Optimal. Leaf size=80 \[ \frac{2^{m-\frac{1}{2}} (d+e x)^m \left (\frac{e x}{d}+1\right )^{\frac{1}{2}-m} \, _2F_1\left (-\frac{1}{2},\frac{3}{2}-m;\frac{1}{2};\frac{d-e x}{2 d}\right )}{d e \sqrt{d^2-e^2 x^2}} \]
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Rubi [A] time = 0.0539989, antiderivative size = 80, 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} \[ \frac{2^{m-\frac{1}{2}} (d+e x)^m \left (\frac{e x}{d}+1\right )^{\frac{1}{2}-m} \, _2F_1\left (-\frac{1}{2},\frac{3}{2}-m;\frac{1}{2};\frac{d-e x}{2 d}\right )}{d e \sqrt{d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 680
Rule 678
Rule 69
Rubi steps
\begin{align*} \int \frac{(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 \frac{\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{1}{2}-m} \sqrt{d^2-d e x}\right ) \int \frac{\left (1+\frac{e x}{d}\right )^{-\frac{3}{2}+m}}{\left (d^2-d e x\right )^{3/2}} \, dx}{\sqrt{d^2-e^2 x^2}}\\ &=\frac{2^{-\frac{1}{2}+m} (d+e x)^m \left (1+\frac{e x}{d}\right )^{\frac{1}{2}-m} \, _2F_1\left (-\frac{1}{2},\frac{3}{2}-m;\frac{1}{2};\frac{d-e x}{2 d}\right )}{d e \sqrt{d^2-e^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0700174, size = 80, normalized size = 1. \[ \frac{2^{m-\frac{1}{2}} (d+e x)^m \left (\frac{e x}{d}+1\right )^{\frac{1}{2}-m} \, _2F_1\left (-\frac{1}{2},\frac{3}{2}-m;\frac{1}{2};\frac{d-e x}{2 d}\right )}{d e \sqrt{d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.483, 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 \frac{{\left (e x + d\right )}^{m}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{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^{4} x^{4} - 2 \, d^{2} e^{2} x^{2} + d^{4}}, 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}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{3}{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 x + d\right )}^{m}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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