Integrand size = 29, antiderivative size = 75 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^4} \, dx=\frac {(A b-a B) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{2 a^2 x^2}-\frac {A \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 a^2 x^3} \]
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Time = 0.03 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {783, 660, 37} \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^4} \, dx=\frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (A b-a B)}{2 a^2 x^2}-\frac {A \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 a^2 x^3} \]
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Rule 37
Rule 660
Rule 783
Rubi steps \begin{align*} \text {integral}& = -\frac {A \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 a^2 x^3}-\frac {\left (2 A b^2-2 a b B\right ) \int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{x^3} \, dx}{2 a b} \\ & = -\frac {A \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 a^2 x^3}-\frac {\left (\left (2 A b^2-2 a b B\right ) \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {a b+b^2 x}{x^3} \, dx}{2 a b \left (a b+b^2 x\right )} \\ & = \frac {(A b-a B) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{2 a^2 x^2}-\frac {A \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 a^2 x^3} \\ \end{align*}
Time = 0.41 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.61 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^4} \, dx=-\frac {\sqrt {(a+b x)^2} (3 b x (A+2 B x)+a (2 A+3 B x))}{6 x^3 (a+b x)} \]
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Result contains higher order function than in optimal. Order 9 vs. order 2.
Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.45
method | result | size |
default | \(-\frac {\operatorname {csgn}\left (b x +a \right ) \left (6 B b \,x^{2}+3 A b x +3 a B x +2 a A \right )}{6 x^{3}}\) | \(34\) |
gosper | \(-\frac {\left (6 B b \,x^{2}+3 A b x +3 a B x +2 a A \right ) \sqrt {\left (b x +a \right )^{2}}}{6 x^{3} \left (b x +a \right )}\) | \(44\) |
risch | \(\frac {\left (-B b \,x^{2}+\left (-\frac {A b}{2}-\frac {B a}{2}\right ) x -\frac {a A}{3}\right ) \sqrt {\left (b x +a \right )^{2}}}{x^{3} \left (b x +a \right )}\) | \(44\) |
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Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.36 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^4} \, dx=-\frac {6 \, B b x^{2} + 2 \, A a + 3 \, {\left (B a + A b\right )} x}{6 \, x^{3}} \]
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\[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^4} \, dx=\int \frac {\left (A + B x\right ) \sqrt {\left (a + b x\right )^{2}}}{x^{4}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 195 vs. \(2 (58) = 116\).
Time = 0.22 (sec) , antiderivative size = 195, normalized size of antiderivative = 2.60 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^4} \, dx=\frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B b^{2}}{2 \, a^{2}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A b^{3}}{2 \, a^{3}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B b}{2 \, a x} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A b^{2}}{2 \, a^{2} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B}{2 \, a^{2} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A b}{2 \, a^{3} x^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A}{3 \, a^{2} x^{3}} \]
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Time = 0.27 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.03 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^4} \, dx=-\frac {{\left (3 \, B a b^{2} - A b^{3}\right )} \mathrm {sgn}\left (b x + a\right )}{6 \, a^{2}} - \frac {6 \, B b x^{2} \mathrm {sgn}\left (b x + a\right ) + 3 \, B a x \mathrm {sgn}\left (b x + a\right ) + 3 \, A b x \mathrm {sgn}\left (b x + a\right ) + 2 \, A a \mathrm {sgn}\left (b x + a\right )}{6 \, x^{3}} \]
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Time = 10.05 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.57 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^4} \, dx=-\frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (2\,A\,a+3\,A\,b\,x+3\,B\,a\,x+6\,B\,b\,x^2\right )}{6\,x^3\,\left (a+b\,x\right )} \]
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