Optimal. Leaf size=39 \[ \frac {(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{4 c^2 e} \]
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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} \begin {gather*} \frac {(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{4 c^2 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 609
Rule 642
Rubi steps
\begin {align*} \int \frac {(d+e x)^4}{\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^2}\\ &=\frac {(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{4 c^2 e}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 0.69 \begin {gather*} \frac {(d+e x)^5}{4 e \sqrt {c (d+e x)^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.56, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(d+e x)^4}{\sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.41, size = 66, normalized size = 1.69 \begin {gather*} \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 (c e x + c d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.72, size = 63, normalized size = 1.62 \begin {gather*} \frac {1}{4} \, \sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}} {\left (\frac {d^{3} e^{\left (-1\right )}}{c} + {\left (x {\left (\frac {x e^{2}}{c} + \frac {3 \, d e}{c}\right )} + \frac {3 \, d^{2}}{c}\right )} x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 60, normalized size = 1.54 \begin {gather*} \frac {\left (e^{3} x^{3}+4 e^{2} x^{2} d +6 d^{2} x e +4 d^{3}\right ) \left (e x +d \right ) x}{4 \sqrt {c \,e^{2} x^{2}+2 c d e x +c \,d^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.49, size = 120, normalized size = 3.08 \begin {gather*} \frac {3 \, d^{2} e x^{2}}{4 \, \sqrt {c}} + \frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} e^{2} x^{3}}{4 \, c} - \frac {3 \, d^{3} x}{2 \, \sqrt {c}} + \frac {3 \, \sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} d e x^{2}}{4 \, c} + \frac {5 \, \sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} d^{3}}{2 \, c e} \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 = -0.03 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^4}{\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^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 )^{4}}{\sqrt {c \left (d + e x\right )^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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