Optimal. Leaf size=76 \[ \frac{b (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{42 a^2 x^6}-\frac{(a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{7 a x^7} \]
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Rubi [A] time = 0.0230251, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {646, 45, 37} \[ \frac{b (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{42 a^2 x^6}-\frac{(a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{7 a x^7} \]
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^8} \, dx &=\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}{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}}{7 a 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}{7 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}}{7 a x^7}+\frac{b (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{42 a^2 x^6}\\ \end{align*}
Mathematica [A] time = 0.0157314, size = 77, normalized size = 1.01 \[ -\frac{\sqrt{(a+b x)^2} \left (84 a^3 b^2 x^2+105 a^2 b^3 x^3+35 a^4 b x+6 a^5+70 a b^4 x^4+21 b^5 x^5\right )}{42 x^7 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.203, size = 74, normalized size = 1. \begin{align*} -{\frac{21\,{b}^{5}{x}^{5}+70\,a{b}^{4}{x}^{4}+105\,{a}^{2}{b}^{3}{x}^{3}+84\,{a}^{3}{b}^{2}{x}^{2}+35\,{a}^{4}bx+6\,{a}^{5}}{42\,{x}^{7} \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 = 1.66511, size = 128, normalized size = 1.68 \begin{align*} -\frac{21 \, b^{5} x^{5} + 70 \, a b^{4} x^{4} + 105 \, a^{2} b^{3} x^{3} + 84 \, a^{3} b^{2} x^{2} + 35 \, a^{4} b x + 6 \, a^{5}}{42 \, 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{5}{2}}}{x^{8}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.38857, size = 146, normalized size = 1.92 \begin{align*} \frac{b^{7} \mathrm{sgn}\left (b x + a\right )}{42 \, a^{2}} - \frac{21 \, b^{5} x^{5} \mathrm{sgn}\left (b x + a\right ) + 70 \, a b^{4} x^{4} \mathrm{sgn}\left (b x + a\right ) + 105 \, a^{2} b^{3} x^{3} \mathrm{sgn}\left (b x + a\right ) + 84 \, a^{3} b^{2} x^{2} \mathrm{sgn}\left (b x + a\right ) + 35 \, a^{4} b x \mathrm{sgn}\left (b x + a\right ) + 6 \, a^{5} \mathrm{sgn}\left (b x + a\right )}{42 \, x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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