Optimal. Leaf size=73 \[ -\frac{2 a^2 A}{5 x^{5/2}}-\frac{2 a^2 B}{3 x^{3/2}}-\frac{4 a A c}{\sqrt{x}}+4 a B c \sqrt{x}+\frac{2}{3} A c^2 x^{3/2}+\frac{2}{5} B c^2 x^{5/2} \]
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Rubi [A] time = 0.0255857, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {766} \[ -\frac{2 a^2 A}{5 x^{5/2}}-\frac{2 a^2 B}{3 x^{3/2}}-\frac{4 a A c}{\sqrt{x}}+4 a B c \sqrt{x}+\frac{2}{3} A c^2 x^{3/2}+\frac{2}{5} B c^2 x^{5/2} \]
Antiderivative was successfully verified.
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Rule 766
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a+c x^2\right )^2}{x^{7/2}} \, dx &=\int \left (\frac{a^2 A}{x^{7/2}}+\frac{a^2 B}{x^{5/2}}+\frac{2 a A c}{x^{3/2}}+\frac{2 a B c}{\sqrt{x}}+A c^2 \sqrt{x}+B c^2 x^{3/2}\right ) \, dx\\ &=-\frac{2 a^2 A}{5 x^{5/2}}-\frac{2 a^2 B}{3 x^{3/2}}-\frac{4 a A c}{\sqrt{x}}+4 a B c \sqrt{x}+\frac{2}{3} A c^2 x^{3/2}+\frac{2}{5} B c^2 x^{5/2}\\ \end{align*}
Mathematica [A] time = 0.0217361, size = 53, normalized size = 0.73 \[ \frac{-2 a^2 (3 A+5 B x)+60 a c x^2 (B x-A)+2 c^2 x^4 (5 A+3 B x)}{15 x^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 54, normalized size = 0.7 \begin{align*} -{\frac{-6\,B{c}^{2}{x}^{5}-10\,A{c}^{2}{x}^{4}-60\,aBc{x}^{3}+60\,aAc{x}^{2}+10\,{a}^{2}Bx+6\,A{a}^{2}}{15}{x}^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02888, size = 73, normalized size = 1. \begin{align*} \frac{2}{5} \, B c^{2} x^{\frac{5}{2}} + \frac{2}{3} \, A c^{2} x^{\frac{3}{2}} + 4 \, B a c \sqrt{x} - \frac{2 \,{\left (30 \, A a c x^{2} + 5 \, B a^{2} x + 3 \, A a^{2}\right )}}{15 \, x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.28849, size = 127, normalized size = 1.74 \begin{align*} \frac{2 \,{\left (3 \, B c^{2} x^{5} + 5 \, A c^{2} x^{4} + 30 \, B a c x^{3} - 30 \, A a c x^{2} - 5 \, B a^{2} x - 3 \, A a^{2}\right )}}{15 \, x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.97412, size = 76, normalized size = 1.04 \begin{align*} - \frac{2 A a^{2}}{5 x^{\frac{5}{2}}} - \frac{4 A a c}{\sqrt{x}} + \frac{2 A c^{2} x^{\frac{3}{2}}}{3} - \frac{2 B a^{2}}{3 x^{\frac{3}{2}}} + 4 B a c \sqrt{x} + \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 = 1.1547, size = 73, normalized size = 1. \begin{align*} \frac{2}{5} \, B c^{2} x^{\frac{5}{2}} + \frac{2}{3} \, A c^{2} x^{\frac{3}{2}} + 4 \, B a c \sqrt{x} - \frac{2 \,{\left (30 \, A a c x^{2} + 5 \, B a^{2} x + 3 \, A a^{2}\right )}}{15 \, x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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