Integrand size = 16, antiderivative size = 67 \[ \int x \sqrt {a+b x} (A+B x) \, dx=-\frac {2 a (A b-a B) (a+b x)^{3/2}}{3 b^3}+\frac {2 (A b-2 a B) (a+b x)^{5/2}}{5 b^3}+\frac {2 B (a+b x)^{7/2}}{7 b^3} \]
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Time = 0.02 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {78} \[ \int x \sqrt {a+b x} (A+B x) \, dx=\frac {2 (a+b x)^{5/2} (A b-2 a B)}{5 b^3}-\frac {2 a (a+b x)^{3/2} (A b-a B)}{3 b^3}+\frac {2 B (a+b x)^{7/2}}{7 b^3} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a (-A b+a B) \sqrt {a+b x}}{b^2}+\frac {(A b-2 a B) (a+b x)^{3/2}}{b^2}+\frac {B (a+b x)^{5/2}}{b^2}\right ) \, dx \\ & = -\frac {2 a (A b-a B) (a+b x)^{3/2}}{3 b^3}+\frac {2 (A b-2 a B) (a+b x)^{5/2}}{5 b^3}+\frac {2 B (a+b x)^{7/2}}{7 b^3} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.73 \[ \int x \sqrt {a+b x} (A+B x) \, dx=\frac {2 (a+b x)^{3/2} \left (8 a^2 B+3 b^2 x (7 A+5 B x)-2 a b (7 A+6 B x)\right )}{105 b^3} \]
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Time = 0.52 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.61
method | result | size |
pseudoelliptic | \(-\frac {4 \left (-\frac {3 x \left (\frac {5 B x}{7}+A \right ) b^{2}}{2}+a \left (\frac {6 B x}{7}+A \right ) b -\frac {4 a^{2} B}{7}\right ) \left (b x +a \right )^{\frac {3}{2}}}{15 b^{3}}\) | \(41\) |
gosper | \(-\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (-15 b^{2} B \,x^{2}-21 A \,b^{2} x +12 B a b x +14 a b A -8 a^{2} B \right )}{105 b^{3}}\) | \(47\) |
derivativedivides | \(\frac {\frac {2 B \left (b x +a \right )^{\frac {7}{2}}}{7}+\frac {2 \left (A b -2 B a \right ) \left (b x +a \right )^{\frac {5}{2}}}{5}-\frac {2 a \left (A b -B a \right ) \left (b x +a \right )^{\frac {3}{2}}}{3}}{b^{3}}\) | \(52\) |
default | \(\frac {\frac {2 B \left (b x +a \right )^{\frac {7}{2}}}{7}+\frac {2 \left (A b -2 B a \right ) \left (b x +a \right )^{\frac {5}{2}}}{5}-\frac {2 a \left (A b -B a \right ) \left (b x +a \right )^{\frac {3}{2}}}{3}}{b^{3}}\) | \(52\) |
trager | \(-\frac {2 \left (-15 b^{3} B \,x^{3}-21 A \,b^{3} x^{2}-3 B a \,b^{2} x^{2}-7 a \,b^{2} A x +4 a^{2} b B x +14 a^{2} b A -8 a^{3} B \right ) \sqrt {b x +a}}{105 b^{3}}\) | \(71\) |
risch | \(-\frac {2 \left (-15 b^{3} B \,x^{3}-21 A \,b^{3} x^{2}-3 B a \,b^{2} x^{2}-7 a \,b^{2} A x +4 a^{2} b B x +14 a^{2} b A -8 a^{3} B \right ) \sqrt {b x +a}}{105 b^{3}}\) | \(71\) |
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none
Time = 0.22 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.06 \[ \int x \sqrt {a+b x} (A+B x) \, dx=\frac {2 \, {\left (15 \, B b^{3} x^{3} + 8 \, B a^{3} - 14 \, A a^{2} b + 3 \, {\left (B a b^{2} + 7 \, A b^{3}\right )} x^{2} - {\left (4 \, B a^{2} b - 7 \, A a b^{2}\right )} x\right )} \sqrt {b x + a}}{105 \, b^{3}} \]
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Time = 0.61 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.22 \[ \int x \sqrt {a+b x} (A+B x) \, dx=\begin {cases} \frac {2 \left (\frac {B \left (a + b x\right )^{\frac {7}{2}}}{7 b} + \frac {\left (a + b x\right )^{\frac {5}{2}} \left (A b - 2 B a\right )}{5 b} + \frac {\left (a + b x\right )^{\frac {3}{2}} \left (- A a b + B a^{2}\right )}{3 b}\right )}{b^{2}} & \text {for}\: b \neq 0 \\\sqrt {a} \left (\frac {A x^{2}}{2} + \frac {B x^{3}}{3}\right ) & \text {otherwise} \end {cases} \]
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none
Time = 0.22 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.81 \[ \int x \sqrt {a+b x} (A+B x) \, dx=\frac {2 \, {\left (15 \, {\left (b x + a\right )}^{\frac {7}{2}} B - 21 \, {\left (2 \, B a - A b\right )} {\left (b x + a\right )}^{\frac {5}{2}} + 35 \, {\left (B a^{2} - A a b\right )} {\left (b x + a\right )}^{\frac {3}{2}}\right )}}{105 \, b^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (56) = 112\).
Time = 0.29 (sec) , antiderivative size = 158, normalized size of antiderivative = 2.36 \[ \int x \sqrt {a+b x} (A+B x) \, dx=\frac {2 \, {\left (\frac {35 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} A a}{b} + \frac {7 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} B a}{b^{2}} + \frac {7 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} A}{b} + \frac {3 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} B}{b^{2}}\right )}}{105 \, b} \]
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Time = 0.06 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.78 \[ \int x \sqrt {a+b x} (A+B x) \, dx=\frac {2\,{\left (a+b\,x\right )}^{3/2}\,\left (35\,B\,a^2+15\,B\,{\left (a+b\,x\right )}^2-35\,A\,a\,b+21\,A\,b\,\left (a+b\,x\right )-42\,B\,a\,\left (a+b\,x\right )\right )}{105\,b^3} \]
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