Integrand size = 22, antiderivative size = 102 \[ \int \frac {\sqrt {a+b x}}{x (c+d x)^{5/2}} \, dx=-\frac {2 d (a+b x)^{3/2}}{3 c (b c-a d) (c+d x)^{3/2}}+\frac {2 \sqrt {a+b x}}{c^2 \sqrt {c+d x}}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{5/2}} \]
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Time = 0.03 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {98, 96, 95, 214} \[ \int \frac {\sqrt {a+b x}}{x (c+d x)^{5/2}} \, dx=-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{5/2}}+\frac {2 \sqrt {a+b x}}{c^2 \sqrt {c+d x}}-\frac {2 d (a+b x)^{3/2}}{3 c (c+d x)^{3/2} (b c-a d)} \]
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Rule 95
Rule 96
Rule 98
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {2 d (a+b x)^{3/2}}{3 c (b c-a d) (c+d x)^{3/2}}+\frac {\int \frac {\sqrt {a+b x}}{x (c+d x)^{3/2}} \, dx}{c} \\ & = -\frac {2 d (a+b x)^{3/2}}{3 c (b c-a d) (c+d x)^{3/2}}+\frac {2 \sqrt {a+b x}}{c^2 \sqrt {c+d x}}+\frac {a \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{c^2} \\ & = -\frac {2 d (a+b x)^{3/2}}{3 c (b c-a d) (c+d x)^{3/2}}+\frac {2 \sqrt {a+b x}}{c^2 \sqrt {c+d x}}+\frac {(2 a) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{c^2} \\ & = -\frac {2 d (a+b x)^{3/2}}{3 c (b c-a d) (c+d x)^{3/2}}+\frac {2 \sqrt {a+b x}}{c^2 \sqrt {c+d x}}-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{5/2}} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.01 \[ \int \frac {\sqrt {a+b x}}{x (c+d x)^{5/2}} \, dx=\frac {2 \sqrt {a+b x} \left (3 b c-3 a d-\frac {c d (a+b x)}{c+d x}\right )}{3 c^2 (b c-a d) \sqrt {c+d x}}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{5/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(429\) vs. \(2(80)=160\).
Time = 1.27 (sec) , antiderivative size = 430, normalized size of antiderivative = 4.22
method | result | size |
default | \(-\frac {\left (3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} d^{3} x^{2}-3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a b c \,d^{2} x^{2}+6 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} c \,d^{2} x -6 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a b \,c^{2} d x +3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} c^{2} d -3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a b \,c^{3}-6 a \,d^{2} x \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+4 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b c d x -8 a c d \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+6 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b \,c^{2}\right ) \sqrt {b x +a}}{3 \left (a d -b c \right ) \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \left (d x +c \right )^{\frac {3}{2}} c^{2}}\) | \(430\) |
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Leaf count of result is larger than twice the leaf count of optimal. 224 vs. \(2 (80) = 160\).
Time = 0.42 (sec) , antiderivative size = 479, normalized size of antiderivative = 4.70 \[ \int \frac {\sqrt {a+b x}}{x (c+d x)^{5/2}} \, dx=\left [\frac {3 \, {\left (b c^{3} - a c^{2} d + {\left (b c d^{2} - a d^{3}\right )} x^{2} + 2 \, {\left (b c^{2} d - a c d^{2}\right )} x\right )} \sqrt {\frac {a}{c}} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c^{2} + {\left (b c^{2} + a c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {a}{c}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (3 \, b c^{2} - 4 \, a c d + {\left (2 \, b c d - 3 \, a d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{6 \, {\left (b c^{5} - a c^{4} d + {\left (b c^{3} d^{2} - a c^{2} d^{3}\right )} x^{2} + 2 \, {\left (b c^{4} d - a c^{3} d^{2}\right )} x\right )}}, \frac {3 \, {\left (b c^{3} - a c^{2} d + {\left (b c d^{2} - a d^{3}\right )} x^{2} + 2 \, {\left (b c^{2} d - a c d^{2}\right )} x\right )} \sqrt {-\frac {a}{c}} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {a}{c}}}{2 \, {\left (a b d x^{2} + a^{2} c + {\left (a b c + a^{2} d\right )} x\right )}}\right ) + 2 \, {\left (3 \, b c^{2} - 4 \, a c d + {\left (2 \, b c d - 3 \, a d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3 \, {\left (b c^{5} - a c^{4} d + {\left (b c^{3} d^{2} - a c^{2} d^{3}\right )} x^{2} + 2 \, {\left (b c^{4} d - a c^{3} d^{2}\right )} x\right )}}\right ] \]
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\[ \int \frac {\sqrt {a+b x}}{x (c+d x)^{5/2}} \, dx=\int \frac {\sqrt {a + b x}}{x \left (c + d x\right )^{\frac {5}{2}}}\, dx \]
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Exception generated. \[ \int \frac {\sqrt {a+b x}}{x (c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (80) = 160\).
Time = 0.35 (sec) , antiderivative size = 243, normalized size of antiderivative = 2.38 \[ \int \frac {\sqrt {a+b x}}{x (c+d x)^{5/2}} \, dx=\frac {2 \, \sqrt {b x + a} {\left (\frac {{\left (2 \, b^{4} c^{3} d^{2} {\left | b \right |} - 3 \, a b^{3} c^{2} d^{3} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{3} c^{5} d - a b^{2} c^{4} d^{2}} + \frac {3 \, {\left (b^{5} c^{4} d {\left | b \right |} - 2 \, a b^{4} c^{3} d^{2} {\left | b \right |} + a^{2} b^{3} c^{2} d^{3} {\left | b \right |}\right )}}{b^{3} c^{5} d - a b^{2} c^{4} d^{2}}\right )}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} - \frac {2 \, \sqrt {b d} a b \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} c^{2} {\left | b \right |}} \]
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Timed out. \[ \int \frac {\sqrt {a+b x}}{x (c+d x)^{5/2}} \, dx=\int \frac {\sqrt {a+b\,x}}{x\,{\left (c+d\,x\right )}^{5/2}} \,d x \]
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