Optimal. Leaf size=80 \[ -\frac{16 b^2 \sqrt{x}}{3 c^3 \sqrt{b x+c x^2}}-\frac{8 b x^{3/2}}{3 c^2 \sqrt{b x+c x^2}}+\frac{2 x^{5/2}}{3 c \sqrt{b x+c x^2}} \]
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Rubi [A] time = 0.0275102, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {656, 648} \[ -\frac{16 b^2 \sqrt{x}}{3 c^3 \sqrt{b x+c x^2}}-\frac{8 b x^{3/2}}{3 c^2 \sqrt{b x+c x^2}}+\frac{2 x^{5/2}}{3 c \sqrt{b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 656
Rule 648
Rubi steps
\begin{align*} \int \frac{x^{7/2}}{\left (b x+c x^2\right )^{3/2}} \, dx &=\frac{2 x^{5/2}}{3 c \sqrt{b x+c x^2}}-\frac{(4 b) \int \frac{x^{5/2}}{\left (b x+c x^2\right )^{3/2}} \, dx}{3 c}\\ &=-\frac{8 b x^{3/2}}{3 c^2 \sqrt{b x+c x^2}}+\frac{2 x^{5/2}}{3 c \sqrt{b x+c x^2}}+\frac{\left (8 b^2\right ) \int \frac{x^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx}{3 c^2}\\ &=-\frac{16 b^2 \sqrt{x}}{3 c^3 \sqrt{b x+c x^2}}-\frac{8 b x^{3/2}}{3 c^2 \sqrt{b x+c x^2}}+\frac{2 x^{5/2}}{3 c \sqrt{b x+c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0184767, size = 41, normalized size = 0.51 \[ \frac{2 \sqrt{x} \left (-8 b^2-4 b c x+c^2 x^2\right )}{3 c^3 \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 44, normalized size = 0.6 \begin{align*} -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -{c}^{2}{x}^{2}+4\,bcx+8\,{b}^{2} \right ) }{3\,{c}^{3}}{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 ({\left (c^{3} x^{2} - b c^{2} x - 2 \, b^{2} c\right )} x^{2} - 2 \,{\left (b c^{2} x^{2} + 2 \, b^{2} c x + b^{3}\right )} x\right )}}{3 \,{\left (c^{4} x^{2} + b c^{3} x\right )} \sqrt{c x + b}} + \int \frac{2 \,{\left (b^{2} c x + b^{3}\right )} x}{{\left (c^{4} x^{3} + 2 \, b c^{3} x^{2} + b^{2} c^{2} 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 = 1.99862, size = 107, normalized size = 1.34 \begin{align*} \frac{2 \,{\left (c^{2} x^{2} - 4 \, b c x - 8 \, b^{2}\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{3 \,{\left (c^{4} x^{2} + b c^{3} 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 = 1.28931, size = 59, normalized size = 0.74 \begin{align*} \frac{16 \, b^{\frac{3}{2}}}{3 \, c^{3}} + \frac{2 \,{\left ({\left (c x + b\right )}^{\frac{3}{2}} - 6 \, \sqrt{c x + b} b - \frac{3 \, b^{2}}{\sqrt{c x + b}}\right )}}{3 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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