Optimal. Leaf size=59 \[ -\frac{2 A b^2}{\sqrt{x}}+\frac{2}{3} c x^{3/2} (A c+2 b B)+2 b \sqrt{x} (2 A c+b B)+\frac{2}{5} B c^2 x^{5/2} \]
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Rubi [A] time = 0.029797, antiderivative size = 59, 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} \[ -\frac{2 A b^2}{\sqrt{x}}+\frac{2}{3} c x^{3/2} (A c+2 b B)+2 b \sqrt{x} (2 A c+b B)+\frac{2}{5} B c^2 x^{5/2} \]
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^{7/2}} \, dx &=\int \left (\frac{A b^2}{x^{3/2}}+\frac{b (b B+2 A c)}{\sqrt{x}}+c (2 b B+A c) \sqrt{x}+B c^2 x^{3/2}\right ) \, dx\\ &=-\frac{2 A b^2}{\sqrt{x}}+2 b (b B+2 A c) \sqrt{x}+\frac{2}{3} c (2 b B+A c) x^{3/2}+\frac{2}{5} B c^2 x^{5/2}\\ \end{align*}
Mathematica [A] time = 0.0151144, size = 54, normalized size = 0.92 \[ \frac{10 A \left (-3 b^2+6 b c x+c^2 x^2\right )+2 B x \left (15 b^2+10 b c x+3 c^2 x^2\right )}{15 \sqrt{x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 52, normalized size = 0.9 \begin{align*} -{\frac{-6\,B{c}^{2}{x}^{3}-10\,A{c}^{2}{x}^{2}-20\,B{x}^{2}bc-60\,Abcx-30\,{b}^{2}Bx+30\,A{b}^{2}}{15}{\frac{1}{\sqrt{x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.998861, size = 69, normalized size = 1.17 \begin{align*} \frac{2}{5} \, B c^{2} x^{\frac{5}{2}} - \frac{2 \, A b^{2}}{\sqrt{x}} + \frac{2}{3} \,{\left (2 \, B b c + A c^{2}\right )} x^{\frac{3}{2}} + 2 \,{\left (B b^{2} + 2 \, A b c\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.76893, size = 122, normalized size = 2.07 \begin{align*} \frac{2 \,{\left (3 \, B c^{2} x^{3} - 15 \, A b^{2} + 5 \,{\left (2 \, B b c + A c^{2}\right )} x^{2} + 15 \,{\left (B b^{2} + 2 \, A b c\right )} x\right )}}{15 \, \sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.61825, size = 75, normalized size = 1.27 \begin{align*} - \frac{2 A b^{2}}{\sqrt{x}} + 4 A b c \sqrt{x} + \frac{2 A c^{2} x^{\frac{3}{2}}}{3} + 2 B b^{2} \sqrt{x} + \frac{4 B b c x^{\frac{3}{2}}}{3} + \frac{2 B c^{2} x^{\frac{5}{2}}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10995, size = 72, normalized size = 1.22 \begin{align*} \frac{2}{5} \, B c^{2} x^{\frac{5}{2}} + \frac{4}{3} \, B b c x^{\frac{3}{2}} + \frac{2}{3} \, A c^{2} x^{\frac{3}{2}} + 2 \, B b^{2} \sqrt{x} + 4 \, A b c \sqrt{x} - \frac{2 \, A b^{2}}{\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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