Optimal. Leaf size=406 \[ \frac{2 a^{5/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (5 \sqrt{a} f+21 \sqrt{b} d\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{35 \sqrt [4]{b} \sqrt{a+b x^4}}-\frac{12 a^{5/4} \sqrt [4]{b} d \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 \sqrt{a+b x^4}}-\frac{1}{2} a^{3/2} e \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )-\frac{\left (a+b x^4\right )^{3/2} \left (3 c-e x^2\right )}{6 x^2}+\frac{1}{4} \sqrt{a+b x^4} \left (2 a e+3 b c x^2\right )+\frac{3}{4} a \sqrt{b} c \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )-\frac{\left (a+b x^4\right )^{3/2} \left (7 d-f x^2\right )}{7 x}+\frac{2}{35} x \sqrt{a+b x^4} \left (5 a f+21 b d x^2\right )+\frac{12 a \sqrt{b} d x \sqrt{a+b x^4}}{5 \left (\sqrt{a}+\sqrt{b} x^2\right )} \]
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Rubi [A] time = 0.826172, antiderivative size = 406, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 15, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ \frac{2 a^{5/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (5 \sqrt{a} f+21 \sqrt{b} d\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{35 \sqrt [4]{b} \sqrt{a+b x^4}}-\frac{12 a^{5/4} \sqrt [4]{b} d \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 \sqrt{a+b x^4}}-\frac{1}{2} a^{3/2} e \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )-\frac{\left (a+b x^4\right )^{3/2} \left (3 c-e x^2\right )}{6 x^2}+\frac{1}{4} \sqrt{a+b x^4} \left (2 a e+3 b c x^2\right )+\frac{3}{4} a \sqrt{b} c \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )-\frac{\left (a+b x^4\right )^{3/2} \left (7 d-f x^2\right )}{7 x}+\frac{2}{35} x \sqrt{a+b x^4} \left (5 a f+21 b d x^2\right )+\frac{12 a \sqrt{b} d x \sqrt{a+b x^4}}{5 \left (\sqrt{a}+\sqrt{b} x^2\right )} \]
Antiderivative was successfully verified.
[In] Int[((c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^(3/2))/x^3,x]
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Rubi in Sympy [A] time = 93.4706, size = 377, normalized size = 0.93 \[ - \frac{12 a^{\frac{5}{4}} \sqrt [4]{b} d \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{5 \sqrt{a + b x^{4}}} + \frac{2 a^{\frac{5}{4}} \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) \left (5 \sqrt{a} f + 21 \sqrt{b} d\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{35 \sqrt [4]{b} \sqrt{a + b x^{4}}} - \frac{a^{\frac{3}{2}} e \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{4}}}{\sqrt{a}} \right )}}{2} + \frac{3 a \sqrt{b} c \operatorname{atanh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a + b x^{4}}} \right )}}{4} + \frac{12 a \sqrt{b} d x \sqrt{a + b x^{4}}}{5 \left (\sqrt{a} + \sqrt{b} x^{2}\right )} + \frac{2 x \sqrt{a + b x^{4}} \left (5 a f + 21 b d x^{2}\right )}{35} + \frac{\sqrt{a + b x^{4}} \left (4 a e + 6 b c x^{2}\right )}{8} - \frac{\left (a + b x^{4}\right )^{\frac{3}{2}} \left (7 d - f x^{2}\right )}{7 x} - \frac{\left (a + b x^{4}\right )^{\frac{3}{2}} \left (3 c - e x^{2}\right )}{6 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((f*x**3+e*x**2+d*x+c)*(b*x**4+a)**(3/2)/x**3,x)
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Mathematica [C] time = 1.15303, size = 326, normalized size = 0.8 \[ \frac{\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \left (-210 a^{3/2} e x^2 \sqrt{a+b x^4} \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )+\left (a+b x^4\right ) \left (-210 a c+20 a x (x (14 e+9 f x)-21 d)+b x^4 \left (105 c+84 d x+70 e x^2+60 f x^3\right )\right )+315 a \sqrt{b} c x^2 \sqrt{a+b x^4} \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )\right )-48 i a^{3/2} x^2 \sqrt{\frac{b x^4}{a}+1} \left (5 \sqrt{a} f-21 i \sqrt{b} d\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )+1008 a^{3/2} \sqrt{b} d x^2 \sqrt{\frac{b x^4}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )}{420 x^2 \sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \sqrt{a+b x^4}} \]
Antiderivative was successfully verified.
