Optimal. Leaf size=51 \[ -\frac{2 (a B+A b)}{\sqrt{x}}-\frac{2 a A}{3 x^{3/2}}+2 \sqrt{x} (A c+b B)+\frac{2}{3} B c x^{3/2} \]
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Rubi [A] time = 0.0229725, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {765} \[ -\frac{2 (a B+A b)}{\sqrt{x}}-\frac{2 a A}{3 x^{3/2}}+2 \sqrt{x} (A c+b B)+\frac{2}{3} B c x^{3/2} \]
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a+b x+c x^2\right )}{x^{5/2}} \, dx &=\int \left (\frac{a A}{x^{5/2}}+\frac{A b+a B}{x^{3/2}}+\frac{b B+A c}{\sqrt{x}}+B c \sqrt{x}\right ) \, dx\\ &=-\frac{2 a A}{3 x^{3/2}}-\frac{2 (A b+a B)}{\sqrt{x}}+2 (b B+A c) \sqrt{x}+\frac{2}{3} B c x^{3/2}\\ \end{align*}
Mathematica [A] time = 0.0428171, size = 42, normalized size = 0.82 \[ \frac{2 x (B x (3 b+c x)-3 A (b-c x))-2 a (A+3 B x)}{3 x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 41, normalized size = 0.8 \begin{align*} -{\frac{-2\,Bc{x}^{3}-6\,Ac{x}^{2}-6\,Bb{x}^{2}+6\,Abx+6\,aBx+2\,aA}{3}{x}^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0285, size = 53, normalized size = 1.04 \begin{align*} \frac{2}{3} \, B c x^{\frac{3}{2}} + 2 \,{\left (B b + A c\right )} \sqrt{x} - \frac{2 \,{\left (A a + 3 \,{\left (B a + A b\right )} x\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.03, size = 90, normalized size = 1.76 \begin{align*} \frac{2 \,{\left (B c x^{3} + 3 \,{\left (B b + A c\right )} x^{2} - A a - 3 \,{\left (B a + A b\right )} x\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.61947, size = 63, normalized size = 1.24 \begin{align*} - \frac{2 A a}{3 x^{\frac{3}{2}}} - \frac{2 A b}{\sqrt{x}} + 2 A c \sqrt{x} - \frac{2 B a}{\sqrt{x}} + 2 B b \sqrt{x} + \frac{2 B c x^{\frac{3}{2}}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19991, size = 55, normalized size = 1.08 \begin{align*} \frac{2}{3} \, B c x^{\frac{3}{2}} + 2 \, B b \sqrt{x} + 2 \, A c \sqrt{x} - \frac{2 \,{\left (3 \, B a x + 3 \, A b x + A a\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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