Integrand size = 24, antiderivative size = 112 \[ \int \frac {a+c x^2}{(d+e x) (f+g x)^{3/2}} \, dx=\frac {2 \left (c f^2+a g^2\right )}{g^2 (e f-d g) \sqrt {f+g x}}+\frac {2 c \sqrt {f+g x}}{e g^2}-\frac {2 \left (c d^2+a e^2\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{e^{3/2} (e f-d g)^{3/2}} \]
[Out]
Time = 0.13 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {912, 1275, 214} \[ \int \frac {a+c x^2}{(d+e x) (f+g x)^{3/2}} \, dx=-\frac {2 \left (a e^2+c d^2\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{e^{3/2} (e f-d g)^{3/2}}+\frac {2 \left (a g^2+c f^2\right )}{g^2 \sqrt {f+g x} (e f-d g)}+\frac {2 c \sqrt {f+g x}}{e g^2} \]
[In]
[Out]
Rule 214
Rule 912
Rule 1275
Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {\frac {c f^2+a g^2}{g^2}-\frac {2 c f 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 )} \, dx,x,\sqrt {f+g x}\right )}{g} \\ & = \frac {2 \text {Subst}\left (\int \left (\frac {c}{e g}+\frac {c f^2+a g^2}{g (-e f+d g) x^2}-\frac {\left (c d^2+a e^2\right ) g}{e (e f-d g) \left (e f-d g-e x^2\right )}\right ) \, dx,x,\sqrt {f+g x}\right )}{g} \\ & = \frac {2 \left (c f^2+a g^2\right )}{g^2 (e f-d g) \sqrt {f+g x}}+\frac {2 c \sqrt {f+g x}}{e g^2}-\frac {\left (2 \left (c d^2+a e^2\right )\right ) \text {Subst}\left (\int \frac {1}{e f-d g-e x^2} \, dx,x,\sqrt {f+g x}\right )}{e (e f-d g)} \\ & = \frac {2 \left (c f^2+a g^2\right )}{g^2 (e f-d g) \sqrt {f+g x}}+\frac {2 c \sqrt {f+g x}}{e g^2}-\frac {2 \left (c d^2+a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{e^{3/2} (e f-d g)^{3/2}} \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.02 \[ \int \frac {a+c x^2}{(d+e x) (f+g x)^{3/2}} \, dx=-\frac {2 \left (a e g^2-c d g (f+g x)+c e f (2 f+g x)\right )}{e g^2 (-e f+d g) \sqrt {f+g x}}-\frac {2 \left (c d^2+a e^2\right ) \arctan \left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {-e f+d g}}\right )}{e^{3/2} (-e f+d g)^{3/2}} \]
[In]
[Out]
Time = 0.48 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00
method | result | size |
derivativedivides | \(\frac {\frac {2 c \sqrt {g x +f}}{e}-\frac {2 g^{2} \left (e^{2} a +c \,d^{2}\right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{\left (d g -e f \right ) e \sqrt {\left (d g -e f \right ) e}}-\frac {2 \left (a \,g^{2}+c \,f^{2}\right )}{\left (d g -e f \right ) \sqrt {g x +f}}}{g^{2}}\) | \(112\) |
default | \(\frac {\frac {2 c \sqrt {g x +f}}{e}-\frac {2 g^{2} \left (e^{2} a +c \,d^{2}\right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{\left (d g -e f \right ) e \sqrt {\left (d g -e f \right ) e}}-\frac {2 \left (a \,g^{2}+c \,f^{2}\right )}{\left (d g -e f \right ) \sqrt {g x +f}}}{g^{2}}\) | \(112\) |
pseudoelliptic | \(\frac {\frac {2 c \sqrt {g x +f}}{e}-\frac {2 g^{2} \left (e^{2} a +c \,d^{2}\right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{\left (d g -e f \right ) e \sqrt {\left (d g -e f \right ) e}}-\frac {2 \left (a \,g^{2}+c \,f^{2}\right )}{\left (d g -e f \right ) \sqrt {g x +f}}}{g^{2}}\) | \(112\) |
risch | \(\frac {2 c \sqrt {g x +f}}{e \,g^{2}}-\frac {2 \left (\frac {g^{2} \left (e^{2} a +c \,d^{2}\right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{\left (d g -e f \right ) \sqrt {\left (d g -e f \right ) e}}+\frac {\left (a \,g^{2}+c \,f^{2}\right ) e}{\left (d g -e f \right ) \sqrt {g x +f}}\right )}{g^{2} e}\) | \(116\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 239 vs. \(2 (98) = 196\).
