Optimal. Leaf size=110 \[ -\frac {e \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (-\frac {1}{2},-p;\frac {1}{2};\frac {e^2 x^2}{d^2}\right )}{x}-\frac {e^2 \left (d^2-e^2 x^2\right )^{p+1} \, _2F_1\left (2,p+1;p+2;1-\frac {e^2 x^2}{d^2}\right )}{2 d^3 (p+1)} \]
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Rubi [A] time = 0.06, 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 = {764, 266, 65, 365, 364} \[ -\frac {e \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (-\frac {1}{2},-p;\frac {1}{2};\frac {e^2 x^2}{d^2}\right )}{x}-\frac {e^2 \left (d^2-e^2 x^2\right )^{p+1} \, _2F_1\left (2,p+1;p+2;1-\frac {e^2 x^2}{d^2}\right )}{2 d^3 (p+1)} \]
Antiderivative was successfully verified.
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Rule 65
Rule 266
Rule 364
Rule 365
Rule 764
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 \operatorname {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 = 0.05, size = 106, normalized size = 0.96 \[ \frac {1}{2} e \left (d^2-e^2 x^2\right )^p \left (\frac {e \left (e^2 x^2-d^2\right ) \, _2F_1\left (2,p+1;p+2;1-\frac {e^2 x^2}{d^2}\right )}{d^3 (p+1)}-\frac {2 \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (-\frac {1}{2},-p;\frac {1}{2};\frac {e^2 x^2}{d^2}\right )}{x}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e x + d\right )} {\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{x^{3}}, x\right ) \]
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 \[ \int \frac {{\left (e x + d\right )} {\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{x^{3}}\,{d x} \]
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 \[ \int \frac {\left (e x +d \right ) \left (-e^{2} x^{2}+d^{2}\right )^{p}}{x^{3}}\, dx \]
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 \[ \int \frac {{\left (e x + d\right )} {\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{x^{3}}\,{d x} \]
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 \[ \int \frac {{\left (d^2-e^2\,x^2\right )}^p\,\left (d+e\,x\right )}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 5.07, size = 85, normalized size = 0.77 \[ - \frac {d e^{2 p} x^{2 p} e^{i \pi p} \Gamma \left (1 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 1 - p \\ 2 - p \end {matrix}\middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{2 x^{2} \Gamma \left (2 - p\right )} - \frac {d^{2 p} e {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - p \\ \frac {1}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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