Optimal. Leaf size=33 \[ \frac {(d+e x)^5}{5 d e \left (d^2-e^2 x^2\right )^{5/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {651} \begin {gather*} \frac {(d+e x)^5}{5 d e \left (d^2-e^2 x^2\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 651
Rubi steps
\begin {align*} \int \frac {(d+e x)^5}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac {(d+e x)^5}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 41, normalized size = 1.24 \begin {gather*} \frac {(d+e x)^3}{5 d e (d-e x)^2 \sqrt {d^2-e^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.44, size = 50, normalized size = 1.52 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (d^2+2 d e x+e^2 x^2\right )}{5 d e (d-e x)^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 100, normalized size = 3.03 \begin {gather*} \frac {e^{3} x^{3} - 3 \, d e^{2} x^{2} + 3 \, d^{2} e x - d^{3} - {\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{5 \, {\left (d e^{4} x^{3} - 3 \, d^{2} e^{3} x^{2} + 3 \, d^{3} e^{2} x - d^{4} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.30, size = 76, normalized size = 2.30 \begin {gather*} -\frac {{\left (d^{4} e^{\left (-1\right )} + {\left (5 \, d^{3} + {\left (10 \, d^{2} e + {\left (x {\left (\frac {x e^{4}}{d} + 5 \, e^{3}\right )} + 10 \, d e^{2}\right )} x\right )} x\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{5 \, {\left (x^{2} e^{2} - d^{2}\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 36, normalized size = 1.09 \begin {gather*} \frac {\left (e x +d \right )^{6} \left (-e x +d \right )}{5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.45, size = 148, normalized size = 4.48 \begin {gather*} \frac {e^{3} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {5 \, d e^{2} x^{3}}{2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {2 \, d^{2} e x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {7 \, d^{3} x}{10 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {d^{4}}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e} + \frac {d x}{10 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}}} + \frac {x}{5 \, \sqrt {-e^{2} x^{2} + d^{2}} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.87, size = 37, normalized size = 1.12 \begin {gather*} \frac {\sqrt {d^2-e^2\,x^2}\,{\left (d+e\,x\right )}^2}{5\,d\,e\,{\left (d-e\,x\right )}^3} \end {gather*}
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 \begin {gather*} \int \frac {\left (d + e x\right )^{5}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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