Optimal. Leaf size=32 \[ -\frac {c^2}{e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 32, 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 = {657, 643}
\begin {gather*} -\frac {c^2}{e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 643
Rule 657
Rubi steps
\begin {align*} \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^5} \, dx &=c^3 \int \frac {d+e x}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx\\ &=-\frac {c^2}{e \sqrt {c d^2+2 c d e x+c e^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 25, normalized size = 0.78 \begin {gather*} -\frac {\left (c (d+e x)^2\right )^{3/2}}{e (d+e x)^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.58, size = 35, normalized size = 1.09
method | result | size |
risch | \(-\frac {c \sqrt {\left (e x +d \right )^{2} c}}{\left (e x +d \right )^{2} e}\) | \(25\) |
gosper | \(-\frac {\left (x^{2} c \,e^{2}+2 c d e x +c \,d^{2}\right )^{\frac {3}{2}}}{\left (e x +d \right )^{4} e}\) | \(35\) |
default | \(-\frac {\left (x^{2} c \,e^{2}+2 c d e x +c \,d^{2}\right )^{\frac {3}{2}}}{\left (e x +d \right )^{4} e}\) | \(35\) |
trager | \(\frac {c x \sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}}{d \left (e x +d \right )^{2}}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.68, size = 46, normalized size = 1.44 \begin {gather*} -\frac {\sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}} c}{x^{2} e^{3} + 2 \, d x e^{2} + d^{2} e} \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 (c \left (d + e x\right )^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{5}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.87, size = 22, normalized size = 0.69 \begin {gather*} -\frac {c^{\frac {3}{2}} e^{\left (-1\right )} \mathrm {sgn}\left (x e + d\right )}{x e + d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.43, size = 35, normalized size = 1.09 \begin {gather*} -\frac {c\,\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}}{e\,{\left (d+e\,x\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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