Optimal. Leaf size=235 \[ -\frac{3 b}{2 a^4 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 b}{3 a^3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b}{4 a^2 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{4 b}{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a+b x}{a^5 x \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 b \log (x) (a+b x)}{a^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 b (a+b x) \log (a+b x)}{a^6 \sqrt{a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.102017, antiderivative size = 235, 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} \[ -\frac{3 b}{2 a^4 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 b}{3 a^3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b}{4 a^2 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{4 b}{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a+b x}{a^5 x \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 b \log (x) (a+b x)}{a^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 b (a+b x) \log (a+b x)}{a^6 \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 646
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^2 \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^2 \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^2}-\frac{5}{a^6 b^4 x}+\frac{1}{a^2 b^3 (a+b x)^5}+\frac{2}{a^3 b^3 (a+b x)^4}+\frac{3}{a^4 b^3 (a+b x)^3}+\frac{4}{a^5 b^3 (a+b x)^2}+\frac{5}{a^6 b^3 (a+b x)}\right ) \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{4 b}{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b}{4 a^2 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 b}{3 a^3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 b}{2 a^4 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a+b x}{a^5 x \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 b (a+b x) \log (x)}{a^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 b (a+b x) \log (a+b x)}{a^6 \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0402118, size = 103, normalized size = 0.44 \[ \frac{-a \left (260 a^2 b^2 x^2+125 a^3 b x+12 a^4+210 a b^3 x^3+60 b^4 x^4\right )-60 b x \log (x) (a+b x)^4+60 b x (a+b x)^4 \log (a+b x)}{12 a^6 x (a+b x)^3 \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.225, size = 199, normalized size = 0.9 \begin{align*} -{\frac{ \left ( 60\,{b}^{5}\ln \left ( x \right ){x}^{5}-60\,\ln \left ( bx+a \right ){x}^{5}{b}^{5}+240\,a{b}^{4}\ln \left ( x \right ){x}^{4}-240\,\ln \left ( bx+a \right ){x}^{4}a{b}^{4}+360\,{a}^{2}{b}^{3}\ln \left ( x \right ){x}^{3}-360\,\ln \left ( bx+a \right ){x}^{3}{a}^{2}{b}^{3}+60\,a{b}^{4}{x}^{4}+240\,{a}^{3}{b}^{2}\ln \left ( x \right ){x}^{2}-240\,\ln \left ( bx+a \right ){x}^{2}{a}^{3}{b}^{2}+210\,{a}^{2}{b}^{3}{x}^{3}+60\,{a}^{4}b\ln \left ( x \right ) x-60\,\ln \left ( bx+a \right ) x{a}^{4}b+260\,{a}^{3}{b}^{2}{x}^{2}+125\,{a}^{4}bx+12\,{a}^{5} \right ) \left ( bx+a \right ) }{12\,x{a}^{6}} \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 = 1.7869, size = 421, normalized size = 1.79 \begin{align*} -\frac{60 \, a b^{4} x^{4} + 210 \, a^{2} b^{3} x^{3} + 260 \, a^{3} b^{2} x^{2} + 125 \, a^{4} b x + 12 \, a^{5} - 60 \,{\left (b^{5} x^{5} + 4 \, a b^{4} x^{4} + 6 \, a^{2} b^{3} x^{3} + 4 \, a^{3} b^{2} x^{2} + a^{4} b x\right )} \log \left (b x + a\right ) + 60 \,{\left (b^{5} x^{5} + 4 \, a b^{4} x^{4} + 6 \, a^{2} b^{3} x^{3} + 4 \, a^{3} b^{2} x^{2} + a^{4} b x\right )} \log \left (x\right )}{12 \,{\left (a^{6} b^{4} x^{5} + 4 \, a^{7} b^{3} x^{4} + 6 \, a^{8} b^{2} x^{3} + 4 \, a^{9} b x^{2} + a^{10} x\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^{2} \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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