Optimal. Leaf size=139 \[ -\frac {2 d e \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (-\frac {1}{2},-p;\frac {1}{2};\frac {e^2 x^2}{d^2}\right )}{x}-\frac {e^2 (1-p) \left (d^2-e^2 x^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;1-\frac {e^2 x^2}{d^2}\right )}{2 d^2 (p+1)}-\frac {\left (d^2-e^2 x^2\right )^{p+1}}{2 x^2} \]
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Rubi [A] time = 0.12, antiderivative size = 139, 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 = {1807, 764, 365, 364, 266, 65} \[ -\frac {2 d e \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (-\frac {1}{2},-p;\frac {1}{2};\frac {e^2 x^2}{d^2}\right )}{x}-\frac {e^2 (1-p) \left (d^2-e^2 x^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;1-\frac {e^2 x^2}{d^2}\right )}{2 d^2 (p+1)}-\frac {\left (d^2-e^2 x^2\right )^{p+1}}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 65
Rule 266
Rule 364
Rule 365
Rule 764
Rule 1807
Rubi steps
\begin {align*} \int \frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^p}{x^3} \, dx &=-\frac {\left (d^2-e^2 x^2\right )^{1+p}}{2 x^2}-\frac {\int \frac {\left (-4 d^3 e-2 d^2 e^2 (1-p) x\right ) \left (d^2-e^2 x^2\right )^p}{x^2} \, dx}{2 d^2}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{1+p}}{2 x^2}+(2 d e) \int \frac {\left (d^2-e^2 x^2\right )^p}{x^2} \, dx+\left (e^2 (1-p)\right ) \int \frac {\left (d^2-e^2 x^2\right )^p}{x} \, dx\\ &=-\frac {\left (d^2-e^2 x^2\right )^{1+p}}{2 x^2}+\frac {1}{2} \left (e^2 (1-p)\right ) \operatorname {Subst}\left (\int \frac {\left (d^2-e^2 x\right )^p}{x} \, dx,x,x^2\right )+\left (2 d 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 {\left (d^2-e^2 x^2\right )^{1+p}}{2 x^2}-\frac {2 d 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 (1-p) \left (d^2-e^2 x^2\right )^{1+p} \, _2F_1\left (1,1+p;2+p;1-\frac {e^2 x^2}{d^2}\right )}{2 d^2 (1+p)}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 131, normalized size = 0.94 \[ \frac {e \left (d^2-e^2 x^2\right )^p \left (\frac {e \left (e^2 x^2-d^2\right ) \left (\, _2F_1\left (1,p+1;p+2;1-\frac {e^2 x^2}{d^2}\right )+\, _2F_1\left (2,p+1;p+2;1-\frac {e^2 x^2}{d^2}\right )\right )}{p+1}-\frac {4 d^3 \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (-\frac {1}{2},-p;\frac {1}{2};\frac {e^2 x^2}{d^2}\right )}{x}\right )}{2 d^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.92, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )} {\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{x^{3}}, x\right ) \]
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 \[ \int \frac {{\left (e x + d\right )}^{2} {\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{x^{3}}\,{d x} \]
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 \[ \int \frac {\left (e x +d \right )^{2} \left (-e^{2} x^{2}+d^{2}\right )^{p}}{x^{3}}\, dx \]
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 \[ \int \frac {{\left (e x + d\right )}^{2} {\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{x^{3}}\,{d x} \]
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 \[ \int \frac {{\left (d^2-e^2\,x^2\right )}^p\,{\left (d+e\,x\right )}^2}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 6.92, size = 139, normalized size = 1.00 \[ - \frac {d^{2} e^{2 p} x^{2 p} e^{i \pi p} \Gamma \left (1 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 1 - p \\ 2 - p \end {matrix}\middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{2 x^{2} \Gamma \left (2 - p\right )} - \frac {2 d d^{2 p} e {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - p \\ \frac {1}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{x} - \frac {e^{2} e^{2 p} x^{2 p} e^{i \pi p} \Gamma \left (- p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - p \\ 1 - p \end {matrix}\middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{2 \Gamma \left (1 - p\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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