Optimal. Leaf size=145 \[ -\frac {\left (d^2-e^2 x^2\right )^{-1+p}}{3 x^3}-\frac {2 e^2 (4-p) \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (-\frac {1}{2},2-p;\frac {1}{2};\frac {e^2 x^2}{d^2}\right )}{3 d^4 x}-\frac {e^3 \left (d^2-e^2 x^2\right )^{-1+p} \, _2F_1\left (2,-1+p;p;1-\frac {e^2 x^2}{d^2}\right )}{d^3 (1-p)} \]
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Rubi [A]
time = 0.11, antiderivative size = 145, 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 = {866, 1821, 778,
272, 67, 372, 371} \begin {gather*} -\frac {\left (d^2-e^2 x^2\right )^{p-1}}{3 x^3}-\frac {2 e^2 (4-p) \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 {1}{2};\frac {e^2 x^2}{d^2}\right )}{3 d^4 x}-\frac {e^3 \left (d^2-e^2 x^2\right )^{p-1} \, _2F_1\left (2,p-1;p;1-\frac {e^2 x^2}{d^2}\right )}{d^3 (1-p)} \end {gather*}
Antiderivative was successfully verified.
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Rule 67
Rule 272
Rule 371
Rule 372
Rule 778
Rule 866
Rule 1821
Rubi steps
\begin {align*} \int \frac {\left (d^2-e^2 x^2\right )^p}{x^4 (d+e x)^2} \, dx &=\int \frac {(d-e x)^2 \left (d^2-e^2 x^2\right )^{-2+p}}{x^4} \, dx\\ &=-\frac {\left (d^2-e^2 x^2\right )^{-1+p}}{3 x^3}-\frac {\int \frac {\left (6 d^3 e-2 d^2 e^2 (4-p) x\right ) \left (d^2-e^2 x^2\right )^{-2+p}}{x^3} \, dx}{3 d^2}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{-1+p}}{3 x^3}-(2 d e) \int \frac {\left (d^2-e^2 x^2\right )^{-2+p}}{x^3} \, dx+\frac {1}{3} \left (2 e^2 (4-p)\right ) \int \frac {\left (d^2-e^2 x^2\right )^{-2+p}}{x^2} \, dx\\ &=-\frac {\left (d^2-e^2 x^2\right )^{-1+p}}{3 x^3}-(d e) \text {Subst}\left (\int \frac {\left (d^2-e^2 x\right )^{-2+p}}{x^2} \, dx,x,x^2\right )+\frac {\left (2 e^2 (4-p) \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 )^{-2+p}}{x^2} \, dx}{3 d^4}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{-1+p}}{3 x^3}-\frac {2 e^2 (4-p) \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 {1}{2};\frac {e^2 x^2}{d^2}\right )}{3 d^4 x}-\frac {e^3 \left (d^2-e^2 x^2\right )^{-1+p} \, _2F_1\left (2,-1+p;p;1-\frac {e^2 x^2}{d^2}\right )}{d^3 (1-p)}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(334\) vs. \(2(145)=290\).
time = 0.54, size = 334, normalized size = 2.30 \begin {gather*} \frac {\left (d^2-e^2 x^2\right )^p \left (-\frac {4 d^4 \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (-\frac {3}{2},-p;-\frac {1}{2};\frac {e^2 x^2}{d^2}\right )}{x^3}-\frac {36 d^2 e^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 {12 d^3 e \left (1-\frac {d^2}{e^2 x^2}\right )^{-p} \, _2F_1\left (1-p,-p;2-p;\frac {d^2}{e^2 x^2}\right )}{(-1+p) x^2}+\frac {3\ 2^{3+p} e^3 (-d+e x) \left (1+\frac {e x}{d}\right )^{-p} \, _2F_1\left (1-p,1+p;2+p;\frac {d-e x}{2 d}\right )}{1+p}+\frac {3\ 2^p e^3 (-d+e x) \left (1+\frac {e x}{d}\right )^{-p} \, _2F_1\left (2-p,1+p;2+p;\frac {d-e x}{2 d}\right )}{1+p}-\frac {24 d e^3 \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}\right )}{12 d^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (-e^{2} x^{2}+d^{2}\right )^{p}}{x^{4} \left (e x +d \right )^{2}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{p}}{x^{4} \left (d + e x\right )^{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (d^2-e^2\,x^2\right )}^p}{x^4\,{\left (d+e\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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