Optimal. Leaf size=67 \[ \frac {(d-e x) \sqrt {d^2-e^2 x^2} (d+e x)^{m+1} \, _2F_1\left (1,m+3;m+\frac {5}{2};\frac {d+e x}{2 d}\right )}{d e (2 m+3)} \]
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Rubi [A] time = 0.05, antiderivative size = 83, normalized size of antiderivative = 1.24, 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} \[ -\frac {2^{m+\frac {3}{2}} \left (d^2-e^2 x^2\right )^{3/2} (d+e x)^m \left (\frac {e x}{d}+1\right )^{-m-\frac {3}{2}} \, _2F_1\left (\frac {3}{2},-m-\frac {1}{2};\frac {5}{2};\frac {d-e x}{2 d}\right )}{3 d e} \]
Antiderivative was successfully verified.
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Rule 69
Rule 678
Rule 680
Rubi steps
\begin {align*} \int (d+e x)^m \sqrt {d^2-e^2 x^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 \sqrt {d^2-e^2 x^2} \, dx\\ &=\frac {\left ((d+e x)^m \left (1+\frac {e x}{d}\right )^{-\frac {3}{2}-m} \left (d^2-e^2 x^2\right )^{3/2}\right ) \int \left (1+\frac {e x}{d}\right )^{\frac {1}{2}+m} \sqrt {d^2-d e x} \, dx}{\left (d^2-d e x\right )^{3/2}}\\ &=-\frac {2^{\frac {3}{2}+m} (d+e x)^m \left (1+\frac {e x}{d}\right )^{-\frac {3}{2}-m} \left (d^2-e^2 x^2\right )^{3/2} \, _2F_1\left (\frac {3}{2},-\frac {1}{2}-m;\frac {5}{2};\frac {d-e x}{2 d}\right )}{3 d e}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 86, normalized size = 1.28 \[ -\frac {2^{m+\frac {3}{2}} (d-e x) \sqrt {d^2-e^2 x^2} (d+e x)^m \left (\frac {e x}{d}+1\right )^{-m-\frac {1}{2}} \, _2F_1\left (\frac {3}{2},-m-\frac {1}{2};\frac {5}{2};\frac {d-e x}{2 d}\right )}{3 e} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.21, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {-e^{2} x^{2} + d^{2}} {\left (e x + d\right )}^{m}, x\right ) \]
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 \[ \int \sqrt {-e^{2} x^{2} + d^{2}} {\left (e x + d\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.80, size = 0, normalized size = 0.00 \[ \int \sqrt {-e^{2} x^{2}+d^{2}}\, \left (e x +d \right )^{m}\, dx \]
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 \[ \int \sqrt {-e^{2} x^{2} + d^{2}} {\left (e x + d\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \sqrt {d^2-e^2\,x^2}\,{\left (d+e\,x\right )}^m \,d x \]
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 \[ \int \sqrt {- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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