Optimal. Leaf size=211 \[ \frac{3\ 2^{p-2} (p+2) \left (d^2-e^2 x^2\right )^{p+1} \left (\frac{e x}{d}+1\right )^{-p-1} \, _2F_1\left (3-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{d^2 e^4 (1-2 p) (3-p) p (p+1)}-\frac{\left (d^2-e^2 x^2\right )^{p+1}}{2 e^4 p (d+e x)^2}-\frac{d (2 p+1) \left (d^2-e^2 x^2\right )^{p+1}}{e^4 (1-2 p) p (d+e x)^3}+\frac{d^2 \left (d^2-e^2 x^2\right )^{p+1}}{2 e^4 (3-p) (d+e x)^4} \]
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Rubi [A] time = 0.375452, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {1639, 793, 678, 69} \[ \frac{3\ 2^{p-2} (p+2) \left (d^2-e^2 x^2\right )^{p+1} \left (\frac{e x}{d}+1\right )^{-p-1} \, _2F_1\left (3-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{d^2 e^4 (1-2 p) (3-p) p (p+1)}-\frac{\left (d^2-e^2 x^2\right )^{p+1}}{2 e^4 p (d+e x)^2}-\frac{d (2 p+1) \left (d^2-e^2 x^2\right )^{p+1}}{e^4 (1-2 p) p (d+e x)^3}+\frac{d^2 \left (d^2-e^2 x^2\right )^{p+1}}{2 e^4 (3-p) (d+e x)^4} \]
Antiderivative was successfully verified.
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Rule 1639
Rule 793
Rule 678
Rule 69
Rubi steps
\begin{align*} \int \frac{x^3 \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}}{2 e^4 p (d+e x)^2}-\frac{\int \frac{\left (d^2-e^2 x^2\right )^p \left (2 d^3 e^2+2 d^2 e^3 (2+p) x+2 d e^4 (1+2 p) x^2\right )}{(d+e x)^4} \, dx}{2 e^5 p}\\ &=-\frac{d (1+2 p) \left (d^2-e^2 x^2\right )^{1+p}}{e^4 (1-2 p) p (d+e x)^3}-\frac{\left (d^2-e^2 x^2\right )^{1+p}}{2 e^4 p (d+e x)^2}-\frac{\int \frac{\left (8 d^3 e^6 (1+p)+2 d^2 e^7 \left (4+3 p+2 p^2\right ) x\right ) \left (d^2-e^2 x^2\right )^p}{(d+e x)^4} \, dx}{2 e^9 (1-2 p) p}\\ &=\frac{d^2 \left (d^2-e^2 x^2\right )^{1+p}}{2 e^4 (3-p) (d+e x)^4}-\frac{d (1+2 p) \left (d^2-e^2 x^2\right )^{1+p}}{e^4 (1-2 p) p (d+e x)^3}-\frac{\left (d^2-e^2 x^2\right )^{1+p}}{2 e^4 p (d+e x)^2}-\frac{\left (6 d^2 (2+p)\right ) \int \frac{\left (d^2-e^2 x^2\right )^p}{(d+e x)^3} \, dx}{e^3 (1-2 p) (3-p) p}\\ &=\frac{d^2 \left (d^2-e^2 x^2\right )^{1+p}}{2 e^4 (3-p) (d+e x)^4}-\frac{d (1+2 p) \left (d^2-e^2 x^2\right )^{1+p}}{e^4 (1-2 p) p (d+e x)^3}-\frac{\left (d^2-e^2 x^2\right )^{1+p}}{2 e^4 p (d+e x)^2}-\frac{\left (6 (2+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^2 e^3 (1-2 p) (3-p) p}\\ &=\frac{d^2 \left (d^2-e^2 x^2\right )^{1+p}}{2 e^4 (3-p) (d+e x)^4}-\frac{d (1+2 p) \left (d^2-e^2 x^2\right )^{1+p}}{e^4 (1-2 p) p (d+e x)^3}-\frac{\left (d^2-e^2 x^2\right )^{1+p}}{2 e^4 p (d+e x)^2}+\frac{3\ 2^{-2+p} (2+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^2 e^4 (1-2 p) (3-p) p (1+p)}\\ \end{align*}
Mathematica [C] time = 0.120592, size = 66, normalized size = 0.31 \[ \frac{x^4 (d-e x)^p (d+e x)^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} F_1\left (4;-p,4-p;5;\frac{e x}{d},-\frac{e x}{d}\right )}{4 d^4} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.732, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{p}}{ \left ( ex+d \right ) ^{4}}}\, 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^{3}}{{\left (e x + d\right )}^{4}}\,{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^{3}}{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 ) \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^{3} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{p}}{\left (d + e x\right )^{4}}\, 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^{3}}{{\left (e x + d\right )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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