Integrand size = 29, antiderivative size = 200 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^2} \, dx=-\frac {a^3 A \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac {3 a b (A b+a B) x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {b^2 (A b+3 a B) x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac {b^3 B x^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac {a^2 (3 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2} \log (x)}{a+b x} \]
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Time = 0.06 (sec) , antiderivative size = 200, 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} \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^2} \, dx=\frac {3 a b x \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{a+b x}+\frac {b^2 x^2 \sqrt {a^2+2 a b x+b^2 x^2} (3 a B+A b)}{2 (a+b x)}+\frac {a^2 \log (x) \sqrt {a^2+2 a b x+b^2 x^2} (a B+3 A b)}{a+b x}+\frac {b^3 B x^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}-\frac {a^3 A \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)} \]
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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 )^3 (A+B x)}{x^2} \, dx}{b^2 \left (a b+b^2 x\right )} \\ & = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (3 a b^4 (A b+a B)+\frac {a^3 A b^3}{x^2}+\frac {a^2 b^3 (3 A b+a B)}{x}+b^5 (A b+3 a B) x+b^6 B x^2\right ) \, dx}{b^2 \left (a b+b^2 x\right )} \\ & = -\frac {a^3 A \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac {3 a b (A b+a B) x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {b^2 (A b+3 a B) x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac {b^3 B x^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac {a^2 (3 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2} \log (x)}{a+b x} \\ \end{align*}
Time = 1.03 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.44 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^2} \, dx=\frac {\sqrt {(a+b x)^2} \left (-6 a^3 A+18 a^2 b B x^2+9 a b^2 x^2 (2 A+B x)+b^3 x^3 (3 A+2 B x)+6 a^2 (3 A b+a B) x \log (x)\right )}{6 x (a+b x)} \]
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Time = 0.21 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.48
method | result | size |
default | \(\frac {\left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} \left (2 x^{4} B \,b^{3}+3 A \,b^{3} x^{3}+9 B a \,b^{2} x^{3}+18 A \ln \left (x \right ) x \,a^{2} b +18 A a \,b^{2} x^{2}+6 B \,a^{3} \ln \left (x \right ) x +18 B \,a^{2} b \,x^{2}-6 A \,a^{3}\right )}{6 \left (b x +a \right )^{3} x}\) | \(96\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, b \left (\frac {1}{3} B \,b^{2} x^{3}+\frac {1}{2} A \,b^{2} x^{2}+\frac {3}{2} B a b \,x^{2}+3 a A b x +3 a^{2} B x \right )}{b x +a}-\frac {a^{3} A \sqrt {\left (b x +a \right )^{2}}}{x \left (b x +a \right )}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (3 A \,a^{2} b +B \,a^{3}\right ) \ln \left (x \right )}{b x +a}\) | \(117\) |
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Time = 0.30 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.38 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^2} \, dx=\frac {2 \, B b^{3} x^{4} - 6 \, A a^{3} + 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 18 \, {\left (B a^{2} b + A a b^{2}\right )} x^{2} + 6 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x \log \left (x\right )}{6 \, x} \]
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\[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^2} \, dx=\int \frac {\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}{x^{2}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 283 vs. \(2 (141) = 282\).
Time = 0.21 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.42 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^2} \, dx=\left (-1\right )^{2 \, b^{2} x + 2 \, a b} B a^{3} \log \left (2 \, b^{2} x + 2 \, a b\right ) + 3 \, \left (-1\right )^{2 \, b^{2} x + 2 \, a b} A a^{2} b \log \left (2 \, b^{2} x + 2 \, a b\right ) - \left (-1\right )^{2 \, a b x + 2 \, a^{2}} B a^{3} \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right ) - 3 \, \left (-1\right )^{2 \, a b x + 2 \, a^{2}} A a^{2} b \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right ) + \frac {1}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B a b x + \frac {3}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A b^{2} x + \frac {3}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B a^{2} + \frac {9}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A a b + \frac {1}{3} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A}{x} \]
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Time = 0.26 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.60 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^2} \, dx=\frac {1}{3} \, B b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, B a b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, A b^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + 3 \, B a^{2} b x \mathrm {sgn}\left (b x + a\right ) + 3 \, A a b^{2} x \mathrm {sgn}\left (b x + a\right ) - \frac {A a^{3} \mathrm {sgn}\left (b x + a\right )}{x} + {\left (B a^{3} \mathrm {sgn}\left (b x + a\right ) + 3 \, A a^{2} b \mathrm {sgn}\left (b x + a\right )\right )} \log \left ({\left | x \right |}\right ) \]
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Timed out. \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^2} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{x^2} \,d x \]
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