Optimal. Leaf size=137 \[ -\frac {e \left (d^2-e^2 x^2\right )^{p-1} \, _2F_1\left (1,p-1;p;1-\frac {e^2 x^2}{d^2}\right )}{d (1-p)}-\frac {\left (d^2-e^2 x^2\right )^{p-1}}{x}+\frac {2 e^2 (2-p) x \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac {1}{2},2-p;\frac {3}{2};\frac {e^2 x^2}{d^2}\right )}{d^4} \]
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Rubi [A] time = 0.16, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {852, 1807, 764, 266, 65, 246, 245} \[ -\frac {e \left (d^2-e^2 x^2\right )^{p-1} \, _2F_1\left (1,p-1;p;1-\frac {e^2 x^2}{d^2}\right )}{d (1-p)}+\frac {2 e^2 (2-p) x \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac {1}{2},2-p;\frac {3}{2};\frac {e^2 x^2}{d^2}\right )}{d^4}-\frac {\left (d^2-e^2 x^2\right )^{p-1}}{x} \]
Antiderivative was successfully verified.
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Rule 65
Rule 245
Rule 246
Rule 266
Rule 764
Rule 852
Rule 1807
Rubi steps
\begin {align*} \int \frac {\left (d^2-e^2 x^2\right )^p}{x^2 (d+e x)^2} \, dx &=\int \frac {(d-e x)^2 \left (d^2-e^2 x^2\right )^{-2+p}}{x^2} \, dx\\ &=-\frac {\left (d^2-e^2 x^2\right )^{-1+p}}{x}-\frac {\int \frac {\left (2 d^3 e-2 d^2 e^2 (2-p) x\right ) \left (d^2-e^2 x^2\right )^{-2+p}}{x} \, dx}{d^2}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{-1+p}}{x}-(2 d e) \int \frac {\left (d^2-e^2 x^2\right )^{-2+p}}{x} \, dx+\left (2 e^2 (2-p)\right ) \int \left (d^2-e^2 x^2\right )^{-2+p} \, dx\\ &=-\frac {\left (d^2-e^2 x^2\right )^{-1+p}}{x}-(d e) \operatorname {Subst}\left (\int \frac {\left (d^2-e^2 x\right )^{-2+p}}{x} \, dx,x,x^2\right )+\frac {\left (2 e^2 (2-p) \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 )^{-2+p} \, dx}{d^4}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{-1+p}}{x}+\frac {2 e^2 (2-p) 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},2-p;\frac {3}{2};\frac {e^2 x^2}{d^2}\right )}{d^4}-\frac {e \left (d^2-e^2 x^2\right )^{-1+p} \, _2F_1\left (1,-1+p;p;1-\frac {e^2 x^2}{d^2}\right )}{d (1-p)}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 223, normalized size = 1.63 \[ \frac {\left (d^2-e^2 x^2\right )^p \left (-\frac {4 d e \left (1-\frac {d^2}{e^2 x^2}\right )^{-p} \, _2F_1\left (-p,-p;1-p;\frac {d^2}{e^2 x^2}\right )}{p}-\frac {4 d^2 \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (-\frac {1}{2},-p;\frac {1}{2};\frac {e^2 x^2}{d^2}\right )}{x}+\frac {e 2^{p+2} (e x-d) \left (\frac {e x}{d}+1\right )^{-p} \, _2F_1\left (1-p,p+1;p+2;\frac {d-e x}{2 d}\right )}{p+1}+\frac {e 2^p (e x-d) \left (\frac {e x}{d}+1\right )^{-p} \, _2F_1\left (2-p,p+1;p+2;\frac {d-e x}{2 d}\right )}{p+1}\right )}{4 d^4} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.82, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{e^{2} x^{4} + 2 \, d e x^{3} + d^{2} x^{2}}, x\right ) \]
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 \[ \int \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{{\left (e x + d\right )}^{2} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {\left (-e^{2} x^{2}+d^{2}\right )^{p}}{\left (e x +d \right )^{2} x^{2}}\, dx \]
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 \[ \int \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{{\left (e x + d\right )}^{2} x^{2}}\,{d x} \]
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 \[ \int \frac {{\left (d^2-e^2\,x^2\right )}^p}{x^2\,{\left (d+e\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{p}}{x^{2} \left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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