Optimal. Leaf size=150 \[ -\frac{2 e x^5 \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac{5}{2},2-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right )}{5 d^3}+\frac{d^4 \left (d^2-e^2 x^2\right )^{p-1}}{e^4 (1-p)}+\frac{3 d^2 \left (d^2-e^2 x^2\right )^p}{2 e^4 p}-\frac{\left (d^2-e^2 x^2\right )^{p+1}}{2 e^4 (p+1)} \]
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Rubi [A] time = 0.164881, antiderivative size = 150, 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} \[ -\frac{2 e x^5 \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac{5}{2},2-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right )}{5 d^3}+\frac{d^4 \left (d^2-e^2 x^2\right )^{p-1}}{e^4 (1-p)}+\frac{3 d^2 \left (d^2-e^2 x^2\right )^p}{2 e^4 p}-\frac{\left (d^2-e^2 x^2\right )^{p+1}}{2 e^4 (p+1)} \]
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^3 \left (d^2-e^2 x^2\right )^p}{(d+e x)^2} \, dx &=\int x^3 (d-e x)^2 \left (d^2-e^2 x^2\right )^{-2+p} \, dx\\ &=\int -2 d e x^4 \left (d^2-e^2 x^2\right )^{-2+p} \, dx+\int x^3 \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 \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^4 \left (d^2-e^2 x^2\right )^{-2+p} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{2 d^4 \left (d^2-e^2 x\right )^{-2+p}}{e^2}-\frac{3 d^2 \left (d^2-e^2 x\right )^{-1+p}}{e^2}+\frac{\left (d^2-e^2 x\right )^p}{e^2}\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^4 \left (1-\frac{e^2 x^2}{d^2}\right )^{-2+p} \, dx}{d^3}\\ &=\frac{d^4 \left (d^2-e^2 x^2\right )^{-1+p}}{e^4 (1-p)}+\frac{3 d^2 \left (d^2-e^2 x^2\right )^p}{2 e^4 p}-\frac{\left (d^2-e^2 x^2\right )^{1+p}}{2 e^4 (1+p)}-\frac{2 e 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^3}\\ \end{align*}
Mathematica [B] time = 0.313731, size = 332, normalized size = 2.21 \[ \frac{2^{p-2} \left (\frac{e x}{d}+1\right )^{-p} \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (-8 d e (p+1) x \left (\frac{e x}{2 d}+\frac{1}{2}\right )^p \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )-6 d (d-e x) \left (1-\frac{e^2 x^2}{d^2}\right )^p \, _2F_1\left (1-p,p+1;p+2;\frac{d-e x}{2 d}\right )+d^2 \left (1-\frac{e^2 x^2}{d^2}\right )^p \, _2F_1\left (2-p,p+1;p+2;\frac{d-e x}{2 d}\right )-d e x \left (1-\frac{e^2 x^2}{d^2}\right )^p \, _2F_1\left (2-p,p+1;p+2;\frac{d-e x}{2 d}\right )-2 d^2 \left (1-\frac{e^2 x^2}{d^2}\right )^p \left (\frac{e x}{2 d}+\frac{1}{2}\right )^p+2 e^2 x^2 \left (1-\frac{e^2 x^2}{d^2}\right )^p \left (\frac{e x}{2 d}+\frac{1}{2}\right )^p+2 d^2 \left (\frac{e x}{2 d}+\frac{1}{2}\right )^p\right )}{e^4 (p+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.705, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3} \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^{3}}{{\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^{3}}{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^{3} \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^{3}}{{\left (e x + d\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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