Optimal. Leaf size=49 \[ \frac{d x \left (\frac{a+b}{d}+\frac{c x^2}{d}\right )^{m+1} \, _2F_1\left (1,m+\frac{3}{2};\frac{3}{2};-\frac{c x^2}{a+b}\right )}{a+b} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0401009, antiderivative size = 57, normalized size of antiderivative = 1.16, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ x \left (\frac{c x^2}{a+b}+1\right )^{-m} \left (\frac{a+b}{d}+\frac{c x^2}{d}\right )^m \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};-\frac{c x^2}{a+b}\right ) \]
Antiderivative was successfully verified.
[In] Int[((a + b + c*x^2)/d)^m,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 3.9821, size = 42, normalized size = 0.86 \[ x \left (\frac{c x^{2}}{d} + \frac{a + b}{d}\right )^{m} \left (\frac{c x^{2}}{a + b} + 1\right )^{- m}{{}_{2}F_{1}\left (\begin{matrix} - m, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{- \frac{c x^{2}}{a + b}} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(((c*x**2+a+b)/d)**m,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0283969, size = 53, normalized size = 1.08 \[ x \left (\frac{c x^2}{a+b}+1\right )^{-m} \left (\frac{a+b+c x^2}{d}\right )^m \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};-\frac{c x^2}{a+b}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[((a + b + c*x^2)/d)^m,x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.055, size = 0, normalized size = 0. \[ \int \left ({\frac{c{x}^{2}+a+b}{d}} \right ) ^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(((c*x^2+a+b)/d)^m,x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \left (\frac{c x^{2} + a + b}{d}\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((c*x^2 + a + b)/d)^m,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\left (\frac{c x^{2} + a + b}{d}\right )^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((c*x^2 + a + b)/d)^m,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (\frac{a + b + c x^{2}}{d}\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((c*x**2+a+b)/d)**m,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \left (\frac{c x^{2} + a + b}{d}\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((c*x^2 + a + b)/d)^m,x, algorithm="giac")
[Out]