Optimal. Leaf size=278 \[ \frac{3 b^2}{a^5 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b^2}{a^4 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b^2}{4 a^3 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{10 b^2}{a^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 b (a+b x)}{a^6 x \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a+b x}{2 a^5 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{15 b^2 \log (x) (a+b x)}{a^7 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{15 b^2 (a+b x) \log (a+b x)}{a^7 \sqrt{a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.114067, antiderivative size = 278, 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} \[ \frac{3 b^2}{a^5 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b^2}{a^4 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b^2}{4 a^3 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{10 b^2}{a^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 b (a+b x)}{a^6 x \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a+b x}{2 a^5 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{15 b^2 \log (x) (a+b x)}{a^7 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{15 b^2 (a+b x) \log (a+b x)}{a^7 \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 646
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^3 \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^3 \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^3}-\frac{5}{a^6 b^4 x^2}+\frac{15}{a^7 b^3 x}-\frac{1}{a^3 b^2 (a+b x)^5}-\frac{3}{a^4 b^2 (a+b x)^4}-\frac{6}{a^5 b^2 (a+b x)^3}-\frac{10}{a^6 b^2 (a+b x)^2}-\frac{15}{a^7 b^2 (a+b x)}\right ) \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{10 b^2}{a^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b^2}{4 a^3 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b^2}{a^4 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 b^2}{a^5 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a+b x}{2 a^5 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 b (a+b x)}{a^6 x \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{15 b^2 (a+b x) \log (x)}{a^7 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{15 b^2 (a+b x) \log (a+b x)}{a^7 \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0433473, size = 121, normalized size = 0.44 \[ \frac{a \left (125 a^3 b^2 x^2+260 a^2 b^3 x^3+12 a^4 b x-2 a^5+210 a b^4 x^4+60 b^5 x^5\right )+60 b^2 x^2 \log (x) (a+b x)^4-60 b^2 x^2 (a+b x)^4 \log (a+b x)}{4 a^7 x^2 (a+b x)^3 \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.231, size = 218, normalized size = 0.8 \begin{align*}{\frac{ \left ( 60\,\ln \left ( x \right ){x}^{6}{b}^{6}-60\,\ln \left ( bx+a \right ){x}^{6}{b}^{6}+240\,\ln \left ( x \right ){x}^{5}a{b}^{5}-240\,\ln \left ( bx+a \right ){x}^{5}a{b}^{5}+360\,\ln \left ( x \right ){x}^{4}{a}^{2}{b}^{4}-360\,\ln \left ( bx+a \right ){x}^{4}{a}^{2}{b}^{4}+60\,{x}^{5}a{b}^{5}+240\,\ln \left ( x \right ){x}^{3}{a}^{3}{b}^{3}-240\,\ln \left ( bx+a \right ){x}^{3}{a}^{3}{b}^{3}+210\,{a}^{2}{x}^{4}{b}^{4}+60\,\ln \left ( x \right ){x}^{2}{a}^{4}{b}^{2}-60\,\ln \left ( bx+a \right ){x}^{2}{a}^{4}{b}^{2}+260\,{a}^{3}{x}^{3}{b}^{3}+125\,{a}^{4}{x}^{2}{b}^{2}+12\,{a}^{5}xb-2\,{a}^{6} \right ) \left ( bx+a \right ) }{4\,{x}^{2}{a}^{7}} \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 = 1.74119, size = 455, normalized size = 1.64 \begin{align*} \frac{60 \, a b^{5} x^{5} + 210 \, a^{2} b^{4} x^{4} + 260 \, a^{3} b^{3} x^{3} + 125 \, a^{4} b^{2} x^{2} + 12 \, a^{5} b x - 2 \, a^{6} - 60 \,{\left (b^{6} x^{6} + 4 \, a b^{5} x^{5} + 6 \, a^{2} b^{4} x^{4} + 4 \, a^{3} b^{3} x^{3} + a^{4} b^{2} x^{2}\right )} \log \left (b x + a\right ) + 60 \,{\left (b^{6} x^{6} + 4 \, a b^{5} x^{5} + 6 \, a^{2} b^{4} x^{4} + 4 \, a^{3} b^{3} x^{3} + a^{4} b^{2} x^{2}\right )} \log \left (x\right )}{4 \,{\left (a^{7} b^{4} x^{6} + 4 \, a^{8} b^{3} x^{5} + 6 \, a^{9} b^{2} x^{4} + 4 \, a^{10} b x^{3} + a^{11} x^{2}\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^{3} \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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