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 = 0.01, 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 = {656, 623}
\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 623
Rule 656
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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Maple [A]
time = 0.59, size = 35, normalized size = 0.90
method | result | size |
risch | \(\frac {\left (e x +d \right )^{5}}{4 \sqrt {\left (e x +d \right )^{2} c}\, e}\) | \(24\) |
default | \(\frac {\left (e x +d \right )^{5}}{4 \sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}\, e}\) | \(35\) |
gosper | \(\frac {x \left (e^{3} x^{3}+4 d \,e^{2} x^{2}+6 d^{2} e x +4 d^{3}\right ) \left (e x +d \right )}{4 \sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}}\) | \(60\) |
trager | \(\frac {x \left (e^{3} x^{3}+4 d \,e^{2} x^{2}+6 d^{2} e x +4 d^{3}\right ) \sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}}{4 c \left (e x +d \right )}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 120 vs.
\(2 (35) = 70\).
time = 0.30, size = 120, normalized size = 3.08 \begin {gather*} \frac {3 \, d^{2} x^{2} e}{4 \, \sqrt {c}} - \frac {3 \, d^{3} x}{2 \, \sqrt {c}} + \frac {\sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}} x^{3} e^{2}}{4 \, c} + \frac {3 \, \sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}} d x^{2} e}{4 \, c} + \frac {5 \, \sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}} d^{3} e^{\left (-1\right )}}{2 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.31, size = 66, normalized size = 1.69 \begin {gather*} \frac {{\left (x^{4} e^{3} + 4 \, d x^{3} e^{2} + 6 \, d^{2} x^{2} e + 4 \, d^{3} x\right )} \sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}}}{4 \, {\left (c x e + c d\right )}} \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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Giac [A]
time = 1.17, size = 57, normalized size = 1.46 \begin {gather*} \frac {\sqrt {c} x^{4} e^{3} + 4 \, \sqrt {c} d x^{3} e^{2} + 6 \, \sqrt {c} d^{2} x^{2} e + 4 \, \sqrt {c} d^{3} x}{4 \, c \mathrm {sgn}\left (x e + d\right )} \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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