Integrand size = 29, antiderivative size = 239 \[ \int \frac {d+e x+f x^2}{(g+h x)^2 \left (a+c x^2\right )^{3/2}} \, dx=-\frac {a (c g (e g-2 d h)+a h (2 f g-e h))-\left (c^2 d g^2+a^2 f h^2-a c \left (f g^2-h (2 e g-d h)\right )\right ) x}{a \left (c g^2+a h^2\right )^2 \sqrt {a+c x^2}}-\frac {h \left (f g^2-e g h+d h^2\right ) \sqrt {a+c x^2}}{\left (c g^2+a h^2\right )^2 (g+h x)}+\frac {\left (a h^2 (2 f g-e h)-c g \left (f g^2-h (2 e g-3 d h)\right )\right ) \text {arctanh}\left (\frac {a h-c g x}{\sqrt {c g^2+a h^2} \sqrt {a+c x^2}}\right )}{\left (c g^2+a h^2\right )^{5/2}} \]
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Time = 0.26 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1661, 821, 739, 212} \[ \int \frac {d+e x+f x^2}{(g+h x)^2 \left (a+c x^2\right )^{3/2}} \, dx=-\frac {a (a h (2 f g-e h)+c g (e g-2 d h))-x \left (a^2 f h^2-a c \left (f g^2-h (2 e g-d h)\right )+c^2 d g^2\right )}{a \sqrt {a+c x^2} \left (a h^2+c g^2\right )^2}-\frac {\text {arctanh}\left (\frac {a h-c g x}{\sqrt {a+c x^2} \sqrt {a h^2+c g^2}}\right ) \left (-a h^2 (2 f g-e h)-c g h (2 e g-3 d h)+c f g^3\right )}{\left (a h^2+c g^2\right )^{5/2}}-\frac {h \sqrt {a+c x^2} \left (d h^2-e g h+f g^2\right )}{(g+h x) \left (a h^2+c g^2\right )^2} \]
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Rule 212
Rule 739
Rule 821
Rule 1661
Rubi steps \begin{align*} \text {integral}& = -\frac {a (c g (e g-2 d h)+a h (2 f g-e h))-\left (c^2 d g^2+a^2 f h^2-a c \left (f g^2-h (2 e g-d h)\right )\right ) x}{a \left (c g^2+a h^2\right )^2 \sqrt {a+c x^2}}-\frac {\int \frac {\frac {a c \left (a h^2 \left (f g^2-d h^2\right )-c \left (f g^4-g^2 h (2 e g-3 d h)\right )\right )}{\left (c g^2+a h^2\right )^2}+\frac {a c h^2 (c g (e g-2 d h)+a h (2 f g-e h)) x}{\left (c g^2+a h^2\right )^2}}{(g+h x)^2 \sqrt {a+c x^2}} \, dx}{a c} \\ & = -\frac {a (c g (e g-2 d h)+a h (2 f g-e h))-\left (c^2 d g^2+a^2 f h^2-a c \left (f g^2-h (2 e g-d h)\right )\right ) x}{a \left (c g^2+a h^2\right )^2 \sqrt {a+c x^2}}-\frac {h \left (f g^2-e g h+d h^2\right ) \sqrt {a+c x^2}}{\left (c g^2+a h^2\right )^2 (g+h x)}-\frac {\left (\frac {a^2 c h^3 (c g (e g-2 d h)+a h (2 f g-e h))}{\left (c g^2+a h^2\right )^2}+\frac {a c^2 g \left (a h^2 \left (f g^2-d h^2\right )-c \left (f g^4-g^2 h (2 e g-3 d h)\right )\right )}{\left (c g^2+a h^2\right )^2}\right ) \int \frac {1}{(g+h x) \sqrt {a+c x^2}} \, dx}{a c \left (c g^2+a h^2\right )} \\ & = -\frac {a (c g (e g-2 d h)+a h (2 f g-e h))-\left (c^2 d g^2+a^2 f h^2-a c \left (f g^2-h (2 e g-d h)\right )\right ) x}{a \left (c g^2+a h^2\right )^2 \sqrt {a+c x^2}}-\frac {h \left (f g^2-e g h+d h^2\right ) \sqrt {a+c x^2}}{\left (c g^2+a h^2\right )^2 (g+h x)}+\frac {\left (\frac {a^2 c h^3 (c g (e g-2 d h)+a h (2 f g-e h))}{\left (c g^2+a h^2\right )^2}+\frac {a c^2 g \left (a h^2 \left (f g^2-d h^2\right )-c \left (f g^4-g^2 h (2 e g-3 d h)\right )\right )}{\left (c g^2+a h^2\right )^2}\right ) \text {Subst}\left (\int \frac {1}{c g^2+a h^2-x^2} \, dx,x,\frac {a h-c g x}{\sqrt {a+c x^2}}\right )}{a c \left (c g^2+a h^2\right )} \\ & = -\frac {a (c g (e g-2 d h)+a h (2 f g-e h))-\left (c^2 d g^2+a^2 f h^2-a c \left (f g^2-h (2 e g-d h)\right )\right ) x}{a \left (c g^2+a h^2\right )^2 \sqrt {a+c x^2}}-\frac {h \left (f g^2-e g h+d h^2\right ) \sqrt {a+c x^2}}{\left (c g^2+a h^2\right )^2 (g+h x)}-\frac {\left (c f g^3-c g h (2 e g-3 d h)-a h^2 (2 f g-e h)\right ) \tanh ^{-1}\left (\frac {a h-c g x}{\sqrt {c g^2+a h^2} \sqrt {a+c x^2}}\right )}{\left (c g^2+a h^2\right )^{5/2}} \\ \end{align*}
