Optimal. Leaf size=194 \[ \frac{1}{2 a^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{1}{3 a^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{1}{4 a (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{1}{a^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\log (x) (a+b x)}{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(a+b x) \log (a+b x)}{a^5 \sqrt{a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.0863049, antiderivative size = 194, 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, 44} \[ \frac{1}{2 a^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{1}{3 a^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{1}{4 a (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{1}{a^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\log (x) (a+b x)}{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(a+b x) \log (a+b x)}{a^5 \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 646
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=\frac{\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac{1}{x \left (a b+b^2 x\right )^5} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{\left (b^4 \left (a b+b^2 x\right )\right ) \int \left (\frac{1}{a^5 b^5 x}-\frac{1}{a b^4 (a+b x)^5}-\frac{1}{a^2 b^4 (a+b x)^4}-\frac{1}{a^3 b^4 (a+b x)^3}-\frac{1}{a^4 b^4 (a+b x)^2}-\frac{1}{a^5 b^4 (a+b x)}\right ) \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{1}{a^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{1}{4 a (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{1}{3 a^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{1}{2 a^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) \log (x)}{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(a+b x) \log (a+b x)}{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0346088, size = 84, normalized size = 0.43 \[ \frac{a \left (52 a^2 b x+25 a^3+42 a b^2 x^2+12 b^3 x^3\right )+12 \log (x) (a+b x)^4-12 (a+b x)^4 \log (a+b x)}{12 a^5 (a+b x)^3 \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.238, size = 173, normalized size = 0.9 \begin{align*}{\frac{ \left ( 12\,\ln \left ( x \right ){x}^{4}{b}^{4}-12\,\ln \left ( bx+a \right ){x}^{4}{b}^{4}+48\,\ln \left ( x \right ){x}^{3}a{b}^{3}-48\,\ln \left ( bx+a \right ){x}^{3}a{b}^{3}+72\,\ln \left ( x \right ){x}^{2}{a}^{2}{b}^{2}-72\,\ln \left ( bx+a \right ){x}^{2}{a}^{2}{b}^{2}+12\,a{b}^{3}{x}^{3}+48\,\ln \left ( x \right ) x{a}^{3}b-48\,\ln \left ( bx+a \right ) x{a}^{3}b+42\,{x}^{2}{a}^{2}{b}^{2}+12\,\ln \left ( x \right ){a}^{4}-12\,{a}^{4}\ln \left ( bx+a \right ) +52\,x{a}^{3}b+25\,{a}^{4} \right ) \left ( bx+a \right ) }{12\,{a}^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{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 = 1.7908, size = 365, normalized size = 1.88 \begin{align*} \frac{12 \, a b^{3} x^{3} + 42 \, a^{2} b^{2} x^{2} + 52 \, a^{3} b x + 25 \, a^{4} - 12 \,{\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}\right )} \log \left (b x + a\right ) + 12 \,{\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}\right )} \log \left (x\right )}{12 \,{\left (a^{5} b^{4} x^{4} + 4 \, a^{6} b^{3} x^{3} + 6 \, a^{7} b^{2} x^{2} + 4 \, a^{8} b x + a^{9}\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{1}{x \left (\left (a + b x\right )^{2}\right )^{\frac{5}{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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