Optimal. Leaf size=59 \[ -\frac{2 a^2 A}{\sqrt{x}}+\frac{2}{3} b x^{3/2} (2 a B+A b)+2 a \sqrt{x} (a B+2 A b)+\frac{2}{5} b^2 B x^{5/2} \]
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Rubi [A] time = 0.027135, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {27, 76} \[ -\frac{2 a^2 A}{\sqrt{x}}+\frac{2}{3} b x^{3/2} (2 a B+A b)+2 a \sqrt{x} (a B+2 A b)+\frac{2}{5} b^2 B x^{5/2} \]
Antiderivative was successfully verified.
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Rule 27
Rule 76
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{x^{3/2}} \, dx &=\int \frac{(a+b x)^2 (A+B x)}{x^{3/2}} \, dx\\ &=\int \left (\frac{a^2 A}{x^{3/2}}+\frac{a (2 A b+a B)}{\sqrt{x}}+b (A b+2 a B) \sqrt{x}+b^2 B x^{3/2}\right ) \, dx\\ &=-\frac{2 a^2 A}{\sqrt{x}}+2 a (2 A b+a B) \sqrt{x}+\frac{2}{3} b (A b+2 a B) x^{3/2}+\frac{2}{5} b^2 B x^{5/2}\\ \end{align*}
Mathematica [A] time = 0.015347, size = 49, normalized size = 0.83 \[ \frac{-30 a^2 (A-B x)+20 a b x (3 A+B x)+2 b^2 x^2 (5 A+3 B x)}{15 \sqrt{x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 52, normalized size = 0.9 \begin{align*} -{\frac{-6\,{b}^{2}B{x}^{3}-10\,A{b}^{2}{x}^{2}-20\,B{x}^{2}ab-60\,aAbx-30\,{a}^{2}Bx+30\,A{a}^{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 = 1.02174, size = 69, normalized size = 1.17 \begin{align*} \frac{2}{5} \, B b^{2} x^{\frac{5}{2}} - \frac{2 \, A a^{2}}{\sqrt{x}} + \frac{2}{3} \,{\left (2 \, B a b + A b^{2}\right )} x^{\frac{3}{2}} + 2 \,{\left (B a^{2} + 2 \, 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 = 1.53494, size = 122, normalized size = 2.07 \begin{align*} \frac{2 \,{\left (3 \, B b^{2} x^{3} - 15 \, A a^{2} + 5 \,{\left (2 \, B a b + A b^{2}\right )} x^{2} + 15 \,{\left (B a^{2} + 2 \, 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 = 0.774173, size = 75, normalized size = 1.27 \begin{align*} - \frac{2 A a^{2}}{\sqrt{x}} + 4 A a b \sqrt{x} + \frac{2 A b^{2} x^{\frac{3}{2}}}{3} + 2 B a^{2} \sqrt{x} + \frac{4 B a b x^{\frac{3}{2}}}{3} + \frac{2 B b^{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 = 1.27874, size = 72, normalized size = 1.22 \begin{align*} \frac{2}{5} \, B b^{2} x^{\frac{5}{2}} + \frac{4}{3} \, B a b x^{\frac{3}{2}} + \frac{2}{3} \, A b^{2} x^{\frac{3}{2}} + 2 \, B a^{2} \sqrt{x} + 4 \, A a b \sqrt{x} - \frac{2 \, A a^{2}}{\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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