Integrand size = 19, antiderivative size = 100 \[ \int \frac {x}{\left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=\frac {3 b}{2 a^2 d \sqrt {a+\frac {b}{c+d x^2}}}+\frac {c+d x^2}{2 a d \sqrt {a+\frac {b}{c+d x^2}}}-\frac {3 b \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{c+d x^2}}}{\sqrt {a}}\right )}{2 a^{5/2} d} \]
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Time = 0.05 (sec) , antiderivative size = 100, 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 = {1605, 248, 44, 53, 65, 214} \[ \int \frac {x}{\left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=-\frac {3 b \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{c+d x^2}}}{\sqrt {a}}\right )}{2 a^{5/2} d}+\frac {3 b}{2 a^2 d \sqrt {a+\frac {b}{c+d x^2}}}+\frac {c+d x^2}{2 a d \sqrt {a+\frac {b}{c+d x^2}}} \]
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Rule 44
Rule 53
Rule 65
Rule 214
Rule 248
Rule 1605
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx,x,c+d x^2\right )}{2 d} \\ & = -\frac {\text {Subst}\left (\int \frac {1}{x^2 (a+b x)^{3/2}} \, dx,x,\frac {1}{c+d x^2}\right )}{2 d} \\ & = \frac {c+d x^2}{2 a d \sqrt {a+\frac {b}{c+d x^2}}}+\frac {(3 b) \text {Subst}\left (\int \frac {1}{x (a+b x)^{3/2}} \, dx,x,\frac {1}{c+d x^2}\right )}{4 a d} \\ & = \frac {3 b}{2 a^2 d \sqrt {a+\frac {b}{c+d x^2}}}+\frac {c+d x^2}{2 a d \sqrt {a+\frac {b}{c+d x^2}}}+\frac {(3 b) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{c+d x^2}\right )}{4 a^2 d} \\ & = \frac {3 b}{2 a^2 d \sqrt {a+\frac {b}{c+d x^2}}}+\frac {c+d x^2}{2 a d \sqrt {a+\frac {b}{c+d x^2}}}+\frac {3 \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{c+d x^2}}\right )}{2 a^2 d} \\ & = \frac {3 b}{2 a^2 d \sqrt {a+\frac {b}{c+d x^2}}}+\frac {c+d x^2}{2 a d \sqrt {a+\frac {b}{c+d x^2}}}-\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{c+d x^2}}}{\sqrt {a}}\right )}{2 a^{5/2} d} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.14 \[ \int \frac {x}{\left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=\frac {\left (c+d x^2\right ) \sqrt {\frac {b+a \left (c+d x^2\right )}{c+d x^2}} \left (3 b+a \left (c+d x^2\right )\right )}{2 a^2 d \left (b+a \left (c+d x^2\right )\right )}-\frac {3 b \text {arctanh}\left (\frac {\sqrt {\frac {b+a \left (c+d x^2\right )}{c+d x^2}}}{\sqrt {a}}\right )}{2 a^{5/2} d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(216\) vs. \(2(84)=168\).
Time = 1.14 (sec) , antiderivative size = 217, normalized size of antiderivative = 2.17
method | result | size |
risch | \(\frac {a d \,x^{2}+a c +b}{2 d \,a^{2} \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}-\frac {b \left (\frac {3 \ln \left (\frac {a c d +\frac {1}{2} b d +a \,d^{2} x^{2}}{\sqrt {a \,d^{2}}}+\sqrt {a \,c^{2}+b c +\left (2 a c d +b d \right ) x^{2}+a \,d^{2} x^{4}}\right )}{2 \sqrt {a \,d^{2}}}-\frac {2 \left (d \,x^{2}+c \right )}{d \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}}\right ) \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}}{2 a^{2} \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right )}\) | \(217\) |
derivativedivides | \(-\frac {\sqrt {\frac {a \left (d \,x^{2}+c \right )+b}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right ) \left (-6 \sqrt {\left (a \left (d \,x^{2}+c \right )+b \right ) \left (d \,x^{2}+c \right )}\, a^{\frac {5}{2}} \left (d \,x^{2}+c \right )^{2}+3 \ln \left (\frac {2 \sqrt {\left (a \left (d \,x^{2}+c \right )+b \right ) \left (d \,x^{2}+c \right )}\, \sqrt {a}+2 a \left (d \,x^{2}+c \right )+b}{2 \sqrt {a}}\right ) a^{2} b \left (d \,x^{2}+c \right )^{2}+4 a^{\frac {3}{2}} {\left (\left (a \left (d \,x^{2}+c \right )+b \right ) \left (d \,x^{2}+c \right )\right )}^{\frac {3}{2}}-12 \sqrt {\left (a \left (d \,x^{2}+c \right )+b \right ) \left (d \,x^{2}+c \right )}\, a^{\frac {3}{2}} b \left (d \,x^{2}+c \right )+6 \ln \left (\frac {2 \sqrt {\left (a \left (d \,x^{2}+c \right )+b \right ) \left (d \,x^{2}+c \right )}\, \sqrt {a}+2 a \left (d \,x^{2}+c \right )+b}{2 \sqrt {a}}\right ) a \,b^{2} \left (d \,x^{2}+c \right )-6 \sqrt {\left (a \left (d \,x^{2}+c \right )+b \right ) \left (d \,x^{2}+c \right )}\, \sqrt {a}\, b^{2}+3 \ln \left (\frac {2 \sqrt {\left (a \left (d \,x^{2}+c \right )+b \right ) \left (d \,x^{2}+c \right )}\, \sqrt {a}+2 a \left (d \,x^{2}+c \right )+b}{2 \sqrt {a}}\right ) b^{3}\right )}{4 d \,a^{\frac {5}{2}} \sqrt {\left (a \left (d \,x^{2}+c \right )+b \right ) \left (d \,x^{2}+c \right )}\, {\left (a \left (d \,x^{2}+c \right )+b \right )}^{2}}\) | \(363\) |
