Optimal. Leaf size=83 \[ -\frac{2^{m+\frac{7}{2}} \left (d^2-e^2 x^2\right )^{7/2} (d+e x)^m \left (\frac{e x}{d}+1\right )^{-m-\frac{7}{2}} \, _2F_1\left (\frac{7}{2},-m-\frac{5}{2};\frac{9}{2};\frac{d-e x}{2 d}\right )}{7 d e} \]
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Rubi [A] time = 0.0475875, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {680, 678, 69} \[ -\frac{2^{m+\frac{7}{2}} \left (d^2-e^2 x^2\right )^{7/2} (d+e x)^m \left (\frac{e x}{d}+1\right )^{-m-\frac{7}{2}} \, _2F_1\left (\frac{7}{2},-m-\frac{5}{2};\frac{9}{2};\frac{d-e x}{2 d}\right )}{7 d e} \]
Antiderivative was successfully verified.
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Rule 680
Rule 678
Rule 69
Rubi steps
\begin{align*} \int (d+e x)^m \left (d^2-e^2 x^2\right )^{5/2} \, dx &=\left ((d+e x)^m \left (1+\frac{e x}{d}\right )^{-m}\right ) \int \left (1+\frac{e x}{d}\right )^m \left (d^2-e^2 x^2\right )^{5/2} \, dx\\ &=\frac{\left ((d+e x)^m \left (1+\frac{e x}{d}\right )^{-\frac{7}{2}-m} \left (d^2-e^2 x^2\right )^{7/2}\right ) \int \left (1+\frac{e x}{d}\right )^{\frac{5}{2}+m} \left (d^2-d e x\right )^{5/2} \, dx}{\left (d^2-d e x\right )^{7/2}}\\ &=-\frac{2^{\frac{7}{2}+m} (d+e x)^m \left (1+\frac{e x}{d}\right )^{-\frac{7}{2}-m} \left (d^2-e^2 x^2\right )^{7/2} \, _2F_1\left (\frac{7}{2},-\frac{5}{2}-m;\frac{9}{2};\frac{d-e x}{2 d}\right )}{7 d e}\\ \end{align*}
Mathematica [C] time = 0.39913, size = 227, normalized size = 2.73 \[ \frac{(d+e x)^m \left (\frac{e x}{d}+1\right )^{-m-\frac{1}{2}} \left (-10 d^2 e^3 x^3 \sqrt{d-e x} \sqrt{d+e x} F_1\left (3;-\frac{1}{2},-m-\frac{1}{2};4;\frac{e x}{d},-\frac{e x}{d}\right )+3 e^5 x^5 \sqrt{d-e x} \sqrt{d+e x} F_1\left (5;-\frac{1}{2},-m-\frac{1}{2};6;\frac{e x}{d},-\frac{e x}{d}\right )-5 d^4 2^{m+\frac{3}{2}} (d-e x) \sqrt{1-\frac{e x}{d}} \sqrt{d^2-e^2 x^2} \, _2F_1\left (\frac{3}{2},-m-\frac{1}{2};\frac{5}{2};\frac{d-e x}{2 d}\right )\right )}{15 e \sqrt{1-\frac{e x}{d}}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.484, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) ^{m} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{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{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}{\left (e x + d\right )}^{m}\,{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 ({\left (e^{4} x^{4} - 2 \, d^{2} e^{2} x^{2} + d^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}{\left (e x + d\right )}^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}{\left (e x + d\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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