Optimal. Leaf size=345 \[ \frac{a^{3/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (3 \sqrt{a} f+5 \sqrt{b} d\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 b^{3/4} \sqrt{a+b x^4}}-\frac{2 a^{5/4} f \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt{a+b x^4}}+\frac{1}{4} \sqrt{a+b x^4} \left (2 c+e x^2\right )-\frac{1}{2} \sqrt{a} c \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )+\frac{1}{15} x \sqrt{a+b x^4} \left (5 d+3 f x^2\right )+\frac{a e \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{4 \sqrt{b}}+\frac{2 a f x \sqrt{a+b x^4}}{5 \sqrt{b} \left (\sqrt{a}+\sqrt{b} x^2\right )} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.6867, antiderivative size = 345, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 13, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.433 \[ \frac{a^{3/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (3 \sqrt{a} f+5 \sqrt{b} d\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 b^{3/4} \sqrt{a+b x^4}}-\frac{2 a^{5/4} f \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt{a+b x^4}}+\frac{1}{4} \sqrt{a+b x^4} \left (2 c+e x^2\right )-\frac{1}{2} \sqrt{a} c \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )+\frac{1}{15} x \sqrt{a+b x^4} \left (5 d+3 f x^2\right )+\frac{a e \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{4 \sqrt{b}}+\frac{2 a f x \sqrt{a+b x^4}}{5 \sqrt{b} \left (\sqrt{a}+\sqrt{b} x^2\right )} \]
Antiderivative was successfully verified.
[In] Int[((c + d*x + e*x^2 + f*x^3)*Sqrt[a + b*x^4])/x,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 69.0769, size = 316, normalized size = 0.92 \[ - \frac{2 a^{\frac{5}{4}} f \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{5 b^{\frac{3}{4}} \sqrt{a + b x^{4}}} + \frac{a^{\frac{3}{4}} \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) \left (3 \sqrt{a} f + 5 \sqrt{b} d\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{15 b^{\frac{3}{4}} \sqrt{a + b x^{4}}} - \frac{\sqrt{a} c \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{4}}}{\sqrt{a}} \right )}}{2} + \frac{a e \operatorname{atanh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a + b x^{4}}} \right )}}{4 \sqrt{b}} + \frac{2 a f x \sqrt{a + b x^{4}}}{5 \sqrt{b} \left (\sqrt{a} + \sqrt{b} x^{2}\right )} + \frac{x \sqrt{a + b x^{4}} \left (5 d + 3 f x^{2}\right )}{15} + \frac{\sqrt{a + b x^{4}} \left (2 c + e x^{2}\right )}{4} \]
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,x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 2.35789, size = 280, normalized size = 0.81 \[ \frac{24 a^{3/2} f \sqrt{\frac{b x^4}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )+\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \left (\sqrt{b} \left (\left (a+b x^4\right ) (30 c+x (20 d+3 x (5 e+4 f x)))-30 \sqrt{a} c \sqrt{a+b x^4} \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )\right )+15 a e \sqrt{a+b x^4} \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )\right )-8 a \sqrt{\frac{b x^4}{a}+1} \left (3 \sqrt{a} f+5 i \sqrt{b} d\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )}{60 \sqrt{b} \sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \sqrt{a+b x^4}} \]
Antiderivative was successfully verified.
[In] Integrate[((c + d*x + e*x^2 + f*x^3)*Sqrt[a + b*x^4])/x,x]
[Out]
_______________________________________________________________________________________
Maple [C] time = 0.019, size = 339, normalized size = 1. \[{\frac{dx}{3}\sqrt{b{x}^{4}+a}}+{\frac{2\,ad}{3}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}+{\frac{f{x}^{3}}{5}\sqrt{b{x}^{4}+a}}+{{\frac{2\,i}{5}}f{a}^{{\frac{3}{2}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}{\frac{1}{\sqrt{b}}}}-{{\frac{2\,i}{5}}f{a}^{{\frac{3}{2}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticE} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}{\frac{1}{\sqrt{b}}}}+{\frac{c}{2}\sqrt{b{x}^{4}+a}}-{\frac{c}{2}\sqrt{a}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{4}+a} \right ) } \right ) }+{\frac{e{x}^{2}}{4}\sqrt{b{x}^{4}+a}}+{\frac{ae}{4}\ln \left ( \sqrt{b}{x}^{2}+\sqrt{b{x}^{4}+a} \right ){\frac{1}{\sqrt{b}}}} \]
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,x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x^{4} + a}{\left (f x^{3} + e x^{2} + d x + c\right )}}{x}\,{d x} \]
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,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{b x^{4} + a}{\left (f x^{3} + e x^{2} + d x + c\right )}}{x}, x\right ) \]
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,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 7.37201, size = 204, normalized size = 0.59 \[ - \frac{\sqrt{a} c \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2} + \frac{\sqrt{a} d x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} + \frac{\sqrt{a} e x^{2} \sqrt{1 + \frac{b x^{4}}{a}}}{4} + \frac{\sqrt{a} f x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{7}{4}\right )} + \frac{a c}{2 \sqrt{b} x^{2} \sqrt{\frac{a}{b x^{4}} + 1}} + \frac{a e \operatorname{asinh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{4 \sqrt{b}} + \frac{\sqrt{b} c x^{2}}{2 \sqrt{\frac{a}{b x^{4}} + 1}} \]
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,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x^{4} + a}{\left (f x^{3} + e x^{2} + d x + c\right )}}{x}\,{d x} \]
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,x, algorithm="giac")
[Out]