Optimal. Leaf size=59 \[ \frac{\left (d^2-e^2 x^2\right )^{5/2} (d+e x)^m \, _2F_1\left (1,m+5;m+\frac{7}{2};\frac{d+e x}{2 d}\right )}{d e (2 m+5)} \]
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Rubi [A] time = 0.0473627, antiderivative size = 83, normalized size of antiderivative = 1.41, 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} \[ -\frac{2^{m+\frac{5}{2}} \left (d^2-e^2 x^2\right )^{5/2} (d+e x)^m \left (\frac{e x}{d}+1\right )^{-m-\frac{5}{2}} \, _2F_1\left (\frac{5}{2},-m-\frac{3}{2};\frac{7}{2};\frac{d-e x}{2 d}\right )}{5 d e} \]
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 )^{3/2} \, 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 )^{3/2} \, dx\\ &=\frac{\left ((d+e x)^m \left (1+\frac{e x}{d}\right )^{-\frac{5}{2}-m} \left (d^2-e^2 x^2\right )^{5/2}\right ) \int \left (1+\frac{e x}{d}\right )^{\frac{3}{2}+m} \left (d^2-d e x\right )^{3/2} \, dx}{\left (d^2-d e x\right )^{5/2}}\\ &=-\frac{2^{\frac{5}{2}+m} (d+e x)^m \left (1+\frac{e x}{d}\right )^{-\frac{5}{2}-m} \left (d^2-e^2 x^2\right )^{5/2} \, _2F_1\left (\frac{5}{2},-\frac{3}{2}-m;\frac{7}{2};\frac{d-e x}{2 d}\right )}{5 d e}\\ \end{align*}
Mathematica [C] time = 0.237686, size = 191, normalized size = 3.24 \[ -\frac{2^m (d+e x)^m \left (\frac{e x}{d}+1\right )^{-2 m-\frac{1}{2}} \left (e^3 x^3 \sqrt{d-e x} \sqrt{d+e x} \left (\frac{e x}{2 d}+\frac{1}{2}\right )^m F_1\left (3;-\frac{1}{2},-m-\frac{1}{2};4;\frac{e x}{d},-\frac{e x}{d}\right )+2 d^2 (d-e x) \sqrt{2-\frac{2 e x}{d}} \sqrt{d^2-e^2 x^2} \left (\frac{e x}{d}+1\right )^m \, _2F_1\left (\frac{3}{2},-m-\frac{1}{2};\frac{5}{2};\frac{d-e x}{2 d}\right )\right )}{3 e \sqrt{1-\frac{e x}{d}}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.479, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) ^{m} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{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{3}{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^{2} x^{2} - d^{2}\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{3}{2}} \left (d + e x\right )^{m}\, 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{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{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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