Optimal. Leaf size=69 \[ -\frac {e}{b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b d-a e}{2 b^2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {654, 621}
\begin {gather*} -\frac {b d-a e}{2 b^2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e}{b^2 \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 621
Rule 654
Rubi steps
\begin {align*} \int \frac {d+e x}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx &=-\frac {e}{b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (2 b^2 d-2 a b e\right ) \int \frac {1}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx}{2 b^2}\\ &=-\frac {e}{b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b d-a e}{2 b^2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 39, normalized size = 0.57 \begin {gather*} \frac {-a e-b (d+2 e x)}{2 b^2 (a+b x) \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.60, size = 32, normalized size = 0.46
method | result | size |
gosper | \(-\frac {\left (b x +a \right ) \left (2 b e x +a e +b d \right )}{2 b^{2} \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}\) | \(32\) |
default | \(-\frac {\left (b x +a \right ) \left (2 b e x +a e +b d \right )}{2 b^{2} \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}\) | \(32\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {e x}{b}-\frac {a e +b d}{2 b^{2}}\right )}{\left (b x +a \right )^{3}}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 58, normalized size = 0.84 \begin {gather*} -\frac {e}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} - \frac {d}{2 \, b^{3} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {a e}{2 \, b^{4} {\left (x + \frac {a}{b}\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.57, size = 39, normalized size = 0.57 \begin {gather*} -\frac {b d + {\left (2 \, b x + a\right )} e}{2 \, {\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\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 {d + e x}{\left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.46, size = 34, normalized size = 0.49 \begin {gather*} -\frac {2 \, b x e + b d + a e}{2 \, {\left (b x + a\right )}^{2} b^{2} \mathrm {sgn}\left (b x + a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.62, size = 42, normalized size = 0.61 \begin {gather*} -\frac {\left (a\,e+b\,d+2\,b\,e\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{2\,b^2\,{\left (a+b\,x\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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