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 = 0.05, 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 69
Rule 678
Rule 680
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*}
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Mathematica [C] time = 0.22, 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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fricas [F] time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (e^{2} x^{2} - d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}} {\left (e x + d\right )}^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.76, size = 0, normalized size = 0.00 \[ \int \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} \left (e x +d \right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (d^2-e^2\,x^2\right )}^{3/2}\,{\left (d+e\,x\right )}^m \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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