Optimal. Leaf size=68 \[ -\frac{(d-e x) (d+e x)^{m+1} \, _2F_1\left (1,m-5;m-\frac{3}{2};\frac{d+e x}{2 d}\right )}{d e (5-2 m) \left (d^2-e^2 x^2\right )^{7/2}} \]
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Rubi [A] time = 0.0566717, antiderivative size = 83, normalized size of antiderivative = 1.22, 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} \[ \frac{2^{m-\frac{5}{2}} (d+e x)^m \left (\frac{e x}{d}+1\right )^{\frac{5}{2}-m} \, _2F_1\left (-\frac{5}{2},\frac{7}{2}-m;-\frac{3}{2};\frac{d-e x}{2 d}\right )}{5 d e \left (d^2-e^2 x^2\right )^{5/2}} \]
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 )^{7/2}} \, 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 )^{7/2}} \, dx\\ &=\frac{\left ((d+e x)^m \left (1+\frac{e x}{d}\right )^{\frac{5}{2}-m} \left (d^2-d e x\right )^{5/2}\right ) \int \frac{\left (1+\frac{e x}{d}\right )^{-\frac{7}{2}+m}}{\left (d^2-d e x\right )^{7/2}} \, dx}{\left (d^2-e^2 x^2\right )^{5/2}}\\ &=\frac{2^{-\frac{5}{2}+m} (d+e x)^m \left (1+\frac{e x}{d}\right )^{\frac{5}{2}-m} \, _2F_1\left (-\frac{5}{2},\frac{7}{2}-m;-\frac{3}{2};\frac{d-e x}{2 d}\right )}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.103493, size = 91, normalized size = 1.34 \[ \frac{2^{m-\frac{5}{2}} (d+e x)^m \left (\frac{e x}{d}+1\right )^{\frac{1}{2}-m} \, _2F_1\left (-\frac{5}{2},\frac{7}{2}-m;-\frac{3}{2};\frac{d-e x}{2 d}\right )}{5 d^3 e (d-e x)^2 \sqrt{d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
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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{7}{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{7}{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^{8} x^{8} - 4 \, d^{2} e^{6} x^{6} + 6 \, d^{4} e^{4} x^{4} - 4 \, d^{6} e^{2} x^{2} + d^{8}}, 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 \frac{{\left (e x + d\right )}^{m}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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