Time = 1.36 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.04 \[ \int \frac {d+e x+f x^2}{(g+h x)^2 \left (a+c x^2\right )^{3/2}} \, dx=\frac {c^2 d g^2 x (g+h x)+a^2 h \left (h (2 e g-d h+e h x)+f \left (-3 g^2-g h x+h^2 x^2\right )\right )+a c \left (-f g^2 x (g+2 h x)+d h \left (2 g^2+g h x-2 h^2 x^2\right )+e g \left (-g^2+g h x+3 h^2 x^2\right )\right )}{a \left (c g^2+a h^2\right )^2 (g+h x) \sqrt {a+c x^2}}-\frac {2 \left (c f g^3+c g h (-2 e g+3 d h)+a h^2 (-2 f g+e h)\right ) \arctan \left (\frac {\sqrt {c} (g+h x)-h \sqrt {a+c x^2}}{\sqrt {-c g^2-a h^2}}\right )}{\left (-c g^2-a h^2\right )^{5/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(871\) vs. \(2(228)=456\).
Time = 0.53 (sec) , antiderivative size = 872, normalized size of antiderivative = 3.65
method | result | size |
default | \(\frac {f x}{h^{2} a \sqrt {c \,x^{2}+a}}+\frac {\left (e h -2 f g \right ) \left (\frac {h^{2}}{\left (a \,h^{2}+c \,g^{2}\right ) \sqrt {\left (x +\frac {g}{h}\right )^{2} c -\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}+\frac {2 c g h \left (2 c \left (x +\frac {g}{h}\right )-\frac {2 c g}{h}\right )}{\left (a \,h^{2}+c \,g^{2}\right ) \left (\frac {4 c \left (a \,h^{2}+c \,g^{2}\right )}{h^{2}}-\frac {4 c^{2} g^{2}}{h^{2}}\right ) \sqrt {\left (x +\frac {g}{h}\right )^{2} c -\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}-\frac {h^{2} \ln \left (\frac {\frac {2 a \,h^{2}+2 c \,g^{2}}{h^{2}}-\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+2 \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}\, \sqrt {\left (x +\frac {g}{h}\right )^{2} c -\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}{x +\frac {g}{h}}\right )}{\left (a \,h^{2}+c \,g^{2}\right ) \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}\right )}{h^{3}}+\frac {\left (d \,h^{2}-e g h +f \,g^{2}\right ) \left (-\frac {h^{2}}{\left (a \,h^{2}+c \,g^{2}\right ) \left (x +\frac {g}{h}\right ) \sqrt {\left (x +\frac {g}{h}\right )^{2} c -\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}+\frac {3 c g h \left (\frac {h^{2}}{\left (a \,h^{2}+c \,g^{2}\right ) \sqrt {\left (x +\frac {g}{h}\right )^{2} c -\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}+\frac {2 c g h \left (2 c \left (x +\frac {g}{h}\right )-\frac {2 c g}{h}\right )}{\left (a \,h^{2}+c \,g^{2}\right ) \left (\frac {4 c \left (a \,h^{2}+c \,g^{2}\right )}{h^{2}}-\frac {4 c^{2} g^{2}}{h^{2}}\right ) \sqrt {\left (x +\frac {g}{h}\right )^{2} c -\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}-\frac {h^{2} \ln \left (\frac {\frac {2 a \,h^{2}+2 c \,g^{2}}{h^{2}}-\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+2 \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}\, \sqrt {\left (x +\frac {g}{h}\right )^{2} c -\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}{x +\frac {g}{h}}\right )}{\left (a \,h^{2}+c \,g^{2}\right ) \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}\right )}{a \,h^{2}+c \,g^{2}}-\frac {4 c \,h^{2} \left (2 c \left (x +\frac {g}{h}\right )-\frac {2 c g}{h}\right )}{\left (a \,h^{2}+c \,g^{2}\right ) \left (\frac {4 c \left (a \,h^{2}+c \,g^{2}\right )}{h^{2}}-\frac {4 c^{2} g^{2}}{h^{2}}\right ) \sqrt {\left (x +\frac {g}{h}\right )^{2} c -\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}\right )}{h^{4}}\) | \(872\) |
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Leaf count of result is larger than twice the leaf count of optimal. 773 vs. \(2 (230) = 460\).
