Optimal. Leaf size=48 \[ \frac{4 b \left (a x+b x^2\right )^{7/2}}{63 a^2 x^7}-\frac{2 \left (a x+b x^2\right )^{7/2}}{9 a x^8} \]
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Rubi [A] time = 0.0173728, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {658, 650} \[ \frac{4 b \left (a x+b x^2\right )^{7/2}}{63 a^2 x^7}-\frac{2 \left (a x+b x^2\right )^{7/2}}{9 a x^8} \]
Antiderivative was successfully verified.
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Rule 658
Rule 650
Rubi steps
\begin{align*} \int \frac{\left (a x+b x^2\right )^{5/2}}{x^8} \, dx &=-\frac{2 \left (a x+b x^2\right )^{7/2}}{9 a x^8}-\frac{(2 b) \int \frac{\left (a x+b x^2\right )^{5/2}}{x^7} \, dx}{9 a}\\ &=-\frac{2 \left (a x+b x^2\right )^{7/2}}{9 a x^8}+\frac{4 b \left (a x+b x^2\right )^{7/2}}{63 a^2 x^7}\\ \end{align*}
Mathematica [A] time = 0.0107597, size = 36, normalized size = 0.75 \[ \frac{2 (a+b x)^3 \sqrt{x (a+b x)} (2 b x-7 a)}{63 a^2 x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 33, normalized size = 0.7 \begin{align*} -{\frac{ \left ( 2\,bx+2\,a \right ) \left ( -2\,bx+7\,a \right ) }{63\,{a}^{2}{x}^{7}} \left ( b{x}^{2}+ax \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 = 1.99, size = 130, normalized size = 2.71 \begin{align*} \frac{2 \,{\left (2 \, b^{4} x^{4} - a b^{3} x^{3} - 15 \, a^{2} b^{2} x^{2} - 19 \, a^{3} b x - 7 \, a^{4}\right )} \sqrt{b x^{2} + a x}}{63 \, a^{2} x^{5}} \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 (x \left (a + b x\right )\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 = 1.21591, size = 301, normalized size = 6.27 \begin{align*} \frac{2 \,{\left (63 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{7} b^{\frac{7}{2}} + 273 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{6} a b^{3} + 567 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{5} a^{2} b^{\frac{5}{2}} + 693 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{4} a^{3} b^{2} + 525 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{3} a^{4} b^{\frac{3}{2}} + 243 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{2} a^{5} b + 63 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} a^{6} \sqrt{b} + 7 \, a^{7}\right )}}{63 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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