Integrand size = 22, antiderivative size = 104 \[ \int \frac {A+C x^2}{\sqrt {a+b x+c x^2}} \, dx=-\frac {3 b C \sqrt {a+b x+c x^2}}{4 c^2}+\frac {C x \sqrt {a+b x+c x^2}}{2 c}+\frac {\left (8 A c^2+3 b^2 C-4 a c C\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{5/2}} \]
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Time = 0.05 (sec) , antiderivative size = 104, 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 = {1675, 654, 635, 212} \[ \int \frac {A+C x^2}{\sqrt {a+b x+c x^2}} \, dx=\frac {\left (-4 a c C+8 A c^2+3 b^2 C\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{5/2}}-\frac {3 b C \sqrt {a+b x+c x^2}}{4 c^2}+\frac {C x \sqrt {a+b x+c x^2}}{2 c} \]
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Rule 212
Rule 635
Rule 654
Rule 1675
Rubi steps \begin{align*} \text {integral}& = \frac {C x \sqrt {a+b x+c x^2}}{2 c}+\frac {\int \frac {2 A c-a C-\frac {3 b C x}{2}}{\sqrt {a+b x+c x^2}} \, dx}{2 c} \\ & = -\frac {3 b C \sqrt {a+b x+c x^2}}{4 c^2}+\frac {C x \sqrt {a+b x+c x^2}}{2 c}+\frac {\left (\frac {3 b^2 C}{2}+2 c (2 A c-a C)\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{4 c^2} \\ & = -\frac {3 b C \sqrt {a+b x+c x^2}}{4 c^2}+\frac {C x \sqrt {a+b x+c x^2}}{2 c}+\frac {\left (\frac {3 b^2 C}{2}+2 c (2 A c-a C)\right ) \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 )}{2 c^2} \\ & = -\frac {3 b C \sqrt {a+b x+c x^2}}{4 c^2}+\frac {C x \sqrt {a+b x+c x^2}}{2 c}+\frac {\left (8 A c^2+3 b^2 C-4 a c C\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{5/2}} \\ \end{align*}
Time = 0.40 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.85 \[ \int \frac {A+C x^2}{\sqrt {a+b x+c x^2}} \, dx=\frac {C (-3 b+2 c x) \sqrt {a+x (b+c x)}}{4 c^2}+\frac {\left (8 A c^2+3 b^2 C-4 a c C\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )}{4 c^{5/2}} \]
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Time = 0.67 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.73
method | result | size |
risch | \(-\frac {C \left (-2 c x +3 b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c^{2}}+\frac {\left (8 A \,c^{2}-4 C a c +3 C \,b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {5}{2}}}\) | \(76\) |
default | \(\frac {A \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{\sqrt {c}}+C \left (\frac {x \sqrt {c \,x^{2}+b x +a}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}-\frac {a \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )\) | \(138\) |
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Time = 0.30 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.95 \[ \int \frac {A+C x^2}{\sqrt {a+b x+c x^2}} \, dx=\left [\frac {{\left (3 \, C b^{2} - 4 \, C a c + 8 \, A c^{2}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (2 \, C c^{2} x - 3 \, C b c\right )} \sqrt {c x^{2} + b x + a}}{16 \, c^{3}}, -\frac {{\left (3 \, C b^{2} - 4 \, C a c + 8 \, A c^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) - 2 \, {\left (2 \, C c^{2} x - 3 \, C b c\right )} \sqrt {c x^{2} + b x + a}}{8 \, c^{3}}\right ] \]
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Time = 0.37 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.82 \[ \int \frac {A+C x^2}{\sqrt {a+b x+c x^2}} \, dx=\begin {cases} \left (- \frac {3 C b}{4 c^{2}} + \frac {C x}{2 c}\right ) \sqrt {a + b x + c x^{2}} + \left (A - \frac {C a}{2 c} + \frac {3 C b^{2}}{8 c^{2}}\right ) \left (\begin {cases} \frac {\log {\left (b + 2 \sqrt {c} \sqrt {a + b x + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: a - \frac {b^{2}}{4 c} \neq 0 \\\frac {\left (\frac {b}{2 c} + x\right ) \log {\left (\frac {b}{2 c} + x \right )}}{\sqrt {c \left (\frac {b}{2 c} + x\right )^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: c \neq 0 \\\frac {2 A \sqrt {a + b x} + \frac {2 C \left (a^{2} \sqrt {a + b x} - \frac {2 a \left (a + b x\right )^{\frac {3}{2}}}{3} + \frac {\left (a + b x\right )^{\frac {5}{2}}}{5}\right )}{b^{2}}}{b} & \text {for}\: b \neq 0 \\\frac {A x + \frac {C x^{3}}{3}}{\sqrt {a}} & \text {otherwise} \end {cases} \]
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Exception generated. \[ \int \frac {A+C x^2}{\sqrt {a+b x+c x^2}} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.29 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.79 \[ \int \frac {A+C x^2}{\sqrt {a+b x+c x^2}} \, dx=\frac {1}{4} \, \sqrt {c x^{2} + b x + a} {\left (\frac {2 \, C x}{c} - \frac {3 \, C b}{c^{2}}\right )} - \frac {{\left (3 \, C b^{2} - 4 \, C a c + 8 \, A c^{2}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{8 \, c^{\frac {5}{2}}} \]
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Timed out. \[ \int \frac {A+C x^2}{\sqrt {a+b x+c x^2}} \, dx=\int \frac {C\,x^2+A}{\sqrt {c\,x^2+b\,x+a}} \,d x \]
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