Optimal. Leaf size=42 \[ \frac {(d+e x) \log (d+e x)}{c e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]
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Rubi [A] time = 0.03, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {642, 608, 31} \begin {gather*} \frac {(d+e x) \log (d+e x)}{c e \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 31
Rule 608
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 )^{3/2}} \, dx &=\frac {\int \frac {1}{\sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx}{c}\\ &=\frac {\left (c d e+c e^2 x\right ) \int \frac {1}{c d e+c e^2 x} \, dx}{c \sqrt {c d^2+2 c d e x+c e^2 x^2}}\\ &=\frac {(d+e x) \log (d+e x)}{c e \sqrt {c d^2+2 c d e x+c e^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 31, normalized size = 0.74 \begin {gather*} \frac {(d+e x) \log (d+e x)}{c e \sqrt {c (d+e x)^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 0.34, size = 136, normalized size = 3.24 \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {x \sqrt {c e^2}}{\sqrt {c} d}-\frac {\sqrt {c d^2+2 c d e x+c e^2 x^2}}{\sqrt {c} d}\right )}{c^{3/2} e}-\frac {\sqrt {c e^2} \log \left (x \left (c d e+c e^2 x\right )-x \sqrt {c e^2} \sqrt {c d^2+2 c d e x+c e^2 x^2}\right )}{2 c^2 e^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 46, normalized size = 1.10 \begin {gather*} \frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} \log \left (e x + d\right )}{c^{2} e^{2} x + c^{2} d e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.68, size = 88, normalized size = 2.10 \begin {gather*} \frac {2 \, {\left (C_{0} d e^{\left (-1\right )} + C_{0} x\right )}}{\sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}}} - \frac {e^{\left (-1\right )} \log \left ({\left | -\sqrt {c} d e^{2} - {\left (\sqrt {c} x e - \sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}}\right )} e^{2} \right |}\right )}{c^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 40, normalized size = 0.95 \begin {gather*} \frac {\left (e x +d \right )^{3} \ln \left (e x +d \right )}{\left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{\frac {3}{2}} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.34, size = 86, normalized size = 2.05 \begin {gather*} -\frac {2 \, d}{\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c e} + \frac {\log \left (x + \frac {d}{e}\right )}{c^{\frac {3}{2}} e} + \frac {2 \, d x}{c^{\frac {3}{2}} e^{2} {\left (x + \frac {d}{e}\right )}^{2}} + \frac {2 \, d^{2}}{c^{\frac {3}{2}} e^{3} {\left (x + \frac {d}{e}\right )}^{2}} \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 = -0.02 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^2}{{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^{3/2}} \,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}}{\left (c \left (d + e x\right )^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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