Optimal. Leaf size=96 \[ -\frac{2 \left (b x+c x^2\right )^{5/2} (4 b B-9 A c)}{63 c^2 x^{3/2}}+\frac{4 b \left (b x+c x^2\right )^{5/2} (4 b B-9 A c)}{315 c^3 x^{5/2}}+\frac{2 B \left (b x+c x^2\right )^{5/2}}{9 c \sqrt{x}} \]
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Rubi [A] time = 0.0791305, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {794, 656, 648} \[ -\frac{2 \left (b x+c x^2\right )^{5/2} (4 b B-9 A c)}{63 c^2 x^{3/2}}+\frac{4 b \left (b x+c x^2\right )^{5/2} (4 b B-9 A c)}{315 c^3 x^{5/2}}+\frac{2 B \left (b x+c x^2\right )^{5/2}}{9 c \sqrt{x}} \]
Antiderivative was successfully verified.
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Rule 794
Rule 656
Rule 648
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (b x+c x^2\right )^{3/2}}{\sqrt{x}} \, dx &=\frac{2 B \left (b x+c x^2\right )^{5/2}}{9 c \sqrt{x}}+\frac{\left (2 \left (\frac{1}{2} (b B-A c)+\frac{5}{2} (-b B+2 A c)\right )\right ) \int \frac{\left (b x+c x^2\right )^{3/2}}{\sqrt{x}} \, dx}{9 c}\\ &=-\frac{2 (4 b B-9 A c) \left (b x+c x^2\right )^{5/2}}{63 c^2 x^{3/2}}+\frac{2 B \left (b x+c x^2\right )^{5/2}}{9 c \sqrt{x}}+\frac{(2 b (4 b B-9 A c)) \int \frac{\left (b x+c x^2\right )^{3/2}}{x^{3/2}} \, dx}{63 c^2}\\ &=\frac{4 b (4 b B-9 A c) \left (b x+c x^2\right )^{5/2}}{315 c^3 x^{5/2}}-\frac{2 (4 b B-9 A c) \left (b x+c x^2\right )^{5/2}}{63 c^2 x^{3/2}}+\frac{2 B \left (b x+c x^2\right )^{5/2}}{9 c \sqrt{x}}\\ \end{align*}
Mathematica [A] time = 0.044869, size = 56, normalized size = 0.58 \[ \frac{2 (x (b+c x))^{5/2} \left (-2 b c (9 A+10 B x)+5 c^2 x (9 A+7 B x)+8 b^2 B\right )}{315 c^3 x^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 59, normalized size = 0.6 \begin{align*} -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -35\,B{c}^{2}{x}^{2}-45\,A{c}^{2}x+20\,Bbcx+18\,Abc-8\,{b}^{2}B \right ) }{315\,{c}^{3}} \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 [B] time = 1.24386, size = 246, normalized size = 2.56 \begin{align*} \frac{2 \,{\left ({\left (15 \, c^{3} x^{3} + 3 \, b c^{2} x^{2} - 4 \, b^{2} c x + 8 \, b^{3}\right )} x^{2} + 7 \,{\left (3 \, b c^{2} x^{3} + b^{2} c x^{2} - 2 \, b^{3} x\right )} x\right )} \sqrt{c x + b} A}{105 \, c^{2} x^{2}} + \frac{2 \,{\left ({\left (35 \, c^{4} x^{4} + 5 \, b c^{3} x^{3} - 6 \, b^{2} c^{2} x^{2} + 8 \, b^{3} c x - 16 \, b^{4}\right )} x^{3} + 3 \,{\left (15 \, b c^{3} x^{4} + 3 \, b^{2} c^{2} x^{3} - 4 \, b^{3} c x^{2} + 8 \, b^{4} x\right )} x^{2}\right )} \sqrt{c x + b} B}{315 \, c^{3} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48641, size = 231, normalized size = 2.41 \begin{align*} \frac{2 \,{\left (35 \, B c^{4} x^{4} + 8 \, B b^{4} - 18 \, A b^{3} c + 5 \,{\left (10 \, B b c^{3} + 9 \, A c^{4}\right )} x^{3} + 3 \,{\left (B b^{2} c^{2} + 24 \, A b c^{3}\right )} x^{2} -{\left (4 \, B b^{3} c - 9 \, A b^{2} c^{2}\right )} x\right )} \sqrt{c x^{2} + b x}}{315 \, c^{3} \sqrt{x}} \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}} \left (A + B x\right )}{\sqrt{x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17294, size = 269, normalized size = 2.8 \begin{align*} \frac{2}{315} \, B c{\left (\frac{16 \, b^{\frac{9}{2}}}{c^{4}} + \frac{35 \,{\left (c x + b\right )}^{\frac{9}{2}} - 135 \,{\left (c x + b\right )}^{\frac{7}{2}} b + 189 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{2} - 105 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{3}}{c^{4}}\right )} - \frac{2}{105} \, B b{\left (\frac{8 \, b^{\frac{7}{2}}}{c^{3}} - \frac{15 \,{\left (c x + b\right )}^{\frac{7}{2}} - 42 \,{\left (c x + b\right )}^{\frac{5}{2}} b + 35 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{2}}{c^{3}}\right )} - \frac{2}{105} \, A c{\left (\frac{8 \, b^{\frac{7}{2}}}{c^{3}} - \frac{15 \,{\left (c x + b\right )}^{\frac{7}{2}} - 42 \,{\left (c x + b\right )}^{\frac{5}{2}} b + 35 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{2}}{c^{3}}\right )} + \frac{2}{15} \, A b{\left (\frac{2 \, b^{\frac{5}{2}}}{c^{2}} + \frac{3 \,{\left (c x + b\right )}^{\frac{5}{2}} - 5 \,{\left (c x + b\right )}^{\frac{3}{2}} b}{c^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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