Optimal. Leaf size=418 \[ -\frac{a^{7/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (7 \sqrt{a} e+5 \sqrt{b} c\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{105 b^{7/4} \sqrt{a+b x^4}}+\frac{2 a^{9/4} e \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 b^{7/4} \sqrt{a+b x^4}}-\frac{a^2 d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{16 b^{3/2}}-\frac{2 a^2 e x \sqrt{a+b x^4}}{15 b^{3/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{\left (a+b x^4\right )^{3/2} \left (8 a f-15 b d x^2\right )}{120 b^2}+\frac{1}{63} x^5 \sqrt{a+b x^4} \left (9 c+7 e x^2\right )+\frac{2 a c x \sqrt{a+b x^4}}{21 b}-\frac{a d x^2 \sqrt{a+b x^4}}{16 b}+\frac{2 a e x^3 \sqrt{a+b x^4}}{45 b}+\frac{f x^4 \left (a+b x^4\right )^{3/2}}{10 b} \]
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Rubi [A] time = 1.07942, 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 \[ -\frac{a^{7/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (7 \sqrt{a} e+5 \sqrt{b} c\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{105 b^{7/4} \sqrt{a+b x^4}}+\frac{2 a^{9/4} e \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 b^{7/4} \sqrt{a+b x^4}}-\frac{a^2 d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{16 b^{3/2}}-\frac{2 a^2 e x \sqrt{a+b x^4}}{15 b^{3/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{\left (a+b x^4\right )^{3/2} \left (8 a f-15 b d x^2\right )}{120 b^2}+\frac{1}{63} x^5 \sqrt{a+b x^4} \left (9 c+7 e x^2\right )+\frac{2 a c x \sqrt{a+b x^4}}{21 b}-\frac{a d x^2 \sqrt{a+b x^4}}{16 b}+\frac{2 a e x^3 \sqrt{a+b x^4}}{45 b}+\frac{f x^4 \left (a+b x^4\right )^{3/2}}{10 b} \]
Antiderivative was successfully verified.
[In] Int[x^4*(c + d*x + e*x^2 + f*x^3)*Sqrt[a + b*x^4],x]
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Rubi in Sympy [A] time = 102.131, size = 386, normalized size = 0.92 \[ \frac{2 a^{\frac{9}{4}} e \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{15 b^{\frac{7}{4}} \sqrt{a + b x^{4}}} - \frac{a^{\frac{7}{4}} \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) \left (7 \sqrt{a} e + 5 \sqrt{b} c\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{105 b^{\frac{7}{4}} \sqrt{a + b x^{4}}} - \frac{a^{2} d \operatorname{atanh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a + b x^{4}}} \right )}}{16 b^{\frac{3}{2}}} - \frac{2 a^{2} e x \sqrt{a + b x^{4}}}{15 b^{\frac{3}{2}} \left (\sqrt{a} + \sqrt{b} x^{2}\right )} + \frac{2 a c x \sqrt{a + b x^{4}}}{21 b} - \frac{a d x^{2} \sqrt{a + b x^{4}}}{16 b} + \frac{2 a e x^{3} \sqrt{a + b x^{4}}}{45 b} + \frac{x^{5} \sqrt{a + b x^{4}} \left (9 c + 7 e x^{2}\right )}{63} + \frac{f x^{4} \left (a + b x^{4}\right )^{\frac{3}{2}}}{10 b} - \frac{\left (a + b x^{4}\right )^{\frac{3}{2}} \left (8 a f - 15 b d x^{2}\right )}{120 b^{2}} \]
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)**(1/2),x)
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Mathematica [C] time = 0.859589, size = 296, normalized size = 0.71 \[ \frac{-672 a^{5/2} \sqrt{b} e \sqrt{\frac{b x^4}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )+\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \left (-\left (a+b x^4\right ) \left (336 a^2 f-a b x (480 c+7 x (45 d+8 x (4 e+3 f x)))-2 b^2 x^5 \left (360 c+7 x \left (45 d+40 e x+36 f x^2\right )\right )\right )-315 a^2 \sqrt{b} d \sqrt{a+b x^4} \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )\right )+96 a^2 \sqrt{b} \sqrt{\frac{b x^4}{a}+1} \left (7 \sqrt{a} e+5 i \sqrt{b} c\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )}{5040 b^2 \sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \sqrt{a+b x^4}} \]
Antiderivative was successfully verified.
[In] Integrate[x^4*(c + d*x + e*x^2 + f*x^3)*Sqrt[a + b*x^4],x]
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Maple [C] time = 0.041, size = 390, normalized size = 0.9 \[{\frac{c{x}^{5}}{7}\sqrt{b{x}^{4}+a}}+{\frac{2\,acx}{21\,b}\sqrt{b{x}^{4}+a}}-{\frac{2\,{a}^{2}c}{21\,b}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}+{\frac{d{x}^{2}}{8\,b} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{2}}}}-{\frac{ad{x}^{2}}{16\,b}\sqrt{b{x}^{4}+a}}-{\frac{{a}^{2}d}{16}\ln \left ( \sqrt{b}{x}^{2}+\sqrt{b{x}^{4}+a} \right ){b}^{-{\frac{3}{2}}}}+{\frac{e{x}^{7}}{9}\sqrt{b{x}^{4}+a}}+{\frac{2\,ae{x}^{3}}{45\,b}\sqrt{b{x}^{4}+a}}-{{\frac{2\,i}{15}}e{a}^{{\frac{5}{2}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}+{{\frac{2\,i}{15}}e{a}^{{\frac{5}{2}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticE} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{\frac{f \left ( -3\,b{x}^{4}+2\,a \right ) }{30\,{b}^{2}} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{2}}}} \]
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)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{b x^{4} + a}{\left (f x^{3} + e x^{2} + d x + c\right )} x^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^4 + a)*(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. \[{\rm integral}\left ({\left (f x^{7} + e x^{6} + d x^{5} + c x^{4}\right )} \sqrt{b x^{4} + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^4 + a)*(f*x^3 + e*x^2 + d*x + c)*x^4,x, algorithm="fricas")
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Sympy [A] time = 10.2662, size = 252, normalized size = 0.6 \[ \frac{a^{\frac{3}{2}} d x^{2}}{16 b \sqrt{1 + \frac{b x^{4}}{a}}} + \frac{\sqrt{a} c x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{9}{4}\right )} + \frac{3 \sqrt{a} d x^{6}}{16 \sqrt{1 + \frac{b x^{4}}{a}}} + \frac{\sqrt{a} e x^{7} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{11}{4}\right )} - \frac{a^{2} d \operatorname{asinh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{16 b^{\frac{3}{2}}} + f \left (\begin{cases} - \frac{a^{2} \sqrt{a + b x^{4}}}{15 b^{2}} + \frac{a x^{4} \sqrt{a + b x^{4}}}{30 b} + \frac{x^{8} \sqrt{a + b x^{4}}}{10} & \text{for}\: b \neq 0 \\\frac{\sqrt{a} x^{8}}{8} & \text{otherwise} \end{cases}\right ) + \frac{b d x^{10}}{8 \sqrt{a} \sqrt{1 + \frac{b x^{4}}{a}}} \]
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)**(1/2),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{b x^{4} + a}{\left (f x^{3} + e x^{2} + d x + c\right )} x^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^4 + a)*(f*x^3 + e*x^2 + d*x + c)*x^4,x, algorithm="giac")
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