Integrand size = 17, antiderivative size = 76 \[ \int \frac {1}{(a+b x)^2 \sqrt {c+d x}} \, dx=-\frac {\sqrt {c+d x}}{(b c-a d) (a+b x)}+\frac {d \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{\sqrt {b} (b c-a d)^{3/2}} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 76, 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 = {44, 65, 214} \[ \int \frac {1}{(a+b x)^2 \sqrt {c+d x}} \, dx=\frac {d \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{\sqrt {b} (b c-a d)^{3/2}}-\frac {\sqrt {c+d x}}{(a+b x) (b c-a d)} \]
[In]
[Out]
Rule 44
Rule 65
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {c+d x}}{(b c-a d) (a+b x)}-\frac {d \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{2 (b c-a d)} \\ & = -\frac {\sqrt {c+d x}}{(b c-a d) (a+b x)}-\frac {\text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{b c-a d} \\ & = -\frac {\sqrt {c+d x}}{(b c-a d) (a+b x)}+\frac {d \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{\sqrt {b} (b c-a d)^{3/2}} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.99 \[ \int \frac {1}{(a+b x)^2 \sqrt {c+d x}} \, dx=\frac {\sqrt {c+d x}}{(-b c+a d) (a+b x)}+\frac {d \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{\sqrt {b} (-b c+a d)^{3/2}} \]
[In]
[Out]
Time = 0.50 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.84
method | result | size |
pseudoelliptic | \(\frac {\frac {\sqrt {d x +c}}{b x +a}+\frac {d \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}}}{a d -b c}\) | \(64\) |
derivativedivides | \(2 d \left (\frac {\sqrt {d x +c}}{2 \left (a d -b c \right ) \left (\left (d x +c \right ) b +a d -b c \right )}+\frac {\arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \left (a d -b c \right ) \sqrt {\left (a d -b c \right ) b}}\right )\) | \(87\) |
default | \(2 d \left (\frac {\sqrt {d x +c}}{2 \left (a d -b c \right ) \left (\left (d x +c \right ) b +a d -b c \right )}+\frac {\arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \left (a d -b c \right ) \sqrt {\left (a d -b c \right ) b}}\right )\) | \(87\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (64) = 128\).
Time = 0.23 (sec) , antiderivative size = 280, normalized size of antiderivative = 3.68 \[ \int \frac {1}{(a+b x)^2 \sqrt {c+d x}} \, dx=\left [-\frac {\sqrt {b^{2} c - a b d} {\left (b d x + a d\right )} \log \left (\frac {b d x + 2 \, b c - a d - 2 \, \sqrt {b^{2} c - a b d} \sqrt {d x + c}}{b x + a}\right ) + 2 \, {\left (b^{2} c - a b d\right )} \sqrt {d x + c}}{2 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2} + {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x\right )}}, -\frac {\sqrt {-b^{2} c + a b d} {\left (b d x + a d\right )} \arctan \left (\frac {\sqrt {-b^{2} c + a b d} \sqrt {d x + c}}{b d x + b c}\right ) + {\left (b^{2} c - a b d\right )} \sqrt {d x + c}}{a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2} + {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x}\right ] \]
[In]
[Out]
\[ \int \frac {1}{(a+b x)^2 \sqrt {c+d x}} \, dx=\int \frac {1}{\left (a + b x\right )^{2} \sqrt {c + d x}}\, dx \]
[In]
[Out]
Exception generated. \[ \int \frac {1}{(a+b x)^2 \sqrt {c+d x}} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.14 \[ \int \frac {1}{(a+b x)^2 \sqrt {c+d x}} \, dx=-\frac {d \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} {\left (b c - a d\right )}} - \frac {\sqrt {d x + c} d}{{\left ({\left (d x + c\right )} b - b c + a d\right )} {\left (b c - a d\right )}} \]
[In]
[Out]
Time = 0.09 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.97 \[ \int \frac {1}{(a+b x)^2 \sqrt {c+d x}} \, dx=\frac {d\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c+d\,x}}{\sqrt {a\,d-b\,c}}\right )}{\sqrt {b}\,{\left (a\,d-b\,c\right )}^{3/2}}+\frac {d\,\sqrt {c+d\,x}}{\left (a\,d-b\,c\right )\,\left (a\,d-b\,c+b\,\left (c+d\,x\right )\right )} \]
[In]
[Out]