Optimal. Leaf size=39 \[ \frac {(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{4 c e} \]
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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} \[ \frac {(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{4 c e} \]
Antiderivative was successfully verified.
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Rule 609
Rule 642
Rubi steps
\begin {align*} \int (d+e x)^2 \sqrt {c d^2+2 c d e x+c e^2 x^2} \, dx &=\frac {\int \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2} \, dx}{c}\\ &=\frac {(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{4 c e}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 0.72 \[ \frac {(d+e x) \left (c (d+e x)^2\right )^{3/2}}{4 c e} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.12, size = 63, normalized size = 1.62 \[ \frac {{\left (e^{3} x^{4} + 4 \, d e^{2} x^{3} + 6 \, d^{2} e x^{2} + 4 \, d^{3} x\right )} \sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{4 \, {\left (e x + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 51, normalized size = 1.31 \[ \frac {1}{4} \, \sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}} {\left (d^{3} e^{\left (-1\right )} + {\left (3 \, d^{2} + {\left (x e^{2} + 3 \, d e\right )} x\right )} x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 62, normalized size = 1.59 \[ \frac {\left (e^{3} x^{3}+4 e^{2} x^{2} d +6 d^{2} x e +4 d^{3}\right ) \sqrt {c \,e^{2} x^{2}+2 c d e x +c \,d^{2}}\, x}{4 e x +4 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 60, normalized size = 1.54 \[ \frac {{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {3}{2}} x}{4 \, c} + \frac {{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {3}{2}} d}{4 \, c e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.70, size = 76, normalized size = 1.95 \[ \frac {\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}\,\left (c\,d^3+e\,x\,\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )+2\,c\,d^2\,e\,x+c\,d\,e^2\,x^2\right )}{4\,c\,e} \]
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 \[ \int \sqrt {c \left (d + e x\right )^{2}} \left (d + e x\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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