Optimal. Leaf size=39 \[ \frac {(d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}}{2 c^2 e} \]
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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} \[ \frac {(d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}}{2 c^2 e} \]
Antiderivative was successfully verified.
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Rule 609
Rule 642
Rubi steps
\begin {align*} \int \frac {(d+e x)^4}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx &=\frac {\int \sqrt {c d^2+2 c d e x+c e^2 x^2} \, dx}{c^2}\\ &=\frac {(d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}}{2 c^2 e}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 33, normalized size = 0.85 \[ \frac {x (d+e x) (2 d+e x)}{2 c \sqrt {c (d+e x)^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.27, size = 48, normalized size = 1.23 \[ \frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} {\left (e x^{2} + 2 \, d x\right )}}{2 \, {\left (c^{2} e x + c^{2} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.53, size = 65, normalized size = 1.67 \[ \frac {4 \, C_{0} d e^{\left (-1\right )} - \frac {2 \, d^{3} e^{\left (-1\right )}}{c} + {\left (x {\left (\frac {x e^{2}}{c} + \frac {3 \, d e}{c}\right )} + 4 \, C_{0}\right )} x}{2 \, \sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 40, normalized size = 1.03 \[ \frac {\left (e x +2 d \right ) \left (e x +d \right )^{3} x}{2 \left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.50, size = 99, normalized size = 2.54 \[ \frac {e^{2} x^{3}}{2 \, \sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c} + \frac {3 \, d e x^{2}}{2 \, \sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c} - \frac {d^{3}}{\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c e} \]
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 \[ \int \frac {{\left (d+e\,x\right )}^4}{{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (d + e x\right )^{4}}{\left (c \left (d + e x\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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