Optimal. Leaf size=77 \[ -\frac{16 c^2 (b+2 c x)}{5 b^4 \sqrt{b x+c x^2}}+\frac{4 c}{5 b^2 x \sqrt{b x+c x^2}}-\frac{2}{5 b x^2 \sqrt{b x+c x^2}} \]
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Rubi [A] time = 0.0244401, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {658, 613} \[ -\frac{16 c^2 (b+2 c x)}{5 b^4 \sqrt{b x+c x^2}}+\frac{4 c}{5 b^2 x \sqrt{b x+c x^2}}-\frac{2}{5 b x^2 \sqrt{b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 658
Rule 613
Rubi steps
\begin{align*} \int \frac{1}{x^2 \left (b x+c x^2\right )^{3/2}} \, dx &=-\frac{2}{5 b x^2 \sqrt{b x+c x^2}}-\frac{(6 c) \int \frac{1}{x \left (b x+c x^2\right )^{3/2}} \, dx}{5 b}\\ &=-\frac{2}{5 b x^2 \sqrt{b x+c x^2}}+\frac{4 c}{5 b^2 x \sqrt{b x+c x^2}}+\frac{\left (8 c^2\right ) \int \frac{1}{\left (b x+c x^2\right )^{3/2}} \, dx}{5 b^2}\\ &=-\frac{2}{5 b x^2 \sqrt{b x+c x^2}}+\frac{4 c}{5 b^2 x \sqrt{b x+c x^2}}-\frac{16 c^2 (b+2 c x)}{5 b^4 \sqrt{b x+c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0122747, size = 49, normalized size = 0.64 \[ -\frac{2 \left (-2 b^2 c x+b^3+8 b c^2 x^2+16 c^3 x^3\right )}{5 b^4 x^2 \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 53, normalized size = 0.7 \begin{align*} -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 16\,{x}^{3}{c}^{3}+8\,b{x}^{2}{c}^{2}-2\,{b}^{2}xc+{b}^{3} \right ) }{5\,x{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 = 1.86682, size = 123, normalized size = 1.6 \begin{align*} -\frac{2 \,{\left (16 \, c^{3} x^{3} + 8 \, b c^{2} x^{2} - 2 \, b^{2} c x + b^{3}\right )} \sqrt{c x^{2} + b x}}{5 \,{\left (b^{4} c x^{4} + b^{5} 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{1}{x^{2} \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{1}{{\left (c x^{2} + b x\right )}^{\frac{3}{2}} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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