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 = 0.05, antiderivative size = 80, normalized size of antiderivative = 1.00, 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 69
Rule 678
Rule 680
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*}
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Mathematica [A] time = 0.07, size = 80, normalized size = 1.00 \[ \frac {2^{m-\frac {1}{2}} (d+e x)^m \left (\frac {e x}{d}+1\right )^{\frac {1}{2}-m} \, _2F_1\left (-\frac {1}{2},\frac {3}{2}-m;\frac {1}{2};\frac {d-e x}{2 d}\right )}{d e \sqrt {d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.19, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-e^{2} x^{2} + d^{2}} {\left (e x + d\right )}^{m}}{e^{4} x^{4} - 2 \, d^{2} e^{2} x^{2} + d^{4}}, x\right ) \]
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 \[ \int \frac {{\left (e x + d\right )}^{m}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {\left (e x +d \right )^{m}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {{\left (e x + d\right )}^{m}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (d+e\,x\right )}^m}{{\left (d^2-e^2\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (d + e x\right )^{m}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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