Optimal. Leaf size=163 \[ \frac{128 c^3 (b+2 c x) (9 b B-10 A c)}{315 b^6 \sqrt{b x+c x^2}}-\frac{32 c^2 (9 b B-10 A c)}{315 b^4 x \sqrt{b x+c x^2}}+\frac{16 c (9 b B-10 A c)}{315 b^3 x^2 \sqrt{b x+c x^2}}-\frac{2 (9 b B-10 A c)}{63 b^2 x^3 \sqrt{b x+c x^2}}-\frac{2 A}{9 b x^4 \sqrt{b x+c x^2}} \]
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Rubi [A] time = 0.143882, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {792, 658, 613} \[ \frac{128 c^3 (b+2 c x) (9 b B-10 A c)}{315 b^6 \sqrt{b x+c x^2}}-\frac{32 c^2 (9 b B-10 A c)}{315 b^4 x \sqrt{b x+c x^2}}+\frac{16 c (9 b B-10 A c)}{315 b^3 x^2 \sqrt{b x+c x^2}}-\frac{2 (9 b B-10 A c)}{63 b^2 x^3 \sqrt{b x+c x^2}}-\frac{2 A}{9 b x^4 \sqrt{b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 792
Rule 658
Rule 613
Rubi steps
\begin{align*} \int \frac{A+B x}{x^4 \left (b x+c x^2\right )^{3/2}} \, dx &=-\frac{2 A}{9 b x^4 \sqrt{b x+c x^2}}+\frac{\left (2 \left (\frac{1}{2} (b B-2 A c)-4 (-b B+A c)\right )\right ) \int \frac{1}{x^3 \left (b x+c x^2\right )^{3/2}} \, dx}{9 b}\\ &=-\frac{2 A}{9 b x^4 \sqrt{b x+c x^2}}-\frac{2 (9 b B-10 A c)}{63 b^2 x^3 \sqrt{b x+c x^2}}-\frac{(8 c (9 b B-10 A c)) \int \frac{1}{x^2 \left (b x+c x^2\right )^{3/2}} \, dx}{63 b^2}\\ &=-\frac{2 A}{9 b x^4 \sqrt{b x+c x^2}}-\frac{2 (9 b B-10 A c)}{63 b^2 x^3 \sqrt{b x+c x^2}}+\frac{16 c (9 b B-10 A c)}{315 b^3 x^2 \sqrt{b x+c x^2}}+\frac{\left (16 c^2 (9 b B-10 A c)\right ) \int \frac{1}{x \left (b x+c x^2\right )^{3/2}} \, dx}{105 b^3}\\ &=-\frac{2 A}{9 b x^4 \sqrt{b x+c x^2}}-\frac{2 (9 b B-10 A c)}{63 b^2 x^3 \sqrt{b x+c x^2}}+\frac{16 c (9 b B-10 A c)}{315 b^3 x^2 \sqrt{b x+c x^2}}-\frac{32 c^2 (9 b B-10 A c)}{315 b^4 x \sqrt{b x+c x^2}}-\frac{\left (64 c^3 (9 b B-10 A c)\right ) \int \frac{1}{\left (b x+c x^2\right )^{3/2}} \, dx}{315 b^4}\\ &=-\frac{2 A}{9 b x^4 \sqrt{b x+c x^2}}-\frac{2 (9 b B-10 A c)}{63 b^2 x^3 \sqrt{b x+c x^2}}+\frac{16 c (9 b B-10 A c)}{315 b^3 x^2 \sqrt{b x+c x^2}}-\frac{32 c^2 (9 b B-10 A c)}{315 b^4 x \sqrt{b x+c x^2}}+\frac{128 c^3 (9 b B-10 A c) (b+2 c x)}{315 b^6 \sqrt{b x+c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0353644, size = 123, normalized size = 0.75 \[ -\frac{2 \left (5 A \left (16 b^3 c^2 x^2-32 b^2 c^3 x^3-10 b^4 c x+7 b^5+128 b c^4 x^4+256 c^5 x^5\right )+9 b B x \left (16 b^2 c^2 x^2-8 b^3 c x+5 b^4-64 b c^3 x^3-128 c^4 x^4\right )\right )}{315 b^6 x^4 \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 134, normalized size = 0.8 \begin{align*} -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 1280\,A{c}^{5}{x}^{5}-1152\,Bb{c}^{4}{x}^{5}+640\,Ab{c}^{4}{x}^{4}-576\,B{b}^{2}{c}^{3}{x}^{4}-160\,A{b}^{2}{c}^{3}{x}^{3}+144\,B{b}^{3}{c}^{2}{x}^{3}+80\,A{b}^{3}{c}^{2}{x}^{2}-72\,B{b}^{4}c{x}^{2}-50\,A{b}^{4}cx+45\,B{b}^{5}x+35\,A{b}^{5} \right ) }{315\,{x}^{3}{b}^{6}} \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 = 1.90241, size = 311, normalized size = 1.91 \begin{align*} -\frac{2 \,{\left (35 \, A b^{5} - 128 \,{\left (9 \, B b c^{4} - 10 \, A c^{5}\right )} x^{5} - 64 \,{\left (9 \, B b^{2} c^{3} - 10 \, A b c^{4}\right )} x^{4} + 16 \,{\left (9 \, B b^{3} c^{2} - 10 \, A b^{2} c^{3}\right )} x^{3} - 8 \,{\left (9 \, B b^{4} c - 10 \, A b^{3} c^{2}\right )} x^{2} + 5 \,{\left (9 \, B b^{5} - 10 \, A b^{4} c\right )} x\right )} \sqrt{c x^{2} + b x}}{315 \,{\left (b^{6} c x^{6} + b^{7} x^{5}\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^{4} \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^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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