Optimal. Leaf size=42 \[ \frac {-a-b x}{2 e (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 39, normalized size of antiderivative = 0.93, number of steps
used = 3, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {784, 21, 32}
\begin {gather*} -\frac {a+b x}{2 e \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 32
Rule 784
Rubi steps
\begin {align*} \int \frac {a+b x}{(d+e x)^3 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx &=\frac {\left (a b+b^2 x\right ) \int \frac {a+b x}{\left (a b+b^2 x\right ) (d+e x)^3} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (a b+b^2 x\right ) \int \frac {1}{(d+e x)^3} \, dx}{b \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {a+b x}{2 e (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 30, normalized size = 0.71 \begin {gather*} -\frac {a+b x}{2 e \sqrt {(a+b x)^2} (d+e x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 27, normalized size = 0.64
method | result | size |
gosper | \(-\frac {b x +a}{2 \left (e x +d \right )^{2} e \sqrt {\left (b x +a \right )^{2}}}\) | \(27\) |
default | \(-\frac {b x +a}{2 \left (e x +d \right )^{2} e \sqrt {\left (b x +a \right )^{2}}}\) | \(27\) |
risch | \(-\frac {\sqrt {\left (b x +a \right )^{2}}}{2 \left (b x +a \right ) \left (e x +d \right )^{2} e}\) | \(29\) |
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: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.69, size = 23, normalized size = 0.55 \begin {gather*} -\frac {1}{2 \, {\left (x^{2} e^{3} + 2 \, d x e^{2} + d^{2} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 26, normalized size = 0.62 \begin {gather*} - \frac {1}{2 d^{2} e + 4 d e^{2} x + 2 e^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.15, size = 18, normalized size = 0.43 \begin {gather*} -\frac {e^{\left (-1\right )} \mathrm {sgn}\left (b x + a\right )}{2 \, {\left (x e + d\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.17, size = 28, normalized size = 0.67 \begin {gather*} -\frac {\sqrt {{\left (a+b\,x\right )}^2}}{2\,e\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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