Integrand size = 27, antiderivative size = 604 \[ \int \frac {\left (a+c x^2\right )^{3/2}}{x^2 \left (d+e x+f x^2\right )} \, dx=-\frac {a e \sqrt {a+c x^2}}{d^2}+\frac {3 c x \sqrt {a+c x^2}}{2 d}+\frac {(2 a e-c d x) \sqrt {a+c x^2}}{2 d^2}-\frac {\left (a+c x^2\right )^{3/2}}{d x}+\frac {3 a \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 d}+\frac {\sqrt {c} (2 c d-3 a f) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 d f}-\frac {\left (4 a c d^2 f^2+c^2 d^2 \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )+a^2 f^2 \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )\right ) \text {arctanh}\left (\frac {2 a f-c \left (e-\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} d^2 f \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )}}+\frac {\left (4 a c d^2 f^2+a^2 f^2 \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )+c^2 d^2 \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )\right ) \text {arctanh}\left (\frac {2 a f-c \left (e+\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} d^2 f \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )}}+\frac {a^{3/2} e \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d^2} \]
[Out]
Time = 1.98 (sec) , antiderivative size = 604, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.519, Rules used = {6860, 283, 201, 223, 212, 272, 52, 65, 214, 1034, 1082, 1094, 1048, 739} \[ \int \frac {\left (a+c x^2\right )^{3/2}}{x^2 \left (d+e x+f x^2\right )} \, dx=\frac {a^{3/2} e \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d^2}-\frac {\left (a^2 f^2 \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )+4 a c d^2 f^2+c^2 d^2 \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )\right ) \text {arctanh}\left (\frac {2 a f-c x \left (e-\sqrt {e^2-4 d f}\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} d^2 f \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}+\frac {\left (a^2 f^2 \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )+4 a c d^2 f^2+c^2 d^2 \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )\right ) \text {arctanh}\left (\frac {2 a f-c x \left (\sqrt {e^2-4 d f}+e\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} d^2 f \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}+\frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) (2 c d-3 a f)}{2 d f}+\frac {3 a \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 d}-\frac {a e \sqrt {a+c x^2}}{d^2}+\frac {\sqrt {a+c x^2} (2 a e-c d x)}{2 d^2}-\frac {\left (a+c x^2\right )^{3/2}}{d x}+\frac {3 c x \sqrt {a+c x^2}}{2 d} \]
[In]
[Out]
Rule 52
Rule 65
Rule 201
Rule 212
Rule 214
Rule 223
Rule 272
Rule 283
Rule 739
Rule 1034
Rule 1048
Rule 1082
Rule 1094
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (a+c x^2\right )^{3/2}}{d x^2}-\frac {e \left (a+c x^2\right )^{3/2}}{d^2 x}+\frac {\left (e^2-d f+e f x\right ) \left (a+c x^2\right )^{3/2}}{d^2 \left (d+e x+f x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {\left (e^2-d f+e f x\right ) \left (a+c x^2\right )^{3/2}}{d+e x+f x^2} \, dx}{d^2}+\frac {\int \frac {\left (a+c x^2\right )^{3/2}}{x^2} \, dx}{d}-\frac {e \int \frac {\left (a+c x^2\right )^{3/2}}{x} \, dx}{d^2} \\ & = \frac {e \left (a+c x^2\right )^{3/2}}{3 d^2}-\frac {\left (a+c x^2\right )^{3/2}}{d x}+\frac {(3 c) \int \sqrt {a+c x^2} \, dx}{d}-\frac {e \text {Subst}\left (\int \frac {(a+c x)^{3/2}}{x} \, dx,x,x^2\right )}{2 d^2}+\frac {\int \frac {\sqrt {a+c x^2} \left (3 a f \left (e^2-d f\right )-3 e f (c d-a f) x-3 c d f^2 x^2\right )}{d+e x+f x^2} \, dx}{3 d^2 f} \\ & = \frac {3 c x \sqrt {a+c x^2}}{2 d}+\frac {(2 a e-c d x) \sqrt {a+c x^2}}{2 d^2}-\frac {\left (a+c x^2\right )^{3/2}}{d x}+\frac {(3 a c) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{2 d}-\frac {(a e) \text {Subst}\left (\int \frac {\sqrt {a+c x}}{x} \, dx,x,x^2\right )}{2 d^2}-\frac {\int \frac {-3 a c f^3 \left (c d^2+2 a e^2-2 a d f\right )+3 a c e f^3 (3 c d-2 a f) x-3 c^2 d f^3 (2 c d-3 a f) x^2}{\sqrt {a+c x^2} \left (d+e x+f x^2\right )} \, dx}{6 c d^2 f^3} \\ & = -\frac {a e \sqrt {a+c x^2}}{d^2}+\frac {3 c x \sqrt {a+c x^2}}{2 d}+\frac {(2 a e-c d x) \sqrt {a+c x^2}}{2 d^2}-\frac {\left (a+c x^2\right )^{3/2}}{d x}+\frac {(3 a c) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{2 d}-\frac {\left (a^2 e\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )}{2 d^2}-\frac {\int \frac {3 c^2 d^2 f^3 (2 c d-3 a f)-3 a c f^4 \left (c d^2+2 a e^2-2 a d f\right )+\left (3 c^2 d e f^3 (2 c d-3 a f)+3 a c e f^4 (3 c d-2 a f)\right ) x}{\sqrt {a+c x^2} \left (d+e x+f x^2\right )} \, dx}{6 c d^2 f^4}+\frac {(c (2 c d-3 a f)) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{2 d f} \\ & = -\frac {a e \sqrt {a+c x^2}}{d^2}+\frac {3 c x \sqrt {a+c x^2}}{2 d}+\frac {(2 a e-c d x) \sqrt {a+c x^2}}{2 d^2}-\frac {\left (a+c x^2\right )^{3/2}}{d x}+\frac {3 a \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 d}-\frac {\left (a^2 e\right ) \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 {(c (2 c d-3 a f)) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{2 d f}-\frac {\left (4 a c d^2 f^2+a^2 f^2 \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )+c^2 d^2 \left (e^2-2 d f+e \sqrt {e^2-4 d f}\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 f \sqrt {e^2-4 d f}}+\frac {\left (4 a c d^2 f^2+c^2 d^2 \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )+a^2 f^2 \left (e^2-2 d f+e \sqrt {e^2-4 d f}\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 f \sqrt {e^2-4 d f}} \\ & = -\frac {a e \sqrt {a+c x^2}}{d^2}+\frac {3 c x \sqrt {a+c x^2}}{2 d}+\frac {(2 a e-c d x) \sqrt {a+c x^2}}{2 d^2}-\frac {\left (a+c x^2\right )^{3/2}}{d x}+\frac {3 a \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 d}+\frac {\sqrt {c} (2 c d-3 a f) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 d f}+\frac {a^{3/2} e \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d^2}+\frac {\left (4 a c d^2 f^2+a^2 f^2 \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )+c^2 d^2 \left (e^2-2 d f+e \sqrt {e^2-4 d f}\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 f \sqrt {e^2-4 d f}}-\frac {\left (4 a c d^2 f^2+c^2 d^2 \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )+a^2 f^2 \left (e^2-2 d f+e \sqrt {e^2-4 d f}\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 f \sqrt {e^2-4 d f}} \\ & = -\frac {a e \sqrt {a+c x^2}}{d^2}+\frac {3 c x \sqrt {a+c x^2}}{2 d}+\frac {(2 a e-c d x) \sqrt {a+c x^2}}{2 d^2}-\frac {\left (a+c x^2\right )^{3/2}}{d x}+\frac {3 a \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 d}+\frac {\sqrt {c} (2 c d-3 a f) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 d f}-\frac {\left (4 a c d^2 f^2+c^2 d^2 \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )+a^2 f^2 \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 f \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 (4 a c d^2 f^2+a^2 f^2 \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )+c^2 d^2 \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 f \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 {a^{3/2} 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 = 0.60 (sec) , antiderivative size = 497, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a+c x^2\right )^{3/2}}{x^2 \left (d+e x+f x^2\right )} \, dx=-\frac {2 a^{3/2} e f x \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+c x^2}}{\sqrt {a}}\right )+d \left (a f \sqrt {a+c x^2}+c^{3/2} d x \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )\right )+x \text {RootSum}\left [a^2 f+2 a \sqrt {c} e \text {$\#$1}+4 c d \text {$\#$1}^2-2 a f \text {$\#$1}^2-2 \sqrt {c} e \text {$\#$1}^3+f \text {$\#$1}^4\&,\frac {-a c^2 d^2 e \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right )+a^3 e f^2 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right )-2 c^{5/2} d^3 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}+4 a c^{3/2} d^2 f \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}+2 a^2 \sqrt {c} e^2 f \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-2 a^2 \sqrt {c} d f^2 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}+c^2 d^2 e \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2-a^2 e f^2 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{a \sqrt {c} e+4 c d \text {$\#$1}-2 a f \text {$\#$1}-3 \sqrt {c} e \text {$\#$1}^2+2 f \text {$\#$1}^3}\&\right ]}{d^2 f x} \]
[In]
[Out]
Time = 0.78 (sec) , antiderivative size = 957, normalized size of antiderivative = 1.58
method | result | size |
risch | \(-\frac {a \sqrt {c \,x^{2}+a}}{d x}+\frac {\frac {c^{\frac {3}{2}} d \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{f}-\frac {4 f \,a^{\frac {3}{2}} 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 (2 a^{2} f^{3} \sqrt {-4 d f +e^{2}}-4 a c d \,f^{2} \sqrt {-4 d f +e^{2}}+2 c^{2} d^{2} f \sqrt {-4 d f +e^{2}}-2 \sqrt {-4 d f +e^{2}}\, c^{2} d \,e^{2}-2 a^{2} e \,f^{3}-4 a c d e \,f^{2}+6 c^{2} d^{2} e f -2 c^{2} d \,e^{3}\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}}\, \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 (-2 a^{2} f^{3} \sqrt {-4 d f +e^{2}}+4 a c d \,f^{2} \sqrt {-4 d f +e^{2}}-2 c^{2} d^{2} f \sqrt {-4 d f +e^{2}}+2 \sqrt {-4 d f +e^{2}}\, c^{2} d \,e^{2}-2 a^{2} e \,f^{3}-4 a c d e \,f^{2}+6 c^{2} d^{2} e f -2 c^{2} d \,e^{3}\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}}\, \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}\) | \(957\) |
default | \(\text {Expression too large to display}\) | \(2494\) |
[In]
[Out]
Timed out. \[ \int \frac {\left (a+c x^2\right )^{3/2}}{x^2 \left (d+e x+f x^2\right )} \, dx=\text {Timed out} \]
[In]
[Out]
Timed out. \[ \int \frac {\left (a+c x^2\right )^{3/2}}{x^2 \left (d+e x+f x^2\right )} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {\left (a+c x^2\right )^{3/2}}{x^2 \left (d+e x+f x^2\right )} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}}}{{\left (f x^{2} + e x + d\right )} x^{2}} \,d x } \]
[In]
[Out]
Exception generated. \[ \int \frac {\left (a+c x^2\right )^{3/2}}{x^2 \left (d+e x+f x^2\right )} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Timed out. \[ \int \frac {\left (a+c x^2\right )^{3/2}}{x^2 \left (d+e x+f x^2\right )} \, dx=\int \frac {{\left (c\,x^2+a\right )}^{3/2}}{x^2\,\left (f\,x^2+e\,x+d\right )} \,d x \]
[In]
[Out]