Optimal. Leaf size=65 \[ \frac {2 (a+b x)^{3/2} (A b-2 a B)}{3 b^3}-\frac {2 a \sqrt {a+b x} (A b-a B)}{b^3}+\frac {2 B (a+b x)^{5/2}}{5 b^3} \]
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Rubi [A] time = 0.02, antiderivative size = 65, 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} \[ \frac {2 (a+b x)^{3/2} (A b-2 a B)}{3 b^3}-\frac {2 a \sqrt {a+b x} (A b-a B)}{b^3}+\frac {2 B (a+b x)^{5/2}}{5 b^3} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int \frac {x (A+B x)}{\sqrt {a+b x}} \, dx &=\int \left (\frac {a (-A b+a B)}{b^2 \sqrt {a+b x}}+\frac {(A b-2 a B) \sqrt {a+b x}}{b^2}+\frac {B (a+b x)^{3/2}}{b^2}\right ) \, dx\\ &=-\frac {2 a (A b-a B) \sqrt {a+b x}}{b^3}+\frac {2 (A b-2 a B) (a+b x)^{3/2}}{3 b^3}+\frac {2 B (a+b x)^{5/2}}{5 b^3}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 48, normalized size = 0.74 \[ \frac {2 \sqrt {a+b x} \left (8 a^2 B-2 a b (5 A+2 B x)+b^2 x (5 A+3 B x)\right )}{15 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 48, normalized size = 0.74 \[ \frac {2 \, {\left (3 \, B b^{2} x^{2} + 8 \, B a^{2} - 10 \, A a b - {\left (4 \, B a b - 5 \, A b^{2}\right )} x\right )} \sqrt {b x + a}}{15 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.27, size = 67, normalized size = 1.03 \[ \frac {2 \, {\left (\frac {5 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} A}{b} + \frac {{\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} B}{b^{2}}\right )}}{15 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 47, normalized size = 0.72 \[ -\frac {2 \sqrt {b x +a}\, \left (-3 B \,b^{2} x^{2}-5 A \,b^{2} x +4 B a b x +10 A a b -8 B \,a^{2}\right )}{15 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.87, size = 54, normalized size = 0.83 \[ \frac {2 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} B - 5 \, {\left (2 \, B a - A b\right )} {\left (b x + a\right )}^{\frac {3}{2}} + 15 \, {\left (B a^{2} - A a b\right )} \sqrt {b x + a}\right )}}{15 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.38, size = 52, normalized size = 0.80 \[ \frac {2\,\sqrt {a+b\,x}\,\left (15\,B\,a^2+3\,B\,{\left (a+b\,x\right )}^2-15\,A\,a\,b+5\,A\,b\,\left (a+b\,x\right )-10\,B\,a\,\left (a+b\,x\right )\right )}{15\,b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 19.47, size = 182, normalized size = 2.80 \[ \begin {cases} \frac {- \frac {2 A a \left (- \frac {a}{\sqrt {a + b x}} - \sqrt {a + b x}\right )}{b} - \frac {2 A \left (\frac {a^{2}}{\sqrt {a + b x}} + 2 a \sqrt {a + b x} - \frac {\left (a + b x\right )^{\frac {3}{2}}}{3}\right )}{b} - \frac {2 B a \left (\frac {a^{2}}{\sqrt {a + b x}} + 2 a \sqrt {a + b x} - \frac {\left (a + b x\right )^{\frac {3}{2}}}{3}\right )}{b^{2}} - \frac {2 B \left (- \frac {a^{3}}{\sqrt {a + b x}} - 3 a^{2} \sqrt {a + b x} + a \left (a + b x\right )^{\frac {3}{2}} - \frac {\left (a + b x\right )^{\frac {5}{2}}}{5}\right )}{b^{2}}}{b} & \text {for}\: b \neq 0 \\\frac {\frac {A x^{2}}{2} + \frac {B x^{3}}{3}}{\sqrt {a}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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