Integrand size = 19, antiderivative size = 80 \[ \int \frac {\left (a+b \sqrt {c+d x}\right )^2}{x^3} \, dx=-\frac {b d \left (b c+a \sqrt {c+d x}\right )}{2 c x}-\frac {\left (a+b \sqrt {c+d x}\right )^2}{2 x^2}+\frac {a b d^2 \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{2 c^{3/2}} \]
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Time = 0.05 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {378, 1412, 835, 12, 653, 213} \[ \int \frac {\left (a+b \sqrt {c+d x}\right )^2}{x^3} \, dx=\frac {a b d^2 \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{2 c^{3/2}}-\frac {\left (a+b \sqrt {c+d x}\right )^2}{2 x^2}-\frac {b d \left (a \sqrt {c+d x}+b c\right )}{2 c x} \]
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Rule 12
Rule 213
Rule 378
Rule 653
Rule 835
Rule 1412
Rubi steps \begin{align*} \text {integral}& = d^2 \text {Subst}\left (\int \frac {\left (a+b \sqrt {x}\right )^2}{(-c+x)^3} \, dx,x,c+d x\right ) \\ & = \left (2 d^2\right ) \text {Subst}\left (\int \frac {x (a+b x)^2}{\left (-c+x^2\right )^3} \, dx,x,\sqrt {c+d x}\right ) \\ & = -\frac {\left (a+b \sqrt {c+d x}\right )^2}{2 x^2}-\frac {d^2 \text {Subst}\left (\int -\frac {2 b c (a+b x)}{\left (-c+x^2\right )^2} \, dx,x,\sqrt {c+d x}\right )}{2 c} \\ & = -\frac {\left (a+b \sqrt {c+d x}\right )^2}{2 x^2}+\left (b d^2\right ) \text {Subst}\left (\int \frac {a+b x}{\left (-c+x^2\right )^2} \, dx,x,\sqrt {c+d x}\right ) \\ & = -\frac {b d \left (b c+a \sqrt {c+d x}\right )}{2 c x}-\frac {\left (a+b \sqrt {c+d x}\right )^2}{2 x^2}-\frac {\left (a b d^2\right ) \text {Subst}\left (\int \frac {1}{-c+x^2} \, dx,x,\sqrt {c+d x}\right )}{2 c} \\ & = -\frac {b d \left (b c+a \sqrt {c+d x}\right )}{2 c x}-\frac {\left (a+b \sqrt {c+d x}\right )^2}{2 x^2}+\frac {a b d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{2 c^{3/2}} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b \sqrt {c+d x}\right )^2}{x^3} \, dx=-\frac {a^2 c+a b \sqrt {c+d x} (2 c+d x)+b^2 c (c+2 d x)}{2 c x^2}+\frac {a b d^2 \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{2 c^{3/2}} \]
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Time = 0.98 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.01
method | result | size |
derivativedivides | \(2 d^{2} \left (-\frac {\frac {a b \left (d x +c \right )^{\frac {3}{2}}}{4 c}+\frac {\left (d x +c \right ) b^{2}}{2}+\frac {a b \sqrt {d x +c}}{4}-\frac {b^{2} c}{4}+\frac {a^{2}}{4}}{d^{2} x^{2}}+\frac {a b \,\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{4 c^{\frac {3}{2}}}\right )\) | \(81\) |
default | \(b^{2} \left (-\frac {c}{2 x^{2}}-\frac {d}{x}\right )+4 a b \,d^{2} \left (-\frac {\frac {\left (d x +c \right )^{\frac {3}{2}}}{8 c}+\frac {\sqrt {d x +c}}{8}}{d^{2} x^{2}}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{8 c^{\frac {3}{2}}}\right )-\frac {a^{2}}{2 x^{2}}\) | \(82\) |
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Time = 0.30 (sec) , antiderivative size = 181, normalized size of antiderivative = 2.26 \[ \int \frac {\left (a+b \sqrt {c+d x}\right )^2}{x^3} \, dx=\left [\frac {a b \sqrt {c} d^{2} x^{2} \log \left (\frac {d x + 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) - 4 \, b^{2} c^{2} d x - 2 \, b^{2} c^{3} - 2 \, a^{2} c^{2} - 2 \, {\left (a b c d x + 2 \, a b c^{2}\right )} \sqrt {d x + c}}{4 \, c^{2} x^{2}}, -\frac {a b \sqrt {-c} d^{2} x^{2} \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c}}{c}\right ) + 2 \, b^{2} c^{2} d x + b^{2} c^{3} + a^{2} c^{2} + {\left (a b c d x + 2 \, a b c^{2}\right )} \sqrt {d x + c}}{2 \, c^{2} x^{2}}\right ] \]
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Time = 90.95 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.68 \[ \int \frac {\left (a+b \sqrt {c+d x}\right )^2}{x^3} \, dx=- \frac {a^{2}}{2 x^{2}} - \frac {a b c}{\sqrt {d} x^{\frac {5}{2}} \sqrt {\frac {c}{d x} + 1}} - \frac {3 a b \sqrt {d}}{2 x^{\frac {3}{2}} \sqrt {\frac {c}{d x} + 1}} - \frac {a b d^{\frac {3}{2}}}{2 c \sqrt {x} \sqrt {\frac {c}{d x} + 1}} + \frac {a b d^{2} \operatorname {asinh}{\left (\frac {\sqrt {c}}{\sqrt {d} \sqrt {x}} \right )}}{2 c^{\frac {3}{2}}} - \frac {b^{2} c}{2 x^{2}} - \frac {b^{2} d}{x} \]
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Time = 0.27 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.41 \[ \int \frac {\left (a+b \sqrt {c+d x}\right )^2}{x^3} \, dx=-\frac {1}{4} \, {\left (\frac {a b \log \left (\frac {\sqrt {d x + c} - \sqrt {c}}{\sqrt {d x + c} + \sqrt {c}}\right )}{c^{\frac {3}{2}}} + \frac {2 \, {\left (2 \, {\left (d x + c\right )} b^{2} c - b^{2} c^{2} + {\left (d x + c\right )}^{\frac {3}{2}} a b + \sqrt {d x + c} a b c + a^{2} c\right )}}{{\left (d x + c\right )}^{2} c - 2 \, {\left (d x + c\right )} c^{2} + c^{3}}\right )} d^{2} \]
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Time = 0.39 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.31 \[ \int \frac {\left (a+b \sqrt {c+d x}\right )^2}{x^3} \, dx=-\frac {\frac {a b d^{3} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{\sqrt {-c} c} + \frac {2 \, {\left (d x + c\right )} b^{2} c d^{3} - b^{2} c^{2} d^{3} + {\left (d x + c\right )}^{\frac {3}{2}} a b d^{3} + \sqrt {d x + c} a b c d^{3} + a^{2} c d^{3}}{c d^{2} x^{2}}}{2 \, d} \]
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Time = 0.08 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \sqrt {c+d x}\right )^2}{x^3} \, dx=\frac {a\,b\,d^2\,\mathrm {atanh}\left (\frac {\sqrt {c+d\,x}}{\sqrt {c}}\right )}{2\,c^{3/2}}-\frac {b^2\,c}{2\,x^2}-\frac {b^2\,d}{x}-\frac {a\,b\,\sqrt {c+d\,x}}{2\,x^2}-\frac {a\,b\,{\left (c+d\,x\right )}^{3/2}}{2\,c\,x^2}-\frac {a^2}{2\,x^2} \]
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