Optimal. Leaf size=110 \[ -\frac {e \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (-\frac {1}{2},-p;\frac {1}{2};\frac {e^2 x^2}{d^2}\right )}{x}-\frac {e^2 \left (d^2-e^2 x^2\right )^{1+p} \, _2F_1\left (2,1+p;2+p;1-\frac {e^2 x^2}{d^2}\right )}{2 d^3 (1+p)} \]
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Rubi [A]
time = 0.04, antiderivative size = 110, 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 = {778, 272, 67,
372, 371} \begin {gather*} -\frac {e \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 {1}{2};\frac {e^2 x^2}{d^2}\right )}{x}-\frac {e^2 \left (d^2-e^2 x^2\right )^{p+1} \, _2F_1\left (2,p+1;p+2;1-\frac {e^2 x^2}{d^2}\right )}{2 d^3 (p+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 67
Rule 272
Rule 371
Rule 372
Rule 778
Rubi steps
\begin {align*} \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^p}{x^3} \, dx &=d \int \frac {\left (d^2-e^2 x^2\right )^p}{x^3} \, dx+e \int \frac {\left (d^2-e^2 x^2\right )^p}{x^2} \, dx\\ &=\frac {1}{2} d \text {Subst}\left (\int \frac {\left (d^2-e^2 x\right )^p}{x^2} \, dx,x,x^2\right )+\left (e \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 )^p}{x^2} \, dx\\ &=-\frac {e \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 {1}{2};\frac {e^2 x^2}{d^2}\right )}{x}-\frac {e^2 \left (d^2-e^2 x^2\right )^{1+p} \, _2F_1\left (2,1+p;2+p;1-\frac {e^2 x^2}{d^2}\right )}{2 d^3 (1+p)}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 106, normalized size = 0.96 \begin {gather*} \frac {1}{2} e \left (d^2-e^2 x^2\right )^p \left (-\frac {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 \left (-d^2+e^2 x^2\right ) \, _2F_1\left (2,1+p;2+p;1-\frac {e^2 x^2}{d^2}\right )}{d^3 (1+p)}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (e x +d \right ) \left (-e^{2} x^{2}+d^{2}\right )^{p}}{x^{3}}\, dx\]
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 [C] Result contains complex when optimal does not.
time = 2.07, size = 85, normalized size = 0.77 \begin {gather*} - \frac {d e^{2 p} x^{2 p} e^{i \pi p} \Gamma \left (1 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 1 - p \\ 2 - p \end {matrix}\middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{2 x^{2} \Gamma \left (2 - p\right )} - \frac {d^{2 p} e {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - p \\ \frac {1}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{x} \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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (d^2-e^2\,x^2\right )}^p\,\left (d+e\,x\right )}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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