Optimal. Leaf size=164 \[ \frac{32 b^2 x^{7/2}}{63 c^3 \sqrt{b x+c x^2}}-\frac{64 b^3 x^{5/2}}{63 c^4 \sqrt{b x+c x^2}}+\frac{256 b^4 x^{3/2}}{63 c^5 \sqrt{b x+c x^2}}+\frac{512 b^5 \sqrt{x}}{63 c^6 \sqrt{b x+c x^2}}-\frac{20 b x^{9/2}}{63 c^2 \sqrt{b x+c x^2}}+\frac{2 x^{11/2}}{9 c \sqrt{b x+c x^2}} \]
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Rubi [A] time = 0.0748355, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {656, 648} \[ \frac{32 b^2 x^{7/2}}{63 c^3 \sqrt{b x+c x^2}}-\frac{64 b^3 x^{5/2}}{63 c^4 \sqrt{b x+c x^2}}+\frac{256 b^4 x^{3/2}}{63 c^5 \sqrt{b x+c x^2}}+\frac{512 b^5 \sqrt{x}}{63 c^6 \sqrt{b x+c x^2}}-\frac{20 b x^{9/2}}{63 c^2 \sqrt{b x+c x^2}}+\frac{2 x^{11/2}}{9 c \sqrt{b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 656
Rule 648
Rubi steps
\begin{align*} \int \frac{x^{13/2}}{\left (b x+c x^2\right )^{3/2}} \, dx &=\frac{2 x^{11/2}}{9 c \sqrt{b x+c x^2}}-\frac{(10 b) \int \frac{x^{11/2}}{\left (b x+c x^2\right )^{3/2}} \, dx}{9 c}\\ &=-\frac{20 b x^{9/2}}{63 c^2 \sqrt{b x+c x^2}}+\frac{2 x^{11/2}}{9 c \sqrt{b x+c x^2}}+\frac{\left (80 b^2\right ) \int \frac{x^{9/2}}{\left (b x+c x^2\right )^{3/2}} \, dx}{63 c^2}\\ &=\frac{32 b^2 x^{7/2}}{63 c^3 \sqrt{b x+c x^2}}-\frac{20 b x^{9/2}}{63 c^2 \sqrt{b x+c x^2}}+\frac{2 x^{11/2}}{9 c \sqrt{b x+c x^2}}-\frac{\left (32 b^3\right ) \int \frac{x^{7/2}}{\left (b x+c x^2\right )^{3/2}} \, dx}{21 c^3}\\ &=-\frac{64 b^3 x^{5/2}}{63 c^4 \sqrt{b x+c x^2}}+\frac{32 b^2 x^{7/2}}{63 c^3 \sqrt{b x+c x^2}}-\frac{20 b x^{9/2}}{63 c^2 \sqrt{b x+c x^2}}+\frac{2 x^{11/2}}{9 c \sqrt{b x+c x^2}}+\frac{\left (128 b^4\right ) \int \frac{x^{5/2}}{\left (b x+c x^2\right )^{3/2}} \, dx}{63 c^4}\\ &=\frac{256 b^4 x^{3/2}}{63 c^5 \sqrt{b x+c x^2}}-\frac{64 b^3 x^{5/2}}{63 c^4 \sqrt{b x+c x^2}}+\frac{32 b^2 x^{7/2}}{63 c^3 \sqrt{b x+c x^2}}-\frac{20 b x^{9/2}}{63 c^2 \sqrt{b x+c x^2}}+\frac{2 x^{11/2}}{9 c \sqrt{b x+c x^2}}-\frac{\left (256 b^5\right ) \int \frac{x^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx}{63 c^5}\\ &=\frac{512 b^5 \sqrt{x}}{63 c^6 \sqrt{b x+c x^2}}+\frac{256 b^4 x^{3/2}}{63 c^5 \sqrt{b x+c x^2}}-\frac{64 b^3 x^{5/2}}{63 c^4 \sqrt{b x+c x^2}}+\frac{32 b^2 x^{7/2}}{63 c^3 \sqrt{b x+c x^2}}-\frac{20 b x^{9/2}}{63 c^2 \sqrt{b x+c x^2}}+\frac{2 x^{11/2}}{9 c \sqrt{b x+c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0357231, size = 75, normalized size = 0.46 \[ \frac{2 \sqrt{x} \left (-32 b^3 c^2 x^2+16 b^2 c^3 x^3+128 b^4 c x+256 b^5-10 b c^4 x^4+7 c^5 x^5\right )}{63 c^6 \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 77, normalized size = 0.5 \begin{align*}{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 7\,{x}^{5}{c}^{5}-10\,b{x}^{4}{c}^{4}+16\,{b}^{2}{x}^{3}{c}^{3}-32\,{b}^{3}{x}^{2}{c}^{2}+128\,{b}^{4}xc+256\,{b}^{5} \right ) }{63\,{c}^{6}}{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 (35 \, c^{6} x^{5} - 5 \, b c^{5} x^{4} + 8 \, b^{2} c^{4} x^{3} - 16 \, b^{3} c^{3} x^{2} + 64 \, b^{4} c^{2} x + 128 \, b^{5} c\right )} x^{5} - 2 \,{\left (5 \, b c^{5} x^{5} - 2 \, b^{2} c^{4} x^{4} + 5 \, b^{3} c^{3} x^{3} - 28 \, b^{4} c^{2} x^{2} - 104 \, b^{5} c x - 64 \, b^{6}\right )} x^{4} + 6 \,{\left (3 \, b^{2} c^{4} x^{5} - 2 \, b^{3} c^{3} x^{4} + 11 \, b^{4} c^{2} x^{3} + 40 \, b^{5} c x^{2} + 24 \, b^{6} x\right )} x^{3} - 42 \,{\left (b^{3} c^{3} x^{5} - 2 \, b^{4} c^{2} x^{4} - 7 \, b^{5} c x^{3} - 4 \, b^{6} x^{2}\right )} x^{2} + 210 \,{\left (b^{4} c^{2} x^{5} + 2 \, b^{5} c x^{4} + b^{6} x^{3}\right )} x\right )}}{315 \,{\left (c^{7} x^{5} + b c^{6} x^{4}\right )} \sqrt{c x + b}} - \int \frac{2 \,{\left (b^{5} c x + b^{6}\right )} x}{{\left (c^{7} x^{3} + 2 \, b c^{6} x^{2} + b^{2} c^{5} 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 = 1.90933, size = 185, normalized size = 1.13 \begin{align*} \frac{2 \,{\left (7 \, c^{5} x^{5} - 10 \, b c^{4} x^{4} + 16 \, b^{2} c^{3} x^{3} - 32 \, b^{3} c^{2} x^{2} + 128 \, b^{4} c x + 256 \, b^{5}\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{63 \,{\left (c^{7} x^{2} + b c^{6} 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 = 1.15987, size = 111, normalized size = 0.68 \begin{align*} -\frac{512 \, b^{\frac{9}{2}}}{63 \, c^{6}} + \frac{2 \,{\left (7 \,{\left (c x + b\right )}^{\frac{9}{2}} - 45 \,{\left (c x + b\right )}^{\frac{7}{2}} b + 126 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{2} - 210 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{3} + 315 \, \sqrt{c x + b} b^{4} + \frac{63 \, b^{5}}{\sqrt{c x + b}}\right )}}{63 \, c^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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