Optimal. Leaf size=148 \[ \frac{1}{7} e x^7 \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{7}{2},-p;\frac{9}{2};\frac{e^2 x^2}{d^2}\right )-\frac{d^5 \left (d^2-e^2 x^2\right )^{p+1}}{2 e^6 (p+1)}+\frac{d^3 \left (d^2-e^2 x^2\right )^{p+2}}{e^6 (p+2)}-\frac{d \left (d^2-e^2 x^2\right )^{p+3}}{2 e^6 (p+3)} \]
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Rubi [A] time = 0.0940171, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {764, 266, 43, 365, 364} \[ \frac{1}{7} e x^7 \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{7}{2},-p;\frac{9}{2};\frac{e^2 x^2}{d^2}\right )-\frac{d^5 \left (d^2-e^2 x^2\right )^{p+1}}{2 e^6 (p+1)}+\frac{d^3 \left (d^2-e^2 x^2\right )^{p+2}}{e^6 (p+2)}-\frac{d \left (d^2-e^2 x^2\right )^{p+3}}{2 e^6 (p+3)} \]
Antiderivative was successfully verified.
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Rule 764
Rule 266
Rule 43
Rule 365
Rule 364
Rubi steps
\begin{align*} \int x^5 (d+e x) \left (d^2-e^2 x^2\right )^p \, dx &=d \int x^5 \left (d^2-e^2 x^2\right )^p \, dx+e \int x^6 \left (d^2-e^2 x^2\right )^p \, dx\\ &=\frac{1}{2} d \operatorname{Subst}\left (\int x^2 \left (d^2-e^2 x\right )^p \, dx,x,x^2\right )+\left (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 )^p \, dx\\ &=\frac{1}{7} 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},-p;\frac{9}{2};\frac{e^2 x^2}{d^2}\right )+\frac{1}{2} d \operatorname{Subst}\left (\int \left (\frac{d^4 \left (d^2-e^2 x\right )^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 )^{2+p}}{e^4}\right ) \, dx,x,x^2\right )\\ &=-\frac{d^5 \left (d^2-e^2 x^2\right )^{1+p}}{2 e^6 (1+p)}+\frac{d^3 \left (d^2-e^2 x^2\right )^{2+p}}{e^6 (2+p)}-\frac{d \left (d^2-e^2 x^2\right )^{3+p}}{2 e^6 (3+p)}+\frac{1}{7} 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},-p;\frac{9}{2};\frac{e^2 x^2}{d^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0985123, size = 132, normalized size = 0.89 \[ \frac{\left (d^2-e^2 x^2\right )^p \left (2 e^7 x^7 \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{7}{2},-p;\frac{9}{2};\frac{e^2 x^2}{d^2}\right )-\frac{7 d \left (d^2-e^2 x^2\right ) \left (2 d^2 e^2 (p+1) x^2+2 d^4+e^4 \left (p^2+3 p+2\right ) x^4\right )}{(p+1) (p+2) (p+3)}\right )}{14 e^6} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.472, size = 0, normalized size = 0. \begin{align*} \int{x}^{5} \left ( ex+d \right ) \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{p}\, 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*} e \int x^{6} e^{\left (p \log \left (e x + d\right ) + p \log \left (-e x + d\right )\right )}\,{d x} + \frac{{\left ({\left (p^{2} + 3 \, p + 2\right )} e^{6} x^{6} -{\left (p^{2} + p\right )} d^{2} e^{4} x^{4} - 2 \, d^{4} e^{2} p x^{2} - 2 \, d^{6}\right )}{\left (-e^{2} x^{2} + d^{2}\right )}^{p} d}{2 \,{\left (p^{3} + 6 \, p^{2} + 11 \, p + 6\right )} e^{6}} \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 ({\left (e x^{6} + d x^{5}\right )}{\left (-e^{2} x^{2} + d^{2}\right )}^{p}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 8.67009, size = 967, normalized size = 6.53 \begin{align*} \text{result too large to display} \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{\left (e x + d\right )}{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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