Optimal. Leaf size=90 \[ \frac{4 c \left (b x+c x^2\right )^{7/2} (11 b B-4 A c)}{693 b^3 x^7}-\frac{2 \left (b x+c x^2\right )^{7/2} (11 b B-4 A c)}{99 b^2 x^8}-\frac{2 A \left (b x+c x^2\right )^{7/2}}{11 b x^9} \]
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Rubi [A] time = 0.088501, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {792, 658, 650} \[ \frac{4 c \left (b x+c x^2\right )^{7/2} (11 b B-4 A c)}{693 b^3 x^7}-\frac{2 \left (b x+c x^2\right )^{7/2} (11 b B-4 A c)}{99 b^2 x^8}-\frac{2 A \left (b x+c x^2\right )^{7/2}}{11 b x^9} \]
Antiderivative was successfully verified.
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Rule 792
Rule 658
Rule 650
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (b x+c x^2\right )^{5/2}}{x^9} \, dx &=-\frac{2 A \left (b x+c x^2\right )^{7/2}}{11 b x^9}+\frac{\left (2 \left (-9 (-b B+A c)+\frac{7}{2} (-b B+2 A c)\right )\right ) \int \frac{\left (b x+c x^2\right )^{5/2}}{x^8} \, dx}{11 b}\\ &=-\frac{2 A \left (b x+c x^2\right )^{7/2}}{11 b x^9}-\frac{2 (11 b B-4 A c) \left (b x+c x^2\right )^{7/2}}{99 b^2 x^8}-\frac{(2 c (11 b B-4 A c)) \int \frac{\left (b x+c x^2\right )^{5/2}}{x^7} \, dx}{99 b^2}\\ &=-\frac{2 A \left (b x+c x^2\right )^{7/2}}{11 b x^9}-\frac{2 (11 b B-4 A c) \left (b x+c x^2\right )^{7/2}}{99 b^2 x^8}+\frac{4 c (11 b B-4 A c) \left (b x+c x^2\right )^{7/2}}{693 b^3 x^7}\\ \end{align*}
Mathematica [A] time = 0.0258829, size = 63, normalized size = 0.7 \[ \frac{2 (b+c x)^3 \sqrt{x (b+c x)} \left (A \left (-63 b^2+28 b c x-8 c^2 x^2\right )+11 b B x (2 c x-7 b)\right )}{693 b^3 x^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 62, normalized size = 0.7 \begin{align*} -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 8\,A{c}^{2}{x}^{2}-22\,B{x}^{2}bc-28\,Abcx+77\,{b}^{2}Bx+63\,A{b}^{2} \right ) }{693\,{x}^{8}{b}^{3}} \left ( c{x}^{2}+bx \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 = 1.89402, size = 286, normalized size = 3.18 \begin{align*} -\frac{2 \,{\left (63 \, A b^{5} - 2 \,{\left (11 \, B b c^{4} - 4 \, A c^{5}\right )} x^{5} +{\left (11 \, B b^{2} c^{3} - 4 \, A b c^{4}\right )} x^{4} + 3 \,{\left (55 \, B b^{3} c^{2} + A b^{2} c^{3}\right )} x^{3} +{\left (209 \, B b^{4} c + 113 \, A b^{3} c^{2}\right )} x^{2} + 7 \,{\left (11 \, B b^{5} + 23 \, A b^{4} c\right )} x\right )} \sqrt{c x^{2} + b x}}{693 \, b^{3} x^{6}} \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 (b + c x\right )\right )^{\frac{5}{2}} \left (A + B x\right )}{x^{9}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15327, size = 663, normalized size = 7.37 \begin{align*} \frac{2 \,{\left (693 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{9} B c^{\frac{7}{2}} + 3003 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{8} B b c^{3} + 924 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{8} A c^{4} + 6237 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{7} B b^{2} c^{\frac{5}{2}} + 4851 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{7} A b c^{\frac{7}{2}} + 7623 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{6} B b^{3} c^{2} + 11781 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{6} A b^{2} c^{3} + 5775 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} B b^{4} c^{\frac{3}{2}} + 16863 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} A b^{3} c^{\frac{5}{2}} + 2673 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} B b^{5} c + 15345 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} A b^{4} c^{2} + 693 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} B b^{6} \sqrt{c} + 9009 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} A b^{5} c^{\frac{3}{2}} + 77 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} B b^{7} + 3311 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} A b^{6} c + 693 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} A b^{7} \sqrt{c} + 63 \, A b^{8}\right )}}{693 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{11}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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