Optimal. Leaf size=80 \[ \frac {b x \sqrt {a^2+2 a b x+b^2 x^2}}{e (a+b x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) \log (d+e x)}{e^2 (a+b x)} \]
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Rubi [A] time = 0.04, 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 = {646, 43} \[ \frac {b x \sqrt {a^2+2 a b x+b^2 x^2}}{e (a+b x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) \log (d+e x)}{e^2 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 43
Rule 646
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 = 0.02, size = 42, normalized size = 0.52 \[ \frac {\sqrt {(a+b x)^2} ((a e-b d) \log (d+e x)+b e x)}{e^2 (a+b x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 25, normalized size = 0.31 \[ \frac {b e x - {\left (b d - a e\right )} \log \left (e x + d\right )}{e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 45, normalized size = 0.56 \[ b x e^{\left (-1\right )} \mathrm {sgn}\left (b x + a\right ) - {\left (b d \mathrm {sgn}\left (b x + a\right ) - a e \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-2\right )} \log \left ({\left | x e + d \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.06, size = 44, normalized size = 0.55 \[ \frac {\left (a e \ln \left (b e x +b d \right )-b d \ln \left (b e x +b d \right )+b e x +a e \right ) \mathrm {csgn}\left (b x +a \right )}{e^{2}} \]
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 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {\sqrt {{\left (a+b\,x\right )}^2}}{d+e\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 20, normalized size = 0.25 \[ \frac {b x}{e} + \frac {\left (a e - b d\right ) \log {\left (d + e x \right )}}{e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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