Optimal. Leaf size=128 \[ -\frac{16 c^2 (b+2 c x) (7 b B-8 A c)}{35 b^5 \sqrt{b x+c x^2}}+\frac{4 c (7 b B-8 A c)}{35 b^3 x \sqrt{b x+c x^2}}-\frac{2 (7 b B-8 A c)}{35 b^2 x^2 \sqrt{b x+c x^2}}-\frac{2 A}{7 b x^3 \sqrt{b x+c x^2}} \]
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Rubi [A] time = 0.114742, antiderivative size = 128, 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, 613} \[ -\frac{16 c^2 (b+2 c x) (7 b B-8 A c)}{35 b^5 \sqrt{b x+c x^2}}+\frac{4 c (7 b B-8 A c)}{35 b^3 x \sqrt{b x+c x^2}}-\frac{2 (7 b B-8 A c)}{35 b^2 x^2 \sqrt{b x+c x^2}}-\frac{2 A}{7 b x^3 \sqrt{b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 792
Rule 658
Rule 613
Rubi steps
\begin{align*} \int \frac{A+B x}{x^3 \left (b x+c x^2\right )^{3/2}} \, dx &=-\frac{2 A}{7 b x^3 \sqrt{b x+c x^2}}+\frac{\left (2 \left (\frac{1}{2} (b B-2 A c)-3 (-b B+A c)\right )\right ) \int \frac{1}{x^2 \left (b x+c x^2\right )^{3/2}} \, dx}{7 b}\\ &=-\frac{2 A}{7 b x^3 \sqrt{b x+c x^2}}-\frac{2 (7 b B-8 A c)}{35 b^2 x^2 \sqrt{b x+c x^2}}-\frac{(6 c (7 b B-8 A c)) \int \frac{1}{x \left (b x+c x^2\right )^{3/2}} \, dx}{35 b^2}\\ &=-\frac{2 A}{7 b x^3 \sqrt{b x+c x^2}}-\frac{2 (7 b B-8 A c)}{35 b^2 x^2 \sqrt{b x+c x^2}}+\frac{4 c (7 b B-8 A c)}{35 b^3 x \sqrt{b x+c x^2}}+\frac{\left (8 c^2 (7 b B-8 A c)\right ) \int \frac{1}{\left (b x+c x^2\right )^{3/2}} \, dx}{35 b^3}\\ &=-\frac{2 A}{7 b x^3 \sqrt{b x+c x^2}}-\frac{2 (7 b B-8 A c)}{35 b^2 x^2 \sqrt{b x+c x^2}}+\frac{4 c (7 b B-8 A c)}{35 b^3 x \sqrt{b x+c x^2}}-\frac{16 c^2 (7 b B-8 A c) (b+2 c x)}{35 b^5 \sqrt{b x+c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0325647, size = 98, normalized size = 0.77 \[ -\frac{2 \left (A \left (16 b^2 c^2 x^2-8 b^3 c x+5 b^4-64 b c^3 x^3-128 c^4 x^4\right )+7 b B x \left (-2 b^2 c x+b^3+8 b c^2 x^2+16 c^3 x^3\right )\right )}{35 b^5 x^3 \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 110, normalized size = 0.9 \begin{align*} -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -128\,A{c}^{4}{x}^{4}+112\,Bb{c}^{3}{x}^{4}-64\,Ab{c}^{3}{x}^{3}+56\,B{b}^{2}{c}^{2}{x}^{3}+16\,A{b}^{2}{c}^{2}{x}^{2}-14\,B{b}^{3}c{x}^{2}-8\,A{b}^{3}cx+7\,{b}^{4}Bx+5\,A{b}^{4} \right ) }{35\,{x}^{2}{b}^{5}} \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 = 1.94433, size = 246, normalized size = 1.92 \begin{align*} -\frac{2 \,{\left (5 \, A b^{4} + 16 \,{\left (7 \, B b c^{3} - 8 \, A c^{4}\right )} x^{4} + 8 \,{\left (7 \, B b^{2} c^{2} - 8 \, A b c^{3}\right )} x^{3} - 2 \,{\left (7 \, B b^{3} c - 8 \, A b^{2} c^{2}\right )} x^{2} +{\left (7 \, B b^{4} - 8 \, A b^{3} c\right )} x\right )} \sqrt{c x^{2} + b x}}{35 \,{\left (b^{5} c x^{5} + b^{6} x^{4}\right )}} \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{A + B x}{x^{3} \left (x \left (b + c x\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B x + A}{{\left (c x^{2} + b x\right )}^{\frac{3}{2}} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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