Optimal. Leaf size=105 \[ \frac {x \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{a+b x}+\frac {a A \log (x) \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {b B x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 (a+b x)} \]
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Rubi [A] time = 0.04, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {770, 76} \[ \frac {x \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{a+b x}+\frac {a A \log (x) \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {b B x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 76
Rule 770
Rubi steps
\begin {align*} \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right ) (A+B x)}{x} \, dx}{a b+b^2 x}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (b (A b+a B)+\frac {a A b}{x}+b^2 B x\right ) \, dx}{a b+b^2 x}\\ &=\frac {(A b+a B) x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {b B x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac {a A \sqrt {a^2+2 a b x+b^2 x^2} \log (x)}{a+b x}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 44, normalized size = 0.42 \[ \frac {\sqrt {(a+b x)^2} (x (2 a B+2 A b+b B x)+2 a A \log (x))}{2 (a+b x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.12, size = 22, normalized size = 0.21 \[ \frac {1}{2} \, B b x^{2} + A a \log \relax (x) + {\left (B a + A b\right )} x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 46, normalized size = 0.44 \[ \frac {1}{2} \, B b x^{2} \mathrm {sgn}\left (b x + a\right ) + B a x \mathrm {sgn}\left (b x + a\right ) + A b x \mathrm {sgn}\left (b x + a\right ) + A a \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (b x + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.07, size = 53, normalized size = 0.50 \[ \frac {\left (B \,b^{2} x^{2}+2 A a b \ln \left (b x \right )+2 A \,b^{2} x +2 B a b x +2 A a b +B \,a^{2}\right ) \mathrm {csgn}\left (b x +a \right )}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 133, normalized size = 1.27 \[ \left (-1\right )^{2 \, b^{2} x + 2 \, a b} A a \log \left (2 \, b^{2} x + 2 \, a b\right ) - \left (-1\right )^{2 \, a b x + 2 \, a^{2}} A a \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right ) + \frac {1}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B x + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B a}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.37, size = 122, normalized size = 1.16 \[ A\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}-A\,\ln \left (a\,b+\frac {a^2}{x}+\frac {\sqrt {a^2}\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x}\right )\,\sqrt {a^2}+\frac {B\,\sqrt {{\left (a+b\,x\right )}^2}\,\left (a+b\,x\right )}{2\,b}+\frac {A\,a\,b\,\ln \left (a\,b+\sqrt {{\left (a+b\,x\right )}^2}\,\sqrt {b^2}+b^2\,x\right )}{\sqrt {b^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 22, normalized size = 0.21 \[ A a \log {\relax (x )} + \frac {B b x^{2}}{2} + x \left (A b + B a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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