Optimal. Leaf size=80 \[ \frac {b x \sqrt {a^2+2 a b x+b^2 x^2}}{e (a+b x)}-\frac {(b d-a e) \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^2 (a+b x)} \]
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Rubi [A]
time = 0.03, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {660, 45}
\begin {gather*} \frac {b x \sqrt {a^2+2 a b x+b^2 x^2}}{e (a+b x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) \log (d+e x)}{e^2 (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}}{d+e x} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {a b+b^2 x}{d+e x} \, dx}{a b+b^2 x}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {b^2}{e}-\frac {b (b d-a e)}{e (d+e x)}\right ) \, dx}{a b+b^2 x}\\ &=\frac {b x \sqrt {a^2+2 a b x+b^2 x^2}}{e (a+b x)}-\frac {(b d-a e) \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^2 (a+b x)}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 42, normalized size = 0.52 \begin {gather*} \frac {\sqrt {(a+b x)^2} (b e x+(-b d+a e) \log (d+e x))}{e^2 (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
3.
time = 0.56, size = 48, normalized size = 0.60
method | result | size |
default | \(\frac {\mathrm {csgn}\left (b x +a \right ) \left (\ln \left (-b e x -b d \right ) a e -\ln \left (-b e x -b d \right ) b d +b e x +a e \right )}{e^{2}}\) | \(48\) |
risch | \(\frac {b x \sqrt {\left (b x +a \right )^{2}}}{e \left (b x +a \right )}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (a e -b d \right ) \ln \left (e x +d \right )}{\left (b x +a \right ) e^{2}}\) | \(58\) |
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 = 3.57, size = 27, normalized size = 0.34 \begin {gather*} {\left (b x e - {\left (b d - a e\right )} \log \left (x e + d\right )\right )} e^{\left (-2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 20, normalized size = 0.25 \begin {gather*} \frac {b x}{e} + \frac {\left (a e - b d\right ) \log {\left (d + e x \right )}}{e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.74, size = 45, normalized size = 0.56 \begin {gather*} b x e^{\left (-1\right )} \mathrm {sgn}\left (b x + a\right ) - {\left (b d \mathrm {sgn}\left (b x + a\right ) - a e \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-2\right )} \log \left ({\left | x e + d \right |}\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 = -0.01 \begin {gather*} \int \frac {\sqrt {{\left (a+b\,x\right )}^2}}{d+e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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