Integrand size = 27, antiderivative size = 116 \[ \int \frac {a+b x+c x^2}{(d+e x) \sqrt {f+g x}} \, dx=\frac {2 (b e g-c (e f+d g)) \sqrt {f+g x}}{e^2 g^2}+\frac {2 c (f+g x)^{3/2}}{3 e g^2}-\frac {2 \left (c d^2-b d e+a e^2\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{e^{5/2} \sqrt {e f-d g}} \]
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Time = 0.11 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {911, 1167, 214} \[ \int \frac {a+b x+c x^2}{(d+e x) \sqrt {f+g x}} \, dx=-\frac {2 \left (a e^2-b d e+c d^2\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{e^{5/2} \sqrt {e f-d g}}+\frac {2 \sqrt {f+g x} (b e g-c (d g+e f))}{e^2 g^2}+\frac {2 c (f+g x)^{3/2}}{3 e g^2} \]
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Rule 214
Rule 911
Rule 1167
Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {\frac {c f^2-b f g+a g^2}{g^2}-\frac {(2 c f-b g) x^2}{g^2}+\frac {c x^4}{g^2}}{\frac {-e f+d g}{g}+\frac {e x^2}{g}} \, dx,x,\sqrt {f+g x}\right )}{g} \\ & = \frac {2 \text {Subst}\left (\int \left (\frac {b e g-c (e f+d g)}{e^2 g}+\frac {c x^2}{e g}+\frac {c d^2-b d e+a e^2}{e^2 \left (d-\frac {e f}{g}+\frac {e x^2}{g}\right )}\right ) \, dx,x,\sqrt {f+g x}\right )}{g} \\ & = \frac {2 (b e g-c (e f+d g)) \sqrt {f+g x}}{e^2 g^2}+\frac {2 c (f+g x)^{3/2}}{3 e g^2}+\frac {\left (2 \left (c d^2-b d e+a e^2\right )\right ) \text {Subst}\left (\int \frac {1}{d-\frac {e f}{g}+\frac {e x^2}{g}} \, dx,x,\sqrt {f+g x}\right )}{e^2 g} \\ & = \frac {2 (b e g-c (e f+d g)) \sqrt {f+g x}}{e^2 g^2}+\frac {2 c (f+g x)^{3/2}}{3 e g^2}-\frac {2 \left (c d^2-b d e+a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{e^{5/2} \sqrt {e f-d g}} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.90 \[ \int \frac {a+b x+c x^2}{(d+e x) \sqrt {f+g x}} \, dx=\frac {2 \sqrt {f+g x} (3 b e g+c (-2 e f-3 d g+e g x))}{3 e^2 g^2}+\frac {2 \left (c d^2+e (-b d+a e)\right ) \arctan \left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {-e f+d g}}\right )}{e^{5/2} \sqrt {-e f+d g}} \]
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Time = 0.54 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.80
method | result | size |
risch | \(\frac {2 \left (c e g x +3 b e g -3 c d g -2 c e f \right ) \sqrt {g x +f}}{3 g^{2} e^{2}}+\frac {2 \left (e^{2} a -b d e +c \,d^{2}\right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{e^{2} \sqrt {\left (d g -e f \right ) e}}\) | \(93\) |
pseudoelliptic | \(\frac {\frac {2 \sqrt {g x +f}\, \left (c e g x +3 b e g -3 c d g -2 c e f \right )}{3}+\frac {2 g^{2} \left (e^{2} a -b d e +c \,d^{2}\right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{\sqrt {\left (d g -e f \right ) e}}}{e^{2} g^{2}}\) | \(94\) |
derivativedivides | \(\frac {\frac {2 \left (\frac {c \left (g x +f \right )^{\frac {3}{2}} e}{3}+b e g \sqrt {g x +f}-c d g \sqrt {g x +f}-c e f \sqrt {g x +f}\right )}{e^{2}}+\frac {2 g^{2} \left (e^{2} a -b d e +c \,d^{2}\right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{e^{2} \sqrt {\left (d g -e f \right ) e}}}{g^{2}}\) | \(115\) |
default | \(\frac {\frac {2 \left (\frac {c \left (g x +f \right )^{\frac {3}{2}} e}{3}+b e g \sqrt {g x +f}-c d g \sqrt {g x +f}-c e f \sqrt {g x +f}\right )}{e^{2}}+\frac {2 g^{2} \left (e^{2} a -b d e +c \,d^{2}\right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{e^{2} \sqrt {\left (d g -e f \right ) e}}}{g^{2}}\) | \(115\) |
