Optimal. Leaf size=63 \[ \frac {2 \sqrt {a+b x} (A b-2 a B)}{b^3}+\frac {2 a (A b-a B)}{b^3 \sqrt {a+b x}}+\frac {2 B (a+b x)^{3/2}}{3 b^3} \]
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Rubi [A] time = 0.02, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {77} \begin {gather*} \frac {2 \sqrt {a+b x} (A b-2 a B)}{b^3}+\frac {2 a (A b-a B)}{b^3 \sqrt {a+b x}}+\frac {2 B (a+b x)^{3/2}}{3 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int \frac {x (A+B x)}{(a+b x)^{3/2}} \, dx &=\int \left (\frac {a (-A b+a B)}{b^2 (a+b x)^{3/2}}+\frac {A b-2 a B}{b^2 \sqrt {a+b x}}+\frac {B \sqrt {a+b x}}{b^2}\right ) \, dx\\ &=\frac {2 a (A b-a B)}{b^3 \sqrt {a+b x}}+\frac {2 (A b-2 a B) \sqrt {a+b x}}{b^3}+\frac {2 B (a+b x)^{3/2}}{3 b^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 47, normalized size = 0.75 \begin {gather*} \frac {2 \left (-8 a^2 B+a (6 A b-4 b B x)+b^2 x (3 A+B x)\right )}{3 b^3 \sqrt {a+b x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.04, size = 55, normalized size = 0.87 \begin {gather*} \frac {2 \left (-3 a^2 B+3 A b (a+b x)+3 a A b-6 a B (a+b x)+B (a+b x)^2\right )}{3 b^3 \sqrt {a+b x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.12, size = 57, normalized size = 0.90 \begin {gather*} \frac {2 \, {\left (B b^{2} x^{2} - 8 \, B a^{2} + 6 \, A a b - {\left (4 \, B a b - 3 \, A b^{2}\right )} x\right )} \sqrt {b x + a}}{3 \, {\left (b^{4} x + a b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.30, size = 69, normalized size = 1.10 \begin {gather*} -\frac {2 \, {\left (B a^{2} - A a b\right )}}{\sqrt {b x + a} b^{3}} + \frac {2 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} B b^{6} - 6 \, \sqrt {b x + a} B a b^{6} + 3 \, \sqrt {b x + a} A b^{7}\right )}}{3 \, b^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 46, normalized size = 0.73 \begin {gather*} \frac {\frac {2}{3} B \,b^{2} x^{2}+2 A \,b^{2} x -\frac {8}{3} B a b x +4 A a b -\frac {16}{3} B \,a^{2}}{\sqrt {b x +a}\, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.89, size = 61, normalized size = 0.97 \begin {gather*} \frac {2 \, {\left (\frac {{\left (b x + a\right )}^{\frac {3}{2}} B - 3 \, {\left (2 \, B a - A b\right )} \sqrt {b x + a}}{b} - \frac {3 \, {\left (B a^{2} - A a b\right )}}{\sqrt {b x + a} b}\right )}}{3 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 52, normalized size = 0.83 \begin {gather*} \frac {2\,B\,{\left (a+b\,x\right )}^2-6\,B\,a^2+6\,A\,a\,b+6\,A\,b\,\left (a+b\,x\right )-12\,B\,a\,\left (a+b\,x\right )}{3\,b^3\,\sqrt {a+b\,x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 10.13, size = 60, normalized size = 0.95 \begin {gather*} \frac {2 B \left (a + b x\right )^{\frac {3}{2}}}{3 b^{3}} - \frac {2 a \left (- A b + B a\right )}{b^{3} \sqrt {a + b x}} + \frac {\sqrt {a + b x} \left (2 A b - 4 B a\right )}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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