Optimal. Leaf size=136 \[ \frac{256 b^4 \left (b x+c x^2\right )^{5/2}}{15015 c^5 x^{5/2}}-\frac{128 b^3 \left (b x+c x^2\right )^{5/2}}{3003 c^4 x^{3/2}}+\frac{32 b^2 \left (b x+c x^2\right )^{5/2}}{429 c^3 \sqrt{x}}-\frac{16 b \sqrt{x} \left (b x+c x^2\right )^{5/2}}{143 c^2}+\frac{2 x^{3/2} \left (b x+c x^2\right )^{5/2}}{13 c} \]
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Rubi [A] time = 0.0560172, 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} \[ \frac{256 b^4 \left (b x+c x^2\right )^{5/2}}{15015 c^5 x^{5/2}}-\frac{128 b^3 \left (b x+c x^2\right )^{5/2}}{3003 c^4 x^{3/2}}+\frac{32 b^2 \left (b x+c x^2\right )^{5/2}}{429 c^3 \sqrt{x}}-\frac{16 b \sqrt{x} \left (b x+c x^2\right )^{5/2}}{143 c^2}+\frac{2 x^{3/2} \left (b x+c x^2\right )^{5/2}}{13 c} \]
Antiderivative was successfully verified.
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Rule 656
Rule 648
Rubi steps
\begin{align*} \int x^{5/2} \left (b x+c x^2\right )^{3/2} \, dx &=\frac{2 x^{3/2} \left (b x+c x^2\right )^{5/2}}{13 c}-\frac{(8 b) \int x^{3/2} \left (b x+c x^2\right )^{3/2} \, dx}{13 c}\\ &=-\frac{16 b \sqrt{x} \left (b x+c x^2\right )^{5/2}}{143 c^2}+\frac{2 x^{3/2} \left (b x+c x^2\right )^{5/2}}{13 c}+\frac{\left (48 b^2\right ) \int \sqrt{x} \left (b x+c x^2\right )^{3/2} \, dx}{143 c^2}\\ &=\frac{32 b^2 \left (b x+c x^2\right )^{5/2}}{429 c^3 \sqrt{x}}-\frac{16 b \sqrt{x} \left (b x+c x^2\right )^{5/2}}{143 c^2}+\frac{2 x^{3/2} \left (b x+c x^2\right )^{5/2}}{13 c}-\frac{\left (64 b^3\right ) \int \frac{\left (b x+c x^2\right )^{3/2}}{\sqrt{x}} \, dx}{429 c^3}\\ &=-\frac{128 b^3 \left (b x+c x^2\right )^{5/2}}{3003 c^4 x^{3/2}}+\frac{32 b^2 \left (b x+c x^2\right )^{5/2}}{429 c^3 \sqrt{x}}-\frac{16 b \sqrt{x} \left (b x+c x^2\right )^{5/2}}{143 c^2}+\frac{2 x^{3/2} \left (b x+c x^2\right )^{5/2}}{13 c}+\frac{\left (128 b^4\right ) \int \frac{\left (b x+c x^2\right )^{3/2}}{x^{3/2}} \, dx}{3003 c^4}\\ &=\frac{256 b^4 \left (b x+c x^2\right )^{5/2}}{15015 c^5 x^{5/2}}-\frac{128 b^3 \left (b x+c x^2\right )^{5/2}}{3003 c^4 x^{3/2}}+\frac{32 b^2 \left (b x+c x^2\right )^{5/2}}{429 c^3 \sqrt{x}}-\frac{16 b \sqrt{x} \left (b x+c x^2\right )^{5/2}}{143 c^2}+\frac{2 x^{3/2} \left (b x+c x^2\right )^{5/2}}{13 c}\\ \end{align*}
Mathematica [A] time = 0.0381884, size = 64, normalized size = 0.47 \[ \frac{2 (x (b+c x))^{5/2} \left (560 b^2 c^2 x^2-320 b^3 c x+128 b^4-840 b c^3 x^3+1155 c^4 x^4\right )}{15015 c^5 x^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 66, normalized size = 0.5 \begin{align*}{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 1155\,{x}^{4}{c}^{4}-840\,b{x}^{3}{c}^{3}+560\,{b}^{2}{x}^{2}{c}^{2}-320\,{b}^{3}xc+128\,{b}^{4} \right ) }{15015\,{c}^{5}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06191, size = 198, normalized size = 1.46 \begin{align*} \frac{2 \,{\left (5 \,{\left (693 \, c^{6} x^{6} + 63 \, b c^{5} x^{5} - 70 \, b^{2} c^{4} x^{4} + 80 \, b^{3} c^{3} x^{3} - 96 \, b^{4} c^{2} x^{2} + 128 \, b^{5} c x - 256 \, b^{6}\right )} x^{5} + 13 \,{\left (315 \, b c^{5} x^{6} + 35 \, b^{2} c^{4} x^{5} - 40 \, b^{3} c^{3} x^{4} + 48 \, b^{4} c^{2} x^{3} - 64 \, b^{5} c x^{2} + 128 \, b^{6} x\right )} x^{4}\right )} \sqrt{c x + b}}{45045 \, c^{5} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01196, size = 198, normalized size = 1.46 \begin{align*} \frac{2 \,{\left (1155 \, c^{6} x^{6} + 1470 \, b c^{5} x^{5} + 35 \, b^{2} c^{4} x^{4} - 40 \, b^{3} c^{3} x^{3} + 48 \, b^{4} c^{2} x^{2} - 64 \, b^{5} c x + 128 \, b^{6}\right )} \sqrt{c x^{2} + b x}}{15015 \, c^{5} \sqrt{x}} \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 = 1.22036, size = 213, normalized size = 1.57 \begin{align*} \frac{2}{9009} \, c{\left (\frac{256 \, b^{\frac{13}{2}}}{c^{6}} + \frac{693 \,{\left (c x + b\right )}^{\frac{13}{2}} - 4095 \,{\left (c x + b\right )}^{\frac{11}{2}} b + 10010 \,{\left (c x + b\right )}^{\frac{9}{2}} b^{2} - 12870 \,{\left (c x + b\right )}^{\frac{7}{2}} b^{3} + 9009 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{4} - 3003 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{5}}{c^{6}}\right )} - \frac{2}{3465} \, b{\left (\frac{128 \, b^{\frac{11}{2}}}{c^{5}} - \frac{315 \,{\left (c x + b\right )}^{\frac{11}{2}} - 1540 \,{\left (c x + b\right )}^{\frac{9}{2}} b + 2970 \,{\left (c x + b\right )}^{\frac{7}{2}} b^{2} - 2772 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{3} + 1155 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{4}}{c^{5}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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