Integrand size = 27, antiderivative size = 165 \[ \int \frac {a+b x+c x^2}{(d+e x)^2 (f+g x)^{3/2}} \, dx=-\frac {2 \left (c f^2-b f g+a g^2\right )}{g (e f-d g)^2 \sqrt {f+g x}}-\frac {\left (c d^2-b d e+a e^2\right ) \sqrt {f+g x}}{e (e f-d g)^2 (d+e x)}+\frac {(c d (4 e f-d g)-e (2 b e f+b d g-3 a e g)) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{e^{3/2} (e f-d g)^{5/2}} \]
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Time = 0.23 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {911, 1273, 464, 214} \[ \int \frac {a+b x+c x^2}{(d+e x)^2 (f+g x)^{3/2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) (c d (4 e f-d g)-e (-3 a e g+b d g+2 b e f))}{e^{3/2} (e f-d g)^{5/2}}-\frac {\sqrt {f+g x} \left (a e^2-b d e+c d^2\right )}{e (d+e x) (e f-d g)^2}-\frac {2 \left (a g^2-b f g+c f^2\right )}{g \sqrt {f+g x} (e f-d g)^2} \]
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Rule 214
Rule 464
Rule 911
Rule 1273
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}}{x^2 \left (\frac {-e f+d g}{g}+\frac {e x^2}{g}\right )^2} \, dx,x,\sqrt {f+g x}\right )}{g} \\ & = -\frac {\left (c d^2-b d e+a e^2\right ) \sqrt {f+g x}}{e (e f-d g)^2 (d+e x)}-\frac {g^3 \text {Subst}\left (\int \frac {\frac {2 e^2 (e f-d g) \left (c f^2-b f g+a g^2\right )}{g^5}-\frac {e \left (e (b d-a e) g^2+c \left (2 e^2 f^2-4 d e f g+d^2 g^2\right )\right ) x^2}{g^5}}{x^2 \left (\frac {-e f+d g}{g}+\frac {e x^2}{g}\right )} \, dx,x,\sqrt {f+g x}\right )}{e^2 (e f-d g)^2} \\ & = -\frac {2 \left (c f^2-b f g+a g^2\right )}{g (e f-d g)^2 \sqrt {f+g x}}-\frac {\left (c d^2-b d e+a e^2\right ) \sqrt {f+g x}}{e (e f-d g)^2 (d+e x)}-\frac {(c d (4 e f-d g)-e (2 b e f+b d g-3 a e g)) \text {Subst}\left (\int \frac {1}{\frac {-e f+d g}{g}+\frac {e x^2}{g}} \, dx,x,\sqrt {f+g x}\right )}{e g (e f-d g)^2} \\ & = -\frac {2 \left (c f^2-b f g+a g^2\right )}{g (e f-d g)^2 \sqrt {f+g x}}-\frac {\left (c d^2-b d e+a e^2\right ) \sqrt {f+g x}}{e (e f-d g)^2 (d+e x)}+\frac {(c d (4 e f-d g)-e (2 b e f+b d g-3 a e g)) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{e^{3/2} (e f-d g)^{5/2}} \\ \end{align*}
Time = 0.58 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.07 \[ \int \frac {a+b x+c x^2}{(d+e x)^2 (f+g x)^{3/2}} \, dx=\frac {-c \left (2 d e f^2+2 e^2 f^2 x+d^2 g (f+g x)\right )+e g (b (3 d f+2 e f x+d g x)-a (e f+2 d g+3 e g x))}{e g (e f-d g)^2 (d+e x) \sqrt {f+g x}}+\frac {(c d (-4 e f+d g)+e (2 b e f+b d g-3 a e g)) \arctan \left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {-e f+d g}}\right )}{e^{3/2} (-e f+d g)^{5/2}} \]
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Time = 0.56 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.06
method | result | size |
derivativedivides | \(\frac {-\frac {2 g \left (\frac {g \left (e^{2} a -b d e +c \,d^{2}\right ) \sqrt {g x +f}}{2 e \left (e \left (g x +f \right )+d g -e f \right )}+\frac {\left (3 a \,e^{2} g -b d e g -2 b \,e^{2} f -c \,d^{2} g +4 c d e f \right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{2 e \sqrt {\left (d g -e f \right ) e}}\right )}{\left (d g -e f \right )^{2}}-\frac {2 \left (a \,g^{2}-b f g +c \,f^{2}\right )}{\left (d g -e f \right )^{2} \sqrt {g x +f}}}{g}\) | \(175\) |
