Optimal. Leaf size=211 \[ -\frac{b (a+b x) (A b-a B)}{a^3 x \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (A b-a B)}{2 a^2 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b^2 \log (x) (a+b x) (A b-a B)}{a^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b^2 (a+b x) (A b-a B) \log (a+b x)}{a^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{A (a+b x)}{3 a x^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.103678, antiderivative size = 211, normalized size of antiderivative = 1., 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} \[ -\frac{b (a+b x) (A b-a B)}{a^3 x \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (A b-a B)}{2 a^2 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b^2 \log (x) (a+b x) (A b-a B)}{a^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b^2 (a+b x) (A b-a B) \log (a+b x)}{a^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{A (a+b x)}{3 a x^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 770
Rule 77
Rubi steps
\begin{align*} \int \frac{A+B x}{x^4 \sqrt{a^2+2 a b x+b^2 x^2}} \, dx &=\frac{\left (a b+b^2 x\right ) \int \frac{A+B x}{x^4 \left (a b+b^2 x\right )} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{\left (a b+b^2 x\right ) \int \left (\frac{A}{a b x^4}+\frac{-A b+a B}{a^2 b x^3}+\frac{A b-a B}{a^3 x^2}+\frac{b (-A b+a B)}{a^4 x}-\frac{b^2 (-A b+a B)}{a^4 (a+b x)}\right ) \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{A (a+b x)}{3 a x^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-a B) (a+b x)}{2 a^2 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b (A b-a B) (a+b x)}{a^3 x \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b^2 (A b-a B) (a+b x) \log (x)}{a^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b^2 (A b-a B) (a+b x) \log (a+b x)}{a^4 \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0523854, size = 102, normalized size = 0.48 \[ -\frac{(a+b x) \left (a \left (a^2 (2 A+3 B x)-3 a b x (A+2 B x)+6 A b^2 x^2\right )+6 b^2 x^3 \log (x) (A b-a B)+6 b^2 x^3 (a B-A b) \log (a+b x)\right )}{6 a^4 x^3 \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 119, normalized size = 0.6 \begin{align*} -{\frac{ \left ( bx+a \right ) \left ( 6\,A\ln \left ( x \right ){x}^{3}{b}^{3}-6\,A\ln \left ( bx+a \right ){x}^{3}{b}^{3}-6\,B\ln \left ( x \right ){x}^{3}a{b}^{2}+6\,B\ln \left ( bx+a \right ){x}^{3}a{b}^{2}+6\,A{x}^{2}a{b}^{2}-6\,B{x}^{2}{a}^{2}b-3\,A{a}^{2}bx+3\,{a}^{3}Bx+2\,A{a}^{3} \right ) }{6\,{a}^{4}{x}^{3}}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57725, size = 203, normalized size = 0.96 \begin{align*} -\frac{6 \,{\left (B a b^{2} - A b^{3}\right )} x^{3} \log \left (b x + a\right ) - 6 \,{\left (B a b^{2} - A b^{3}\right )} x^{3} \log \left (x\right ) + 2 \, A a^{3} - 6 \,{\left (B a^{2} b - A a b^{2}\right )} x^{2} + 3 \,{\left (B a^{3} - A a^{2} b\right )} x}{6 \, a^{4} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.8091, size = 165, normalized size = 0.78 \begin{align*} \frac{- 2 A a^{2} + x^{2} \left (- 6 A b^{2} + 6 B a b\right ) + x \left (3 A a b - 3 B a^{2}\right )}{6 a^{3} x^{3}} + \frac{b^{2} \left (- A b + B a\right ) \log{\left (x + \frac{- A a b^{3} + B a^{2} b^{2} - a b^{2} \left (- A b + B a\right )}{- 2 A b^{4} + 2 B a b^{3}} \right )}}{a^{4}} - \frac{b^{2} \left (- A b + B a\right ) \log{\left (x + \frac{- A a b^{3} + B a^{2} b^{2} + a b^{2} \left (- A b + B a\right )}{- 2 A b^{4} + 2 B a b^{3}} \right )}}{a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21276, size = 207, normalized size = 0.98 \begin{align*} \frac{{\left (B a b^{2} \mathrm{sgn}\left (b x + a\right ) - A b^{3} \mathrm{sgn}\left (b x + a\right )\right )} \log \left ({\left | x \right |}\right )}{a^{4}} - \frac{{\left (B a b^{3} \mathrm{sgn}\left (b x + a\right ) - A b^{4} \mathrm{sgn}\left (b x + a\right )\right )} \log \left ({\left | b x + a \right |}\right )}{a^{4} b} - \frac{2 \, A a^{3} \mathrm{sgn}\left (b x + a\right ) - 6 \,{\left (B a^{2} b \mathrm{sgn}\left (b x + a\right ) - A a b^{2} \mathrm{sgn}\left (b x + a\right )\right )} x^{2} + 3 \,{\left (B a^{3} \mathrm{sgn}\left (b x + a\right ) - A a^{2} b \mathrm{sgn}\left (b x + a\right )\right )} x}{6 \, a^{4} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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