Optimal. Leaf size=71 \[ -\frac {a \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac {b \sqrt {a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)} \]
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Rubi [A]
time = 0.01, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {660, 45}
\begin {gather*} -\frac {b \sqrt {a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}-\frac {a \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 660
Rubi steps
\begin {align*} \int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{x^4} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {a b+b^2 x}{x^4} \, dx}{a b+b^2 x}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {a b}{x^4}+\frac {b^2}{x^3}\right ) \, dx}{a b+b^2 x}\\ &=-\frac {a \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac {b \sqrt {a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 33, normalized size = 0.46 \begin {gather*} -\frac {\sqrt {(a+b x)^2} (2 a+3 b x)}{6 x^3 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
2.
time = 0.14, size = 20, normalized size = 0.28
method | result | size |
default | \(-\frac {\mathrm {csgn}\left (b x +a \right ) \left (3 b x +2 a \right )}{6 x^{3}}\) | \(20\) |
risch | \(\frac {\left (-\frac {b x}{2}-\frac {a}{3}\right ) \sqrt {\left (b x +a \right )^{2}}}{x^{3} \left (b x +a \right )}\) | \(29\) |
gosper | \(-\frac {\left (3 b x +2 a \right ) \sqrt {\left (b x +a \right )^{2}}}{6 x^{3} \left (b x +a \right )}\) | \(30\) |
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. 109 vs.
\(2 (45) = 90\).
time = 0.26, size = 109, normalized size = 1.54 \begin {gather*} -\frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{3}}{2 \, a^{3}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}}{2 \, a^{2} x} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b}{2 \, a^{3} x^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}}}{3 \, a^{2} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.14, size = 13, normalized size = 0.18 \begin {gather*} -\frac {3 \, b x + 2 \, a}{6 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 14, normalized size = 0.20 \begin {gather*} \frac {- 2 a - 3 b x}{6 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.56, size = 40, normalized size = 0.56 \begin {gather*} \frac {b^{3} \mathrm {sgn}\left (b x + a\right )}{6 \, a^{2}} - \frac {3 \, b x \mathrm {sgn}\left (b x + a\right ) + 2 \, a \mathrm {sgn}\left (b x + a\right )}{6 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.16, size = 29, normalized size = 0.41 \begin {gather*} -\frac {\left (2\,a+3\,b\,x\right )\,\sqrt {{\left (a+b\,x\right )}^2}}{6\,x^3\,\left (a+b\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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