Optimal. Leaf size=116 \[ -\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{168 a^3 x^6}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{28 a^2 x^7}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{8 a x^8} \]
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Rubi [A] time = 0.0329352, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {646, 45, 37} \[ -\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{168 a^3 x^6}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{28 a^2 x^7}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{8 a x^8} \]
Antiderivative was successfully verified.
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Rule 646
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^9} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{\left (a b+b^2 x\right )^5}{x^9} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac{(a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{8 a x^8}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{\left (a b+b^2 x\right )^5}{x^8} \, dx}{4 a b^3 \left (a b+b^2 x\right )}\\ &=-\frac{(a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{8 a x^8}+\frac{b (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{28 a^2 x^7}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{\left (a b+b^2 x\right )^5}{x^7} \, dx}{28 a^2 b^2 \left (a b+b^2 x\right )}\\ &=-\frac{(a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{8 a x^8}+\frac{b (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{28 a^2 x^7}-\frac{b^2 (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{168 a^3 x^6}\\ \end{align*}
Mathematica [A] time = 0.015702, size = 77, normalized size = 0.66 \[ -\frac{\sqrt{(a+b x)^2} \left (280 a^3 b^2 x^2+336 a^2 b^3 x^3+120 a^4 b x+21 a^5+210 a b^4 x^4+56 b^5 x^5\right )}{168 x^8 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.184, size = 74, normalized size = 0.6 \begin{align*} -{\frac{56\,{b}^{5}{x}^{5}+210\,a{b}^{4}{x}^{4}+336\,{a}^{2}{b}^{3}{x}^{3}+280\,{a}^{3}{b}^{2}{x}^{2}+120\,{a}^{4}bx+21\,{a}^{5}}{168\,{x}^{8} \left ( bx+a \right ) ^{5}} \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 = 1.61849, size = 135, normalized size = 1.16 \begin{align*} -\frac{56 \, b^{5} x^{5} + 210 \, a b^{4} x^{4} + 336 \, a^{2} b^{3} x^{3} + 280 \, a^{3} b^{2} x^{2} + 120 \, a^{4} b x + 21 \, a^{5}}{168 \, x^{8}} \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^{9}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.41711, size = 146, normalized size = 1.26 \begin{align*} -\frac{b^{8} \mathrm{sgn}\left (b x + a\right )}{168 \, a^{3}} - \frac{56 \, b^{5} x^{5} \mathrm{sgn}\left (b x + a\right ) + 210 \, a b^{4} x^{4} \mathrm{sgn}\left (b x + a\right ) + 336 \, a^{2} b^{3} x^{3} \mathrm{sgn}\left (b x + a\right ) + 280 \, a^{3} b^{2} x^{2} \mathrm{sgn}\left (b x + a\right ) + 120 \, a^{4} b x \mathrm{sgn}\left (b x + a\right ) + 21 \, a^{5} \mathrm{sgn}\left (b x + a\right )}{168 \, x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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