Integrand size = 17, antiderivative size = 110 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^3} \, dx=-\frac {\sqrt {c+d x}}{2 b (a+b x)^2}-\frac {d \sqrt {c+d x}}{4 b (b c-a d) (a+b x)}+\frac {d^2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{4 b^{3/2} (b c-a d)^{3/2}} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {43, 44, 65, 214} \[ \int \frac {\sqrt {c+d x}}{(a+b x)^3} \, dx=\frac {d^2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{4 b^{3/2} (b c-a d)^{3/2}}-\frac {d \sqrt {c+d x}}{4 b (a+b x) (b c-a d)}-\frac {\sqrt {c+d x}}{2 b (a+b x)^2} \]
[In]
[Out]
Rule 43
Rule 44
Rule 65
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {c+d x}}{2 b (a+b x)^2}+\frac {d \int \frac {1}{(a+b x)^2 \sqrt {c+d x}} \, dx}{4 b} \\ & = -\frac {\sqrt {c+d x}}{2 b (a+b x)^2}-\frac {d \sqrt {c+d x}}{4 b (b c-a d) (a+b x)}-\frac {d^2 \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{8 b (b c-a d)} \\ & = -\frac {\sqrt {c+d x}}{2 b (a+b x)^2}-\frac {d \sqrt {c+d x}}{4 b (b c-a d) (a+b x)}-\frac {d \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{4 b (b c-a d)} \\ & = -\frac {\sqrt {c+d x}}{2 b (a+b x)^2}-\frac {d \sqrt {c+d x}}{4 b (b c-a d) (a+b x)}+\frac {d^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{4 b^{3/2} (b c-a d)^{3/2}} \\ \end{align*}
Time = 0.37 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^3} \, dx=\frac {\frac {\sqrt {b} \sqrt {c+d x} (2 b c-a d+b d x)}{(-b c+a d) (a+b x)^2}+\frac {d^2 \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{(-b c+a d)^{3/2}}}{4 b^{3/2}} \]
[In]
[Out]
Time = 0.51 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.79
method | result | size |
pseudoelliptic | \(\frac {d^{2} \left (-\frac {\sqrt {d x +c}\, \left (-b d x +a d -2 b c \right )}{d^{2} \left (b x +a \right )^{2}}+\frac {\arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}}\right )}{4 \left (a d -b c \right ) b}\) | \(87\) |
derivativedivides | \(2 d^{2} \left (\frac {\frac {\left (d x +c \right )^{\frac {3}{2}}}{8 a d -8 b c}-\frac {\sqrt {d x +c}}{8 b}}{\left (\left (d x +c \right ) b +a d -b c \right )^{2}}+\frac {\arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{8 \left (a d -b c \right ) b \sqrt {\left (a d -b c \right ) b}}\right )\) | \(106\) |
default | \(2 d^{2} \left (\frac {\frac {\left (d x +c \right )^{\frac {3}{2}}}{8 a d -8 b c}-\frac {\sqrt {d x +c}}{8 b}}{\left (\left (d x +c \right ) b +a d -b c \right )^{2}}+\frac {\arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{8 \left (a d -b c \right ) b \sqrt {\left (a d -b c \right ) b}}\right )\) | \(106\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (90) = 180\).
Time = 0.24 (sec) , antiderivative size = 456, normalized size of antiderivative = 4.15 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^3} \, dx=\left [-\frac {{\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \sqrt {b^{2} c - a b d} \log \left (\frac {b d x + 2 \, b c - a d - 2 \, \sqrt {b^{2} c - a b d} \sqrt {d x + c}}{b x + a}\right ) + 2 \, {\left (2 \, b^{3} c^{2} - 3 \, a b^{2} c d + a^{2} b d^{2} + {\left (b^{3} c d - a b^{2} d^{2}\right )} x\right )} \sqrt {d x + c}}{8 \, {\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2} + {\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} x^{2} + 2 \, {\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} x\right )}}, -\frac {{\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \sqrt {-b^{2} c + a b d} \arctan \left (\frac {\sqrt {-b^{2} c + a b d} \sqrt {d x + c}}{b d x + b c}\right ) + {\left (2 \, b^{3} c^{2} - 3 \, a b^{2} c d + a^{2} b d^{2} + {\left (b^{3} c d - a b^{2} d^{2}\right )} x\right )} \sqrt {d x + c}}{4 \, {\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2} + {\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} x^{2} + 2 \, {\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} x\right )}}\right ] \]
[In]
[Out]
\[ \int \frac {\sqrt {c+d x}}{(a+b x)^3} \, dx=\int \frac {\sqrt {c + d x}}{\left (a + b x\right )^{3}}\, dx \]
[In]
[Out]
Exception generated. \[ \int \frac {\sqrt {c+d x}}{(a+b x)^3} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.15 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^3} \, dx=-\frac {d^{2} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{4 \, {\left (b^{2} c - a b d\right )} \sqrt {-b^{2} c + a b d}} - \frac {{\left (d x + c\right )}^{\frac {3}{2}} b d^{2} + \sqrt {d x + c} b c d^{2} - \sqrt {d x + c} a d^{3}}{4 \, {\left (b^{2} c - a b d\right )} {\left ({\left (d x + c\right )} b - b c + a d\right )}^{2}} \]
[In]
[Out]
Time = 0.35 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.23 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^3} \, dx=\frac {d^2\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c+d\,x}}{\sqrt {a\,d-b\,c}}\right )}{4\,b^{3/2}\,{\left (a\,d-b\,c\right )}^{3/2}}-\frac {\frac {d^2\,\sqrt {c+d\,x}}{4\,b}-\frac {d^2\,{\left (c+d\,x\right )}^{3/2}}{4\,\left (a\,d-b\,c\right )}}{b^2\,{\left (c+d\,x\right )}^2-\left (2\,b^2\,c-2\,a\,b\,d\right )\,\left (c+d\,x\right )+a^2\,d^2+b^2\,c^2-2\,a\,b\,c\,d} \]
[In]
[Out]