Optimal. Leaf size=140 \[ \frac {A b-a B}{2 a b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A}{a^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A \log (x) (a+b x)}{a^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A (a+b x) \log (a+b x)}{a^3 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.09, antiderivative size = 140, 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, 77} \begin {gather*} \frac {A b-a B}{2 a b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A}{a^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A \log (x) (a+b x)}{a^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A (a+b x) \log (a+b x)}{a^3 \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rule 770
Rubi steps
\begin {align*} \int \frac {A+B x}{x \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx &=\frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \frac {A+B x}{x \left (a b+b^2 x\right )^3} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \left (\frac {A}{a^3 b^3 x}+\frac {-A b+a B}{a b^3 (a+b x)^3}-\frac {A}{a^2 b^2 (a+b x)^2}-\frac {A}{a^3 b^2 (a+b x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {A}{a^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{2 a b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A (a+b x) \log (x)}{a^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A (a+b x) \log (a+b x)}{a^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 80, normalized size = 0.57 \begin {gather*} \frac {a \left (a^2 (-B)+3 a A b+2 A b^2 x\right )+2 A b \log (x) (a+b x)^2-2 A b (a+b x)^2 \log (a+b x)}{2 a^3 b (a+b x) \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.76, size = 241, normalized size = 1.72 \begin {gather*} \frac {2 A \tanh ^{-1}\left (\frac {\sqrt {b^2} x}{a}-\frac {\sqrt {a^2+2 a b x+b^2 x^2}}{a}\right )}{a^3}+\frac {a^4 b B-a^3 A b^2+a^2 b^3 B x^2+\sqrt {b^2} \sqrt {a^2+2 a b x+b^2 x^2} \left (a^3 B-a^2 A b-a^2 b B x+a A b^2 x+2 A b^3 x^2\right )-3 a A b^4 x^2-2 A b^5 x^3}{a^2 b \sqrt {b^2} x^2 \left (2 a^2 b^2+4 a b^3 x+2 b^4 x^2\right )+a^2 b x^2 \left (-2 a b^3-2 b^4 x\right ) \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 109, normalized size = 0.78 \begin {gather*} \frac {2 \, A a b^{2} x - B a^{3} + 3 \, A a^{2} b - 2 \, {\left (A b^{3} x^{2} + 2 \, A a b^{2} x + A a^{2} b\right )} \log \left (b x + a\right ) + 2 \, {\left (A b^{3} x^{2} + 2 \, A a b^{2} x + A a^{2} b\right )} \log \relax (x)}{2 \, {\left (a^{3} b^{3} x^{2} + 2 \, a^{4} b^{2} x + a^{5} b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 116, normalized size = 0.83 \begin {gather*} -\frac {\left (-2 A \,b^{3} x^{2} \ln \relax (x )+2 A \,b^{3} x^{2} \ln \left (b x +a \right )-4 A a \,b^{2} x \ln \relax (x )+4 A a \,b^{2} x \ln \left (b x +a \right )-2 A \,a^{2} b \ln \relax (x )+2 A \,a^{2} b \ln \left (b x +a \right )-2 A a \,b^{2} x -3 A \,a^{2} b +B \,a^{3}\right ) \left (b x +a \right )}{2 \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} a^{3} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 96, normalized size = 0.69 \begin {gather*} -\frac {\left (-1\right )^{2 \, a b x + 2 \, a^{2}} A \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right )}{a^{3}} + \frac {A}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2}} - \frac {B}{2 \, b^{3} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {A}{2 \, a b^{2} {\left (x + \frac {a}{b}\right )}^{2}} \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 {A+B\,x}{x\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,d x \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 {A + B x}{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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