[In] Integrate[((c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^(3/2))/x^3,x]
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Maple [C] time = 0.027, size = 409, normalized size = 1. \[{\frac{bf{x}^{5}}{7}\sqrt{b{x}^{4}+a}}+{\frac{3\,afx}{7}\sqrt{b{x}^{4}+a}}+{\frac{4\,{a}^{2}f}{7}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}+{\frac{{x}^{2}bc}{4}\sqrt{b{x}^{4}+a}}+{\frac{3\,ac}{4}\sqrt{b}\ln \left ( \sqrt{b}{x}^{2}+\sqrt{b{x}^{4}+a} \right ) }-{\frac{ac}{2\,{x}^{2}}\sqrt{b{x}^{4}+a}}-{\frac{ad}{x}\sqrt{b{x}^{4}+a}}+{\frac{{x}^{3}bd}{5}\sqrt{b{x}^{4}+a}}+{{\frac{12\,i}{5}}d{a}^{{\frac{3}{2}}}\sqrt{b}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{{\frac{12\,i}{5}}d{a}^{{\frac{3}{2}}}\sqrt{b}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticE} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{\frac{e}{2}{a}^{{\frac{3}{2}}}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{4}+a} \right ) } \right ) }+{\frac{{x}^{4}be}{6}\sqrt{b{x}^{4}+a}}+{\frac{2\,ae}{3}\sqrt{b{x}^{4}+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(3/2)/x^3,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{3}{2}}{\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{3}}\,{d x} \]
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^3,x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b f x^{7} + b e x^{6} + b d x^{5} + b c x^{4} + a f x^{3} + a e x^{2} + a d x + a c\right )} \sqrt{b x^{4} + a}}{x^{3}}, x\right ) \]
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^3,x, algorithm="fricas")
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Sympy [A] time = 13.3853, size = 377, normalized size = 0.93 \[ - \frac{a^{\frac{3}{2}} c}{2 x^{2} \sqrt{1 + \frac{b x^{4}}{a}}} + \frac{a^{\frac{3}{2}} d \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4} \\ \frac{3}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 x \Gamma \left (\frac{3}{4}\right )} - \frac{a^{\frac{3}{2}} e \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2} + \frac{a^{\frac{3}{2}} f x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} + \frac{\sqrt{a} b c x^{2} \sqrt{1 + \frac{b x^{4}}{a}}}{4} - \frac{\sqrt{a} b c x^{2}}{2 \sqrt{1 + \frac{b x^{4}}{a}}} + \frac{\sqrt{a} b d x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{7}{4}\right )} + \frac{\sqrt{a} b f x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{9}{4}\right )} + \frac{a^{2} e}{2 \sqrt{b} x^{2} \sqrt{\frac{a}{b x^{4}} + 1}} + \frac{3 a \sqrt{b} c \operatorname{asinh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{4} + \frac{a \sqrt{b} e x^{2}}{2 \sqrt{\frac{a}{b x^{4}} + 1}} + b e \left (\begin{cases} \frac{\sqrt{a} x^{4}}{4} & \text{for}\: b = 0 \\\frac{\left (a + b x^{4}\right )^{\frac{3}{2}}}{6 b} & \text{otherwise} \end{cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x**3+e*x**2+d*x+c)*(b*x**4+a)**(3/2)/x**3,x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{3}{2}}{\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{3}}\,{d x} \]
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^3,x, algorithm="giac")
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