Time = 0.28 (sec) , antiderivative size = 492, normalized size of antiderivative = 4.39 \[ \int \frac {a+c x^2}{(d+e x) (f+g x)^{3/2}} \, dx=\left [-\frac {{\left ({\left (c d^{2} + a e^{2}\right )} g^{3} x + {\left (c d^{2} + a e^{2}\right )} f g^{2}\right )} \sqrt {e^{2} f - d e g} \log \left (\frac {e g x + 2 \, e f - d g + 2 \, \sqrt {e^{2} f - d e g} \sqrt {g x + f}}{e x + d}\right ) - 2 \, {\left (2 \, c e^{3} f^{3} - 3 \, c d e^{2} f^{2} g - a d e^{2} g^{3} + {\left (c d^{2} e + a e^{3}\right )} f g^{2} + {\left (c e^{3} f^{2} g - 2 \, c d e^{2} f g^{2} + c d^{2} e g^{3}\right )} x\right )} \sqrt {g x + f}}{e^{4} f^{3} g^{2} - 2 \, d e^{3} f^{2} g^{3} + d^{2} e^{2} f g^{4} + {\left (e^{4} f^{2} g^{3} - 2 \, d e^{3} f g^{4} + d^{2} e^{2} g^{5}\right )} x}, \frac {2 \, {\left ({\left ({\left (c d^{2} + a e^{2}\right )} g^{3} x + {\left (c d^{2} + a e^{2}\right )} f g^{2}\right )} \sqrt {-e^{2} f + d e g} \arctan \left (\frac {\sqrt {-e^{2} f + d e g} \sqrt {g x + f}}{e g x + e f}\right ) + {\left (2 \, c e^{3} f^{3} - 3 \, c d e^{2} f^{2} g - a d e^{2} g^{3} + {\left (c d^{2} e + a e^{3}\right )} f g^{2} + {\left (c e^{3} f^{2} g - 2 \, c d e^{2} f g^{2} + c d^{2} e g^{3}\right )} x\right )} \sqrt {g x + f}\right )}}{e^{4} f^{3} g^{2} - 2 \, d e^{3} f^{2} g^{3} + d^{2} e^{2} f g^{4} + {\left (e^{4} f^{2} g^{3} - 2 \, d e^{3} f g^{4} + d^{2} e^{2} g^{5}\right )} x}\right ] \]
[In]
[Out]
Time = 4.86 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.35 \[ \int \frac {a+c x^2}{(d+e x) (f+g x)^{3/2}} \, dx=\begin {cases} \frac {2 \left (\frac {c \sqrt {f + g x}}{e g} - \frac {a g^{2} + c f^{2}}{g \sqrt {f + g x} \left (d g - e f\right )} - \frac {g \left (a e^{2} + c d^{2}\right ) \operatorname {atan}{\left (\frac {\sqrt {f + g x}}{\sqrt {\frac {d g - e f}{e}}} \right )}}{e^{2} \sqrt {\frac {d g - e f}{e}} \left (d g - e f\right )}\right )}{g} & \text {for}\: g \neq 0 \\\frac {- \frac {c d x}{e^{2}} + \frac {c x^{2}}{2 e} + \frac {\left (a e^{2} + c d^{2}\right ) \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x \right )}}{e} & \text {otherwise} \end {cases}\right )}{e^{2}}}{f^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
[In]
[Out]
Exception generated. \[ \int \frac {a+c x^2}{(d+e x) (f+g x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.02 \[ \int \frac {a+c x^2}{(d+e x) (f+g x)^{3/2}} \, dx=\frac {2 \, {\left (c d^{2} + a e^{2}\right )} \arctan \left (\frac {\sqrt {g x + f} e}{\sqrt {-e^{2} f + d e g}}\right )}{{\left (e^{2} f - d e g\right )} \sqrt {-e^{2} f + d e g}} + \frac {2 \, {\left (c f^{2} + a g^{2}\right )}}{{\left (e f g^{2} - d g^{3}\right )} \sqrt {g x + f}} + \frac {2 \, \sqrt {g x + f} c}{e g^{2}} \]
[In]
[Out]
Time = 12.20 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.26 \[ \int \frac {a+c x^2}{(d+e x) (f+g x)^{3/2}} \, dx=\frac {2\,\mathrm {atan}\left (\frac {2\,\sqrt {f+g\,x}\,\left (c\,d^2+a\,e^2\right )\,\left (e^2\,f-d\,e\,g\right )}{\sqrt {e}\,\left (2\,c\,d^2+2\,a\,e^2\right )\,{\left (d\,g-e\,f\right )}^{3/2}}\right )\,\left (c\,d^2+a\,e^2\right )}{e^{3/2}\,{\left (d\,g-e\,f\right )}^{3/2}}+\frac {2\,c\,\sqrt {f+g\,x}}{e\,g^2}-\frac {2\,\left (c\,e\,f^2+a\,e\,g^2\right )}{e\,g^2\,\sqrt {f+g\,x}\,\left (d\,g-e\,f\right )} \]
[In]
[Out]