Optimal. Leaf size=104 \[ -\frac {e x \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {1}{2},1-p;\frac {3}{2};\frac {e^2 x^2}{d^2}\right )}{d^2}-\frac {\left (d^2-e^2 x^2\right )^p \, _2F_1\left (1,p;p+1;1-\frac {e^2 x^2}{d^2}\right )}{2 d p} \]
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Rubi [A] time = 0.07, antiderivative size = 104, 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, 266, 65, 246, 245} \[ -\frac {e x \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {1}{2},1-p;\frac {3}{2};\frac {e^2 x^2}{d^2}\right )}{d^2}-\frac {\left (d^2-e^2 x^2\right )^p \, _2F_1\left (1,p;p+1;1-\frac {e^2 x^2}{d^2}\right )}{2 d p} \]
Antiderivative was successfully verified.
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Rule 65
Rule 245
Rule 246
Rule 266
Rule 764
Rule 850
Rubi steps
\begin {align*} \int \frac {\left (d^2-e^2 x^2\right )^p}{x (d+e x)} \, dx &=\int \frac {(d-e x) \left (d^2-e^2 x^2\right )^{-1+p}}{x} \, dx\\ &=d \int \frac {\left (d^2-e^2 x^2\right )^{-1+p}}{x} \, dx-e \int \left (d^2-e^2 x^2\right )^{-1+p} \, dx\\ &=\frac {1}{2} d \operatorname {Subst}\left (\int \frac {\left (d^2-e^2 x\right )^{-1+p}}{x} \, dx,x,x^2\right )-\frac {\left (e \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p}\right ) \int \left (1-\frac {e^2 x^2}{d^2}\right )^{-1+p} \, dx}{d^2}\\ &=-\frac {e x \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 {3}{2};\frac {e^2 x^2}{d^2}\right )}{d^2}-\frac {\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 p}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 151, normalized size = 1.45 \[ \frac {2^{p-1} \left (1-\frac {d^2}{e^2 x^2}\right )^{-p} \left (\frac {e x}{d}+1\right )^{-p} \left (d^2-e^2 x^2\right )^p \left (p (d-e x) \left (1-\frac {d^2}{e^2 x^2}\right )^p \, _2F_1\left (1-p,p+1;p+2;\frac {d-e x}{2 d}\right )+d (p+1) \left (\frac {e x}{2 d}+\frac {1}{2}\right )^p \, _2F_1\left (-p,-p;1-p;\frac {d^2}{e^2 x^2}\right )\right )}{d^2 p (p+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{e x^{2} + d x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{{\left (e x + d\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[ \int \frac {\left (-e^{2} x^{2}+d^{2}\right )^{p}}{\left (e x +d \right ) x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{{\left (e x + d\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (d^2-e^2\,x^2\right )}^p}{x\,\left (d+e\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 6.90, size = 355, normalized size = 3.41 \[ \begin {cases} - \frac {0^{p} d^{2 p} \log {\left (\frac {d^{2}}{e^{2} x^{2}} - 1 \right )}}{2 d} - \frac {0^{p} d^{2 p} \operatorname {acoth}{\left (\frac {d}{e x} \right )}}{d} + \frac {d e^{2 p} p x^{2 p} e^{i \pi p} \Gamma \relax (p) \Gamma \left (1 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, 1 - p \\ 2 - p \end {matrix}\middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{2 e^{2} x^{2} \Gamma \left (2 - p\right ) \Gamma \left (p + 1\right )} - \frac {e^{2 p} p x^{2 p} e^{i \pi p} \Gamma \relax (p) \Gamma \left (\frac {1}{2} - p\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, \frac {1}{2} - p \\ \frac {3}{2} - p \end {matrix}\middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{2 e x \Gamma \left (\frac {3}{2} - p\right ) \Gamma \left (p + 1\right )} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {0^{p} d^{2 p} \log {\left (- \frac {d^{2}}{e^{2} x^{2}} + 1 \right )}}{2 d} - \frac {0^{p} d^{2 p} \operatorname {atanh}{\left (\frac {d}{e x} \right )}}{d} + \frac {d e^{2 p} p x^{2 p} e^{i \pi p} \Gamma \relax (p) \Gamma \left (1 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, 1 - p \\ 2 - p \end {matrix}\middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{2 e^{2} x^{2} \Gamma \left (2 - p\right ) \Gamma \left (p + 1\right )} - \frac {e^{2 p} p x^{2 p} e^{i \pi p} \Gamma \relax (p) \Gamma \left (\frac {1}{2} - p\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, \frac {1}{2} - p \\ \frac {3}{2} - p \end {matrix}\middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{2 e x \Gamma \left (\frac {3}{2} - p\right ) \Gamma \left (p + 1\right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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