Optimal. Leaf size=61 \[ 2 A b^2 \sqrt{x}+\frac{2}{5} c x^{5/2} (A c+2 b B)+\frac{2}{3} b x^{3/2} (2 A c+b B)+\frac{2}{7} B c^2 x^{7/2} \]
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Rubi [A] time = 0.030016, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {765} \[ 2 A b^2 \sqrt{x}+\frac{2}{5} c x^{5/2} (A c+2 b B)+\frac{2}{3} b x^{3/2} (2 A c+b B)+\frac{2}{7} B c^2 x^{7/2} \]
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (b x+c x^2\right )^2}{x^{5/2}} \, dx &=\int \left (\frac{A b^2}{\sqrt{x}}+b (b B+2 A c) \sqrt{x}+c (2 b B+A c) x^{3/2}+B c^2 x^{5/2}\right ) \, dx\\ &=2 A b^2 \sqrt{x}+\frac{2}{3} b (b B+2 A c) x^{3/2}+\frac{2}{5} c (2 b B+A c) x^{5/2}+\frac{2}{7} B c^2 x^{7/2}\\ \end{align*}
Mathematica [A] time = 0.0166796, size = 54, normalized size = 0.89 \[ \frac{2}{105} \sqrt{x} \left (7 A \left (15 b^2+10 b c x+3 c^2 x^2\right )+B x \left (35 b^2+42 b c x+15 c^2 x^2\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 52, normalized size = 0.9 \begin{align*}{\frac{30\,B{c}^{2}{x}^{3}+42\,A{c}^{2}{x}^{2}+84\,B{x}^{2}bc+140\,Abcx+70\,{b}^{2}Bx+210\,A{b}^{2}}{105}\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00272, size = 69, normalized size = 1.13 \begin{align*} \frac{2}{7} \, B c^{2} x^{\frac{7}{2}} + 2 \, A b^{2} \sqrt{x} + \frac{2}{5} \,{\left (2 \, B b c + A c^{2}\right )} x^{\frac{5}{2}} + \frac{2}{3} \,{\left (B b^{2} + 2 \, A b c\right )} x^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.78917, size = 127, normalized size = 2.08 \begin{align*} \frac{2}{105} \,{\left (15 \, B c^{2} x^{3} + 105 \, A b^{2} + 21 \,{\left (2 \, B b c + A c^{2}\right )} x^{2} + 35 \,{\left (B b^{2} + 2 \, A b c\right )} x\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.34169, size = 78, normalized size = 1.28 \begin{align*} 2 A b^{2} \sqrt{x} + \frac{4 A b c x^{\frac{3}{2}}}{3} + \frac{2 A c^{2} x^{\frac{5}{2}}}{5} + \frac{2 B b^{2} x^{\frac{3}{2}}}{3} + \frac{4 B b c x^{\frac{5}{2}}}{5} + \frac{2 B c^{2} x^{\frac{7}{2}}}{7} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12841, size = 72, normalized size = 1.18 \begin{align*} \frac{2}{7} \, B c^{2} x^{\frac{7}{2}} + \frac{4}{5} \, B b c x^{\frac{5}{2}} + \frac{2}{5} \, A c^{2} x^{\frac{5}{2}} + \frac{2}{3} \, B b^{2} x^{\frac{3}{2}} + \frac{4}{3} \, A b c x^{\frac{3}{2}} + 2 \, A b^{2} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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