Integrand size = 17, antiderivative size = 69 \[ \int \frac {1}{(a+b x) (c+d x)^{3/2}} \, dx=\frac {2}{(b c-a d) \sqrt {c+d x}}-\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{(b c-a d)^{3/2}} \]
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Time = 0.03 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {53, 65, 214} \[ \int \frac {1}{(a+b x) (c+d x)^{3/2}} \, dx=\frac {2}{\sqrt {c+d x} (b c-a d)}-\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{(b c-a d)^{3/2}} \]
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Rule 53
Rule 65
Rule 214
Rubi steps \begin{align*} \text {integral}& = \frac {2}{(b c-a d) \sqrt {c+d x}}+\frac {b \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{b c-a d} \\ & = \frac {2}{(b c-a d) \sqrt {c+d x}}+\frac {(2 b) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{d (b c-a d)} \\ & = \frac {2}{(b c-a d) \sqrt {c+d x}}-\frac {2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{(b c-a d)^{3/2}} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(a+b x) (c+d x)^{3/2}} \, dx=\frac {2}{(b c-a d) \sqrt {c+d x}}-\frac {2 \sqrt {b} \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{(-b c+a d)^{3/2}} \]
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Time = 0.26 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.99
method | result | size |
derivativedivides | \(-\frac {2 b \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{\left (a d -b c \right ) \sqrt {\left (a d -b c \right ) b}}-\frac {2}{\left (a d -b c \right ) \sqrt {d x +c}}\) | \(68\) |
default | \(-\frac {2 b \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{\left (a d -b c \right ) \sqrt {\left (a d -b c \right ) b}}-\frac {2}{\left (a d -b c \right ) \sqrt {d x +c}}\) | \(68\) |
pseudoelliptic | \(-\frac {2 \left (b \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right ) \sqrt {d x +c}+\sqrt {\left (a d -b c \right ) b}\right )}{\left (a d -b c \right ) \sqrt {\left (a d -b c \right ) b}\, \sqrt {d x +c}}\) | \(76\) |
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Time = 0.23 (sec) , antiderivative size = 214, normalized size of antiderivative = 3.10 \[ \int \frac {1}{(a+b x) (c+d x)^{3/2}} \, dx=\left [-\frac {{\left (d x + c\right )} \sqrt {\frac {b}{b c - a d}} \log \left (\frac {b d x + 2 \, b c - a d + 2 \, {\left (b c - a d\right )} \sqrt {d x + c} \sqrt {\frac {b}{b c - a d}}}{b x + a}\right ) - 2 \, \sqrt {d x + c}}{b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x}, -\frac {2 \, {\left ({\left (d x + c\right )} \sqrt {-\frac {b}{b c - a d}} \arctan \left (-\frac {{\left (b c - a d\right )} \sqrt {d x + c} \sqrt {-\frac {b}{b c - a d}}}{b d x + b c}\right ) - \sqrt {d x + c}\right )}}{b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x}\right ] \]
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Time = 2.88 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.19 \[ \int \frac {1}{(a+b x) (c+d x)^{3/2}} \, dx=\begin {cases} \frac {2 \left (- \frac {d}{\sqrt {c + d x} \left (a d - b c\right )} - \frac {d \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d - b c}{b}}} \right )}}{\sqrt {\frac {a d - b c}{b}} \left (a d - b c\right )}\right )}{d} & \text {for}\: d \neq 0 \\\frac {\begin {cases} \frac {x}{a} & \text {for}\: b = 0 \\\frac {\log {\left (a + b x \right )}}{b} & \text {otherwise} \end {cases}}{c^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
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Exception generated. \[ \int \frac {1}{(a+b x) (c+d x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.32 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(a+b x) (c+d x)^{3/2}} \, dx=\frac {2 \, b \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} {\left (b c - a d\right )}} + \frac {2}{{\left (b c - a d\right )} \sqrt {d x + c}} \]
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Time = 0.29 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.83 \[ \int \frac {1}{(a+b x) (c+d x)^{3/2}} \, dx=-\frac {2}{\left (a\,d-b\,c\right )\,\sqrt {c+d\,x}}-\frac {2\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c+d\,x}}{\sqrt {a\,d-b\,c}}\right )}{{\left (a\,d-b\,c\right )}^{3/2}} \]
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