Integrand size = 31, antiderivative size = 116 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^{5/2}} \, dx=-\frac {2 a A \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^{3/2} (a+b x)}-\frac {2 (A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{\sqrt {x} (a+b x)}+\frac {2 b B \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x} \]
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Time = 0.03 (sec) , antiderivative size = 116, 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} \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^{5/2}} \, dx=-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{\sqrt {x} (a+b x)}-\frac {2 a A \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^{3/2} (a+b x)}+\frac {2 b B \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x} \]
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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)}{x^{5/2}} \, dx}{a b+b^2 x} \\ & = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {a A b}{x^{5/2}}+\frac {b (A b+a B)}{x^{3/2}}+\frac {b^2 B}{\sqrt {x}}\right ) \, dx}{a b+b^2 x} \\ & = -\frac {2 a A \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^{3/2} (a+b x)}-\frac {2 (A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{\sqrt {x} (a+b x)}+\frac {2 b B \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.40 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^{5/2}} \, dx=-\frac {2 \sqrt {(a+b x)^2} (3 b x (A-B x)+a (A+3 B x))}{3 x^{3/2} (a+b x)} \]
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Result contains higher order function than in optimal. Order 9 vs. order 2.
Time = 0.04 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.28
method | result | size |
default | \(-\frac {2 \,\operatorname {csgn}\left (b x +a \right ) \left (-3 B b \,x^{2}+3 A b x +3 a B x +a A \right )}{3 x^{\frac {3}{2}}}\) | \(33\) |
gosper | \(-\frac {2 \left (-3 B b \,x^{2}+3 A b x +3 a B x +a A \right ) \sqrt {\left (b x +a \right )^{2}}}{3 x^{\frac {3}{2}} \left (b x +a \right )}\) | \(43\) |
risch | \(-\frac {2 \left (-3 B b \,x^{2}+3 A b x +3 a B x +a A \right ) \sqrt {\left (b x +a \right )^{2}}}{3 x^{\frac {3}{2}} \left (b x +a \right )}\) | \(43\) |
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Time = 0.38 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.23 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^{5/2}} \, dx=\frac {2 \, {\left (3 \, B b x^{2} - A a - 3 \, {\left (B a + A b\right )} x\right )}}{3 \, x^{\frac {3}{2}}} \]
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Timed out. \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^{5/2}} \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.28 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^{5/2}} \, dx=\frac {2 \, {\left (b x^{2} - a x\right )} B}{x^{\frac {3}{2}}} - \frac {2 \, {\left (3 \, b x^{2} + a x\right )} A}{3 \, x^{\frac {5}{2}}} \]
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Time = 0.26 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.44 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^{5/2}} \, dx=2 \, B b \sqrt {x} \mathrm {sgn}\left (b x + a\right ) - \frac {2 \, {\left (3 \, B a x \mathrm {sgn}\left (b x + a\right ) + 3 \, A b x \mathrm {sgn}\left (b x + a\right ) + A a \mathrm {sgn}\left (b x + a\right )\right )}}{3 \, x^{\frac {3}{2}}} \]
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Time = 10.06 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.47 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^{5/2}} \, dx=-\frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (\frac {2\,A\,a}{3\,b}-2\,B\,x^2+\frac {x\,\left (6\,A\,b+6\,B\,a\right )}{3\,b}\right )}{x^{5/2}+\frac {a\,x^{3/2}}{b}} \]
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