Optimal. Leaf size=83 \[ -\frac {\left (d^2-e^2 x^2\right )^{1+p}}{2 e (1+p)}+d x \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};\frac {e^2 x^2}{d^2}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {655, 252, 251}
\begin {gather*} d 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},-p;\frac {3}{2};\frac {e^2 x^2}{d^2}\right )-\frac {\left (d^2-e^2 x^2\right )^{p+1}}{2 e (p+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 251
Rule 252
Rule 655
Rubi steps
\begin {align*} \int (d+e x) \left (d^2-e^2 x^2\right )^p \, dx &=-\frac {\left (d^2-e^2 x^2\right )^{1+p}}{2 e (1+p)}+d \int \left (d^2-e^2 x^2\right )^p \, dx\\ &=-\frac {\left (d^2-e^2 x^2\right )^{1+p}}{2 e (1+p)}+\left (d \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 )^p \, dx\\ &=-\frac {\left (d^2-e^2 x^2\right )^{1+p}}{2 e (1+p)}+d 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},-p;\frac {3}{2};\frac {e^2 x^2}{d^2}\right )\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 83, normalized size = 1.00 \begin {gather*} -\frac {\left (d^2-e^2 x^2\right )^{1+p}}{2 e (1+p)}+d 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},-p;\frac {3}{2};\frac {e^2 x^2}{d^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \left (e x +d \right ) \left (-e^{2} x^{2}+d^{2}\right )^{p}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.48, size = 82, normalized size = 0.99 \begin {gather*} d d^{2 p} x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, - p \\ \frac {3}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )} + e \left (\begin {cases} \frac {x^{2} \left (d^{2}\right )^{p}}{2} & \text {for}\: e^{2} = 0 \\- \frac {\begin {cases} \frac {\left (d^{2} - e^{2} x^{2}\right )^{p + 1}}{p + 1} & \text {for}\: p \neq -1 \\\log {\left (d^{2} - e^{2} x^{2} \right )} & \text {otherwise} \end {cases}}{2 e^{2}} & \text {otherwise} \end {cases}\right ) \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.35, size = 78, normalized size = 0.94 \begin {gather*} \frac {d\,x\,{\left (d^2-e^2\,x^2\right )}^p\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},-p;\ \frac {3}{2};\ \frac {e^2\,x^2}{d^2}\right )}{{\left (1-\frac {e^2\,x^2}{d^2}\right )}^p}-\frac {{\left (d^2-e^2\,x^2\right )}^{p+1}}{2\,e\,\left (p+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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