Optimal. Leaf size=59 \[ \frac{\left (d^2-e^2 x^2\right )^{9/2} (d+e x)^m \, _2F_1\left (1,m+9;m+\frac{11}{2};\frac{d+e x}{2 d}\right )}{d e (2 m+9)} \]
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Rubi [A] time = 0.0490376, 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} \[ -\frac{2^{m+\frac{9}{2}} \left (d^2-e^2 x^2\right )^{9/2} (d+e x)^m \left (\frac{e x}{d}+1\right )^{-m-\frac{9}{2}} \, _2F_1\left (\frac{9}{2},-m-\frac{7}{2};\frac{11}{2};\frac{d-e x}{2 d}\right )}{9 d e} \]
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 )^{7/2} \, 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 )^{7/2} \, dx\\ &=\frac{\left ((d+e x)^m \left (1+\frac{e x}{d}\right )^{-\frac{9}{2}-m} \left (d^2-e^2 x^2\right )^{9/2}\right ) \int \left (1+\frac{e x}{d}\right )^{\frac{7}{2}+m} \left (d^2-d e x\right )^{7/2} \, dx}{\left (d^2-d e x\right )^{9/2}}\\ &=-\frac{2^{\frac{9}{2}+m} (d+e x)^m \left (1+\frac{e x}{d}\right )^{-\frac{9}{2}-m} \left (d^2-e^2 x^2\right )^{9/2} \, _2F_1\left (\frac{9}{2},-\frac{7}{2}-m;\frac{11}{2};\frac{d-e x}{2 d}\right )}{9 d e}\\ \end{align*}
Mathematica [C] time = 0.516095, size = 347, normalized size = 5.88 \[ \frac{(d+e x)^m \left (\frac{e x}{d}+1\right )^{-m-\frac{1}{2}} \left (-105 d^4 e^3 x^3 \sqrt{d-e x} \sqrt{d+e x} F_1\left (3;-\frac{1}{2},-m-\frac{1}{2};4;\frac{e x}{d},-\frac{e x}{d}\right )+63 d^2 e^5 x^5 \sqrt{d-e x} \sqrt{d+e x} F_1\left (5;-\frac{1}{2},-m-\frac{1}{2};6;\frac{e x}{d},-\frac{e x}{d}\right )-15 e^7 x^7 \sqrt{d-e x} \sqrt{d+e x} F_1\left (7;-\frac{1}{2},-m-\frac{1}{2};8;\frac{e x}{d},-\frac{e x}{d}\right )-35 d^7 2^{m+\frac{3}{2}} \sqrt{1-\frac{e x}{d}} \sqrt{d^2-e^2 x^2} \, _2F_1\left (\frac{3}{2},-m-\frac{1}{2};\frac{5}{2};\frac{d-e x}{2 d}\right )+35 d^6 e 2^{m+\frac{3}{2}} x \sqrt{1-\frac{e x}{d}} \sqrt{d^2-e^2 x^2} \, _2F_1\left (\frac{3}{2},-m-\frac{1}{2};\frac{5}{2};\frac{d-e x}{2 d}\right )\right )}{105 e \sqrt{1-\frac{e x}{d}}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.506, 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{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{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^{6} x^{6} - 3 \, d^{2} e^{4} x^{4} + 3 \, d^{4} e^{2} x^{2} - d^{6}\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{7}{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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