Integrand size = 27, antiderivative size = 452 \[ \int \frac {x^2 \sqrt {a+c x^2}}{d+e x+f x^2} \, dx=-\frac {(2 e-f x) \sqrt {a+c x^2}}{2 f^2}+\frac {\left (a f^2+2 c \left (e^2-d f\right )\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 \sqrt {c} f^3}-\frac {\left (e \left (e-\sqrt {e^2-4 d f}\right ) \left (a f^2+c \left (e^2-2 d f\right )\right )-2 d f \left (a f^2+c \left (e^2-d f\right )\right )\right ) \text {arctanh}\left (\frac {2 a f-c \left (e-\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} f^3 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )}}+\frac {\left (e \left (e+\sqrt {e^2-4 d f}\right ) \left (a f^2+c \left (e^2-2 d f\right )\right )-2 d f \left (a f^2+c \left (e^2-d f\right )\right )\right ) \text {arctanh}\left (\frac {2 a f-c \left (e+\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} f^3 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )}} \]
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Time = 1.30 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1083, 1094, 223, 212, 1048, 739} \[ \int \frac {x^2 \sqrt {a+c x^2}}{d+e x+f x^2} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (a f^2+2 c \left (e^2-d f\right )\right )}{2 \sqrt {c} f^3}-\frac {\left (e \left (e-\sqrt {e^2-4 d f}\right ) \left (a f^2+c \left (e^2-2 d f\right )\right )-2 d f \left (a f^2+c \left (e^2-d f\right )\right )\right ) \text {arctanh}\left (\frac {2 a f-c x \left (e-\sqrt {e^2-4 d f}\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} f^3 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}+\frac {\left (e \left (\sqrt {e^2-4 d f}+e\right ) \left (a f^2+c \left (e^2-2 d f\right )\right )-2 d f \left (a f^2+c \left (e^2-d f\right )\right )\right ) \text {arctanh}\left (\frac {2 a f-c x \left (\sqrt {e^2-4 d f}+e\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} f^3 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}-\frac {\sqrt {a+c x^2} (2 e-f x)}{2 f^2} \]
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Rule 212
Rule 223
Rule 739
Rule 1048
Rule 1083
Rule 1094
Rubi steps \begin{align*} \text {integral}& = -\frac {(2 e-f x) \sqrt {a+c x^2}}{2 f^2}-\frac {\int \frac {a c d f-c e (2 c d-a f) x-c \left (a f^2+2 c \left (e^2-d f\right )\right ) x^2}{\sqrt {a+c x^2} \left (d+e x+f x^2\right )} \, dx}{2 c f^2} \\ & = -\frac {(2 e-f x) \sqrt {a+c x^2}}{2 f^2}-\frac {\int \frac {a c d f^2+c d \left (a f^2+2 c \left (e^2-d f\right )\right )+\left (-c e f (2 c d-a f)+c e \left (a f^2+2 c \left (e^2-d f\right )\right )\right ) x}{\sqrt {a+c x^2} \left (d+e x+f x^2\right )} \, dx}{2 c f^3}+\frac {\left (a f^2+2 c \left (e^2-d f\right )\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{2 f^3} \\ & = -\frac {(2 e-f x) \sqrt {a+c x^2}}{2 f^2}+\frac {\left (a f^2+2 c \left (e^2-d f\right )\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{2 f^3}+\frac {\left (e \left (e-\sqrt {e^2-4 d f}\right ) \left (a f^2+c \left (e^2-2 d f\right )\right )-2 d f \left (a f^2+c \left (e^2-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}{f^3 \sqrt {e^2-4 d f}}-\frac {\left (e \left (e+\sqrt {e^2-4 d f}\right ) \left (a f^2+c \left (e^2-2 d f\right )\right )-2 d f \left (a f^2+c \left (e^2-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}{f^3 \sqrt {e^2-4 d f}} \\ & = -\frac {(2 e-f x) \sqrt {a+c x^2}}{2 f^2}+\frac {\left (a f^2+2 c \left (e^2-d f\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 \sqrt {c} f^3}-\frac {\left (e \left (e-\sqrt {e^2-4 d f}\right ) \left (a f^2+c \left (e^2-2 d f\right )\right )-2 d f \left (a f^2+c \left (e^2-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 )}{f^3 \sqrt {e^2-4 d f}}+\frac {\left (e \left (e+\sqrt {e^2-4 d f}\right ) \left (a f^2+c \left (e^2-2 d f\right )\right )-2 d f \left (a f^2+c \left (e^2-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 )}{f^3 \sqrt {e^2-4 d f}} \\ & = -\frac {(2 e-f x) \sqrt {a+c x^2}}{2 f^2}+\frac {\left (a f^2+2 c \left (e^2-d f\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 \sqrt {c} f^3}-\frac {\left (e \left (e-\sqrt {e^2-4 d f}\right ) \left (a f^2+c \left (e^2-2 d f\right )\right )-2 d f \left (a f^2+c \left (e^2-d f\right )\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} f^3 \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 {\left (e \left (e+\sqrt {e^2-4 d f}\right ) \left (a f^2+c \left (e^2-2 d f\right )\right )-2 d f \left (a f^2+c \left (e^2-d f\right )\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} f^3 \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 )}} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.69 (sec) , antiderivative size = 615, normalized size of antiderivative = 1.36 \[ \int \frac {x^2 \sqrt {a+c x^2}}{d+e x+f x^2} \, dx=\frac {f (-2 e+f x) \sqrt {a+c x^2}+\frac {2 \left (a f^2+2 c \left (e^2-d f\right )\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+c x^2}}\right )}{\sqrt {c}}-2 \text {RootSum}\left [c^2 d+2 \sqrt {a} c e \text {$\#$1}-2 c d \text {$\#$1}^2+4 a f \text {$\#$1}^2-2 \sqrt {a} e \text {$\#$1}^3+d \text {$\#$1}^4\&,\frac {c^2 d e^2 \log (x)-c^2 d^2 f \log (x)+a c d f^2 \log (x)-c^2 d e^2 \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right )+c^2 d^2 f \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right )-a c d f^2 \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right )+2 \sqrt {a} c e^3 \log (x) \text {$\#$1}-4 \sqrt {a} c d e f \log (x) \text {$\#$1}+2 a^{3/2} e f^2 \log (x) \text {$\#$1}-2 \sqrt {a} c e^3 \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}+4 \sqrt {a} c d e f \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}-2 a^{3/2} e f^2 \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}-c d e^2 \log (x) \text {$\#$1}^2+c d^2 f \log (x) \text {$\#$1}^2-a d f^2 \log (x) \text {$\#$1}^2+c d e^2 \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2-c d^2 f \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2+a d f^2 \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-\sqrt {a} c e+2 c d \text {$\#$1}-4 a f \text {$\#$1}+3 \sqrt {a} e \text {$\#$1}^2-2 d \text {$\#$1}^3}\&\right ]}{2 f^3} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(841\) vs. \(2(403)=806\).
Time = 0.82 (sec) , antiderivative size = 842, normalized size of antiderivative = 1.86
method | result | size |
risch | \(-\frac {\left (-f x +2 e \right ) \sqrt {c \,x^{2}+a}}{2 f^{2}}+\frac {\frac {\left (a \,f^{2}-2 c d f +2 c \,e^{2}\right ) \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{f \sqrt {c}}-\frac {\left (-2 a e \,f^{2} \sqrt {-4 d f +e^{2}}+4 c d e f \sqrt {-4 d f +e^{2}}-2 c \,e^{3} \sqrt {-4 d f +e^{2}}-4 a d \,f^{3}+2 a \,e^{2} f^{2}+4 c \,d^{2} f^{2}-8 c d \,e^{2} f +2 c \,e^{4}\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 )}{2 f^{2} \sqrt {-4 d f +e^{2}}\, \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 (-2 a e \,f^{2} \sqrt {-4 d f +e^{2}}+4 c d e f \sqrt {-4 d f +e^{2}}-2 c \,e^{3} \sqrt {-4 d f +e^{2}}+4 a d \,f^{3}-2 a \,e^{2} f^{2}-4 c \,d^{2} f^{2}+8 c d \,e^{2} f -2 c \,e^{4}\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 )}{2 f^{2} \sqrt {-4 d f +e^{2}}\, \sqrt {\frac {\sqrt {-4 d f +e^{2}}\, c e +2 a \,f^{2}-2 c d f +c \,e^{2}}{f^{2}}}}}{2 f^{2}}\) | \(842\) |
default | \(\text {Expression too large to display}\) | \(1302\) |
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Timed out. \[ \int \frac {x^2 \sqrt {a+c x^2}}{d+e x+f x^2} \, dx=\text {Timed out} \]
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\[ \int \frac {x^2 \sqrt {a+c x^2}}{d+e x+f x^2} \, dx=\int \frac {x^{2} \sqrt {a + c x^{2}}}{d + e x + f x^{2}}\, dx \]
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Exception generated. \[ \int \frac {x^2 \sqrt {a+c x^2}}{d+e x+f x^2} \, dx=\text {Exception raised: ValueError} \]
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Exception generated. \[ \int \frac {x^2 \sqrt {a+c x^2}}{d+e x+f x^2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^2 \sqrt {a+c x^2}}{d+e x+f x^2} \, dx=\int \frac {x^2\,\sqrt {c\,x^2+a}}{f\,x^2+e\,x+d} \,d x \]
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