Optimal. Leaf size=194 \[ -\frac{2 e (3 p+4) x^5 \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac{5}{2},3-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right )}{5 d^4 (2 p+1)}+\frac{e x^5 \left (d^2-e^2 x^2\right )^{p-2}}{2 p+1}+\frac{2 d^5 \left (d^2-e^2 x^2\right )^{p-2}}{e^4 (2-p)}-\frac{7 d^3 \left (d^2-e^2 x^2\right )^{p-1}}{2 e^4 (1-p)}-\frac{3 d \left (d^2-e^2 x^2\right )^p}{2 e^4 p} \]
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Rubi [A] time = 0.205627, antiderivative size = 194, 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, 459, 365, 364} \[ -\frac{2 e (3 p+4) x^5 \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac{5}{2},3-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right )}{5 d^4 (2 p+1)}+\frac{e x^5 \left (d^2-e^2 x^2\right )^{p-2}}{2 p+1}+\frac{2 d^5 \left (d^2-e^2 x^2\right )^{p-2}}{e^4 (2-p)}-\frac{7 d^3 \left (d^2-e^2 x^2\right )^{p-1}}{2 e^4 (1-p)}-\frac{3 d \left (d^2-e^2 x^2\right )^p}{2 e^4 p} \]
Antiderivative was successfully verified.
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Rule 852
Rule 1652
Rule 446
Rule 77
Rule 459
Rule 365
Rule 364
Rubi steps
\begin{align*} \int \frac{x^3 \left (d^2-e^2 x^2\right )^p}{(d+e x)^3} \, dx &=\int x^3 (d-e x)^3 \left (d^2-e^2 x^2\right )^{-3+p} \, dx\\ &=\int x^3 \left (d^2-e^2 x^2\right )^{-3+p} \left (d^3+3 d e^2 x^2\right ) \, dx+\int x^4 \left (d^2-e^2 x^2\right )^{-3+p} \left (-3 d^2 e-e^3 x^2\right ) \, dx\\ &=\frac{e x^5 \left (d^2-e^2 x^2\right )^{-2+p}}{1+2 p}+\frac{1}{2} \operatorname{Subst}\left (\int x \left (d^2-e^2 x\right )^{-3+p} \left (d^3+3 d e^2 x\right ) \, dx,x,x^2\right )-\frac{\left (2 d^2 e (4+3 p)\right ) \int x^4 \left (d^2-e^2 x^2\right )^{-3+p} \, dx}{1+2 p}\\ &=\frac{e x^5 \left (d^2-e^2 x^2\right )^{-2+p}}{1+2 p}+\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{4 d^5 \left (d^2-e^2 x\right )^{-3+p}}{e^2}-\frac{7 d^3 \left (d^2-e^2 x\right )^{-2+p}}{e^2}+\frac{3 d \left (d^2-e^2 x\right )^{-1+p}}{e^2}\right ) \, dx,x,x^2\right )-\frac{\left (2 e (4+3 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 )^{-3+p} \, dx}{d^4 (1+2 p)}\\ &=\frac{2 d^5 \left (d^2-e^2 x^2\right )^{-2+p}}{e^4 (2-p)}+\frac{e x^5 \left (d^2-e^2 x^2\right )^{-2+p}}{1+2 p}-\frac{7 d^3 \left (d^2-e^2 x^2\right )^{-1+p}}{2 e^4 (1-p)}-\frac{3 d \left (d^2-e^2 x^2\right )^p}{2 e^4 p}-\frac{2 e (4+3 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},3-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right )}{5 d^4 (1+2 p)}\\ \end{align*}
Mathematica [C] time = 0.121968, size = 66, normalized size = 0.34 \[ \frac{x^4 (d-e x)^p (d+e x)^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} F_1\left (4;-p,3-p;5;\frac{e x}{d},-\frac{e x}{d}\right )}{4 d^3} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.671, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{p}}{ \left ( ex+d \right ) ^{3}}}\, 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 )}^{3}}\,{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^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}, 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 )^{3}}\, 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 )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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