Optimal. Leaf size=125 \[ -\frac{16 c^2 \left (b x+c x^2\right )^{5/2} (11 b B-6 A c)}{3465 b^4 x^5}+\frac{8 c \left (b x+c x^2\right )^{5/2} (11 b B-6 A c)}{693 b^3 x^6}-\frac{2 \left (b x+c x^2\right )^{5/2} (11 b B-6 A c)}{99 b^2 x^7}-\frac{2 A \left (b x+c x^2\right )^{5/2}}{11 b x^8} \]
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Rubi [A] time = 0.118227, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {792, 658, 650} \[ -\frac{16 c^2 \left (b x+c x^2\right )^{5/2} (11 b B-6 A c)}{3465 b^4 x^5}+\frac{8 c \left (b x+c x^2\right )^{5/2} (11 b B-6 A c)}{693 b^3 x^6}-\frac{2 \left (b x+c x^2\right )^{5/2} (11 b B-6 A c)}{99 b^2 x^7}-\frac{2 A \left (b x+c x^2\right )^{5/2}}{11 b x^8} \]
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 )^{3/2}}{x^8} \, dx &=-\frac{2 A \left (b x+c x^2\right )^{5/2}}{11 b x^8}+\frac{\left (2 \left (-8 (-b B+A c)+\frac{5}{2} (-b B+2 A c)\right )\right ) \int \frac{\left (b x+c x^2\right )^{3/2}}{x^7} \, dx}{11 b}\\ &=-\frac{2 A \left (b x+c x^2\right )^{5/2}}{11 b x^8}-\frac{2 (11 b B-6 A c) \left (b x+c x^2\right )^{5/2}}{99 b^2 x^7}-\frac{(4 c (11 b B-6 A c)) \int \frac{\left (b x+c x^2\right )^{3/2}}{x^6} \, dx}{99 b^2}\\ &=-\frac{2 A \left (b x+c x^2\right )^{5/2}}{11 b x^8}-\frac{2 (11 b B-6 A c) \left (b x+c x^2\right )^{5/2}}{99 b^2 x^7}+\frac{8 c (11 b B-6 A c) \left (b x+c x^2\right )^{5/2}}{693 b^3 x^6}+\frac{\left (8 c^2 (11 b B-6 A c)\right ) \int \frac{\left (b x+c x^2\right )^{3/2}}{x^5} \, dx}{693 b^3}\\ &=-\frac{2 A \left (b x+c x^2\right )^{5/2}}{11 b x^8}-\frac{2 (11 b B-6 A c) \left (b x+c x^2\right )^{5/2}}{99 b^2 x^7}+\frac{8 c (11 b B-6 A c) \left (b x+c x^2\right )^{5/2}}{693 b^3 x^6}-\frac{16 c^2 (11 b B-6 A c) \left (b x+c x^2\right )^{5/2}}{3465 b^4 x^5}\\ \end{align*}
Mathematica [A] time = 0.0338788, size = 79, normalized size = 0.63 \[ -\frac{2 (x (b+c x))^{5/2} \left (3 A \left (-70 b^2 c x+105 b^3+40 b c^2 x^2-16 c^3 x^3\right )+11 b B x \left (35 b^2-20 b c x+8 c^2 x^2\right )\right )}{3465 b^4 x^8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 86, normalized size = 0.7 \begin{align*} -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -48\,A{x}^{3}{c}^{3}+88\,B{x}^{3}b{c}^{2}+120\,A{x}^{2}b{c}^{2}-220\,B{x}^{2}{b}^{2}c-210\,A{b}^{2}cx+385\,{b}^{3}Bx+315\,A{b}^{3} \right ) }{3465\,{x}^{7}{b}^{4}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{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 = 1.84999, size = 296, normalized size = 2.37 \begin{align*} -\frac{2 \,{\left (315 \, A b^{5} + 8 \,{\left (11 \, B b c^{4} - 6 \, A c^{5}\right )} x^{5} - 4 \,{\left (11 \, B b^{2} c^{3} - 6 \, A b c^{4}\right )} x^{4} + 3 \,{\left (11 \, B b^{3} c^{2} - 6 \, A b^{2} c^{3}\right )} x^{3} + 5 \,{\left (110 \, B b^{4} c + 3 \, A b^{3} c^{2}\right )} x^{2} + 35 \,{\left (11 \, B b^{5} + 12 \, A b^{4} c\right )} x\right )} \sqrt{c x^{2} + b x}}{3465 \, b^{4} 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{3}{2}} \left (A + B x\right )}{x^{8}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17536, size = 582, normalized size = 4.66 \begin{align*} \frac{2 \,{\left (4620 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{8} B c^{3} + 17325 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{7} B b c^{\frac{5}{2}} + 6930 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{7} A c^{\frac{7}{2}} + 28413 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{6} B b^{2} c^{2} + 30492 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{6} A b c^{3} + 25410 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} B b^{3} c^{\frac{3}{2}} + 58905 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} A b^{2} c^{\frac{5}{2}} + 12870 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} B b^{4} c + 63855 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} A b^{3} c^{2} + 3465 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} B b^{5} \sqrt{c} + 41580 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} A b^{4} c^{\frac{3}{2}} + 385 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} B b^{6} + 16170 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} A b^{5} c + 3465 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} A b^{6} \sqrt{c} + 315 \, A b^{7}\right )}}{3465 \,{\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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