Optimal. Leaf size=133 \[ \frac{a^3}{2 b^4 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 a^2}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x (a+b x)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 a (a+b x) \log (a+b x)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.0580191, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {646, 43} \[ \frac{a^3}{2 b^4 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 a^2}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x (a+b x)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 a (a+b x) \log (a+b x)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 646
Rule 43
Rubi steps
\begin{align*} \int \frac{x^3}{\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{x^3}{\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{1}{b^6}-\frac{a^3}{b^6 (a+b x)^3}+\frac{3 a^2}{b^6 (a+b x)^2}-\frac{3 a}{b^6 (a+b x)}\right ) \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{3 a^2}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a^3}{2 b^4 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x (a+b x)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 a (a+b x) \log (a+b x)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.02345, size = 71, normalized size = 0.53 \[ \frac{-4 a^2 b x-5 a^3+4 a b^2 x^2-6 a (a+b x)^2 \log (a+b x)+2 b^3 x^3}{2 b^4 (a+b x) \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.224, size = 89, normalized size = 0.7 \begin{align*} -{\frac{ \left ( 6\,\ln \left ( bx+a \right ){x}^{2}a{b}^{2}-2\,{b}^{3}{x}^{3}+12\,\ln \left ( bx+a \right ) x{a}^{2}b-4\,a{b}^{2}{x}^{2}+6\,{a}^{3}\ln \left ( bx+a \right ) +4\,b{a}^{2}x+5\,{a}^{3} \right ) \left ( bx+a \right ) }{2\,{b}^{4}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.18906, size = 180, normalized size = 1.35 \begin{align*} \frac{x^{2}}{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} - \frac{3 \, a \log \left (x + \frac{a}{b}\right )}{{\left (b^{2}\right )}^{\frac{3}{2}} b} - \frac{9 \, a^{3} b}{2 \,{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x + \frac{a}{b}\right )}^{2}} - \frac{6 \, a^{2} x}{{\left (b^{2}\right )}^{\frac{5}{2}}{\left (x + \frac{a}{b}\right )}^{2}} + \frac{2 \, a^{2}}{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{4}} - \frac{a^{3}}{{\left (b^{2}\right )}^{\frac{3}{2}} b^{3}{\left (x + \frac{a}{b}\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67851, size = 176, normalized size = 1.32 \begin{align*} \frac{2 \, b^{3} x^{3} + 4 \, a b^{2} x^{2} - 4 \, a^{2} b x - 5 \, a^{3} - 6 \,{\left (a b^{2} x^{2} + 2 \, a^{2} b x + a^{3}\right )} \log \left (b x + a\right )}{2 \,{\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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