}\sqrt{b{x}^{4}+a}}-{\frac{4\,{a}^{3}c}{77\,b}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}+{\frac{{a}^{2}d{x}^{2}}{32\,b}\sqrt{b{x}^{4}+a}}-{\frac{{a}^{3}d}{32}\ln \left ( \sqrt{b}{x}^{2}+\sqrt{b{x}^{4}+a} \right ){b}^{-{\frac{3}{2}}}}+{\frac{bd{x}^{10}}{12}\sqrt{b{x}^{4}+a}}+{\frac{7\,ad{x}^{6}}{48}\sqrt{b{x}^{4}+a}}+{\frac{be{x}^{11}}{13}\sqrt{b{x}^{4}+a}}+{\frac{5\,ae{x}^{7}}{39}\sqrt{b{x}^{4}+a}}+{\frac{4\,{a}^{2}e{x}^{3}}{195\,b}\sqrt{b{x}^{4}+a}}-{{\frac{4\,i}{65}}e{a}^{{\frac{7}{2}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}+{{\frac{4\,i}{65}}e{a}^{{\frac{7}{2}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticE} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{\frac{f \left ( -5\,b{x}^{4}+2\,a \right ) \left ({b}^{2}{x}^{8}+2\,ab{x}^{4}+{a}^{2} \right ) }{70\,{b}^{2}}\sqrt{b{x}^{4}+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4*(f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(3/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{4} + a\right )}^{\frac{3}{2}}{\left (f x^{3} + e x^{2} + d x + c\right )} x^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(3/2)*(f*x^3 + e*x^2 + d*x + c)*x^4,x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b f x^{11} + b e x^{10} + b d x^{9} + b c x^{8} + a f x^{7} + a e x^{6} + a d x^{5} + a c x^{4}\right )} \sqrt{b x^{4} + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(3/2)*(f*x^3 + e*x^2 + d*x + c)*x^4,x, algorithm="fricas")
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Sympy [A] time = 28.9211, size = 462, normalized size = 0.97 \[ \frac{a^{\frac{5}{2}} d x^{2}}{32 b \sqrt{1 + \frac{b x^{4}}{a}}} + \frac{a^{\frac{3}{2}} c x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{9}{4}\right )} + \frac{17 a^{\frac{3}{2}} d x^{6}}{96 \sqrt{1 + \frac{b x^{4}}{a}}} + \frac{a^{\frac{3}{2}} e x^{7} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{11}{4}\right )} + \frac{\sqrt{a} b c x^{9} \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{13}{4}\right )} + \frac{11 \sqrt{a} b d x^{10}}{48 \sqrt{1 + \frac{b x^{4}}{a}}} + \frac{\sqrt{a} b e x^{11} \Gamma \left (\frac{11}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{11}{4} \\ \frac{15}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{15}{4}\right )} - \frac{a^{3} d \operatorname{asinh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{32 b^{\frac{3}{2}}} + a f \left (\begin{cases} - \frac{a^{2} \sqrt{a + b x^{4}}}{15 b^{2}} + \frac{a x^{4} \sqrt{a + b x^{4}}}{30 b} + \frac{x^{8} \sqrt{a + b x^{4}}}{10} & \text{for}\: b \neq 0 \\\frac{\sqrt{a} x^{8}}{8} & \text{otherwise} \end{cases}\right ) + b f \left (\begin{cases} \frac{4 a^{3} \sqrt{a + b x^{4}}}{105 b^{3}} - \frac{2 a^{2} x^{4} \sqrt{a + b x^{4}}}{105 b^{2}} + \frac{a x^{8} \sqrt{a + b x^{4}}}{70 b} + \frac{x^{12} \sqrt{a + b x^{4}}}{14} & \text{for}\: b \neq 0 \\\frac{\sqrt{a} x^{12}}{12} & \text{otherwise} \end{cases}\right ) + \frac{b^{2} d x^{14}}{12 \sqrt{a} \sqrt{1 + \frac{b x^{4}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4*(f*x**3+e*x**2+d*x+c)*(b*x**4+a)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{4} + a\right )}^{\frac{3}{2}}{\left (f x^{3} + e x^{2} + d x + c\right )} x^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(3/2)*(f*x^3 + e*x^2 + d*x + c)*x^4,x, algorithm="giac")
[Out]