Optimal. Leaf size=360 \[ \frac{b^{3/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (5 \sqrt{a} e+3 \sqrt{b} c\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 a^{3/4} \sqrt{a+b x^4}}-\frac{2 b^{5/4} c \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 a^{3/4} \sqrt{a+b x^4}}+\frac{2 b^{3/2} c x \sqrt{a+b x^4}}{5 a \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{1}{60} \sqrt{a+b x^4} \left (\frac{12 c}{x^5}+\frac{15 d}{x^4}+\frac{20 e}{x^3}+\frac{30 f}{x^2}\right )-\frac{2 b c \sqrt{a+b x^4}}{5 a x}-\frac{b d \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{4 \sqrt{a}}+\frac{1}{2} \sqrt{b} f \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right ) \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.818206, antiderivative size = 360, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 14, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467 \[ \frac{b^{3/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (5 \sqrt{a} e+3 \sqrt{b} c\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 a^{3/4} \sqrt{a+b x^4}}-\frac{2 b^{5/4} c \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 a^{3/4} \sqrt{a+b x^4}}+\frac{2 b^{3/2} c x \sqrt{a+b x^4}}{5 a \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{1}{60} \sqrt{a+b x^4} \left (\frac{12 c}{x^5}+\frac{15 d}{x^4}+\frac{20 e}{x^3}+\frac{30 f}{x^2}\right )-\frac{2 b c \sqrt{a+b x^4}}{5 a x}-\frac{b d \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{4 \sqrt{a}}+\frac{1}{2} \sqrt{b} f \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right ) \]
Antiderivative was successfully verified.
[In] Int[((c + d*x + e*x^2 + f*x^3)*Sqrt[a + b*x^4])/x^6,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
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)**(1/2)/x**6,x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 1.16428, size = 314, normalized size = 0.87 \[ \frac{24 \sqrt{a} b^{3/2} c x^5 \sqrt{\frac{b x^4}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )-\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \left (\left (a+b x^4\right ) \left (12 a c+5 a x \left (3 d+4 e x+6 f x^2\right )+24 b c x^4\right )+15 \sqrt{a} b d x^5 \sqrt{a+b x^4} \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )-30 a \sqrt{b} f x^5 \sqrt{a+b x^4} \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )\right )-8 i \sqrt{a} b x^5 \sqrt{\frac{b x^4}{a}+1} \left (5 \sqrt{a} e-3 i \sqrt{b} c\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )}{60 a x^5 \sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \sqrt{a+b x^4}} \]
Antiderivative was successfully verified.
[In] Integrate[((c + d*x + e*x^2 + f*x^3)*Sqrt[a + b*x^4])/x^6,x]
[Out]
_______________________________________________________________________________________
Maple [C] time = 0.027, size = 404, normalized size = 1.1 \[ -{\frac{c}{5\,{x}^{5}}\sqrt{b{x}^{4}+a}}-{\frac{2\,bc}{5\,ax}\sqrt{b{x}^{4}+a}}+{{\frac{2\,i}{5}}c{b}^{{\frac{3}{2}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{{\frac{2\,i}{5}}c{b}^{{\frac{3}{2}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticE} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{\frac{d}{4\,a{x}^{4}} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{2}}}}-{\frac{bd}{4}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{4}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}+{\frac{bd}{4\,a}\sqrt{b{x}^{4}+a}}-{\frac{e}{3\,{x}^{3}}\sqrt{b{x}^{4}+a}}+{\frac{2\,be}{3}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{\frac{f}{2\,a{x}^{2}} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{2}}}}+{\frac{bf{x}^{2}}{2\,a}\sqrt{b{x}^{4}+a}}+{\frac{f}{2}\sqrt{b}\ln \left ( \sqrt{b}{x}^{2}+\sqrt{b{x}^{4}+a} \right ) } \]
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)^(1/2)/x^6,x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x^{4} + a}{\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{6}}\,{d x} \]
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^6,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{b x^{4} + a}{\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{6}}, x\right ) \]
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^6,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 8.72219, size = 216, normalized size = 0.6 \[ \frac{\sqrt{a} c \Gamma \left (- \frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, - \frac{1}{2} \\ - \frac{1}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{5} \Gamma \left (- \frac{1}{4}\right )} + \frac{\sqrt{a} e \Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, - \frac{1}{2} \\ \frac{1}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{3} \Gamma \left (\frac{1}{4}\right )} - \frac{\sqrt{a} f}{2 x^{2} \sqrt{1 + \frac{b x^{4}}{a}}} - \frac{\sqrt{b} d \sqrt{\frac{a}{b x^{4}} + 1}}{4 x^{2}} + \frac{\sqrt{b} f \operatorname{asinh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2} - \frac{b d \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{4 \sqrt{a}} - \frac{b f x^{2}}{2 \sqrt{a} \sqrt{1 + \frac{b x^{4}}{a}}} \]
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)**(1/2)/x**6,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x^{4} + a}{\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{6}}\,{d x} \]
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^6,x, algorithm="giac")
[Out]