Integrand size = 27, antiderivative size = 382 \[ \int \frac {\sqrt {a+c x^2}}{x^2 \left (d+e x+f x^2\right )} \, dx=-\frac {\sqrt {a+c x^2}}{d x}-\frac {f \left (2 c d^2+a \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )\right ) \text {arctanh}\left (\frac {2 a f-c \left (e-\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} d^2 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )}}+\frac {f \left (2 c d^2+a \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )\right ) \text {arctanh}\left (\frac {2 a f-c \left (e+\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} d^2 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )}}+\frac {\sqrt {a} e \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d^2} \]
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Time = 0.92 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {6860, 283, 223, 212, 272, 52, 65, 214, 1034, 1094, 1048, 739} \[ \int \frac {\sqrt {a+c x^2}}{x^2 \left (d+e x+f x^2\right )} \, dx=-\frac {f \left (a \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )+2 c d^2\right ) \text {arctanh}\left (\frac {2 a f-c x \left (e-\sqrt {e^2-4 d f}\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} d^2 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}+\frac {f \left (a \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )+2 c d^2\right ) \text {arctanh}\left (\frac {2 a f-c x \left (\sqrt {e^2-4 d f}+e\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} d^2 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}+\frac {\sqrt {a} e \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d^2}-\frac {\sqrt {a+c x^2}}{d x} \]
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Rule 52
Rule 65
Rule 212
Rule 214
Rule 223
Rule 272
Rule 283
Rule 739
Rule 1034
Rule 1048
Rule 1094
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\sqrt {a+c x^2}}{d x^2}-\frac {e \sqrt {a+c x^2}}{d^2 x}+\frac {\left (e^2-d f+e f x\right ) \sqrt {a+c x^2}}{d^2 \left (d+e x+f x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {\left (e^2-d f+e f x\right ) \sqrt {a+c x^2}}{d+e x+f x^2} \, dx}{d^2}+\frac {\int \frac {\sqrt {a+c x^2}}{x^2} \, dx}{d}-\frac {e \int \frac {\sqrt {a+c x^2}}{x} \, dx}{d^2} \\ & = \frac {e \sqrt {a+c x^2}}{d^2}-\frac {\sqrt {a+c x^2}}{d x}+\frac {c \int \frac {1}{\sqrt {a+c x^2}} \, dx}{d}-\frac {e \text {Subst}\left (\int \frac {\sqrt {a+c x}}{x} \, dx,x,x^2\right )}{2 d^2}+\frac {\int \frac {a f \left (e^2-d f\right )-e f (c d-a f) x-c d f^2 x^2}{\sqrt {a+c x^2} \left (d+e x+f x^2\right )} \, dx}{d^2 f} \\ & = -\frac {\sqrt {a+c x^2}}{d x}-\frac {c \int \frac {1}{\sqrt {a+c x^2}} \, dx}{d}+\frac {c \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{d}-\frac {(a e) \text {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )}{2 d^2}+\frac {\int \frac {c d^2 f^2+a f^2 \left (e^2-d f\right )+\left (c d e f^2-e f^2 (c d-a f)\right ) x}{\sqrt {a+c x^2} \left (d+e x+f x^2\right )} \, dx}{d^2 f^2} \\ & = -\frac {\sqrt {a+c x^2}}{d x}+\frac {\sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{d}-\frac {c \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{d}-\frac {(a e) \text {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )}{c d^2}-\frac {\left (f \left (2 c d^2+a \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )\right )\right ) \int \frac {1}{\left (e+\sqrt {e^2-4 d f}+2 f x\right ) \sqrt {a+c x^2}} \, dx}{d^2 \sqrt {e^2-4 d f}}+\frac {\left (f \left (2 c d^2+a \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )\right )\right ) \int \frac {1}{\left (e-\sqrt {e^2-4 d f}+2 f x\right ) \sqrt {a+c x^2}} \, dx}{d^2 \sqrt {e^2-4 d f}} \\ & = -\frac {\sqrt {a+c x^2}}{d x}+\frac {\sqrt {a} e \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d^2}+\frac {\left (f \left (2 c d^2+a \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )\right )\right ) \text {Subst}\left (\int \frac {1}{4 a f^2+c \left (e+\sqrt {e^2-4 d f}\right )^2-x^2} \, dx,x,\frac {2 a f-c \left (e+\sqrt {e^2-4 d f}\right ) x}{\sqrt {a+c x^2}}\right )}{d^2 \sqrt {e^2-4 d f}}-\frac {\left (f \left (2 c d^2+a \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )\right )\right ) \text {Subst}\left (\int \frac {1}{4 a f^2+c \left (e-\sqrt {e^2-4 d f}\right )^2-x^2} \, dx,x,\frac {2 a f-c \left (e-\sqrt {e^2-4 d f}\right ) x}{\sqrt {a+c x^2}}\right )}{d^2 \sqrt {e^2-4 d f}} \\ & = -\frac {\sqrt {a+c x^2}}{d x}-\frac {f \left (2 c d^2+a \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c \left (e-\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} d^2 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )}}+\frac {f \left (2 c d^2+a \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c \left (e+\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} d^2 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )}}+\frac {\sqrt {a} e \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d^2} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.38 (sec) , antiderivative size = 467, normalized size of antiderivative = 1.22 \[ \int \frac {\sqrt {a+c x^2}}{x^2 \left (d+e x+f x^2\right )} \, dx=-\frac {d \sqrt {a+c x^2}-\sqrt {a} e x \log (x)+\sqrt {a} e x \log \left (-\sqrt {a}+\sqrt {a+c x^2}\right )+x \text {RootSum}\left [c^2 d+2 \sqrt {a} c e \text {$\#$1}-2 c d \text {$\#$1}^2+4 a f \text {$\#$1}^2-2 \sqrt {a} e \text {$\#$1}^3+d \text {$\#$1}^4\&,\frac {-c^2 d^2 \log (x)-a c e^2 \log (x)+a c d f \log (x)+c^2 d^2 \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right )+a c e^2 \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right )-a c d f \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right )-2 a^{3/2} e f \log (x) \text {$\#$1}+2 a^{3/2} e f \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}+c d^2 \log (x) \text {$\#$1}^2+a e^2 \log (x) \text {$\#$1}^2-a d f \log (x) \text {$\#$1}^2-c d^2 \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2-a e^2 \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2+a d f \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-\sqrt {a} c e+2 c d \text {$\#$1}-4 a f \text {$\#$1}+3 \sqrt {a} e \text {$\#$1}^2-2 d \text {$\#$1}^3}\&\right ]}{d^2 x} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(775\) vs. \(2(337)=674\).
