Optimal. Leaf size=103 \[ \frac{128 c^3 (b+2 c x)}{35 b^5 \sqrt{b x+c x^2}}-\frac{32 c^2}{35 b^3 x \sqrt{b x+c x^2}}+\frac{16 c}{35 b^2 x^2 \sqrt{b x+c x^2}}-\frac{2}{7 b x^3 \sqrt{b x+c x^2}} \]
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Rubi [A] time = 0.0394125, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {658, 613} \[ \frac{128 c^3 (b+2 c x)}{35 b^5 \sqrt{b x+c x^2}}-\frac{32 c^2}{35 b^3 x \sqrt{b x+c x^2}}+\frac{16 c}{35 b^2 x^2 \sqrt{b x+c x^2}}-\frac{2}{7 b x^3 \sqrt{b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 658
Rule 613
Rubi steps
\begin{align*} \int \frac{1}{x^3 \left (b x+c x^2\right )^{3/2}} \, dx &=-\frac{2}{7 b x^3 \sqrt{b x+c x^2}}-\frac{(8 c) \int \frac{1}{x^2 \left (b x+c x^2\right )^{3/2}} \, dx}{7 b}\\ &=-\frac{2}{7 b x^3 \sqrt{b x+c x^2}}+\frac{16 c}{35 b^2 x^2 \sqrt{b x+c x^2}}+\frac{\left (48 c^2\right ) \int \frac{1}{x \left (b x+c x^2\right )^{3/2}} \, dx}{35 b^2}\\ &=-\frac{2}{7 b x^3 \sqrt{b x+c x^2}}+\frac{16 c}{35 b^2 x^2 \sqrt{b x+c x^2}}-\frac{32 c^2}{35 b^3 x \sqrt{b x+c x^2}}-\frac{\left (64 c^3\right ) \int \frac{1}{\left (b x+c x^2\right )^{3/2}} \, dx}{35 b^3}\\ &=-\frac{2}{7 b x^3 \sqrt{b x+c x^2}}+\frac{16 c}{35 b^2 x^2 \sqrt{b x+c x^2}}-\frac{32 c^2}{35 b^3 x \sqrt{b x+c x^2}}+\frac{128 c^3 (b+2 c x)}{35 b^5 \sqrt{b x+c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0144757, size = 62, normalized size = 0.6 \[ \frac{2 \left (-16 b^2 c^2 x^2+8 b^3 c x-5 b^4+64 b c^3 x^3+128 c^4 x^4\right )}{35 b^5 x^3 \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 66, normalized size = 0.6 \begin{align*} -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -128\,{c}^{4}{x}^{4}-64\,{x}^{3}{c}^{3}b+16\,{c}^{2}{x}^{2}{b}^{2}-8\,cx{b}^{3}+5\,{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 = 1.93629, size = 151, normalized size = 1.47 \begin{align*} \frac{2 \,{\left (128 \, c^{4} x^{4} + 64 \, b c^{3} x^{3} - 16 \, b^{2} c^{2} x^{2} + 8 \, b^{3} c x - 5 \, b^{4}\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{1}{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{1}{{\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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