Optimal. Leaf size=67 \[ \frac{(d-e x) (d+e x)^{m+1} \, _2F_1\left (1,m+1;m+\frac{3}{2};\frac{d+e x}{2 d}\right )}{d e (2 m+1) \sqrt{d^2-e^2 x^2}} \]
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Rubi [A] time = 0.0451431, antiderivative size = 81, normalized size of antiderivative = 1.21, 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} \[ -\frac{2^{m+\frac{1}{2}} \sqrt{d^2-e^2 x^2} (d+e x)^m \left (\frac{e x}{d}+1\right )^{-m-\frac{1}{2}} \, _2F_1\left (\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{d-e x}{2 d}\right )}{d e} \]
Antiderivative was successfully verified.
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Rule 680
Rule 678
Rule 69
Rubi steps
\begin{align*} \int \frac{(d+e x)^m}{\sqrt{d^2-e^2 x^2}} \, dx &=\left ((d+e x)^m \left (1+\frac{e x}{d}\right )^{-m}\right ) \int \frac{\left (1+\frac{e x}{d}\right )^m}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=\frac{\left ((d+e x)^m \left (1+\frac{e x}{d}\right )^{-\frac{1}{2}-m} \sqrt{d^2-e^2 x^2}\right ) \int \frac{\left (1+\frac{e x}{d}\right )^{-\frac{1}{2}+m}}{\sqrt{d^2-d e x}} \, dx}{\sqrt{d^2-d e x}}\\ &=-\frac{2^{\frac{1}{2}+m} (d+e x)^m \left (1+\frac{e x}{d}\right )^{-\frac{1}{2}-m} \sqrt{d^2-e^2 x^2} \, _2F_1\left (\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{d-e x}{2 d}\right )}{d e}\\ \end{align*}
Mathematica [A] time = 0.0816501, size = 84, normalized size = 1.25 \[ -\frac{2^{m+\frac{1}{2}} (d-e x) (d+e x)^m \left (\frac{e x}{d}+1\right )^{\frac{1}{2}-m} \, _2F_1\left (\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{d-e x}{2 d}\right )}{e \sqrt{d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.484, size = 0, normalized size = 0. \begin{align*} \int{ \left ( ex+d \right ) ^{m}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{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 x + d\right )}^{m}}{\sqrt{-e^{2} x^{2} + d^{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{\sqrt{-e^{2} x^{2} + d^{2}}{\left (e x + d\right )}^{m}}{e^{2} x^{2} - 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{\left (d + e x\right )^{m}}{\sqrt{- \left (- d + e x\right ) \left (d + e x\right )}}\, 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 x + d\right )}^{m}}{\sqrt{-e^{2} x^{2} + d^{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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