Optimal. Leaf size=90 \[ -\frac {e x^3 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {3}{2},1-p;\frac {5}{2};\frac {e^2 x^2}{d^2}\right )}{3 d^2}-\frac {d \left (d^2-e^2 x^2\right )^p}{2 e^2 p} \]
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Rubi [A] time = 0.06, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {785, 764, 261, 365, 364} \[ -\frac {e x^3 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {3}{2},1-p;\frac {5}{2};\frac {e^2 x^2}{d^2}\right )}{3 d^2}-\frac {d \left (d^2-e^2 x^2\right )^p}{2 e^2 p} \]
Antiderivative was successfully verified.
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Rule 261
Rule 364
Rule 365
Rule 764
Rule 785
Rubi steps
\begin {align*} \int \frac {x \left (d^2-e^2 x^2\right )^p}{d+e x} \, dx &=\frac {\int x \left (d^2 e-d e^2 x\right ) \left (d^2-e^2 x^2\right )^{-1+p} \, dx}{d e}\\ &=d \int x \left (d^2-e^2 x^2\right )^{-1+p} \, dx-e \int x^2 \left (d^2-e^2 x^2\right )^{-1+p} \, dx\\ &=-\frac {d \left (d^2-e^2 x^2\right )^p}{2 e^2 p}-\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 x^2 \left (1-\frac {e^2 x^2}{d^2}\right )^{-1+p} \, dx}{d^2}\\ &=-\frac {d \left (d^2-e^2 x^2\right )^p}{2 e^2 p}-\frac {e x^3 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {3}{2},1-p;\frac {5}{2};\frac {e^2 x^2}{d^2}\right )}{3 d^2}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 147, normalized size = 1.63 \[ \frac {2^{p-1} \left (\frac {e x}{d}+1\right )^{-p} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \left (2 e (p+1) x \left (\frac {e x}{2 d}+\frac {1}{2}\right )^p \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};\frac {e^2 x^2}{d^2}\right )+(d-e x) \left (1-\frac {e^2 x^2}{d^2}\right )^p \, _2F_1\left (1-p,p+1;p+2;\frac {d-e x}{2 d}\right )\right )}{e^2 (p+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.82, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x}{e x + d}, x\right ) \]
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 \[ \int \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {x \left (-e^{2} x^{2}+d^{2}\right )^{p}}{e x +d}\, dx \]
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 \[ \int \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x}{e x + d}\,{d x} \]
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 \[ \int \frac {x\,{\left (d^2-e^2\,x^2\right )}^p}{d+e\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 7.00, size = 427, normalized size = 4.74 \[ \begin {cases} \frac {0^{p} d d^{2 p} \log {\left (\frac {d^{2}}{e^{2} x^{2}} \right )}}{2 e^{2}} - \frac {0^{p} d d^{2 p} \log {\left (\frac {d^{2}}{e^{2} x^{2}} - 1 \right )}}{2 e^{2}} - \frac {0^{p} d d^{2 p} \operatorname {acoth}{\left (\frac {d}{e x} \right )}}{e^{2}} + \frac {0^{p} d^{2 p} x}{e} - \frac {e^{2 p} p x x^{2 p} e^{i \pi p} \Gamma \relax (p) \Gamma \left (- p - \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, - p - \frac {1}{2} \\ \frac {1}{2} - p \end {matrix}\middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{2 e \Gamma \left (\frac {1}{2} - p\right ) \Gamma \left (p + 1\right )} - \frac {d^{2 p} x^{2} \Gamma \relax (p) \Gamma \left (1 - p\right ) {{}_{3}F_{2}\left (\begin {matrix} 2, 1, 1 - p \\ 2, 2 \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 d \Gamma \left (- p\right ) \Gamma \left (p + 1\right )} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {0^{p} d d^{2 p} \log {\left (\frac {d^{2}}{e^{2} x^{2}} \right )}}{2 e^{2}} - \frac {0^{p} d d^{2 p} \log {\left (- \frac {d^{2}}{e^{2} x^{2}} + 1 \right )}}{2 e^{2}} - \frac {0^{p} d d^{2 p} \operatorname {atanh}{\left (\frac {d}{e x} \right )}}{e^{2}} + \frac {0^{p} d^{2 p} x}{e} - \frac {e^{2 p} p x x^{2 p} e^{i \pi p} \Gamma \relax (p) \Gamma \left (- p - \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, - p - \frac {1}{2} \\ \frac {1}{2} - p \end {matrix}\middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{2 e \Gamma \left (\frac {1}{2} - p\right ) \Gamma \left (p + 1\right )} - \frac {d^{2 p} x^{2} \Gamma \relax (p) \Gamma \left (1 - p\right ) {{}_{3}F_{2}\left (\begin {matrix} 2, 1, 1 - p \\ 2, 2 \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 d \Gamma \left (- p\right ) \Gamma \left (p + 1\right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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