default | \(\frac {\sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right ) \left (-3 \ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}+b d}{2 \sqrt {a \,d^{2}}}\right ) a b \,d^{2} x^{2}+2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}\, a d \,x^{2}-3 \ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}+b d}{2 \sqrt {a \,d^{2}}}\right ) a b c d +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}\, a c -3 \ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}+b d}{2 \sqrt {a \,d^{2}}}\right ) b^{2} d +4 \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}\, \sqrt {a \,d^{2}}\, b +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}\, b \right )}{4 a^{2} d \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}\, \sqrt {a \,d^{2}}\, \left (a d \,x^{2}+a c +b \right )}\) | \(478\) |
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Leaf count of result is larger than twice the leaf count of optimal. 178 vs. \(2 (84) = 168\).
Time = 0.34 (sec) , antiderivative size = 395, normalized size of antiderivative = 3.95 \[ \int \frac {x}{\left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=\left [\frac {3 \, {\left (a b d x^{2} + a b c + b^{2}\right )} \sqrt {a} \log \left (8 \, a^{2} d^{2} x^{4} + 8 \, a^{2} c^{2} + 8 \, {\left (2 \, a^{2} c + a b\right )} d x^{2} + 8 \, a b c + b^{2} - 4 \, {\left (2 \, a d^{2} x^{4} + {\left (4 \, a c + b\right )} d x^{2} + 2 \, a c^{2} + b c\right )} \sqrt {a} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}\right ) + 4 \, {\left (a^{2} d^{2} x^{4} + a^{2} c^{2} + {\left (2 \, a^{2} c + 3 \, a b\right )} d x^{2} + 3 \, a b c\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{8 \, {\left (a^{4} d^{2} x^{2} + {\left (a^{4} c + a^{3} b\right )} d\right )}}, \frac {3 \, {\left (a b d x^{2} + a b c + b^{2}\right )} \sqrt {-a} \arctan \left (\frac {{\left (2 \, a d x^{2} + 2 \, a c + b\right )} \sqrt {-a} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{2 \, {\left (a^{2} d x^{2} + a^{2} c + a b\right )}}\right ) + 2 \, {\left (a^{2} d^{2} x^{4} + a^{2} c^{2} + {\left (2 \, a^{2} c + 3 \, a b\right )} d x^{2} + 3 \, a b c\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{4 \, {\left (a^{4} d^{2} x^{2} + {\left (a^{4} c + a^{3} b\right )} d\right )}}\right ] \]
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\[ \int \frac {x}{\left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=\int \frac {x}{\left (\frac {a c + a d x^{2} + b}{c + d x^{2}}\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.61 \[ \int \frac {x}{\left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=\frac {2 \, a b - \frac {3 \, {\left (a d x^{2} + a c + b\right )} b}{d x^{2} + c}}{2 \, {\left (a^{3} d \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} - a^{2} d \left (\frac {a d x^{2} + a c + b}{d x^{2} + c}\right )^{\frac {3}{2}}\right )}} + \frac {3 \, b \log \left (-\frac {\sqrt {a} - \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{\sqrt {a} + \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}\right )}{4 \, a^{\frac {5}{2}} d} \]
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Exception generated. \[ \int \frac {x}{\left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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Time = 18.32 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.61 \[ \int \frac {x}{\left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=\frac {{\left (\frac {a\,\left (d\,x^2+c\right )}{b}+1\right )}^{3/2}\,\left (d\,x^2+c\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {3}{2},\frac {5}{2};\ \frac {7}{2};\ -\frac {a\,\left (d\,x^2+c\right )}{b}\right )}{5\,d\,{\left (a+\frac {b}{d\,x^2+c}\right )}^{3/2}} \]
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