default | \(\frac {-\frac {2 g \left (\frac {g \left (e^{2} a -b d e +c \,d^{2}\right ) \sqrt {g x +f}}{2 e \left (e \left (g x +f \right )+d g -e f \right )}+\frac {\left (3 a \,e^{2} g -b d e g -2 b \,e^{2} f -c \,d^{2} g +4 c d e f \right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{2 e \sqrt {\left (d g -e f \right ) e}}\right )}{\left (d g -e f \right )^{2}}-\frac {2 \left (a \,g^{2}-b f g +c \,f^{2}\right )}{\left (d g -e f \right )^{2} \sqrt {g x +f}}}{g}\) | \(175\) |
pseudoelliptic | \(-\frac {2 \left (\frac {3 \sqrt {g x +f}\, \left (e x +d \right ) g \left (\left (a g -\frac {2 b f}{3}\right ) e^{2}-\frac {d \left (b g -4 c f \right ) e}{3}-\frac {c \,d^{2} g}{3}\right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{2}+\left (\left (\frac {3 a \,g^{2} x}{2}+\frac {f \left (-2 b x +a \right ) g}{2}+c \,f^{2} x \right ) e^{2}+d \left (\left (-\frac {b x}{2}+a \right ) g^{2}-\frac {3 b f g}{2}+c \,f^{2}\right ) e +\frac {c \,d^{2} g \left (g x +f \right )}{2}\right ) \sqrt {\left (d g -e f \right ) e}\right )}{\sqrt {g x +f}\, \sqrt {\left (d g -e f \right ) e}\, g \left (e x +d \right ) \left (d g -e f \right )^{2} e}\) | \(193\) |
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Leaf count of result is larger than twice the leaf count of optimal. 537 vs. \(2 (151) = 302\).
Time = 0.50 (sec) , antiderivative size = 1088, normalized size of antiderivative = 6.59 \[ \int \frac {a+b x+c x^2}{(d+e x)^2 (f+g x)^{3/2}} \, dx=\left [\frac {{\left (2 \, {\left (2 \, c d^{2} e - b d e^{2}\right )} f^{2} g - {\left (c d^{3} + b d^{2} e - 3 \, a d e^{2}\right )} f g^{2} + {\left (2 \, {\left (2 \, c d e^{2} - b e^{3}\right )} f g^{2} - {\left (c d^{2} e + b d e^{2} - 3 \, a e^{3}\right )} g^{3}\right )} x^{2} + {\left (2 \, {\left (2 \, c d e^{2} - b e^{3}\right )} f^{2} g + 3 \, {\left (c d^{2} e - b d e^{2} + a e^{3}\right )} f g^{2} - {\left (c d^{3} + b d^{2} e - 3 \, a d e^{2}\right )} g^{3}\right )} x\right )} \sqrt {e^{2} f - d e g} \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 d e^{3} f^{3} - 2 \, a d^{2} e^{2} g^{3} - {\left (c d^{2} e^{2} + 3 \, b d e^{3} - a e^{4}\right )} f^{2} g - {\left (c d^{3} e - 3 \, b d^{2} e^{2} - a d e^{3}\right )} f g^{2} + {\left (2 \, c e^{4} f^{3} - 2 \, {\left (c d e^{3} + b e^{4}\right )} f^{2} g + {\left (c d^{2} e^{2} + b d e^{3} + 3 \, a e^{4}\right )} f g^{2} - {\left (c d^{3} e - b d^{2} e^{2} + 3 \, a d e^{3}\right )} g^{3}\right )} x\right )} \sqrt {g x + f}}{2 \, {\left (d e^{5} f^{4} g - 3 \, d^{2} e^{4} f^{3} g^{2} + 3 \, d^{3} e^{3} f^{2} g^{3} - d^{4} e^{2} f g^{4} + {\left (e^{6} f^{3} g^{2} - 3 \, d e^{5} f^{2} g^{3} + 3 \, d^{2} e^{4} f g^{4} - d^{3} e^{3} g^{5}\right )} x^{2} + {\left (e^{6} f^{4} g - 2 \, d e^{5} f^{3} g^{2} + 2 \, d^{3} e^{3} f g^{4} - d^{4} e^{2} g^{5}\right )} x\right )}}, -\frac {{\left (2 \, {\left (2 \, c d^{2} e - b d e^{2}\right )} f^{2} g - {\left (c d^{3} + b d^{2} e - 3 \, a d e^{2}\right )} f g^{2} + {\left (2 \, {\left (2 \, c d e^{2} - b e^{3}\right )} f g^{2} - {\left (c d^{2} e + b d