Optimal. Leaf size=42 \[ \frac {2 (A b-a B) (a+b x)^{3/2}}{3 b^2}+\frac {2 B (a+b x)^{5/2}}{5 b^2} \]
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Rubi [A]
time = 0.01, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {45}
\begin {gather*} \frac {2 (a+b x)^{3/2} (A b-a B)}{3 b^2}+\frac {2 B (a+b x)^{5/2}}{5 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int \sqrt {a+b x} (A+B x) \, dx &=\int \left (\frac {(A b-a B) \sqrt {a+b x}}{b}+\frac {B (a+b x)^{3/2}}{b}\right ) \, dx\\ &=\frac {2 (A b-a B) (a+b x)^{3/2}}{3 b^2}+\frac {2 B (a+b x)^{5/2}}{5 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 30, normalized size = 0.71 \begin {gather*} \frac {2 (a+b x)^{3/2} (5 A b-2 a B+3 b B x)}{15 b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 34, normalized size = 0.81
method | result | size |
gosper | \(\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (3 b B x +5 A b -2 B a \right )}{15 b^{2}}\) | \(27\) |
derivativedivides | \(\frac {\frac {2 B \left (b x +a \right )^{\frac {5}{2}}}{5}+\frac {2 \left (A b -B a \right ) \left (b x +a \right )^{\frac {3}{2}}}{3}}{b^{2}}\) | \(34\) |
default | \(\frac {\frac {2 B \left (b x +a \right )^{\frac {5}{2}}}{5}+\frac {2 \left (A b -B a \right ) \left (b x +a \right )^{\frac {3}{2}}}{3}}{b^{2}}\) | \(34\) |
trager | \(\frac {2 \left (3 b^{2} B \,x^{2}+5 b^{2} A x +a b B x +5 a b A -2 a^{2} B \right ) \sqrt {b x +a}}{15 b^{2}}\) | \(46\) |
risch | \(\frac {2 \left (3 b^{2} B \,x^{2}+5 b^{2} A x +a b B x +5 a b A -2 a^{2} B \right ) \sqrt {b x +a}}{15 b^{2}}\) | \(46\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 33, normalized size = 0.79 \begin {gather*} \frac {2 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} B - 5 \, {\left (B a - A b\right )} {\left (b x + a\right )}^{\frac {3}{2}}\right )}}{15 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.82, size = 46, normalized size = 1.10 \begin {gather*} \frac {2 \, {\left (3 \, B b^{2} x^{2} - 2 \, B a^{2} + 5 \, A a b + {\left (B a b + 5 \, A b^{2}\right )} x\right )} \sqrt {b x + a}}{15 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.90, size = 36, normalized size = 0.86 \begin {gather*} \frac {2 \left (\frac {B \left (a + b x\right )^{\frac {5}{2}}}{5 b} + \frac {\left (a + b x\right )^{\frac {3}{2}} \left (A b - B a\right )}{3 b}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 100 vs.
\(2 (34) = 68\).
time = 2.65, size = 100, normalized size = 2.38 \begin {gather*} \frac {2 \, {\left (15 \, \sqrt {b x + a} A a + 5 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} A + \frac {5 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} B 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}\right )}}{15 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 29, normalized size = 0.69 \begin {gather*} \frac {2\,{\left (a+b\,x\right )}^{3/2}\,\left (5\,A\,b-5\,B\,a+3\,B\,\left (a+b\,x\right )\right )}{15\,b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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