Integrand size = 19, antiderivative size = 76 \[ \int \frac {c+\frac {d}{x}}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=\frac {3 b c-2 a d}{a^2 \sqrt {a+\frac {b}{x}}}+\frac {c x}{a \sqrt {a+\frac {b}{x}}}-\frac {(3 b c-2 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{5/2}} \]
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Time = 0.03 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {382, 79, 53, 65, 214} \[ \int \frac {c+\frac {d}{x}}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) (3 b c-2 a d)}{a^{5/2}}+\frac {3 b c-2 a d}{a^2 \sqrt {a+\frac {b}{x}}}+\frac {c x}{a \sqrt {a+\frac {b}{x}}} \]
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Rule 53
Rule 65
Rule 79
Rule 214
Rule 382
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {c+d x}{x^2 (a+b x)^{3/2}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {c x}{a \sqrt {a+\frac {b}{x}}}-\frac {\left (-\frac {3 b c}{2}+a d\right ) \text {Subst}\left (\int \frac {1}{x (a+b x)^{3/2}} \, dx,x,\frac {1}{x}\right )}{a} \\ & = \frac {3 b c-2 a d}{a^2 \sqrt {a+\frac {b}{x}}}+\frac {c x}{a \sqrt {a+\frac {b}{x}}}+\frac {(3 b c-2 a d) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )}{2 a^2} \\ & = \frac {3 b c-2 a d}{a^2 \sqrt {a+\frac {b}{x}}}+\frac {c x}{a \sqrt {a+\frac {b}{x}}}+\frac {(3 b c-2 a d) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x}}\right )}{a^2 b} \\ & = \frac {3 b c-2 a d}{a^2 \sqrt {a+\frac {b}{x}}}+\frac {c x}{a \sqrt {a+\frac {b}{x}}}-\frac {(3 b c-2 a d) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{5/2}} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.92 \[ \int \frac {c+\frac {d}{x}}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=\frac {\sqrt {a+\frac {b}{x}} x (3 b c-2 a d+a c x)}{a^2 (b+a x)}+\frac {(-3 b c+2 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{5/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(158\) vs. \(2(66)=132\).
Time = 0.17 (sec) , antiderivative size = 159, normalized size of antiderivative = 2.09
method | result | size |
risch | \(\frac {c \left (a x +b \right )}{a^{2} \sqrt {\frac {a x +b}{x}}}+\frac {\left (2 \sqrt {a}\, d \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right )-\frac {3 b c \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right )}{\sqrt {a}}-\frac {4 \left (a d -b c \right ) \sqrt {a \left (x +\frac {b}{a}\right )^{2}-b \left (x +\frac {b}{a}\right )}}{a \left (x +\frac {b}{a}\right )}\right ) \sqrt {x \left (a x +b \right )}}{2 a^{2} x \sqrt {\frac {a x +b}{x}}}\) | \(159\) |
default | \(-\frac {\sqrt {\frac {a x +b}{x}}\, x \left (4 a^{\frac {7}{2}} \sqrt {x \left (a x +b \right )}\, d \,x^{2}-6 a^{\frac {5}{2}} \sqrt {x \left (a x +b \right )}\, b c \,x^{2}-4 a^{\frac {5}{2}} \left (x \left (a x +b \right )\right )^{\frac {3}{2}} d +8 a^{\frac {5}{2}} \sqrt {x \left (a x +b \right )}\, b d x +4 a^{\frac {3}{2}} \left (x \left (a x +b \right )\right )^{\frac {3}{2}} b c -12 a^{\frac {3}{2}} \sqrt {x \left (a x +b \right )}\, b^{2} c x -2 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{3} b d \,x^{2}+3 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{2} b^{2} c \,x^{2}+4 a^{\frac {3}{2}} \sqrt {x \left (a x +b \right )}\, b^{2} d -4 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{2} b^{2} d x +6 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a \,b^{3} c x -6 \sqrt {a}\, \sqrt {x \left (a x +b \right )}\, b^{3} c -2 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a \,b^{3} d +3 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) b^{4} c \right )}{2 a^{\frac {5}{2}} \sqrt {x \left (a x +b \right )}\, b \left (a x +b \right )^{2}}\) | \(387\) |
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none
Time = 0.26 (sec) , antiderivative size = 210, normalized size of antiderivative = 2.76 \[ \int \frac {c+\frac {d}{x}}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=\left [-\frac {{\left (3 \, b^{2} c - 2 \, a b d + {\left (3 \, a b c - 2 \, a^{2} d\right )} x\right )} \sqrt {a} \log \left (2 \, a x + 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) - 2 \, {\left (a^{2} c x^{2} + {\left (3 \, a b c - 2 \, a^{2} d\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{2 \, {\left (a^{4} x + a^{3} b\right )}}, \frac {{\left (3 \, b^{2} c - 2 \, a b d + {\left (3 \, a b c - 2 \, a^{2} d\right )} x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) + {\left (a^{2} c x^{2} + {\left (3 \, a b c - 2 \, a^{2} d\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{a^{4} x + a^{3} b}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 224 vs. \(2 (65) = 130\).
