Optimal. Leaf size=36 \[ \frac {(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{4 e} \]
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Rubi [A] time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {609} \[ \frac {(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{4 e} \]
Antiderivative was successfully verified.
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Rule 609
Rubi steps
\begin {align*} \int \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2} \, dx &=\frac {(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{4 e}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 25, normalized size = 0.69 \[ \frac {(d+e x) \left (c (d+e x)^2\right )^{3/2}}{4 e} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.15, size = 67, normalized size = 1.86 \[ \frac {{\left (c e^{3} x^{4} + 4 \, c d e^{2} x^{3} + 6 \, c d^{2} e x^{2} + 4 \, c d^{3} x\right )} \sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{4 \, {\left (e x + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 55, normalized size = 1.53 \[ \frac {1}{4} \, {\left (c d^{3} e^{\left (-1\right )} + {\left (3 \, c d^{2} + {\left (c x e^{2} + 3 \, c d e\right )} x\right )} x\right )} \sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 62, normalized size = 1.72 \[ \frac {\left (e^{3} x^{3}+4 e^{2} x^{2} d +6 d^{2} x e +4 d^{3}\right ) \left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{\frac {3}{2}} x}{4 \left (e x +d \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.40, size = 54, normalized size = 1.50 \[ \frac {1}{4} \, {\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {3}{2}} x + \frac {{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {3}{2}} d}{4 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.41, size = 36, normalized size = 1.00 \[ \frac {\left (x\,e^2+d\,e\right )\,{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^{3/2}}{4\,e^2} \]
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 \[ \int \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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