Optimal. Leaf size=100 \[ \frac{32 b^3 \left (a x+b x^2\right )^{7/2}}{3003 a^4 x^7}-\frac{16 b^2 \left (a x+b x^2\right )^{7/2}}{429 a^3 x^8}+\frac{12 b \left (a x+b x^2\right )^{7/2}}{143 a^2 x^9}-\frac{2 \left (a x+b x^2\right )^{7/2}}{13 a x^{10}} \]
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Rubi [A] time = 0.0415272, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {658, 650} \[ \frac{32 b^3 \left (a x+b x^2\right )^{7/2}}{3003 a^4 x^7}-\frac{16 b^2 \left (a x+b x^2\right )^{7/2}}{429 a^3 x^8}+\frac{12 b \left (a x+b x^2\right )^{7/2}}{143 a^2 x^9}-\frac{2 \left (a x+b x^2\right )^{7/2}}{13 a x^{10}} \]
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^{10}} \, dx &=-\frac{2 \left (a x+b x^2\right )^{7/2}}{13 a x^{10}}-\frac{(6 b) \int \frac{\left (a x+b x^2\right )^{5/2}}{x^9} \, dx}{13 a}\\ &=-\frac{2 \left (a x+b x^2\right )^{7/2}}{13 a x^{10}}+\frac{12 b \left (a x+b x^2\right )^{7/2}}{143 a^2 x^9}+\frac{\left (24 b^2\right ) \int \frac{\left (a x+b x^2\right )^{5/2}}{x^8} \, dx}{143 a^2}\\ &=-\frac{2 \left (a x+b x^2\right )^{7/2}}{13 a x^{10}}+\frac{12 b \left (a x+b x^2\right )^{7/2}}{143 a^2 x^9}-\frac{16 b^2 \left (a x+b x^2\right )^{7/2}}{429 a^3 x^8}-\frac{\left (16 b^3\right ) \int \frac{\left (a x+b x^2\right )^{5/2}}{x^7} \, dx}{429 a^3}\\ &=-\frac{2 \left (a x+b x^2\right )^{7/2}}{13 a x^{10}}+\frac{12 b \left (a x+b x^2\right )^{7/2}}{143 a^2 x^9}-\frac{16 b^2 \left (a x+b x^2\right )^{7/2}}{429 a^3 x^8}+\frac{32 b^3 \left (a x+b x^2\right )^{7/2}}{3003 a^4 x^7}\\ \end{align*}
Mathematica [A] time = 0.0143286, size = 58, normalized size = 0.58 \[ \frac{2 (a+b x)^3 \sqrt{x (a+b x)} \left (126 a^2 b x-231 a^3-56 a b^2 x^2+16 b^3 x^3\right )}{3003 a^4 x^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 55, normalized size = 0.6 \begin{align*} -{\frac{ \left ( 2\,bx+2\,a \right ) \left ( -16\,{b}^{3}{x}^{3}+56\,a{b}^{2}{x}^{2}-126\,bx{a}^{2}+231\,{a}^{3} \right ) }{3003\,{x}^{9}{a}^{4}} \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 = 1.944, size = 185, normalized size = 1.85 \begin{align*} \frac{2 \,{\left (16 \, b^{6} x^{6} - 8 \, a b^{5} x^{5} + 6 \, a^{2} b^{4} x^{4} - 5 \, a^{3} b^{3} x^{3} - 371 \, a^{4} b^{2} x^{2} - 567 \, a^{5} b x - 231 \, a^{6}\right )} \sqrt{b x^{2} + a x}}{3003 \, a^{4} 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 (x \left (a + b x\right )\right )^{\frac{5}{2}}}{x^{10}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.28221, size = 379, normalized size = 3.79 \begin{align*} \frac{2 \,{\left (6006 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{9} b^{\frac{9}{2}} + 36036 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{8} a b^{4} + 99099 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{7} a^{2} b^{\frac{7}{2}} + 161733 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{6} a^{3} b^{3} + 171171 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{5} a^{4} b^{\frac{5}{2}} + 121121 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{4} a^{5} b^{2} + 57057 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{3} a^{6} b^{\frac{3}{2}} + 17199 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{2} a^{7} b + 3003 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} a^{8} \sqrt{b} + 231 \, a^{9}\right )}}{3003 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{13}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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