Optimal. Leaf size=184 \[ \frac{2 (p+4) x^5 \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac{5}{2},2-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right )}{5 d^2 (2 p+3)}-\frac{x^5 \left (d^2-e^2 x^2\right )^{p-1}}{2 p+3}-\frac{d^5 \left (d^2-e^2 x^2\right )^{p-1}}{e^5 (1-p)}-\frac{2 d^3 \left (d^2-e^2 x^2\right )^p}{e^5 p}+\frac{d \left (d^2-e^2 x^2\right )^{p+1}}{e^5 (p+1)} \]
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Rubi [A] time = 0.205062, antiderivative size = 184, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {852, 1652, 459, 365, 364, 12, 266, 43} \[ \frac{2 (p+4) x^5 \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac{5}{2},2-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right )}{5 d^2 (2 p+3)}-\frac{x^5 \left (d^2-e^2 x^2\right )^{p-1}}{2 p+3}-\frac{d^5 \left (d^2-e^2 x^2\right )^{p-1}}{e^5 (1-p)}-\frac{2 d^3 \left (d^2-e^2 x^2\right )^p}{e^5 p}+\frac{d \left (d^2-e^2 x^2\right )^{p+1}}{e^5 (p+1)} \]
Antiderivative was successfully verified.
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Rule 852
Rule 1652
Rule 459
Rule 365
Rule 364
Rule 12
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^4 \left (d^2-e^2 x^2\right )^p}{(d+e x)^2} \, dx &=\int x^4 (d-e x)^2 \left (d^2-e^2 x^2\right )^{-2+p} \, dx\\ &=\int -2 d e x^5 \left (d^2-e^2 x^2\right )^{-2+p} \, dx+\int x^4 \left (d^2-e^2 x^2\right )^{-2+p} \left (d^2+e^2 x^2\right ) \, dx\\ &=-\frac{x^5 \left (d^2-e^2 x^2\right )^{-1+p}}{3+2 p}-(2 d e) \int x^5 \left (d^2-e^2 x^2\right )^{-2+p} \, dx+\frac{\left (2 d^2 (4+p)\right ) \int x^4 \left (d^2-e^2 x^2\right )^{-2+p} \, dx}{3+2 p}\\ &=-\frac{x^5 \left (d^2-e^2 x^2\right )^{-1+p}}{3+2 p}-(d e) \operatorname{Subst}\left (\int x^2 \left (d^2-e^2 x\right )^{-2+p} \, dx,x,x^2\right )+\frac{\left (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 x^4 \left (1-\frac{e^2 x^2}{d^2}\right )^{-2+p} \, dx}{d^2 (3+2 p)}\\ &=-\frac{x^5 \left (d^2-e^2 x^2\right )^{-1+p}}{3+2 p}+\frac{2 (4+p) x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{5}{2},2-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right )}{5 d^2 (3+2 p)}-(d e) \operatorname{Subst}\left (\int \left (\frac{d^4 \left (d^2-e^2 x\right )^{-2+p}}{e^4}-\frac{2 d^2 \left (d^2-e^2 x\right )^{-1+p}}{e^4}+\frac{\left (d^2-e^2 x\right )^p}{e^4}\right ) \, dx,x,x^2\right )\\ &=-\frac{d^5 \left (d^2-e^2 x^2\right )^{-1+p}}{e^5 (1-p)}-\frac{x^5 \left (d^2-e^2 x^2\right )^{-1+p}}{3+2 p}-\frac{2 d^3 \left (d^2-e^2 x^2\right )^p}{e^5 p}+\frac{d \left (d^2-e^2 x^2\right )^{1+p}}{e^5 (1+p)}+\frac{2 (4+p) x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{5}{2},2-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right )}{5 d^2 (3+2 p)}\\ \end{align*}
Mathematica [C] time = 0.125365, size = 66, normalized size = 0.36 \[ \frac{x^5 (d-e x)^p (d+e x)^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} F_1\left (5;-p,2-p;6;\frac{e x}{d},-\frac{e x}{d}\right )}{5 d^2} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.694, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{4} \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^{4}}{{\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^{4}}{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^{4} \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^{4}}{{\left (e x + d\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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