Time = 14.55 (sec) , antiderivative size = 224, normalized size of antiderivative = 2.95 \[ \int \frac {c+\frac {d}{x}}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=c \left (\frac {x^{\frac {3}{2}}}{a \sqrt {b} \sqrt {\frac {a x}{b} + 1}} + \frac {3 \sqrt {b} \sqrt {x}}{a^{2} \sqrt {\frac {a x}{b} + 1}} - \frac {3 b \operatorname {asinh}{\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}} \right )}}{a^{\frac {5}{2}}}\right ) + d \left (- \frac {2 a^{3} x \sqrt {1 + \frac {b}{a x}}}{a^{\frac {9}{2}} x + a^{\frac {7}{2}} b} - \frac {a^{3} x \log {\left (\frac {b}{a x} \right )}}{a^{\frac {9}{2}} x + a^{\frac {7}{2}} b} + \frac {2 a^{3} x \log {\left (\sqrt {1 + \frac {b}{a x}} + 1 \right )}}{a^{\frac {9}{2}} x + a^{\frac {7}{2}} b} - \frac {a^{2} b \log {\left (\frac {b}{a x} \right )}}{a^{\frac {9}{2}} x + a^{\frac {7}{2}} b} + \frac {2 a^{2} b \log {\left (\sqrt {1 + \frac {b}{a x}} + 1 \right )}}{a^{\frac {9}{2}} x + a^{\frac {7}{2}} b}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 144 vs. \(2 (66) = 132\).
Time = 0.35 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.89 \[ \int \frac {c+\frac {d}{x}}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=\frac {1}{2} \, c {\left (\frac {2 \, {\left (3 \, {\left (a + \frac {b}{x}\right )} b - 2 \, a b\right )}}{{\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{2} - \sqrt {a + \frac {b}{x}} a^{3}} + \frac {3 \, b \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{a^{\frac {5}{2}}}\right )} - d {\left (\frac {\log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{a^{\frac {3}{2}}} + \frac {2}{\sqrt {a + \frac {b}{x}} a}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 149 vs. \(2 (66) = 132\).
Time = 0.32 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.96 \[ \int \frac {c+\frac {d}{x}}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=-\frac {{\left (3 \, b c \log \left ({\left | b \right |}\right ) - 2 \, a d \log \left ({\left | b \right |}\right ) + 4 \, b c - 4 \, a d\right )} \mathrm {sgn}\left (x\right )}{2 \, a^{\frac {5}{2}}} + \frac {\sqrt {a x^{2} + b x} c}{a^{2} \mathrm {sgn}\left (x\right )} + \frac {{\left (3 \, b c - 2 \, a d\right )} \log \left ({\left | 2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} + b \right |}\right )}{2 \, a^{\frac {5}{2}} \mathrm {sgn}\left (x\right )} + \frac {2 \, {\left (\sqrt {a} b^{2} c - a^{\frac {3}{2}} b d\right )}}{{\left ({\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} + b\right )} a^{3} \mathrm {sgn}\left (x\right )} \]
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Time = 6.29 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.93 \[ \int \frac {c+\frac {d}{x}}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=\frac {2\,d\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {2\,d}{a\,\sqrt {a+\frac {b}{x}}}+\frac {2\,c\,x\,{\left (\frac {a\,x}{b}+1\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (\frac {3}{2},\frac {5}{2};\ \frac {7}{2};\ -\frac {a\,x}{b}\right )}{5\,{\left (a+\frac {b}{x}\right )}^{3/2}} \]
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