Optimal. Leaf size=179 \[ -\frac{2 e x^7 \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac{7}{2},2-p;\frac{9}{2};\frac{e^2 x^2}{d^2}\right )}{7 d^3}+\frac{d^6 \left (d^2-e^2 x^2\right )^{p-1}}{e^6 (1-p)}+\frac{5 d^4 \left (d^2-e^2 x^2\right )^p}{2 e^6 p}-\frac{2 d^2 \left (d^2-e^2 x^2\right )^{p+1}}{e^6 (p+1)}+\frac{\left (d^2-e^2 x^2\right )^{p+2}}{2 e^6 (p+2)} \]
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Rubi [A] time = 0.188684, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {852, 1652, 446, 77, 12, 365, 364} \[ -\frac{2 e x^7 \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac{7}{2},2-p;\frac{9}{2};\frac{e^2 x^2}{d^2}\right )}{7 d^3}+\frac{d^6 \left (d^2-e^2 x^2\right )^{p-1}}{e^6 (1-p)}+\frac{5 d^4 \left (d^2-e^2 x^2\right )^p}{2 e^6 p}-\frac{2 d^2 \left (d^2-e^2 x^2\right )^{p+1}}{e^6 (p+1)}+\frac{\left (d^2-e^2 x^2\right )^{p+2}}{2 e^6 (p+2)} \]
Antiderivative was successfully verified.
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Rule 852
Rule 1652
Rule 446
Rule 77
Rule 12
Rule 365
Rule 364
Rubi steps
\begin{align*} \int \frac{x^5 \left (d^2-e^2 x^2\right )^p}{(d+e x)^2} \, dx &=\int x^5 (d-e x)^2 \left (d^2-e^2 x^2\right )^{-2+p} \, dx\\ &=\int -2 d e x^6 \left (d^2-e^2 x^2\right )^{-2+p} \, dx+\int x^5 \left (d^2-e^2 x^2\right )^{-2+p} \left (d^2+e^2 x^2\right ) \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int x^2 \left (d^2-e^2 x\right )^{-2+p} \left (d^2+e^2 x\right ) \, dx,x,x^2\right )-(2 d e) \int x^6 \left (d^2-e^2 x^2\right )^{-2+p} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{2 d^6 \left (d^2-e^2 x\right )^{-2+p}}{e^4}-\frac{5 d^4 \left (d^2-e^2 x\right )^{-1+p}}{e^4}+\frac{4 d^2 \left (d^2-e^2 x\right )^p}{e^4}-\frac{\left (d^2-e^2 x\right )^{1+p}}{e^4}\right ) \, 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 x^6 \left (1-\frac{e^2 x^2}{d^2}\right )^{-2+p} \, dx}{d^3}\\ &=\frac{d^6 \left (d^2-e^2 x^2\right )^{-1+p}}{e^6 (1-p)}+\frac{5 d^4 \left (d^2-e^2 x^2\right )^p}{2 e^6 p}-\frac{2 d^2 \left (d^2-e^2 x^2\right )^{1+p}}{e^6 (1+p)}+\frac{\left (d^2-e^2 x^2\right )^{2+p}}{2 e^6 (2+p)}-\frac{2 e x^7 \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{7}{2},2-p;\frac{9}{2};\frac{e^2 x^2}{d^2}\right )}{7 d^3}\\ \end{align*}
Mathematica [C] time = 0.168477, size = 66, normalized size = 0.37 \[ \frac{x^6 (d-e x)^p (d+e x)^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} F_1\left (6;-p,2-p;7;\frac{e x}{d},-\frac{e x}{d}\right )}{6 d^2} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.705, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{5} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{p}}{ \left ( ex+d \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{5}}{{\left (e x + d\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{5}}{e^{2} x^{2} + 2 \, d e x + d^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{p}}{\left (d + e x\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{5}}{{\left (e x + d\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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