Optimal. Leaf size=91 \[ -\frac{2 a^2 (A b-a B)}{3 b^4 (a+b x)^{3/2}}+\frac{2 a (2 A b-3 a B)}{b^4 \sqrt{a+b x}}+\frac{2 \sqrt{a+b x} (A b-3 a B)}{b^4}+\frac{2 B (a+b x)^{3/2}}{3 b^4} \]
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Rubi [A] time = 0.0384888, 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} \[ -\frac{2 a^2 (A b-a B)}{3 b^4 (a+b x)^{3/2}}+\frac{2 a (2 A b-3 a B)}{b^4 \sqrt{a+b x}}+\frac{2 \sqrt{a+b x} (A b-3 a B)}{b^4}+\frac{2 B (a+b x)^{3/2}}{3 b^4} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin{align*} \int \frac{x^2 (A+B x)}{(a+b x)^{5/2}} \, dx &=\int \left (-\frac{a^2 (-A b+a B)}{b^3 (a+b x)^{5/2}}+\frac{a (-2 A b+3 a B)}{b^3 (a+b x)^{3/2}}+\frac{A b-3 a B}{b^3 \sqrt{a+b x}}+\frac{B \sqrt{a+b x}}{b^3}\right ) \, dx\\ &=-\frac{2 a^2 (A b-a B)}{3 b^4 (a+b x)^{3/2}}+\frac{2 a (2 A b-3 a B)}{b^4 \sqrt{a+b x}}+\frac{2 (A b-3 a B) \sqrt{a+b x}}{b^4}+\frac{2 B (a+b x)^{3/2}}{3 b^4}\\ \end{align*}
Mathematica [A] time = 0.0459144, size = 63, normalized size = 0.69 \[ \frac{2 \left (8 a^2 b (A-3 B x)-16 a^3 B-6 a b^2 x (B x-2 A)+b^3 x^2 (3 A+B x)\right )}{3 b^4 (a+b x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 70, normalized size = 0.8 \begin{align*}{\frac{2\,{b}^{3}B{x}^{3}+6\,A{x}^{2}{b}^{3}-12\,B{x}^{2}a{b}^{2}+24\,a{b}^{2}Ax-48\,{a}^{2}bBx+16\,Ab{a}^{2}-32\,B{a}^{3}}{3\,{b}^{4}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07362, size = 109, normalized size = 1.2 \begin{align*} \frac{2 \,{\left (\frac{{\left (b x + a\right )}^{\frac{3}{2}} B - 3 \,{\left (3 \, B a - A b\right )} \sqrt{b x + a}}{b} + \frac{B a^{3} - A a^{2} b - 3 \,{\left (3 \, B a^{2} - 2 \, A a b\right )}{\left (b x + a\right )}}{{\left (b x + a\right )}^{\frac{3}{2}} b}\right )}}{3 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.30696, size = 193, normalized size = 2.12 \begin{align*} \frac{2 \,{\left (B b^{3} x^{3} - 16 \, B a^{3} + 8 \, A a^{2} b - 3 \,{\left (2 \, B a b^{2} - A b^{3}\right )} x^{2} - 12 \,{\left (2 \, B a^{2} b - A a b^{2}\right )} x\right )} \sqrt{b x + a}}{3 \,{\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.83897, size = 299, normalized size = 3.29 \begin{align*} \begin{cases} \frac{16 A a^{2} b}{3 a b^{4} \sqrt{a + b x} + 3 b^{5} x \sqrt{a + b x}} + \frac{24 A a b^{2} x}{3 a b^{4} \sqrt{a + b x} + 3 b^{5} x \sqrt{a + b x}} + \frac{6 A b^{3} x^{2}}{3 a b^{4} \sqrt{a + b x} + 3 b^{5} x \sqrt{a + b x}} - \frac{32 B a^{3}}{3 a b^{4} \sqrt{a + b x} + 3 b^{5} x \sqrt{a + b x}} - \frac{48 B a^{2} b x}{3 a b^{4} \sqrt{a + b x} + 3 b^{5} x \sqrt{a + b x}} - \frac{12 B a b^{2} x^{2}}{3 a b^{4} \sqrt{a + b x} + 3 b^{5} x \sqrt{a + b x}} + \frac{2 B b^{3} x^{3}}{3 a b^{4} \sqrt{a + b x} + 3 b^{5} x \sqrt{a + b x}} & \text{for}\: b \neq 0 \\\frac{\frac{A x^{3}}{3} + \frac{B x^{4}}{4}}{a^{\frac{5}{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19464, size = 124, normalized size = 1.36 \begin{align*} -\frac{2 \,{\left (9 \,{\left (b x + a\right )} B a^{2} - B a^{3} - 6 \,{\left (b x + a\right )} A a b + A a^{2} b\right )}}{3 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{4}} + \frac{2 \,{\left ({\left (b x + a\right )}^{\frac{3}{2}} B b^{8} - 9 \, \sqrt{b x + a} B a b^{8} + 3 \, \sqrt{b x + a} A b^{9}\right )}}{3 \, b^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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