e^{2} - 3 \, a e^{3}\right )} g^{3}\right )} x^{2} + {\left (2 \, {\left (2 \, c d e^{2} - b e^{3}\right )} f^{2} g + 3 \, {\left (c d^{2} e - b d e^{2} + a e^{3}\right )} f g^{2} - {\left (c d^{3} + b d^{2} e - 3 \, a d e^{2}\right )} g^{3}\right )} x\right )} \sqrt {-e^{2} f + d e g} \arctan \left (\frac {\sqrt {-e^{2} f + d e g} \sqrt {g x + f}}{e g x + e f}\right ) + {\left (2 \, c d e^{3} f^{3} - 2 \, a d^{2} e^{2} g^{3} - {\left (c d^{2} e^{2} + 3 \, b d e^{3} - a e^{4}\right )} f^{2} g - {\left (c d^{3} e - 3 \, b d^{2} e^{2} - a d e^{3}\right )} f g^{2} + {\left (2 \, c e^{4} f^{3} - 2 \, {\left (c d e^{3} + b e^{4}\right )} f^{2} g + {\left (c d^{2} e^{2} + b d e^{3} + 3 \, a e^{4}\right )} f g^{2} - {\left (c d^{3} e - b d^{2} e^{2} + 3 \, a d e^{3}\right )} g^{3}\right )} x\right )} \sqrt {g x + f}}{d e^{5} f^{4} g - 3 \, d^{2} e^{4} f^{3} g^{2} + 3 \, d^{3} e^{3} f^{2} g^{3} - d^{4} e^{2} f g^{4} + {\left (e^{6} f^{3} g^{2} - 3 \, d e^{5} f^{2} g^{3} + 3 \, d^{2} e^{4} f g^{4} - d^{3} e^{3} g^{5}\right )} x^{2} + {\left (e^{6} f^{4} g - 2 \, d e^{5} f^{3} g^{2} + 2 \, d^{3} e^{3} f g^{4} - d^{4} e^{2} g^{5}\right )} x}\right ] \]
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Timed out. \[ \int \frac {a+b x+c x^2}{(d+e x)^2 (f+g x)^{3/2}} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {a+b x+c x^2}{(d+e x)^2 (f+g x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.30 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.73 \[ \int \frac {a+b x+c x^2}{(d+e x)^2 (f+g x)^{3/2}} \, dx=-\frac {{\left (4 \, c d e f - 2 \, b e^{2} f - c d^{2} g - b d e g + 3 \, a e^{2} g\right )} \arctan \left (\frac {\sqrt {g x + f} e}{\sqrt {-e^{2} f + d e g}}\right )}{{\left (e^{3} f^{2} - 2 \, d e^{2} f g + d^{2} e g^{2}\right )} \sqrt {-e^{2} f + d e g}} - \frac {2 \, {\left (g x + f\right )} c e^{2} f^{2} - 2 \, c e^{2} f^{3} - 2 \, {\left (g x + f\right )} b e^{2} f g + 2 \, c d e f^{2} g + 2 \, b e^{2} f^{2} g + {\left (g x + f\right )} c d^{2} g^{2} - {\left (g x + f\right )} b d e g^{2} + 3 \, {\left (g x + f\right )} a e^{2} g^{2} - 2 \, b d e f g^{2} - 2 \, a e^{2} f g^{2} + 2 \, a d e g^{3}}{{\left (e^{3} f^{2} g - 2 \, d e^{2} f g^{2} + d^{2} e g^{3}\right )} {\left ({\left (g x + f\right )}^{\frac {3}{2}} e - \sqrt {g x + f} e f + \sqrt {g x + f} d g\right )}} \]
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Time = 0.31 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.32 \[ \int \frac {a+b x+c x^2}{(d+e x)^2 (f+g x)^{3/2}} \, dx=\frac {\mathrm {atan}\left (\frac {\sqrt {f+g\,x}\,\left (d^2\,e\,g^2-2\,d\,e^2\,f\,g+e^3\,f^2\right )}{\sqrt {e}\,{\left (d\,g-e\,f\right )}^{5/2}}\right )\,\left (2\,b\,e^2\,f-3\,a\,e^2\,g+c\,d^2\,g+b\,d\,e\,g-4\,c\,d\,e\,f\right )}{e^{3/2}\,{\left (d\,g-e\,f\right )}^{5/2}}-\frac {\frac {2\,\left (c\,f^2-b\,f\,g+a\,g^2\right )}{d\,g-e\,f}+\frac {\left (f+g\,x\right )\,\left (c\,d^2\,g^2-b\,d\,e\,g^2+2\,c\,e^2\,f^2-2\,b\,e^2\,f\,g+3\,a\,e^2\,g^2\right )}{e\,{\left (d\,g-e\,f\right )}^2}}{\sqrt {f+g\,x}\,\left (d\,g^2-e\,f\,g\right )+e\,g\,{\left (f+g\,x\right )}^{3/2}} \]
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