Optimal. Leaf size=74 \[ -\frac{16 c^2 \left (b x+c x^2\right )^{5/2}}{315 b^3 x^5}+\frac{8 c \left (b x+c x^2\right )^{5/2}}{63 b^2 x^6}-\frac{2 \left (b x+c x^2\right )^{5/2}}{9 b x^7} \]
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Rubi [A] time = 0.0270773, antiderivative size = 74, 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, 650} \[ -\frac{16 c^2 \left (b x+c x^2\right )^{5/2}}{315 b^3 x^5}+\frac{8 c \left (b x+c x^2\right )^{5/2}}{63 b^2 x^6}-\frac{2 \left (b x+c x^2\right )^{5/2}}{9 b x^7} \]
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^7} \, dx &=-\frac{2 \left (b x+c x^2\right )^{5/2}}{9 b x^7}-\frac{(4 c) \int \frac{\left (b x+c x^2\right )^{3/2}}{x^6} \, dx}{9 b}\\ &=-\frac{2 \left (b x+c x^2\right )^{5/2}}{9 b x^7}+\frac{8 c \left (b x+c x^2\right )^{5/2}}{63 b^2 x^6}+\frac{\left (8 c^2\right ) \int \frac{\left (b x+c x^2\right )^{3/2}}{x^5} \, dx}{63 b^2}\\ &=-\frac{2 \left (b x+c x^2\right )^{5/2}}{9 b x^7}+\frac{8 c \left (b x+c x^2\right )^{5/2}}{63 b^2 x^6}-\frac{16 c^2 \left (b x+c x^2\right )^{5/2}}{315 b^3 x^5}\\ \end{align*}
Mathematica [A] time = 0.0157896, size = 40, normalized size = 0.54 \[ -\frac{2 (x (b+c x))^{5/2} \left (35 b^2-20 b c x+8 c^2 x^2\right )}{315 b^3 x^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 44, normalized size = 0.6 \begin{align*} -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 8\,{c}^{2}{x}^{2}-20\,bcx+35\,{b}^{2} \right ) }{315\,{x}^{6}{b}^{3}} \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 = 1.93022, size = 135, normalized size = 1.82 \begin{align*} -\frac{2 \,{\left (8 \, c^{4} x^{4} - 4 \, b c^{3} x^{3} + 3 \, b^{2} c^{2} x^{2} + 50 \, b^{3} c x + 35 \, b^{4}\right )} \sqrt{c x^{2} + b x}}{315 \, b^{3} x^{5}} \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^{7}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.23412, size = 262, normalized size = 3.54 \begin{align*} \frac{2 \,{\left (420 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{6} c^{3} + 1575 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} b c^{\frac{5}{2}} + 2583 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} b^{2} c^{2} + 2310 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} b^{3} c^{\frac{3}{2}} + 1170 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} b^{4} c + 315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} b^{5} \sqrt{c} + 35 \, b^{6}\right )}}{315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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