Integrand size = 29, antiderivative size = 114 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^6} \, dx=-\frac {a A \sqrt {a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac {(A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac {b B \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)} \]
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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} \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^6} \, dx=-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{4 x^4 (a+b x)}-\frac {a A \sqrt {a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac {b B \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)} \]
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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^6} \, 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^6}+\frac {b (A b+a B)}{x^5}+\frac {b^2 B}{x^4}\right ) \, dx}{a b+b^2 x} \\ & = -\frac {a A \sqrt {a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac {(A b+a B) \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}}{3 x^3 (a+b x)} \\ \end{align*}
Time = 0.32 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.43 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^6} \, dx=-\frac {\sqrt {(a+b x)^2} (5 b x (3 A+4 B x)+3 a (4 A+5 B x))}{60 x^5 (a+b x)} \]
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Result contains higher order function than in optimal. Order 9 vs. order 2.
Time = 0.43 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.30
method | result | size |
default | \(-\frac {\operatorname {csgn}\left (b x +a \right ) \left (20 B b \,x^{2}+15 A b x +15 a B x +12 a A \right )}{60 x^{5}}\) | \(34\) |
gosper | \(-\frac {\left (20 B b \,x^{2}+15 A b x +15 a B x +12 a A \right ) \sqrt {\left (b x +a \right )^{2}}}{60 x^{5} \left (b x +a \right )}\) | \(44\) |
risch | \(\frac {\left (-\frac {B b \,x^{2}}{3}+\left (-\frac {A b}{4}-\frac {B a}{4}\right ) x -\frac {a A}{5}\right ) \sqrt {\left (b x +a \right )^{2}}}{x^{5} \left (b x +a \right )}\) | \(44\) |
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Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.24 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^6} \, dx=-\frac {20 \, B b x^{2} + 12 \, A a + 15 \, {\left (B a + A b\right )} x}{60 \, x^{5}} \]
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\[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^6} \, dx=\int \frac {\left (A + B x\right ) \sqrt {\left (a + b x\right )^{2}}}{x^{6}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 315 vs. \(2 (75) = 150\).
Time = 0.22 (sec) , antiderivative size = 315, normalized size of antiderivative = 2.76 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^6} \, dx=\frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B b^{4}}{2 \, a^{4}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A b^{5}}{2 \, a^{5}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B b^{3}}{2 \, a^{3} x} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A b^{4}}{2 \, a^{4} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B b^{2}}{2 \, a^{4} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A b^{3}}{2 \, a^{5} x^{2}} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B b}{12 \, a^{3} x^{3}} - \frac {9 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A b^{2}}{20 \, a^{4} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B}{4 \, a^{2} x^{4}} + \frac {7 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A b}{20 \, a^{3} x^{4}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A}{5 \, a^{2} x^{5}} \]
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Time = 0.27 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.68 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^6} \, dx=-\frac {{\left (5 \, B a b^{4} - 3 \, A b^{5}\right )} \mathrm {sgn}\left (b x + a\right )}{60 \, a^{4}} - \frac {20 \, B b x^{2} \mathrm {sgn}\left (b x + a\right ) + 15 \, B a x \mathrm {sgn}\left (b x + a\right ) + 15 \, A b x \mathrm {sgn}\left (b x + a\right ) + 12 \, A a \mathrm {sgn}\left (b x + a\right )}{60 \, x^{5}} \]
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Time = 9.97 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.38 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^6} \, dx=-\frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (12\,A\,a+15\,A\,b\,x+15\,B\,a\,x+20\,B\,b\,x^2\right )}{60\,x^5\,\left (a+b\,x\right )} \]
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