Optimal. Leaf size=151 \[ -\frac{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}-\frac{a^2 b \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^6 (a+b x)}-\frac{3 a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)} \]
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Rubi [A] time = 0.0372512, antiderivative size = 151, 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} \[ -\frac{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}-\frac{a^2 b \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^6 (a+b x)}-\frac{3 a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)} \]
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 )^{3/2}}{x^8} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{\left (a b+b^2 x\right )^3}{x^8} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (\frac{a^3 b^3}{x^8}+\frac{3 a^2 b^4}{x^7}+\frac{3 a b^5}{x^6}+\frac{b^6}{x^5}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=-\frac{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}-\frac{a^2 b \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^6 (a+b x)}-\frac{3 a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0118509, size = 55, normalized size = 0.36 \[ -\frac{\sqrt{(a+b x)^2} \left (70 a^2 b x+20 a^3+84 a b^2 x^2+35 b^3 x^3\right )}{140 x^7 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.184, size = 52, normalized size = 0.3 \begin{align*} -{\frac{35\,{b}^{3}{x}^{3}+84\,a{b}^{2}{x}^{2}+70\,b{a}^{2}x+20\,{a}^{3}}{140\,{x}^{7} \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{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 = 1.73816, size = 84, normalized size = 0.56 \begin{align*} -\frac{35 \, b^{3} x^{3} + 84 \, a b^{2} x^{2} + 70 \, a^{2} b x + 20 \, a^{3}}{140 \, x^{7}} \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{3}{2}}}{x^{8}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25297, size = 100, normalized size = 0.66 \begin{align*} \frac{b^{7} \mathrm{sgn}\left (b x + a\right )}{140 \, a^{4}} - \frac{35 \, b^{3} x^{3} \mathrm{sgn}\left (b x + a\right ) + 84 \, a b^{2} x^{2} \mathrm{sgn}\left (b x + a\right ) + 70 \, a^{2} b x \mathrm{sgn}\left (b x + a\right ) + 20 \, a^{3} \mathrm{sgn}\left (b x + a\right )}{140 \, x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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