Optimal. Leaf size=91 \[ -\frac{2 a^2 (A b-a B)}{b^4 \sqrt{a+b x}}+\frac{2 (a+b x)^{3/2} (A b-3 a B)}{3 b^4}-\frac{2 a \sqrt{a+b x} (2 A b-3 a B)}{b^4}+\frac{2 B (a+b x)^{5/2}}{5 b^4} \]
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Rubi [A] time = 0.0342253, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {77} \[ -\frac{2 a^2 (A b-a B)}{b^4 \sqrt{a+b x}}+\frac{2 (a+b x)^{3/2} (A b-3 a B)}{3 b^4}-\frac{2 a \sqrt{a+b x} (2 A b-3 a B)}{b^4}+\frac{2 B (a+b x)^{5/2}}{5 b^4} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin{align*} \int \frac{x^2 (A+B x)}{(a+b x)^{3/2}} \, dx &=\int \left (-\frac{a^2 (-A b+a B)}{b^3 (a+b x)^{3/2}}+\frac{a (-2 A b+3 a B)}{b^3 \sqrt{a+b x}}+\frac{(A b-3 a B) \sqrt{a+b x}}{b^3}+\frac{B (a+b x)^{3/2}}{b^3}\right ) \, dx\\ &=-\frac{2 a^2 (A b-a B)}{b^4 \sqrt{a+b x}}-\frac{2 a (2 A b-3 a B) \sqrt{a+b x}}{b^4}+\frac{2 (A b-3 a B) (a+b x)^{3/2}}{3 b^4}+\frac{2 B (a+b x)^{5/2}}{5 b^4}\\ \end{align*}
Mathematica [A] time = 0.0421447, size = 67, normalized size = 0.74 \[ \frac{2 \left (-8 a^2 b (5 A-3 B x)+48 a^3 B-2 a b^2 x (10 A+3 B x)+b^3 x^2 (5 A+3 B x)\right )}{15 b^4 \sqrt{a+b x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 71, normalized size = 0.8 \begin{align*} -{\frac{-6\,{b}^{3}B{x}^{3}-10\,A{x}^{2}{b}^{3}+12\,B{x}^{2}a{b}^{2}+40\,a{b}^{2}Ax-48\,{a}^{2}bBx+80\,Ab{a}^{2}-96\,B{a}^{3}}{15\,{b}^{4}}{\frac{1}{\sqrt{bx+a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17494, size = 115, normalized size = 1.26 \begin{align*} \frac{2 \,{\left (\frac{3 \,{\left (b x + a\right )}^{\frac{5}{2}} B - 5 \,{\left (3 \, B a - A b\right )}{\left (b x + a\right )}^{\frac{3}{2}} + 15 \,{\left (3 \, B a^{2} - 2 \, A a b\right )} \sqrt{b x + a}}{b} + \frac{15 \,{\left (B a^{3} - A a^{2} b\right )}}{\sqrt{b x + a} b}\right )}}{15 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.21701, size = 178, normalized size = 1.96 \begin{align*} \frac{2 \,{\left (3 \, B b^{3} x^{3} + 48 \, B a^{3} - 40 \, A a^{2} b -{\left (6 \, B a b^{2} - 5 \, A b^{3}\right )} x^{2} + 4 \,{\left (6 \, B a^{2} b - 5 \, A a b^{2}\right )} x\right )} \sqrt{b x + a}}{15 \,{\left (b^{5} x + a b^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 11.1527, size = 88, normalized size = 0.97 \begin{align*} \frac{2 B \left (a + b x\right )^{\frac{5}{2}}}{5 b^{4}} + \frac{2 a^{2} \left (- A b + B a\right )}{b^{4} \sqrt{a + b x}} + \frac{\left (a + b x\right )^{\frac{3}{2}} \left (2 A b - 6 B a\right )}{3 b^{4}} + \frac{\sqrt{a + b x} \left (- 4 A a b + 6 B a^{2}\right )}{b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.205, size = 138, normalized size = 1.52 \begin{align*} \frac{2 \,{\left (B a^{3} - A a^{2} b\right )}}{\sqrt{b x + a} b^{4}} + \frac{2 \,{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} B b^{16} - 15 \,{\left (b x + a\right )}^{\frac{3}{2}} B a b^{16} + 45 \, \sqrt{b x + a} B a^{2} b^{16} + 5 \,{\left (b x + a\right )}^{\frac{3}{2}} A b^{17} - 30 \, \sqrt{b x + a} A a b^{17}\right )}}{15 \, b^{20}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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