Optimal. Leaf size=125 \[ -\frac{16 c^2 \left (b x+c x^2\right )^{3/2} (3 b B-2 A c)}{315 b^4 x^3}+\frac{8 c \left (b x+c x^2\right )^{3/2} (3 b B-2 A c)}{105 b^3 x^4}-\frac{2 \left (b x+c x^2\right )^{3/2} (3 b B-2 A c)}{21 b^2 x^5}-\frac{2 A \left (b x+c x^2\right )^{3/2}}{9 b x^6} \]
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Rubi [A] time = 0.10906, 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} \[ -\frac{16 c^2 \left (b x+c x^2\right )^{3/2} (3 b B-2 A c)}{315 b^4 x^3}+\frac{8 c \left (b x+c x^2\right )^{3/2} (3 b B-2 A c)}{105 b^3 x^4}-\frac{2 \left (b x+c x^2\right )^{3/2} (3 b B-2 A c)}{21 b^2 x^5}-\frac{2 A \left (b x+c x^2\right )^{3/2}}{9 b x^6} \]
Antiderivative was successfully verified.
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Rule 792
Rule 658
Rule 650
Rubi steps
\begin{align*} \int \frac{(A+B x) \sqrt{b x+c x^2}}{x^6} \, dx &=-\frac{2 A \left (b x+c x^2\right )^{3/2}}{9 b x^6}+\frac{\left (2 \left (-6 (-b B+A c)+\frac{3}{2} (-b B+2 A c)\right )\right ) \int \frac{\sqrt{b x+c x^2}}{x^5} \, dx}{9 b}\\ &=-\frac{2 A \left (b x+c x^2\right )^{3/2}}{9 b x^6}-\frac{2 (3 b B-2 A c) \left (b x+c x^2\right )^{3/2}}{21 b^2 x^5}-\frac{(4 c (3 b B-2 A c)) \int \frac{\sqrt{b x+c x^2}}{x^4} \, dx}{21 b^2}\\ &=-\frac{2 A \left (b x+c x^2\right )^{3/2}}{9 b x^6}-\frac{2 (3 b B-2 A c) \left (b x+c x^2\right )^{3/2}}{21 b^2 x^5}+\frac{8 c (3 b B-2 A c) \left (b x+c x^2\right )^{3/2}}{105 b^3 x^4}+\frac{\left (8 c^2 (3 b B-2 A c)\right ) \int \frac{\sqrt{b x+c x^2}}{x^3} \, dx}{105 b^3}\\ &=-\frac{2 A \left (b x+c x^2\right )^{3/2}}{9 b x^6}-\frac{2 (3 b B-2 A c) \left (b x+c x^2\right )^{3/2}}{21 b^2 x^5}+\frac{8 c (3 b B-2 A c) \left (b x+c x^2\right )^{3/2}}{105 b^3 x^4}-\frac{16 c^2 (3 b B-2 A c) \left (b x+c x^2\right )^{3/2}}{315 b^4 x^3}\\ \end{align*}
Mathematica [A] time = 0.0327115, size = 78, normalized size = 0.62 \[ -\frac{2 (x (b+c x))^{3/2} \left (A \left (-30 b^2 c x+35 b^3+24 b c^2 x^2-16 c^3 x^3\right )+3 b B x \left (15 b^2-12 b c x+8 c^2 x^2\right )\right )}{315 b^4 x^6} \]
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 ( -16\,A{x}^{3}{c}^{3}+24\,B{x}^{3}b{c}^{2}+24\,A{x}^{2}b{c}^{2}-36\,B{x}^{2}{b}^{2}c-30\,A{b}^{2}cx+45\,{b}^{3}Bx+35\,A{b}^{3} \right ) }{315\,{x}^{5}{b}^{4}}\sqrt{c{x}^{2}+bx}} \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 = 1.8555, size = 231, normalized size = 1.85 \begin{align*} -\frac{2 \,{\left (35 \, A b^{4} + 8 \,{\left (3 \, B b c^{3} - 2 \, A c^{4}\right )} x^{4} - 4 \,{\left (3 \, B b^{2} c^{2} - 2 \, A b c^{3}\right )} x^{3} + 3 \,{\left (3 \, B b^{3} c - 2 \, A b^{2} c^{2}\right )} x^{2} + 5 \,{\left (9 \, B b^{4} + A b^{3} c\right )} x\right )} \sqrt{c x^{2} + b x}}{315 \, b^{4} 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{\sqrt{x \left (b + c x\right )} \left (A + B x\right )}{x^{6}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.181, size = 420, normalized size = 3.36 \begin{align*} \frac{2 \,{\left (420 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{6} B c^{2} + 945 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} B b c^{\frac{3}{2}} + 630 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} A c^{\frac{5}{2}} + 819 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} B b^{2} c + 1764 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} A b c^{2} + 315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} B b^{3} \sqrt{c} + 1995 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} A b^{2} c^{\frac{3}{2}} + 45 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} B b^{4} + 1125 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} A b^{3} c + 315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} A b^{4} \sqrt{c} + 35 \, A b^{5}\right )}}{315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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