Integrand size = 31, antiderivative size = 118 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{\sqrt {x}} \, dx=\frac {2 a A \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {2 (A b+a B) x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac {2 b B x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 (a+b x)} \]
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Time = 0.03 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {784, 77} \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{\sqrt {x}} \, dx=\frac {2 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{3 (a+b x)}+\frac {2 a A \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {2 b B x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 (a+b x)} \]
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Rule 77
Rule 784
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right ) (A+B x)}{\sqrt {x}} \, dx}{a b+b^2 x} \\ & = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {a A b}{\sqrt {x}}+b (A b+a B) \sqrt {x}+b^2 B x^{3/2}\right ) \, dx}{a b+b^2 x} \\ & = \frac {2 a A \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {2 (A b+a B) x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac {2 b B x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 (a+b x)} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.42 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{\sqrt {x}} \, dx=\frac {2 \sqrt {x} \sqrt {(a+b x)^2} (5 a (3 A+B x)+b x (5 A+3 B x))}{15 (a+b x)} \]
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Result contains higher order function than in optimal. Order 9 vs. order 2.
Time = 0.08 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.29
method | result | size |
default | \(\frac {2 \,\operatorname {csgn}\left (b x +a \right ) \sqrt {x}\, \left (3 B b \,x^{2}+5 A b x +5 a B x +15 a A \right )}{15}\) | \(34\) |
gosper | \(\frac {2 \sqrt {x}\, \left (3 B b \,x^{2}+5 A b x +5 a B x +15 a A \right ) \sqrt {\left (b x +a \right )^{2}}}{15 \left (b x +a \right )}\) | \(44\) |
risch | \(\frac {2 \sqrt {x}\, \left (3 B b \,x^{2}+5 A b x +5 a B x +15 a A \right ) \sqrt {\left (b x +a \right )^{2}}}{15 \left (b x +a \right )}\) | \(44\) |
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Time = 0.34 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.23 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{\sqrt {x}} \, dx=\frac {2}{15} \, {\left (3 \, B b x^{2} + 15 \, A a + 5 \, {\left (B a + A b\right )} x\right )} \sqrt {x} \]
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\[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{\sqrt {x}} \, dx=\int \frac {\left (A + B x\right ) \sqrt {\left (a + b x\right )^{2}}}{\sqrt {x}}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.29 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{\sqrt {x}} \, dx=\frac {2}{15} \, {\left (3 \, b x^{2} + 5 \, a x\right )} B \sqrt {x} + \frac {2 \, {\left (b x^{2} + 3 \, a x\right )} A}{3 \, \sqrt {x}} \]
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Time = 0.27 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.45 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{\sqrt {x}} \, dx=\frac {2}{5} \, B b x^{\frac {5}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{3} \, B a x^{\frac {3}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{3} \, A b x^{\frac {3}{2}} \mathrm {sgn}\left (b x + a\right ) + 2 \, A a \sqrt {x} \mathrm {sgn}\left (b x + a\right ) \]
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Time = 10.06 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.47 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{\sqrt {x}} \, dx=\frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (\frac {2\,B\,x^3}{5}+\frac {x^2\,\left (10\,A\,b+10\,B\,a\right )}{15\,b}+\frac {2\,A\,a\,x}{b}\right )}{x^{3/2}+\frac {a\,\sqrt {x}}{b}} \]
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