Optimal. Leaf size=39 \[ \frac {(d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}}{2 c e} \]
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Rubi [A] time = 0.02, antiderivative size = 39, 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, 609} \begin {gather*} \frac {(d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}}{2 c e} \end {gather*}
Antiderivative was successfully verified.
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Rule 609
Rule 642
Rubi steps
\begin {align*} \int \frac {(d+e x)^2}{\sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx &=\frac {\int \sqrt {c d^2+2 c d e x+c e^2 x^2} \, dx}{c}\\ &=\frac {(d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}}{2 c e}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 30, normalized size = 0.77 \begin {gather*} \frac {x (d+e x) (2 d+e x)}{2 \sqrt {c (d+e x)^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.34, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(d+e x)^2}{\sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.40, size = 44, normalized size = 1.13 \begin {gather*} \frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} {\left (e x^{2} + 2 \, d x\right )}}{2 \, {\left (c e x + c d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 37, normalized size = 0.95 \begin {gather*} \frac {1}{2} \, \sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}} {\left (\frac {d e^{\left (-1\right )}}{c} + \frac {x}{c}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 38, normalized size = 0.97 \begin {gather*} \frac {\left (e x +2 d \right ) \left (e x +d \right ) x}{2 \sqrt {c \,e^{2} x^{2}+2 c d e x +c \,d^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 48, normalized size = 1.23 \begin {gather*} \frac {e x^{2}}{2 \, \sqrt {c}} - \frac {d x}{\sqrt {c}} + \frac {2 \, \sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} d}{c e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^2}{\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}} \,d x \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}}{\sqrt {c \left (d + e x\right )^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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