Optimal. Leaf size=136 \[ \frac{32 b^2 x^{5/2}}{35 c^3 \sqrt{b x+c x^2}}-\frac{128 b^3 x^{3/2}}{35 c^4 \sqrt{b x+c x^2}}-\frac{256 b^4 \sqrt{x}}{35 c^5 \sqrt{b x+c x^2}}-\frac{16 b x^{7/2}}{35 c^2 \sqrt{b x+c x^2}}+\frac{2 x^{9/2}}{7 c \sqrt{b x+c x^2}} \]
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Rubi [A] time = 0.0567674, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {656, 648} \[ \frac{32 b^2 x^{5/2}}{35 c^3 \sqrt{b x+c x^2}}-\frac{128 b^3 x^{3/2}}{35 c^4 \sqrt{b x+c x^2}}-\frac{256 b^4 \sqrt{x}}{35 c^5 \sqrt{b x+c x^2}}-\frac{16 b x^{7/2}}{35 c^2 \sqrt{b x+c x^2}}+\frac{2 x^{9/2}}{7 c \sqrt{b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 656
Rule 648
Rubi steps
\begin{align*} \int \frac{x^{11/2}}{\left (b x+c x^2\right )^{3/2}} \, dx &=\frac{2 x^{9/2}}{7 c \sqrt{b x+c x^2}}-\frac{(8 b) \int \frac{x^{9/2}}{\left (b x+c x^2\right )^{3/2}} \, dx}{7 c}\\ &=-\frac{16 b x^{7/2}}{35 c^2 \sqrt{b x+c x^2}}+\frac{2 x^{9/2}}{7 c \sqrt{b x+c x^2}}+\frac{\left (48 b^2\right ) \int \frac{x^{7/2}}{\left (b x+c x^2\right )^{3/2}} \, dx}{35 c^2}\\ &=\frac{32 b^2 x^{5/2}}{35 c^3 \sqrt{b x+c x^2}}-\frac{16 b x^{7/2}}{35 c^2 \sqrt{b x+c x^2}}+\frac{2 x^{9/2}}{7 c \sqrt{b x+c x^2}}-\frac{\left (64 b^3\right ) \int \frac{x^{5/2}}{\left (b x+c x^2\right )^{3/2}} \, dx}{35 c^3}\\ &=-\frac{128 b^3 x^{3/2}}{35 c^4 \sqrt{b x+c x^2}}+\frac{32 b^2 x^{5/2}}{35 c^3 \sqrt{b x+c x^2}}-\frac{16 b x^{7/2}}{35 c^2 \sqrt{b x+c x^2}}+\frac{2 x^{9/2}}{7 c \sqrt{b x+c x^2}}+\frac{\left (128 b^4\right ) \int \frac{x^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx}{35 c^4}\\ &=-\frac{256 b^4 \sqrt{x}}{35 c^5 \sqrt{b x+c x^2}}-\frac{128 b^3 x^{3/2}}{35 c^4 \sqrt{b x+c x^2}}+\frac{32 b^2 x^{5/2}}{35 c^3 \sqrt{b x+c x^2}}-\frac{16 b x^{7/2}}{35 c^2 \sqrt{b x+c x^2}}+\frac{2 x^{9/2}}{7 c \sqrt{b x+c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0291625, size = 64, normalized size = 0.47 \[ \frac{2 \sqrt{x} \left (16 b^2 c^2 x^2-64 b^3 c x-128 b^4-8 b c^3 x^3+5 c^4 x^4\right )}{35 c^5 \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 66, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -5\,{x}^{4}{c}^{4}+8\,b{x}^{3}{c}^{3}-16\,{b}^{2}{x}^{2}{c}^{2}+64\,{b}^{3}xc+128\,{b}^{4} \right ) }{35\,{c}^{5}}{x}^{{\frac{3}{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] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2 \,{\left (3 \,{\left (5 \, c^{5} x^{4} - b c^{4} x^{3} + 2 \, b^{2} c^{3} x^{2} - 8 \, b^{3} c^{2} x - 16 \, b^{4} c\right )} x^{4} - 2 \,{\left (3 \, b c^{4} x^{4} - 2 \, b^{2} c^{3} x^{3} + 11 \, b^{3} c^{2} x^{2} + 40 \, b^{4} c x + 24 \, b^{5}\right )} x^{3} + 14 \,{\left (b^{2} c^{3} x^{4} - 2 \, b^{3} c^{2} x^{3} - 7 \, b^{4} c x^{2} - 4 \, b^{5} x\right )} x^{2} - 70 \,{\left (b^{3} c^{2} x^{4} + 2 \, b^{4} c x^{3} + b^{5} x^{2}\right )} x\right )}}{105 \,{\left (c^{6} x^{4} + b c^{5} x^{3}\right )} \sqrt{c x + b}} + \int \frac{2 \,{\left (b^{4} c x + b^{5}\right )} x}{{\left (c^{6} x^{3} + 2 \, b c^{5} x^{2} + b^{2} c^{4} x\right )} \sqrt{c x + b}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.98118, size = 159, normalized size = 1.17 \begin{align*} \frac{2 \,{\left (5 \, c^{4} x^{4} - 8 \, b c^{3} x^{3} + 16 \, b^{2} c^{2} x^{2} - 64 \, b^{3} c x - 128 \, b^{4}\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{35 \,{\left (c^{6} x^{2} + b c^{5} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17735, size = 95, normalized size = 0.7 \begin{align*} \frac{256 \, b^{\frac{7}{2}}}{35 \, c^{5}} + \frac{2 \,{\left (5 \,{\left (c x + b\right )}^{\frac{7}{2}} - 28 \,{\left (c x + b\right )}^{\frac{5}{2}} b + 70 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{2} - 140 \, \sqrt{c x + b} b^{3} - \frac{35 \, b^{4}}{\sqrt{c x + b}}\right )}}{35 \, c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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