Optimal. Leaf size=69 \[ -\frac{2 \sqrt{c+d x^4} F_1\left (-\frac{1}{8};1,-\frac{1}{2};\frac{7}{8};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{a e \sqrt{e x} \sqrt{\frac{d x^4}{c}+1}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.390963, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107 \[ -\frac{2 \sqrt{c+d x^4} F_1\left (-\frac{1}{8};1,-\frac{1}{2};\frac{7}{8};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{a e \sqrt{e x} \sqrt{\frac{d x^4}{c}+1}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[c + d*x^4]/((e*x)^(3/2)*(a + b*x^4)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 45.4652, size = 60, normalized size = 0.87 \[ - \frac{2 \sqrt{c + d x^{4}} \operatorname{appellf_{1}}{\left (- \frac{1}{8},- \frac{1}{2},1,\frac{7}{8},- \frac{d x^{4}}{c},- \frac{b x^{4}}{a} \right )}}{a e \sqrt{e x} \sqrt{1 + \frac{d x^{4}}{c}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**4+c)**(1/2)/(e*x)**(3/2)/(b*x**4+a),x)
[Out]
_______________________________________________________________________________________
Mathematica [B] time = 1.00336, size = 348, normalized size = 5.04 \[ \frac{2 x \left (\frac{75 c x^4 (b c-4 a d) F_1\left (\frac{7}{8};\frac{1}{2},1;\frac{15}{8};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )}{\left (a+b x^4\right ) \left (4 x^4 \left (2 b c F_1\left (\frac{15}{8};\frac{1}{2},2;\frac{23}{8};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )+a d F_1\left (\frac{15}{8};\frac{3}{2},1;\frac{23}{8};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )-15 a c F_1\left (\frac{7}{8};\frac{1}{2},1;\frac{15}{8};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )}-\frac{161 b c d x^8 F_1\left (\frac{15}{8};\frac{1}{2},1;\frac{23}{8};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )}{\left (a+b x^4\right ) \left (4 x^4 \left (2 b c F_1\left (\frac{23}{8};\frac{1}{2},2;\frac{31}{8};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )+a d F_1\left (\frac{23}{8};\frac{3}{2},1;\frac{31}{8};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )-23 a c F_1\left (\frac{15}{8};\frac{1}{2},1;\frac{23}{8};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )}-\frac{35 \left (c+d x^4\right )}{a}\right )}{35 (e x)^{3/2} \sqrt{c+d x^4}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[Sqrt[c + d*x^4]/((e*x)^(3/2)*(a + b*x^4)),x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.125, size = 0, normalized size = 0. \[ \int{\frac{1}{b{x}^{4}+a}\sqrt{d{x}^{4}+c} \left ( ex \right ) ^{-{\frac{3}{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^4+c)^(1/2)/(e*x)^(3/2)/(b*x^4+a),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d x^{4} + c}}{{\left (b x^{4} + a\right )} \left (e x\right )^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*(e*x)^(3/2)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{d x^{4} + c}}{{\left (b e x^{5} + a e x\right )} \sqrt{e x}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*(e*x)^(3/2)),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c + d x^{4}}}{\left (e x\right )^{\frac{3}{2}} \left (a + b x^{4}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x**4+c)**(1/2)/(e*x)**(3/2)/(b*x**4+a),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d x^{4} + c}}{{\left (b x^{4} + a\right )} \left (e x\right )^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*(e*x)^(3/2)),x, algorithm="giac")
[Out]