Optimal. Leaf size=128 \[ -\frac {\left (d^2-e^2 x^2\right )^p \, _2F_1\left (1,p;p+1;1-\frac {e^2 x^2}{d^2}\right )}{2 d^2 p}+\frac {\left (d^2-e^2 x^2\right )^{p-1}}{1-p}-\frac {2 e x \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac {1}{2},2-p;\frac {3}{2};\frac {e^2 x^2}{d^2}\right )}{d^3} \]
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Rubi [A] time = 0.12, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {852, 1652, 446, 79, 65, 12, 246, 245} \[ -\frac {2 e x \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac {1}{2},2-p;\frac {3}{2};\frac {e^2 x^2}{d^2}\right )}{d^3}-\frac {\left (d^2-e^2 x^2\right )^p \, _2F_1\left (1,p;p+1;1-\frac {e^2 x^2}{d^2}\right )}{2 d^2 p}+\frac {\left (d^2-e^2 x^2\right )^{p-1}}{1-p} \]
Antiderivative was successfully verified.
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Rule 12
Rule 65
Rule 79
Rule 245
Rule 246
Rule 446
Rule 852
Rule 1652
Rubi steps
\begin {align*} \int \frac {\left (d^2-e^2 x^2\right )^p}{x (d+e x)^2} \, dx &=\int \frac {(d-e x)^2 \left (d^2-e^2 x^2\right )^{-2+p}}{x} \, dx\\ &=\int -2 d e \left (d^2-e^2 x^2\right )^{-2+p} \, dx+\int \frac {\left (d^2-e^2 x^2\right )^{-2+p} \left (d^2+e^2 x^2\right )}{x} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\left (d^2-e^2 x\right )^{-2+p} \left (d^2+e^2 x\right )}{x} \, dx,x,x^2\right )-(2 d e) \int \left (d^2-e^2 x^2\right )^{-2+p} \, dx\\ &=\frac {\left (d^2-e^2 x^2\right )^{-1+p}}{1-p}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {\left (d^2-e^2 x\right )^{-1+p}}{x} \, dx,x,x^2\right )-\frac {\left (2 e \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^3}\\ &=\frac {\left (d^2-e^2 x^2\right )^{-1+p}}{1-p}-\frac {2 e 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^3}-\frac {\left (d^2-e^2 x^2\right )^p \, _2F_1\left (1,p;1+p;1-\frac {e^2 x^2}{d^2}\right )}{2 d^2 p}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 201, normalized size = 1.57 \[ \frac {2^{p-2} \left (1-\frac {d^2}{e^2 x^2}\right )^{-p} \left (\frac {e x}{d}+1\right )^{-p} \left (d^2-e^2 x^2\right )^p \left (2 p (d-e x) \left (1-\frac {d^2}{e^2 x^2}\right )^p \, _2F_1\left (1-p,p+1;p+2;\frac {d-e x}{2 d}\right )+p (d-e x) \left (1-\frac {d^2}{e^2 x^2}\right )^p \, _2F_1\left (2-p,p+1;p+2;\frac {d-e x}{2 d}\right )+2 d (p+1) \left (\frac {e x}{2 d}+\frac {1}{2}\right )^p \, _2F_1\left (-p,-p;1-p;\frac {d^2}{e^2 x^2}\right )\right )}{d^3 p (p+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{e^{2} x^{3} + 2 \, d e x^{2} + d^{2} x}, x\right ) \]
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 \[ \int \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{{\left (e x + d\right )}^{2} x}\,{d x} \]
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 \[ \int \frac {\left (-e^{2} x^{2}+d^{2}\right )^{p}}{\left (e x +d \right )^{2} x}\, dx \]
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 \[ \int \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{{\left (e x + d\right )}^{2} x}\,{d x} \]
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 \[ \int \frac {{\left (d^2-e^2\,x^2\right )}^p}{x\,{\left (d+e\,x\right )}^2} \,d x \]
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 \[ \int \frac {\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{p}}{x \left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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