Optimal. Leaf size=96 \[ \frac{16 c (b+2 c x) (5 b B-8 A c)}{15 b^5 \sqrt{b x+c x^2}}-\frac{2 (b+2 c x) (5 b B-8 A c)}{15 b^3 \left (b x+c x^2\right )^{3/2}}-\frac{2 A}{5 b x \left (b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.0675235, antiderivative size = 96, 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, 614, 613} \[ \frac{16 c (b+2 c x) (5 b B-8 A c)}{15 b^5 \sqrt{b x+c x^2}}-\frac{2 (b+2 c x) (5 b B-8 A c)}{15 b^3 \left (b x+c x^2\right )^{3/2}}-\frac{2 A}{5 b x \left (b x+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 792
Rule 614
Rule 613
Rubi steps
\begin{align*} \int \frac{A+B x}{x \left (b x+c x^2\right )^{5/2}} \, dx &=-\frac{2 A}{5 b x \left (b x+c x^2\right )^{3/2}}+\frac{\left (2 \left (b B-A c-\frac{3}{2} (-b B+2 A c)\right )\right ) \int \frac{1}{\left (b x+c x^2\right )^{5/2}} \, dx}{5 b}\\ &=-\frac{2 A}{5 b x \left (b x+c x^2\right )^{3/2}}-\frac{2 (5 b B-8 A c) (b+2 c x)}{15 b^3 \left (b x+c x^2\right )^{3/2}}-\frac{(8 c (5 b B-8 A c)) \int \frac{1}{\left (b x+c x^2\right )^{3/2}} \, dx}{15 b^3}\\ &=-\frac{2 A}{5 b x \left (b x+c x^2\right )^{3/2}}-\frac{2 (5 b B-8 A c) (b+2 c x)}{15 b^3 \left (b x+c x^2\right )^{3/2}}+\frac{16 c (5 b B-8 A c) (b+2 c x)}{15 b^5 \sqrt{b x+c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0339313, size = 98, normalized size = 1.02 \[ -\frac{2 \left (A \left (48 b^2 c^2 x^2-8 b^3 c x+3 b^4+192 b c^3 x^3+128 c^4 x^4\right )+5 b B x \left (-6 b^2 c x+b^3-24 b c^2 x^2-16 c^3 x^3\right )\right )}{15 b^5 x (x (b+c x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 107, normalized size = 1.1 \begin{align*} -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 128\,A{c}^{4}{x}^{4}-80\,Bb{c}^{3}{x}^{4}+192\,Ab{c}^{3}{x}^{3}-120\,B{b}^{2}{c}^{2}{x}^{3}+48\,A{b}^{2}{c}^{2}{x}^{2}-30\,B{b}^{3}c{x}^{2}-8\,A{b}^{3}cx+5\,{b}^{4}Bx+3\,A{b}^{4} \right ) }{15\,{b}^{5}} \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 = 1.98385, size = 269, normalized size = 2.8 \begin{align*} -\frac{2 \,{\left (3 \, A b^{4} - 16 \,{\left (5 \, B b c^{3} - 8 \, A c^{4}\right )} x^{4} - 24 \,{\left (5 \, B b^{2} c^{2} - 8 \, A b c^{3}\right )} x^{3} - 6 \,{\left (5 \, B b^{3} c - 8 \, A b^{2} c^{2}\right )} x^{2} +{\left (5 \, B b^{4} - 8 \, A b^{3} c\right )} x\right )} \sqrt{c x^{2} + b x}}{15 \,{\left (b^{5} c^{2} x^{5} + 2 \, b^{6} c x^{4} + b^{7} x^{3}\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 \left (x \left (b + c x\right )\right )^{\frac{5}{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{5}{2}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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