Optimal. Leaf size=106 \[ \frac{e \left (d^2-e^2 x^2\right )^p \, _2F_1\left (1,p;p+1;1-\frac{e^2 x^2}{d^2}\right )}{2 d^2 p}-\frac{\left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (-\frac{1}{2},1-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right )}{d x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0742381, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {850, 764, 365, 364, 266, 65} \[ \frac{e \left (d^2-e^2 x^2\right )^p \, _2F_1\left (1,p;p+1;1-\frac{e^2 x^2}{d^2}\right )}{2 d^2 p}-\frac{\left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (-\frac{1}{2},1-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right )}{d x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 850
Rule 764
Rule 365
Rule 364
Rule 266
Rule 65
Rubi steps
\begin{align*} \int \frac{\left (d^2-e^2 x^2\right )^p}{x^2 (d+e x)} \, dx &=\int \frac{(d-e x) \left (d^2-e^2 x^2\right )^{-1+p}}{x^2} \, dx\\ &=d \int \frac{\left (d^2-e^2 x^2\right )^{-1+p}}{x^2} \, dx-e \int \frac{\left (d^2-e^2 x^2\right )^{-1+p}}{x} \, dx\\ &=-\left (\frac{1}{2} e \operatorname{Subst}\left (\int \frac{\left (d^2-e^2 x\right )^{-1+p}}{x} \, dx,x,x^2\right )\right )+\frac{\left (\left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p}\right ) \int \frac{\left (1-\frac{e^2 x^2}{d^2}\right )^{-1+p}}{x^2} \, dx}{d}\\ &=-\frac{\left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (-\frac{1}{2},1-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right )}{d x}+\frac{e \left (d^2-e^2 x^2\right )^p \, _2F_1\left (1,p;1+p;1-\frac{e^2 x^2}{d^2}\right )}{2 d^2 p}\\ \end{align*}
Mathematica [A] time = 0.190567, size = 167, normalized size = 1.58 \[ \frac{\left (d^2-e^2 x^2\right )^p \left (-\frac{d e \left (1-\frac{d^2}{e^2 x^2}\right )^{-p} \, _2F_1\left (-p,-p;1-p;\frac{d^2}{e^2 x^2}\right )}{p}-\frac{2 d^2 \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (-\frac{1}{2},-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right )}{x}+\frac{e 2^p (e x-d) \left (\frac{e x}{d}+1\right )^{-p} \, _2F_1\left (1-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{p+1}\right )}{2 d^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.671, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{p}}{{x}^{2} \left ( ex+d \right ) }}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{{\left (e x + d\right )} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{e x^{3} + d x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 8.34782, size = 452, normalized size = 4.26 \begin{align*} \begin{cases} - \frac{0^{p} d^{2 p}}{d x} - \frac{0^{p} d^{2 p} e \log{\left (\frac{e^{2} x^{2}}{d^{2}} \right )}}{2 d^{2}} + \frac{0^{p} d^{2 p} e \log{\left (-1 + \frac{e^{2} x^{2}}{d^{2}} \right )}}{2 d^{2}} + \frac{0^{p} d^{2 p} e \operatorname{acoth}{\left (\frac{e x}{d} \right )}}{d^{2}} + \frac{d e^{2 p} p x^{2 p} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (\frac{3}{2} - p\right ){{}_{2}F_{1}\left (\begin{matrix} 1 - p, \frac{3}{2} - p \\ \frac{5}{2} - p \end{matrix}\middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{2 e^{2} x^{3} \Gamma \left (\frac{5}{2} - p\right ) \Gamma \left (p + 1\right )} - \frac{e^{2 p} p x^{2 p} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (1 - p\right ){{}_{2}F_{1}\left (\begin{matrix} 1 - p, 1 - p \\ 2 - p \end{matrix}\middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{2 e x^{2} \Gamma \left (2 - p\right ) \Gamma \left (p + 1\right )} & \text{for}\: \frac{\left |{e^{2} x^{2}}\right |}{\left |{d^{2}}\right |} > 1 \\- \frac{0^{p} d^{2 p}}{d x} - \frac{0^{p} d^{2 p} e \log{\left (\frac{e^{2} x^{2}}{d^{2}} \right )}}{2 d^{2}} + \frac{0^{p} d^{2 p} e \log{\left (1 - \frac{e^{2} x^{2}}{d^{2}} \right )}}{2 d^{2}} + \frac{0^{p} d^{2 p} e \operatorname{atanh}{\left (\frac{e x}{d} \right )}}{d^{2}} + \frac{d e^{2 p} p x^{2 p} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (\frac{3}{2} - p\right ){{}_{2}F_{1}\left (\begin{matrix} 1 - p, \frac{3}{2} - p \\ \frac{5}{2} - p \end{matrix}\middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{2 e^{2} x^{3} \Gamma \left (\frac{5}{2} - p\right ) \Gamma \left (p + 1\right )} - \frac{e^{2 p} p x^{2 p} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (1 - p\right ){{}_{2}F_{1}\left (\begin{matrix} 1 - p, 1 - p \\ 2 - p \end{matrix}\middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{2 e x^{2} \Gamma \left (2 - p\right ) \Gamma \left (p + 1\right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{{\left (e x + d\right )} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]