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Time = 0.45 (sec) , antiderivative size = 341, normalized size of antiderivative = 2.94 \[ \int \frac {a+b x+c x^2}{(d+e x) \sqrt {f+g x}} \, dx=\left [\frac {3 \, {\left (c d^{2} - b d e + a e^{2}\right )} \sqrt {e^{2} f - d e g} g^{2} \log \left (\frac {e g x + 2 \, e f - d g - 2 \, \sqrt {e^{2} f - d e g} \sqrt {g x + f}}{e x + d}\right ) - 2 \, {\left (2 \, c e^{3} f^{2} + {\left (c d e^{2} - 3 \, b e^{3}\right )} f g - 3 \, {\left (c d^{2} e - b d e^{2}\right )} g^{2} - {\left (c e^{3} f g - c d e^{2} g^{2}\right )} x\right )} \sqrt {g x + f}}{3 \, {\left (e^{4} f g^{2} - d e^{3} g^{3}\right )}}, \frac {2 \, {\left (3 \, {\left (c d^{2} - b d e + a e^{2}\right )} \sqrt {-e^{2} f + d e g} g^{2} \arctan \left (\frac {\sqrt {-e^{2} f + d e g} \sqrt {g x + f}}{e g x + e f}\right ) - {\left (2 \, c e^{3} f^{2} + {\left (c d e^{2} - 3 \, b e^{3}\right )} f g - 3 \, {\left (c d^{2} e - b d e^{2}\right )} g^{2} - {\left (c e^{3} f g - c d e^{2} g^{2}\right )} x\right )} \sqrt {g x + f}\right )}}{3 \, {\left (e^{4} f g^{2} - d e^{3} g^{3}\right )}}\right ] \]
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Time = 2.58 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.38 \[ \int \frac {a+b x+c x^2}{(d+e x) \sqrt {f+g x}} \, dx=\begin {cases} \frac {2 \left (\frac {c \left (f + g x\right )^{\frac {3}{2}}}{3 e g} + \frac {\sqrt {f + g x} \left (b e g - c d g - c e f\right )}{e^{2} g} + \frac {g \left (a e^{2} - b d e + c d^{2}\right ) \operatorname {atan}{\left (\frac {\sqrt {f + g x}}{\sqrt {\frac {d g - e f}{e}}} \right )}}{e^{3} \sqrt {\frac {d g - e f}{e}}}\right )}{g} & \text {for}\: g \neq 0 \\\frac {\frac {c x^{2}}{2 e} + \frac {x \left (b e - c d\right )}{e^{2}} + \frac {\left (a e^{2} - b d e + c d^{2}\right ) \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x \right )}}{e} & \text {otherwise} \end {cases}\right )}{e^{2}}}{\sqrt {f}} & \text {otherwise} \end {cases} \]
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Exception generated. \[ \int \frac {a+b x+c x^2}{(d+e x) \sqrt {f+g x}} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.28 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.12 \[ \int \frac {a+b x+c x^2}{(d+e x) \sqrt {f+g x}} \, dx=\frac {2 \, {\left (c d^{2} - b d e + a e^{2}\right )} \arctan \left (\frac {\sqrt {g x + f} e}{\sqrt {-e^{2} f + d e g}}\right )}{\sqrt {-e^{2} f + d e g} e^{2}} + \frac {2 \, {\left ({\left (g x + f\right )}^{\frac {3}{2}} c e^{2} g^{4} - 3 \, \sqrt {g x + f} c e^{2} f g^{4} - 3 \, \sqrt {g x + f} c d e g^{5} + 3 \, \sqrt {g x + f} b e^{2} g^{5}\right )}}{3 \, e^{3} g^{6}} \]
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Time = 0.15 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.01 \[ \int \frac {a+b x+c x^2}{(d+e x) \sqrt {f+g x}} \, dx=\sqrt {f+g\,x}\,\left (\frac {2\,b\,g-4\,c\,f}{e\,g^2}-\frac {2\,c\,\left (d\,g^3-e\,f\,g^2\right )}{e^2\,g^4}\right )+\frac {2\,\mathrm {atan}\left (\frac {\sqrt {e}\,\sqrt {f+g\,x}}{\sqrt {d\,g-e\,f}}\right )\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{e^{5/2}\,\sqrt {d\,g-e\,f}}+\frac {2\,c\,{\left (f+g\,x\right )}^{3/2}}{3\,e\,g^2} \]
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