Optimal. Leaf size=163 \[ \frac {\left (d^2-e^2 x^2\right )^{p+1}}{e^3 (1-2 p) (d+e x)^3}-\frac {d \left (d^2-e^2 x^2\right )^{p+1}}{2 e^3 (3-p) (d+e x)^4}-\frac {2^{p-3} (p+7) \left (d^2-e^2 x^2\right )^{p+1} \left (\frac {e x}{d}+1\right )^{-p-1} \, _2F_1\left (3-p,p+1;p+2;\frac {d-e x}{2 d}\right )}{d^3 e^3 (1-2 p) (3-p) (p+1)} \]
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Rubi [A] time = 0.19, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {1639, 793, 678, 69} \[ -\frac {2^{p-3} (p+7) \left (d^2-e^2 x^2\right )^{p+1} \left (\frac {e x}{d}+1\right )^{-p-1} \, _2F_1\left (3-p,p+1;p+2;\frac {d-e x}{2 d}\right )}{d^3 e^3 (1-2 p) (3-p) (p+1)}+\frac {\left (d^2-e^2 x^2\right )^{p+1}}{e^3 (1-2 p) (d+e x)^3}-\frac {d \left (d^2-e^2 x^2\right )^{p+1}}{2 e^3 (3-p) (d+e x)^4} \]
Antiderivative was successfully verified.
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Rule 69
Rule 678
Rule 793
Rule 1639
Rubi steps
\begin {align*} \int \frac {x^2 \left (d^2-e^2 x^2\right )^p}{(d+e x)^4} \, dx &=\frac {\left (d^2-e^2 x^2\right )^{1+p}}{e^3 (1-2 p) (d+e x)^3}+\frac {\int \frac {\left (3 d^2 e^2+2 d e^3 (1+p) x\right ) \left (d^2-e^2 x^2\right )^p}{(d+e x)^4} \, dx}{e^4 (1-2 p)}\\ &=-\frac {d \left (d^2-e^2 x^2\right )^{1+p}}{2 e^3 (3-p) (d+e x)^4}+\frac {\left (d^2-e^2 x^2\right )^{1+p}}{e^3 (1-2 p) (d+e x)^3}+\frac {(d (7+p)) \int \frac {\left (d^2-e^2 x^2\right )^p}{(d+e x)^3} \, dx}{e^2 (1-2 p) (3-p)}\\ &=-\frac {d \left (d^2-e^2 x^2\right )^{1+p}}{2 e^3 (3-p) (d+e x)^4}+\frac {\left (d^2-e^2 x^2\right )^{1+p}}{e^3 (1-2 p) (d+e x)^3}+\frac {\left ((7+p) (d-e x)^{-1-p} \left (1+\frac {e x}{d}\right )^{-1-p} \left (d^2-e^2 x^2\right )^{1+p}\right ) \int (d-e x)^p \left (1+\frac {e x}{d}\right )^{-3+p} \, dx}{d^3 e^2 (1-2 p) (3-p)}\\ &=-\frac {d \left (d^2-e^2 x^2\right )^{1+p}}{2 e^3 (3-p) (d+e x)^4}+\frac {\left (d^2-e^2 x^2\right )^{1+p}}{e^3 (1-2 p) (d+e x)^3}-\frac {2^{-3+p} (7+p) \left (1+\frac {e x}{d}\right )^{-1-p} \left (d^2-e^2 x^2\right )^{1+p} \, _2F_1\left (3-p,1+p;2+p;\frac {d-e x}{2 d}\right )}{d^3 e^3 (1-2 p) (3-p) (1+p)}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 130, normalized size = 0.80 \[ -\frac {2^{p-4} (d-e x) \left (\frac {e x}{d}+1\right )^{-p} \left (d^2-e^2 x^2\right )^p \left (4 \, _2F_1\left (2-p,p+1;p+2;\frac {d-e x}{2 d}\right )-4 \, _2F_1\left (3-p,p+1;p+2;\frac {d-e x}{2 d}\right )+\, _2F_1\left (4-p,p+1;p+2;\frac {d-e x}{2 d}\right )\right )}{d^2 e^3 (p+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.93, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{2}}{e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}}, x\right ) \]
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 \[ \int \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{2}}{{\left (e x + d\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.11, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{p}}{\left (e x +d \right )^{4}}\, dx \]
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 \[ \int \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{2}}{{\left (e x + d\right )}^{4}}\,{d x} \]
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 \[ \int \frac {x^2\,{\left (d^2-e^2\,x^2\right )}^p}{{\left (d+e\,x\right )}^4} \,d x \]
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 \[ \int \frac {x^{2} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{p}}{\left (d + e x\right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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