Integrand size = 29, antiderivative size = 114 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^5} \, dx=-\frac {a A \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac {(A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac {b B \sqrt {a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)} \]
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Time = 0.03 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {784, 77} \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^5} \, dx=-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{3 x^3 (a+b x)}-\frac {a A \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac {b B \sqrt {a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)} \]
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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} \, 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}+\frac {b (A b+a B)}{x^4}+\frac {b^2 B}{x^3}\right ) \, dx}{a b+b^2 x} \\ & = -\frac {a A \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac {(A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac {b B \sqrt {a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.41 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^5} \, dx=-\frac {\sqrt {(a+b x)^2} \left (3 a A+4 A b x+4 a B x+6 b B x^2\right )}{12 x^4 (a+b x)} \]
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Result contains higher order function than in optimal. Order 9 vs. order 2.
Time = 0.29 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.30
method | result | size |
default | \(-\frac {\operatorname {csgn}\left (b x +a \right ) \left (6 B b \,x^{2}+4 A b x +4 a B x +3 a A \right )}{12 x^{4}}\) | \(34\) |
gosper | \(-\frac {\left (6 B b \,x^{2}+4 A b x +4 a B x +3 a A \right ) \sqrt {\left (b x +a \right )^{2}}}{12 x^{4} \left (b x +a \right )}\) | \(44\) |
risch | \(\frac {\left (-\frac {B b \,x^{2}}{2}+\left (-\frac {A b}{3}-\frac {B a}{3}\right ) x -\frac {a A}{4}\right ) \sqrt {\left (b x +a \right )^{2}}}{x^{4} \left (b x +a \right )}\) | \(44\) |
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Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.24 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^5} \, dx=-\frac {6 \, B b x^{2} + 3 \, A a + 4 \, {\left (B a + A b\right )} x}{12 \, x^{4}} \]
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\[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^5} \, dx=\int \frac {\left (A + B x\right ) \sqrt {\left (a + b x\right )^{2}}}{x^{5}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 255 vs. \(2 (75) = 150\).
Time = 0.22 (sec) , antiderivative size = 255, normalized size of antiderivative = 2.24 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^5} \, dx=-\frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B b^{3}}{2 \, a^{3}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A b^{4}}{2 \, a^{4}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B b^{2}}{2 \, a^{2} x} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A b^{3}}{2 \, a^{3} x} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B b}{2 \, a^{3} x^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A b^{2}}{2 \, a^{4} x^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B}{3 \, a^{2} x^{3}} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A b}{12 \, a^{3} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A}{4 \, a^{2} x^{4}} \]
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Time = 0.26 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.68 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^5} \, dx=\frac {{\left (2 \, B a b^{3} - A b^{4}\right )} \mathrm {sgn}\left (b x + a\right )}{12 \, a^{3}} - \frac {6 \, B b x^{2} \mathrm {sgn}\left (b x + a\right ) + 4 \, B a x \mathrm {sgn}\left (b x + a\right ) + 4 \, A b x \mathrm {sgn}\left (b x + a\right ) + 3 \, A a \mathrm {sgn}\left (b x + a\right )}{12 \, x^{4}} \]
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Time = 10.04 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.38 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^5} \, dx=-\frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (3\,A\,a+4\,A\,b\,x+4\,B\,a\,x+6\,B\,b\,x^2\right )}{12\,x^4\,\left (a+b\,x\right )} \]
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