Optimal. Leaf size=77 \[ \frac{B (e x)^{m+1}}{d e (m+1)}-\frac{(e x)^{m+1} (B c-A d) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )}{c d e (m+1)} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.11575, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{B (e x)^{m+1}}{d e (m+1)}-\frac{(e x)^{m+1} (B c-A d) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )}{c d e (m+1)} \]
Antiderivative was successfully verified.
[In] Int[((e*x)^m*(A + B*x^2))/(c + d*x^2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 14.052, size = 56, normalized size = 0.73 \[ \frac{B \left (e x\right )^{m + 1}}{d e \left (m + 1\right )} + \frac{\left (e x\right )^{m + 1} \left (A d - B c\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{d x^{2}}{c}} \right )}}{c d e \left (m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x)**m*(B*x**2+A)/(d*x**2+c),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0690555, size = 58, normalized size = 0.75 \[ -\frac{x (e x)^m \left ((B c-A d) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )-B c\right )}{c d (m+1)} \]
Antiderivative was successfully verified.
[In] Integrate[((e*x)^m*(A + B*x^2))/(c + d*x^2),x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.066, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex \right ) ^{m} \left ( B{x}^{2}+A \right ) }{d{x}^{2}+c}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^m*(B*x^2+A)/(d*x^2+c),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{2} + A\right )} \left (e x\right )^{m}}{d x^{2} + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*(e*x)^m/(d*x^2 + c),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B x^{2} + A\right )} \left (e x\right )^{m}}{d x^{2} + c}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*(e*x)^m/(d*x^2 + c),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 7.96214, size = 204, normalized size = 2.65 \[ \frac{A e^{m} m x x^{m} \Phi \left (\frac{d x^{2} e^{i \pi }}{c}, 1, \frac{m}{2} + \frac{1}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{1}{2}\right )}{4 c \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )} + \frac{A e^{m} x x^{m} \Phi \left (\frac{d x^{2} e^{i \pi }}{c}, 1, \frac{m}{2} + \frac{1}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{1}{2}\right )}{4 c \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )} + \frac{B e^{m} m x^{3} x^{m} \Phi \left (\frac{d x^{2} e^{i \pi }}{c}, 1, \frac{m}{2} + \frac{3}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )}{4 c \Gamma \left (\frac{m}{2} + \frac{5}{2}\right )} + \frac{3 B e^{m} x^{3} x^{m} \Phi \left (\frac{d x^{2} e^{i \pi }}{c}, 1, \frac{m}{2} + \frac{3}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )}{4 c \Gamma \left (\frac{m}{2} + \frac{5}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x)**m*(B*x**2+A)/(d*x**2+c),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{2} + A\right )} \left (e x\right )^{m}}{d x^{2} + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*(e*x)^m/(d*x^2 + c),x, algorithm="giac")
[Out]