Optimal. Leaf size=71 \[ \frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} \, _2F_1\left (\frac{3}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right )}{c e (m+1) \sqrt{c+d x^4}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0695755, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} \, _2F_1\left (\frac{3}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right )}{c e (m+1) \sqrt{c+d x^4}} \]
Antiderivative was successfully verified.
[In] Int[(e*x)^m/(c + d*x^4)^(3/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 8.23275, size = 58, normalized size = 0.82 \[ \frac{\left (e x\right )^{m + 1} \sqrt{c + d x^{4}}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{m}{4} + \frac{1}{4} \\ \frac{m}{4} + \frac{5}{4} \end{matrix}\middle |{- \frac{d x^{4}}{c}} \right )}}{c^{2} e \sqrt{1 + \frac{d x^{4}}{c}} \left (m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x)**m/(d*x**4+c)**(3/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0354781, size = 67, normalized size = 0.94 \[ \frac{x \sqrt{\frac{d x^4}{c}+1} (e x)^m \, _2F_1\left (\frac{3}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right )}{c (m+1) \sqrt{c+d x^4}} \]
Antiderivative was successfully verified.
[In] Integrate[(e*x)^m/(c + d*x^4)^(3/2),x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.031, size = 0, normalized size = 0. \[ \int{ \left ( ex \right ) ^{m} \left ( d{x}^{4}+c \right ) ^{-{\frac{3}{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^m/(d*x^4+c)^(3/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (e x\right )^{m}}{{\left (d x^{4} + c\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x)^m/(d*x^4 + c)^(3/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\left (e x\right )^{m}}{{\left (d x^{4} + c\right )}^{\frac{3}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x)^m/(d*x^4 + c)^(3/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 2.79688, size = 56, normalized size = 0.79 \[ \frac{e^{m} x x^{m} \Gamma \left (\frac{m}{4} + \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{m}{4} + \frac{1}{4} \\ \frac{m}{4} + \frac{5}{4} \end{matrix}\middle |{\frac{d x^{4} e^{i \pi }}{c}} \right )}}{4 c^{\frac{3}{2}} \Gamma \left (\frac{m}{4} + \frac{5}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x)**m/(d*x**4+c)**(3/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (e x\right )^{m}}{{\left (d x^{4} + c\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x)^m/(d*x^4 + c)^(3/2),x, algorithm="giac")
[Out]