Time = 1.82 (sec) , antiderivative size = 1573, normalized size of antiderivative = 6.58 \[ \int \frac {d+e x+f x^2}{(g+h x)^2 \left (a+c x^2\right )^{3/2}} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {d+e x+f x^2}{(g+h x)^2 \left (a+c x^2\right )^{3/2}} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1085 vs. \(2 (230) = 460\).
Time = 0.26 (sec) , antiderivative size = 1085, normalized size of antiderivative = 4.54 \[ \int \frac {d+e x+f x^2}{(g+h x)^2 \left (a+c x^2\right )^{3/2}} \, dx=\frac {3 \, c^{2} f g^{4} x}{\sqrt {c x^{2} + a} a c^{2} g^{4} h^{2} + 2 \, \sqrt {c x^{2} + a} a^{2} c g^{2} h^{4} + \sqrt {c x^{2} + a} a^{3} h^{6}} - \frac {3 \, c^{2} e g^{3} x}{\sqrt {c x^{2} + a} a c^{2} g^{4} h + 2 \, \sqrt {c x^{2} + a} a^{2} c g^{2} h^{3} + \sqrt {c x^{2} + a} a^{3} h^{5}} + \frac {3 \, c^{2} d g^{2} x}{\sqrt {c x^{2} + a} a c^{2} g^{4} + 2 \, \sqrt {c x^{2} + a} a^{2} c g^{2} h^{2} + \sqrt {c x^{2} + a} a^{3} h^{4}} + \frac {3 \, c f g^{3}}{\sqrt {c x^{2} + a} c^{2} g^{4} h + 2 \, \sqrt {c x^{2} + a} a c g^{2} h^{3} + \sqrt {c x^{2} + a} a^{2} h^{5}} - \frac {4 \, c f g^{2} x}{\sqrt {c x^{2} + a} a c g^{2} h^{2} + \sqrt {c x^{2} + a} a^{2} h^{4}} - \frac {3 \, c e g^{2}}{\sqrt {c x^{2} + a} c^{2} g^{4} + 2 \, \sqrt {c x^{2} + a} a c g^{2} h^{2} + \sqrt {c x^{2} + a} a^{2} h^{4}} + \frac {3 \, c e g x}{\sqrt {c x^{2} + a} a c g^{2} h + \sqrt {c x^{2} + a} a^{2} h^{3}} + \frac {3 \, c d g}{\frac {\sqrt {c x^{2} + a} c^{2} g^{4}}{h} + 2 \, \sqrt {c x^{2} + a} a c g^{2} h + \sqrt {c x^{2} + a} a^{2} h^{3}} - \frac {f g^{2}}{\sqrt {c x^{2} + a} c g^{2} h^{2} x + \sqrt {c x^{2} + a} a h^{4} x + \sqrt {c x^{2} + a} c g^{3} h + \sqrt {c x^{2} + a} a g h^{3}} - \frac {2 \, c d x}{\sqrt {c x^{2} + a} a c g^{2} + \sqrt {c x^{2} + a} a^{2} h^{2}} + \frac {e g}{\sqrt {c x^{2} + a} c g^{2} h x + \sqrt {c x^{2} + a} a h^{3} x + \sqrt {c x^{2} + a} c g^{3} + \sqrt {c x^{2} + a} a g h^{2}} - \frac {2 \, f g}{\sqrt {c x^{2} + a} c g^{2} h + \sqrt {c x^{2} + a} a h^{3}} - \frac {d}{\sqrt {c x^{2} + a} c g^{2} x + \sqrt {c x^{2} + a} a h^{2} x + \frac {\sqrt {c x^{2} + a} c g^{3}}{h} + \sqrt {c x^{2} + a} a g h} + \frac {e}{\sqrt {c x^{2} + a} c g^{2} + \sqrt {c x^{2} + a} a h^{2}} + \frac {f x}{\sqrt {c x^{2} + a} a h^{2}} + \frac {3 \, c f g^{3} \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{{\left (a + \frac {c g^{2}}{h^{2}}\right )}^{\frac {5}{2}} h^{5}} - \frac {3 \, c e g^{2} \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{{\left (a + \frac {c g^{2}}{h^{2}}\right )}^{\frac {5}{2}} h^{4}} + \frac {3 \, c d g \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{{\left (a + \frac {c g^{2}}{h^{2}}\right )}^{\frac {5}{2}} h^{3}} - \frac {2 \, f g \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{{\left (a + \frac {c g^{2}}{h^{2}}\right )}^{\frac {3}{2}} h^{3}} + \frac {e \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{{\left (a + \frac {c g^{2}}{h^{2}}\right )}^{\frac {3}{2}} h^{2}} \]
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\[ \int \frac {d+e x+f x^2}{(g+h x)^2 \left (a+c x^2\right )^{3/2}} \, dx=\int { \frac {f x^{2} + e x + d}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} {\left (h x + g\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {d+e x+f x^2}{(g+h x)^2 \left (a+c x^2\right )^{3/2}} \, dx=\int \frac {f\,x^2+e\,x+d}{{\left (g+h\,x\right )}^2\,{\left (c\,x^2+a\right )}^{3/2}} \,d x \]
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