Optimal. Leaf size=205 \[ \frac{a^5}{4 b^6 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 a^4}{3 b^6 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 a^3}{b^6 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{10 a^2}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 a (a+b x) \log (a+b x)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.0930691, antiderivative size = 205, 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} \[ \frac{a^5}{4 b^6 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 a^4}{3 b^6 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 a^3}{b^6 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{10 a^2}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 a (a+b x) \log (a+b x)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 646
Rule 43
Rubi steps
\begin{align*} \int \frac{x^5}{\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{x^5}{\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}{b^{10}}-\frac{a^5}{b^{10} (a+b x)^5}+\frac{5 a^4}{b^{10} (a+b x)^4}-\frac{10 a^3}{b^{10} (a+b x)^3}+\frac{10 a^2}{b^{10} (a+b x)^2}-\frac{5 a}{b^{10} (a+b x)}\right ) \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{10 a^2}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a^5}{4 b^6 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 a^4}{3 b^6 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 a^3}{b^6 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 a (a+b x) \log (a+b x)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0294583, size = 93, normalized size = 0.45 \[ \frac{-252 a^3 b^2 x^2-48 a^2 b^3 x^3-248 a^4 b x-77 a^5+48 a b^4 x^4-60 a (a+b x)^4 \log (a+b x)+12 b^5 x^5}{12 b^6 (a+b x)^3 \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.221, size = 145, normalized size = 0.7 \begin{align*} -{\frac{ \left ( 60\,\ln \left ( bx+a \right ){x}^{4}a{b}^{4}-12\,{b}^{5}{x}^{5}+240\,\ln \left ( bx+a \right ){x}^{3}{a}^{2}{b}^{3}-48\,a{b}^{4}{x}^{4}+360\,\ln \left ( bx+a \right ){x}^{2}{a}^{3}{b}^{2}+48\,{a}^{2}{b}^{3}{x}^{3}+240\,\ln \left ( bx+a \right ) x{a}^{4}b+252\,{a}^{3}{b}^{2}{x}^{2}+60\,\ln \left ( bx+a \right ){a}^{5}+248\,{a}^{4}bx+77\,{a}^{5} \right ) \left ( bx+a \right ) }{12\,{b}^{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 [A] time = 1.25088, size = 153, normalized size = 0.75 \begin{align*} \frac{12 \, b^{5} x^{5} + 48 \, a b^{4} x^{4} - 48 \, a^{2} b^{3} x^{3} - 252 \, a^{3} b^{2} x^{2} - 248 \, a^{4} b x - 77 \, a^{5}}{12 \,{\left (b^{10} x^{4} + 4 \, a b^{9} x^{3} + 6 \, a^{2} b^{8} x^{2} + 4 \, a^{3} b^{7} x + a^{4} b^{6}\right )}} - \frac{5 \, a \log \left (b x + a\right )}{b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71691, size = 320, normalized size = 1.56 \begin{align*} \frac{12 \, b^{5} x^{5} + 48 \, a b^{4} x^{4} - 48 \, a^{2} b^{3} x^{3} - 252 \, a^{3} b^{2} x^{2} - 248 \, a^{4} b x - 77 \, a^{5} - 60 \,{\left (a b^{4} x^{4} + 4 \, a^{2} b^{3} x^{3} + 6 \, a^{3} b^{2} x^{2} + 4 \, a^{4} b x + a^{5}\right )} \log \left (b x + a\right )}{12 \,{\left (b^{10} x^{4} + 4 \, a b^{9} x^{3} + 6 \, a^{2} b^{8} x^{2} + 4 \, a^{3} b^{7} x + a^{4} b^{6}\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^{5}}{\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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