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 = 0.07, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, 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 65
Rule 266
Rule 364
Rule 365
Rule 764
Rule 850
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 = 0.18, 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]
________________________________________________________________________________________
fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{e x^{3} + d x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{{\left (e x + d\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {\left (-e^{2} x^{2}+d^{2}\right )^{p}}{\left (e x +d \right ) x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{{\left (e x + d\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (d^2-e^2\,x^2\right )}^p}{x^2\,\left (d+e\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 7.86, size = 450, normalized size = 4.25 \[ \begin {cases} - \frac {0^{p} d^{2 p}}{d x} - \frac {0^{p} d^{2 p} e \log {\left (\frac {e^{2} x^{2}}{d^{2}} \right )}}{2 d^{2}} + \frac {0^{p} d^{2 p} e \log {\left (-1 + \frac {e^{2} x^{2}}{d^{2}} \right )}}{2 d^{2}} + \frac {0^{p} d^{2 p} e \operatorname {acoth}{\left (\frac {e x}{d} \right )}}{d^{2}} + \frac {d e^{2 p} p x^{2 p} e^{i \pi p} \Gamma \relax (p) \Gamma \left (\frac {3}{2} - p\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, \frac {3}{2} - p \\ \frac {5}{2} - p \end {matrix}\middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{2 e^{2} x^{3} \Gamma \left (\frac {5}{2} - p\right ) \Gamma \left (p + 1\right )} - \frac {e^{2 p} p x^{2 p} e^{i \pi p} \Gamma \relax (p) \Gamma \left (1 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, 1 - p \\ 2 - p \end {matrix}\middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{2 e x^{2} \Gamma \left (2 - p\right ) \Gamma \left (p + 1\right )} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\- \frac {0^{p} d^{2 p}}{d x} - \frac {0^{p} d^{2 p} e \log {\left (\frac {e^{2} x^{2}}{d^{2}} \right )}}{2 d^{2}} + \frac {0^{p} d^{2 p} e \log {\left (1 - \frac {e^{2} x^{2}}{d^{2}} \right )}}{2 d^{2}} + \frac {0^{p} d^{2 p} e \operatorname {atanh}{\left (\frac {e x}{d} \right )}}{d^{2}} + \frac {d e^{2 p} p x^{2 p} e^{i \pi p} \Gamma \relax (p) \Gamma \left (\frac {3}{2} - p\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, \frac {3}{2} - p \\ \frac {5}{2} - p \end {matrix}\middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{2 e^{2} x^{3} \Gamma \left (\frac {5}{2} - p\right ) \Gamma \left (p + 1\right )} - \frac {e^{2 p} p x^{2 p} e^{i \pi p} \Gamma \relax (p) \Gamma \left (1 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, 1 - p \\ 2 - p \end {matrix}\middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{2 e x^{2} \Gamma \left (2 - p\right ) \Gamma \left (p + 1\right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________