Optimal. Leaf size=220 \[ -\frac{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac{10 a^3 b^2 x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{5 a^2 b^3 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{5 a b^4 x^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac{b^5 x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac{5 a^4 b \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]
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Rubi [A] time = 0.0504543, antiderivative size = 220, 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} \[ -\frac{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac{10 a^3 b^2 x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{5 a^2 b^3 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{5 a b^4 x^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac{b^5 x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac{5 a^4 b \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]
Antiderivative was successfully verified.
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Rule 646
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^2} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{\left (a b+b^2 x\right )^5}{x^2} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (10 a^3 b^7+\frac{a^5 b^5}{x^2}+\frac{5 a^4 b^6}{x}+10 a^2 b^8 x+5 a b^9 x^2+b^{10} x^3\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac{10 a^3 b^2 x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{5 a^2 b^3 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{5 a b^4 x^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac{b^5 x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac{5 a^4 b \sqrt{a^2+2 a b x+b^2 x^2} \log (x)}{a+b x}\\ \end{align*}
Mathematica [A] time = 0.0254742, size = 79, normalized size = 0.36 \[ \frac{\sqrt{(a+b x)^2} \left (120 a^3 b^2 x^2+60 a^2 b^3 x^3+60 a^4 b x \log (x)-12 a^5+20 a b^4 x^4+3 b^5 x^5\right )}{12 x (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.224, size = 76, normalized size = 0.4 \begin{align*}{\frac{3\,{b}^{5}{x}^{5}+20\,a{b}^{4}{x}^{4}+60\,{a}^{2}{b}^{3}{x}^{3}+60\,{a}^{4}b\ln \left ( x \right ) x+120\,{a}^{3}{b}^{2}{x}^{2}-12\,{a}^{5}}{12\, \left ( bx+a \right ) ^{5}x} \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 = 1.66037, size = 134, normalized size = 0.61 \begin{align*} \frac{3 \, b^{5} x^{5} + 20 \, a b^{4} x^{4} + 60 \, a^{2} b^{3} x^{3} + 120 \, a^{3} b^{2} x^{2} + 60 \, a^{4} b x \log \left (x\right ) - 12 \, a^{5}}{12 \, x} \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{\left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18102, size = 123, normalized size = 0.56 \begin{align*} \frac{1}{4} \, b^{5} x^{4} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{3} \, a b^{4} x^{3} \mathrm{sgn}\left (b x + a\right ) + 5 \, a^{2} b^{3} x^{2} \mathrm{sgn}\left (b x + a\right ) + 10 \, a^{3} b^{2} x \mathrm{sgn}\left (b x + a\right ) + 5 \, a^{4} b \log \left ({\left | x \right |}\right ) \mathrm{sgn}\left (b x + a\right ) - \frac{a^{5} \mathrm{sgn}\left (b x + a\right )}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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