Optimal. Leaf size=174 \[ -\frac{d \left (d^2-e^2 x^2\right )^{p-2}}{4 x^4}+\frac{e \left (d^2-e^2 x^2\right )^{p-2}}{x^3}+\frac{2 e^3 (4-p) \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (-\frac{1}{2},3-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right )}{d^6 x}+\frac{e^4 (10-p) \left (d^2-e^2 x^2\right )^{p-2} \, _2F_1\left (2,p-2;p-1;1-\frac{e^2 x^2}{d^2}\right )}{4 d^3 (2-p)} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.519324, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28 \[ -\frac{d \left (d^2-e^2 x^2\right )^{p-2}}{4 x^4}+\frac{e \left (d^2-e^2 x^2\right )^{p-2}}{x^3}+\frac{2 e^3 (4-p) \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (-\frac{1}{2},3-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right )}{d^6 x}+\frac{e^4 (10-p) \left (d^2-e^2 x^2\right )^{p-2} \, _2F_1\left (2,p-2;p-1;1-\frac{e^2 x^2}{d^2}\right )}{4 d^3 (2-p)} \]
Antiderivative was successfully verified.
[In] Int[(d^2 - e^2*x^2)^p/(x^5*(d + e*x)^3),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 74.2036, size = 192, normalized size = 1.1 \[ \frac{3 e^{4} \left (d^{2} - e^{2} x^{2}\right )^{p - 2}{{}_{2}F_{1}\left (\begin{matrix} 2, p - 2 \\ p - 1 \end{matrix}\middle |{1 - \frac{e^{2} x^{2}}{d^{2}}} \right )}}{2 d^{3} \left (- p + 2\right )} + \frac{e^{4} \left (d^{2} - e^{2} x^{2}\right )^{p - 2}{{}_{2}F_{1}\left (\begin{matrix} 3, p - 2 \\ p - 1 \end{matrix}\middle |{1 - \frac{e^{2} x^{2}}{d^{2}}} \right )}}{2 d^{3} \left (- p + 2\right )} + \frac{e \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 3, - \frac{3}{2} \\ - \frac{1}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{d^{4} x^{3}} + \frac{e^{3} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 3, - \frac{1}{2} \\ \frac{1}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{d^{6} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-e**2*x**2+d**2)**p/x**5/(e*x+d)**3,x)
[Out]
_______________________________________________________________________________________
Mathematica [B] time = 1.01305, size = 446, normalized size = 2.56 \[ \frac{\left (d^2-e^2 x^2\right )^p \left (\frac{60 d e^4 \left (1-\frac{d^2}{e^2 x^2}\right )^{-p} \, _2F_1\left (-p,-p;1-p;\frac{d^2}{e^2 x^2}\right )}{p}+\frac{80 d^2 e^3 \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (-\frac{1}{2},-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right )}{x}+\frac{4 d^5 \left (1-\frac{d^2}{e^2 x^2}\right )^{-p} \, _2F_1\left (2-p,-p;3-p;\frac{d^2}{e^2 x^2}\right )}{(p-2) x^4}+\frac{8 d^4 e \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (-\frac{3}{2},-p;-\frac{1}{2};\frac{e^2 x^2}{d^2}\right )}{x^3}+\frac{24 d^3 e^2 \left (1-\frac{d^2}{e^2 x^2}\right )^{-p} \, _2F_1\left (1-p,-p;2-p;\frac{d^2}{e^2 x^2}\right )}{(p-1) x^2}+\frac{15 e^4 2^{p+2} (d-e x) \left (\frac{e x}{d}+1\right )^{-p} \, _2F_1\left (1-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{p+1}+\frac{5 e^4 2^{p+1} (d-e x) \left (\frac{e x}{d}+1\right )^{-p} \, _2F_1\left (2-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{p+1}+\frac{e^4 2^p (d-e x) \left (\frac{e x}{d}+1\right )^{-p} \, _2F_1\left (3-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{p+1}\right )}{8 d^8} \]
Antiderivative was successfully verified.
[In] Integrate[(d^2 - e^2*x^2)^p/(x^5*(d + e*x)^3),x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.109, size = 0, normalized size = 0. \[ \int{\frac{ \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{p}}{{x}^{5} \left ( ex+d \right ) ^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-e^2*x^2+d^2)^p/x^5/(e*x+d)^3,x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{{\left (e x + d\right )}^{3} x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e^2*x^2 + d^2)^p/((e*x + d)^3*x^5),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{e^{3} x^{8} + 3 \, d e^{2} x^{7} + 3 \, d^{2} e x^{6} + d^{3} x^{5}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e^2*x^2 + d^2)^p/((e*x + d)^3*x^5),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{p}}{x^{5} \left (d + e x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e**2*x**2+d**2)**p/x**5/(e*x+d)**3,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{{\left (e x + d\right )}^{3} x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e^2*x^2 + d^2)^p/((e*x + d)^3*x^5),x, algorithm="giac")
[Out]