Optimal. Leaf size=41 \[ -\frac {1}{2 c^2 e (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]
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Rubi [A] time = 0.02, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {642, 607} \begin {gather*} -\frac {1}{2 c^2 e (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 607
Rule 642
Rubi steps
\begin {align*} \int \frac {(d+e x)^2}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx &=\frac {\int \frac {1}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx}{c}\\ &=-\frac {1}{2 c^2 e (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 0.68 \begin {gather*} -\frac {d+e x}{2 c e \left (c (d+e x)^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 0.62, size = 156, normalized size = 3.80 \begin {gather*} \frac {\sqrt {c e^2} (d-e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}+c d^2 e+c e^3 x^2}{c e x^2 \sqrt {c e^2} \left (2 c^2 d^2 e^2+4 c^2 d e^3 x+2 c^2 e^4 x^2\right )+c e x^2 \left (-2 c^2 d e^3-2 c^2 e^4 x\right ) \sqrt {c d^2+2 c d e x+c e^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.38, size = 69, normalized size = 1.68 \begin {gather*} -\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{2 \, {\left (c^{3} e^{4} x^{3} + 3 \, c^{3} d e^{3} x^{2} + 3 \, c^{3} d^{2} e^{2} x + c^{3} d^{3} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.47, size = 70, normalized size = 1.71 \begin {gather*} \frac {4 \, C_{0} d^{3} e^{\left (-3\right )} + {\left (12 \, C_{0} d^{2} e^{\left (-2\right )} + 4 \, {\left (3 \, C_{0} d e^{\left (-1\right )} + C_{0} x\right )} x - \frac {1}{c}\right )} x - \frac {d e^{\left (-1\right )}}{c}}{2 \, {\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 35, normalized size = 0.85 \begin {gather*} -\frac {\left (e x +d \right )^{3}}{2 \left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{\frac {5}{2}} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 67, normalized size = 1.63 \begin {gather*} -\frac {2 \, d}{3 \, {\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {3}{2}} c e} - \frac {1}{2 \, c^{\frac {5}{2}} e^{3} {\left (x + \frac {d}{e}\right )}^{2}} + \frac {2 \, d}{3 \, c^{\frac {5}{2}} e^{4} {\left (x + \frac {d}{e}\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.49, size = 37, normalized size = 0.90 \begin {gather*} -\frac {\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}}{2\,c^3\,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 )^{2}}{\left (c \left (d + e x\right )^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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