Optimal. Leaf size=51 \[ 2 \sqrt{x} (a B+A b)-\frac{2 a A}{\sqrt{x}}+\frac{2}{3} x^{3/2} (A c+b B)+\frac{2}{5} B c x^{5/2} \]
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Rubi [A] time = 0.024403, 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} \[ 2 \sqrt{x} (a B+A b)-\frac{2 a A}{\sqrt{x}}+\frac{2}{3} x^{3/2} (A c+b B)+\frac{2}{5} B c x^{5/2} \]
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^{3/2}} \, dx &=\int \left (\frac{a A}{x^{3/2}}+\frac{A b+a B}{\sqrt{x}}+(b B+A c) \sqrt{x}+B c x^{3/2}\right ) \, dx\\ &=-\frac{2 a A}{\sqrt{x}}+2 (A b+a B) \sqrt{x}+\frac{2}{3} (b B+A c) x^{3/2}+\frac{2}{5} B c x^{5/2}\\ \end{align*}
Mathematica [A] time = 0.0422304, size = 44, normalized size = 0.86 \[ \frac{2 x (5 A (3 b+c x)+B x (5 b+3 c x))-30 a (A-B x)}{15 \sqrt{x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 42, normalized size = 0.8 \begin{align*} -{\frac{-6\,Bc{x}^{3}-10\,Ac{x}^{2}-10\,Bb{x}^{2}-30\,Abx-30\,aBx+30\,aA}{15}{\frac{1}{\sqrt{x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.18409, size = 53, normalized size = 1.04 \begin{align*} \frac{2}{5} \, B c x^{\frac{5}{2}} + \frac{2}{3} \,{\left (B b + A c\right )} x^{\frac{3}{2}} - \frac{2 \, A a}{\sqrt{x}} + 2 \,{\left (B a + A b\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.19802, size = 100, normalized size = 1.96 \begin{align*} \frac{2 \,{\left (3 \, B c x^{3} + 5 \,{\left (B b + A c\right )} x^{2} - 15 \, A a + 15 \,{\left (B a + A b\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 = 1.3948, size = 65, normalized size = 1.27 \begin{align*} - \frac{2 A a}{\sqrt{x}} + 2 A b \sqrt{x} + \frac{2 A c x^{\frac{3}{2}}}{3} + 2 B a \sqrt{x} + \frac{2 B b x^{\frac{3}{2}}}{3} + \frac{2 B c x^{\frac{5}{2}}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19673, size = 58, normalized size = 1.14 \begin{align*} \frac{2}{5} \, B c x^{\frac{5}{2}} + \frac{2}{3} \, B b x^{\frac{3}{2}} + \frac{2}{3} \, A c x^{\frac{3}{2}} + 2 \, B a \sqrt{x} + 2 \, A b \sqrt{x} - \frac{2 \, A a}{\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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