Optimal. Leaf size=38 \[ -\frac {1}{2 e (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]
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Rubi [A] time = 0.02, antiderivative size = 38, 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 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 {1}{(d+e x)^2 \sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx &=c \int \frac {1}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx\\ &=-\frac {1}{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 = 26, normalized size = 0.68 \begin {gather*} -\frac {c (d+e x)}{2 e \left (c (d+e x)^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 0.71, size = 209, normalized size = 5.50 \begin {gather*} \frac {c \left (c d^4 e-15 c d^2 e^3 x^2-32 c d e^4 x^3-16 c e^5 x^4\right )+c \sqrt {c e^2} \left (d^3-d^2 e x+16 d e^2 x^2+16 e^3 x^3\right ) \sqrt {c d^2+2 c d e x+c e^2 x^2}}{d^2 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 )+d^2 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 = 0.41, size = 61, normalized size = 1.61 \begin {gather*} -\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{2 \, {\left (c e^{4} x^{3} + 3 \, c d e^{3} x^{2} + 3 \, c d^{2} e^{2} x + c d^{3} e\right )}} \end {gather*}
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 \begin {gather*} \mathit {sage}_{0} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 35, normalized size = 0.92 \begin {gather*} -\frac {1}{2 \left (e x +d \right ) \sqrt {c \,e^{2} x^{2}+2 c d e x +c \,d^{2}}\, e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.45, size = 33, normalized size = 0.87 \begin {gather*} -\frac {1}{2 \, {\left (\sqrt {c} e^{3} x^{2} + 2 \, \sqrt {c} d e^{2} x + \sqrt {c} d^{2} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.45, size = 37, normalized size = 0.97 \begin {gather*} -\frac {\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}}{2\,c\,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 {1}{\sqrt {c \left (d + e x\right )^{2}} \left (d + e x\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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