Integrand size = 20, antiderivative size = 68 \[ \int \frac {d+e x}{\sqrt {a+b x+c x^2}} \, dx=\frac {e \sqrt {a+b x+c x^2}}{c}+\frac {(2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 c^{3/2}} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {654, 635, 212} \[ \int \frac {d+e x}{\sqrt {a+b x+c x^2}} \, dx=\frac {(2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 c^{3/2}}+\frac {e \sqrt {a+b x+c x^2}}{c} \]
[In]
[Out]
Rule 212
Rule 635
Rule 654
Rubi steps \begin{align*} \text {integral}& = \frac {e \sqrt {a+b x+c x^2}}{c}+\frac {(2 c d-b e) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{2 c} \\ & = \frac {e \sqrt {a+b x+c x^2}}{c}+\frac {(2 c d-b e) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{c} \\ & = \frac {e \sqrt {a+b x+c x^2}}{c}+\frac {(2 c d-b e) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 c^{3/2}} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.97 \[ \int \frac {d+e x}{\sqrt {a+b x+c x^2}} \, dx=\frac {e \sqrt {a+x (b+c x)}}{c}+\frac {(2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{2 c^{3/2}} \]
[In]
[Out]
Time = 0.33 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.84
method | result | size |
risch | \(\frac {e \sqrt {c \,x^{2}+b x +a}}{c}-\frac {\left (b e -2 c d \right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\) | \(57\) |
default | \(\frac {d \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{\sqrt {c}}+e \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )\) | \(82\) |
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 165, normalized size of antiderivative = 2.43 \[ \int \frac {d+e x}{\sqrt {a+b x+c x^2}} \, dx=\left [\frac {4 \, \sqrt {c x^{2} + b x + a} c e - {\left (2 \, c d - b e\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} + 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right )}{4 \, c^{2}}, \frac {2 \, \sqrt {c x^{2} + b x + a} c e - {\left (2 \, c d - b e\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right )}{2 \, c^{2}}\right ] \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 146 vs. \(2 (56) = 112\).
Time = 0.42 (sec) , antiderivative size = 146, normalized size of antiderivative = 2.15 \[ \int \frac {d+e x}{\sqrt {a+b x+c x^2}} \, dx=\begin {cases} \left (- \frac {b e}{2 c} + d\right ) \left (\begin {cases} \frac {\log {\left (b + 2 \sqrt {c} \sqrt {a + b x + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: a - \frac {b^{2}}{4 c} \neq 0 \\\frac {\left (\frac {b}{2 c} + x\right ) \log {\left (\frac {b}{2 c} + x \right )}}{\sqrt {c \left (\frac {b}{2 c} + x\right )^{2}}} & \text {otherwise} \end {cases}\right ) + \frac {e \sqrt {a + b x + c x^{2}}}{c} & \text {for}\: c \neq 0 \\\frac {2 d \sqrt {a + b x} + \frac {2 e \left (- a \sqrt {a + b x} + \frac {\left (a + b x\right )^{\frac {3}{2}}}{3}\right )}{b}}{b} & \text {for}\: b \neq 0 \\\frac {d x + \frac {e x^{2}}{2}}{\sqrt {a}} & \text {otherwise} \end {cases} \]
[In]
[Out]
Exception generated. \[ \int \frac {d+e x}{\sqrt {a+b x+c x^2}} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.90 \[ \int \frac {d+e x}{\sqrt {a+b x+c x^2}} \, dx=\frac {\sqrt {c x^{2} + b x + a} e}{c} - \frac {{\left (2 \, c d - b e\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{2 \, c^{\frac {3}{2}}} \]
[In]
[Out]
Time = 10.31 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.18 \[ \int \frac {d+e x}{\sqrt {a+b x+c x^2}} \, dx=\frac {e\,\sqrt {c\,x^2+b\,x+a}}{c}+\frac {d\,\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )}{\sqrt {c}}-\frac {b\,e\,\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )}{2\,c^{3/2}} \]
[In]
[Out]