Time = 0.76 (sec) , antiderivative size = 776, normalized size of antiderivative = 2.03
method | result | size |
risch | \(-\frac {\sqrt {c \,x^{2}+a}}{d x}-\frac {\frac {4 f \sqrt {a}\, e \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right )}{\left (-e +\sqrt {-4 d f +e^{2}}\right ) \left (e +\sqrt {-4 d f +e^{2}}\right )}-\frac {\left (f a \sqrt {-4 d f +e^{2}}-c d \sqrt {-4 d f +e^{2}}+a e f +c d e \right ) \sqrt {2}\, \ln \left (\frac {\frac {-\sqrt {-4 d f +e^{2}}\, c e +2 a \,f^{2}-2 c d f +c \,e^{2}}{f^{2}}-\frac {c \left (e -\sqrt {-4 d f +e^{2}}\right ) \left (x -\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}{f}+\frac {\sqrt {2}\, \sqrt {\frac {-\sqrt {-4 d f +e^{2}}\, c e +2 a \,f^{2}-2 c d f +c \,e^{2}}{f^{2}}}\, \sqrt {4 {\left (x -\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}^{2} c -\frac {4 c \left (e -\sqrt {-4 d f +e^{2}}\right ) \left (x -\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}{f}+\frac {-2 \sqrt {-4 d f +e^{2}}\, c e +4 a \,f^{2}-4 c d f +2 c \,e^{2}}{f^{2}}}}{2}}{x -\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}}\right )}{\sqrt {-4 d f +e^{2}}\, \left (-e +\sqrt {-4 d f +e^{2}}\right ) \sqrt {\frac {-\sqrt {-4 d f +e^{2}}\, c e +2 a \,f^{2}-2 c d f +c \,e^{2}}{f^{2}}}}+\frac {\left (f a \sqrt {-4 d f +e^{2}}-c d \sqrt {-4 d f +e^{2}}-a e f -c d e \right ) \sqrt {2}\, \ln \left (\frac {\frac {\sqrt {-4 d f +e^{2}}\, c e +2 a \,f^{2}-2 c d f +c \,e^{2}}{f^{2}}-\frac {c \left (e +\sqrt {-4 d f +e^{2}}\right ) \left (x +\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}{f}+\frac {\sqrt {2}\, \sqrt {\frac {\sqrt {-4 d f +e^{2}}\, c e +2 a \,f^{2}-2 c d f +c \,e^{2}}{f^{2}}}\, \sqrt {4 {\left (x +\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}^{2} c -\frac {4 c \left (e +\sqrt {-4 d f +e^{2}}\right ) \left (x +\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}{f}+\frac {2 \sqrt {-4 d f +e^{2}}\, c e +4 a \,f^{2}-4 c d f +2 c \,e^{2}}{f^{2}}}}{2}}{x +\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}}\right )}{\sqrt {-4 d f +e^{2}}\, \left (e +\sqrt {-4 d f +e^{2}}\right ) \sqrt {\frac {\sqrt {-4 d f +e^{2}}\, c e +2 a \,f^{2}-2 c d f +c \,e^{2}}{f^{2}}}}}{d}\) | \(776\) |
default | \(\text {Expression too large to display}\) | \(1415\) |
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Leaf count of result is larger than twice the leaf count of optimal. 2656 vs. \(2 (335) = 670\).
Time = 176.36 (sec) , antiderivative size = 5324, normalized size of antiderivative = 13.94 \[ \int \frac {\sqrt {a+c x^2}}{x^2 \left (d+e x+f x^2\right )} \, dx=\text {Too large to display} \]
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\[ \int \frac {\sqrt {a+c x^2}}{x^2 \left (d+e x+f x^2\right )} \, dx=\int \frac {\sqrt {a + c x^{2}}}{x^{2} \left (d + e x + f x^{2}\right )}\, dx \]
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\[ \int \frac {\sqrt {a+c x^2}}{x^2 \left (d+e x+f x^2\right )} \, dx=\int { \frac {\sqrt {c x^{2} + a}}{{\left (f x^{2} + e x + d\right )} x^{2}} \,d x } \]
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Exception generated. \[ \int \frac {\sqrt {a+c x^2}}{x^2 \left (d+e x+f x^2\right )} \, dx=\text {Exception raised: AttributeError} \]
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Timed out. \[ \int \frac {\sqrt {a+c x^2}}{x^2 \left (d+e x+f x^2\right )} \, dx=\int \frac {\sqrt {c\,x^2+a}}{x^2\,\left (f\,x^2+e\,x+d\right )} \,d x \]
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