Optimal. Leaf size=48 \[ \frac{4 c \left (b x+c x^2\right )^{5/2}}{35 b^2 x^5}-\frac{2 \left (b x+c x^2\right )^{5/2}}{7 b x^6} \]
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Rubi [A] time = 0.0164416, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {658, 650} \[ \frac{4 c \left (b x+c x^2\right )^{5/2}}{35 b^2 x^5}-\frac{2 \left (b x+c x^2\right )^{5/2}}{7 b x^6} \]
Antiderivative was successfully verified.
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Rule 658
Rule 650
Rubi steps
\begin{align*} \int \frac{\left (b x+c x^2\right )^{3/2}}{x^6} \, dx &=-\frac{2 \left (b x+c x^2\right )^{5/2}}{7 b x^6}-\frac{(2 c) \int \frac{\left (b x+c x^2\right )^{3/2}}{x^5} \, dx}{7 b}\\ &=-\frac{2 \left (b x+c x^2\right )^{5/2}}{7 b x^6}+\frac{4 c \left (b x+c x^2\right )^{5/2}}{35 b^2 x^5}\\ \end{align*}
Mathematica [A] time = 0.0120312, size = 29, normalized size = 0.6 \[ \frac{2 (x (b+c x))^{5/2} (2 c x-5 b)}{35 b^2 x^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 33, normalized size = 0.7 \begin{align*} -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -2\,cx+5\,b \right ) }{35\,{x}^{5}{b}^{2}} \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 = 2.06845, size = 105, normalized size = 2.19 \begin{align*} \frac{2 \,{\left (2 \, c^{3} x^{3} - b c^{2} x^{2} - 8 \, b^{2} c x - 5 \, b^{3}\right )} \sqrt{c x^{2} + b x}}{35 \, b^{2} x^{4}} \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{\left (x \left (b + c x\right )\right )^{\frac{3}{2}}}{x^{6}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.25156, size = 223, normalized size = 4.65 \begin{align*} \frac{2 \,{\left (35 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} c^{\frac{5}{2}} + 105 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} b c^{2} + 140 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} b^{2} c^{\frac{3}{2}} + 98 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} b^{3} c + 35 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} b^{4} \sqrt{c} + 5 \, b^{5}\right )}}{35 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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