Integrand size = 28, antiderivative size = 353 \[ \int \frac {\sqrt {a+b x+c x^2}}{x^3 \left (d-f x^2\right )} \, dx=-\frac {(2 a+b x) \sqrt {a+b x+c x^2}}{4 a d x^2}+\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{8 a^{3/2} d}-\frac {\sqrt {a} f \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{d^2}-\frac {\sqrt {f} \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \text {arctanh}\left (\frac {b \sqrt {d}-2 a \sqrt {f}+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x}{2 \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 d^2}+\frac {\sqrt {f} \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \text {arctanh}\left (\frac {b \sqrt {d}+2 a \sqrt {f}+\left (2 c \sqrt {d}+b \sqrt {f}\right ) x}{2 \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 d^2} \]
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Time = 0.59 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {6857, 734, 738, 212, 748, 857, 635, 1035, 1092, 1047} \[ \int \frac {\sqrt {a+b x+c x^2}}{x^3 \left (d-f x^2\right )} \, dx=\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{8 a^{3/2} d}-\frac {\sqrt {a} f \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{d^2}-\frac {\sqrt {f} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d} \text {arctanh}\left (\frac {-2 a \sqrt {f}+x \left (2 c \sqrt {d}-b \sqrt {f}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )}{2 d^2}+\frac {\sqrt {f} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d} \text {arctanh}\left (\frac {2 a \sqrt {f}+x \left (b \sqrt {f}+2 c \sqrt {d}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )}{2 d^2}-\frac {(2 a+b x) \sqrt {a+b x+c x^2}}{4 a d x^2} \]
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Rule 212
Rule 635
Rule 734
Rule 738
Rule 748
Rule 857
Rule 1035
Rule 1047
Rule 1092
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\sqrt {a+b x+c x^2}}{d x^3}+\frac {f \sqrt {a+b x+c x^2}}{d^2 x}+\frac {f^2 x \sqrt {a+b x+c x^2}}{d^2 \left (d-f x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {\sqrt {a+b x+c x^2}}{x^3} \, dx}{d}+\frac {f \int \frac {\sqrt {a+b x+c x^2}}{x} \, dx}{d^2}+\frac {f^2 \int \frac {x \sqrt {a+b x+c x^2}}{d-f x^2} \, dx}{d^2} \\ & = -\frac {(2 a+b x) \sqrt {a+b x+c x^2}}{4 a d x^2}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx}{8 a d}-\frac {f \int \frac {-2 a-b x}{x \sqrt {a+b x+c x^2}} \, dx}{2 d^2}+\frac {f \int \frac {\frac {b d}{2}+(c d+a f) x+\frac {1}{2} b f x^2}{\sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx}{d^2} \\ & = -\frac {(2 a+b x) \sqrt {a+b x+c x^2}}{4 a d x^2}-\frac {\int \frac {-b d f-f (c d+a f) x}{\sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx}{d^2}+\frac {\left (b^2-4 a c\right ) \text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b x}{\sqrt {a+b x+c x^2}}\right )}{4 a d}+\frac {(a f) \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx}{d^2} \\ & = -\frac {(2 a+b x) \sqrt {a+b x+c x^2}}{4 a d x^2}+\frac {\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{8 a^{3/2} d}-\frac {(2 a f) \text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b x}{\sqrt {a+b x+c x^2}}\right )}{d^2}+\frac {\left (f \left (c d-b \sqrt {d} \sqrt {f}+a f\right )\right ) \int \frac {1}{\left (-\sqrt {d} \sqrt {f}-f x\right ) \sqrt {a+b x+c x^2}} \, dx}{2 d^2}+\frac {\left (f \left (c d+b \sqrt {d} \sqrt {f}+a f\right )\right ) \int \frac {1}{\left (\sqrt {d} \sqrt {f}-f x\right ) \sqrt {a+b x+c x^2}} \, dx}{2 d^2} \\ & = -\frac {(2 a+b x) \sqrt {a+b x+c x^2}}{4 a d x^2}+\frac {\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{8 a^{3/2} d}-\frac {\sqrt {a} f \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{d^2}-\frac {\left (f \left (c d-b \sqrt {d} \sqrt {f}+a f\right )\right ) \text {Subst}\left (\int \frac {1}{4 c d f-4 b \sqrt {d} f^{3/2}+4 a f^2-x^2} \, dx,x,\frac {b \sqrt {d} \sqrt {f}-2 a f-\left (-2 c \sqrt {d} \sqrt {f}+b f\right ) x}{\sqrt {a+b x+c x^2}}\right )}{d^2}-\frac {\left (f \left (c d+b \sqrt {d} \sqrt {f}+a f\right )\right ) \text {Subst}\left (\int \frac {1}{4 c d f+4 b \sqrt {d} f^{3/2}+4 a f^2-x^2} \, dx,x,\frac {-b \sqrt {d} \sqrt {f}-2 a f-\left (2 c \sqrt {d} \sqrt {f}+b f\right ) x}{\sqrt {a+b x+c x^2}}\right )}{d^2} \\ & = -\frac {(2 a+b x) \sqrt {a+b x+c x^2}}{4 a d x^2}+\frac {\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{8 a^{3/2} d}-\frac {\sqrt {a} f \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{d^2}-\frac {\sqrt {f} \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \tanh ^{-1}\left (\frac {b \sqrt {d}-2 a \sqrt {f}+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x}{2 \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 d^2}+\frac {\sqrt {f} \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \tanh ^{-1}\left (\frac {b \sqrt {d}+2 a \sqrt {f}+\left (2 c \sqrt {d}+b \sqrt {f}\right ) x}{2 \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 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 = 381, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {a+b x+c x^2}}{x^3 \left (d-f x^2\right )} \, dx=\frac {-\frac {d (2 a+b x) \sqrt {a+x (b+c x)}}{a x^2}+\frac {\left (b^2 d-4 a (c d+2 a f)\right ) \text {arctanh}\left (\frac {-\sqrt {c} x+\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{a^{3/2}}-2 f \text {RootSum}\left [b^2 d-a^2 f-4 b \sqrt {c} d \text {$\#$1}+4 c d \text {$\#$1}^2+2 a f \text {$\#$1}^2-f \text {$\#$1}^4\&,\frac {b^2 d \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-a c d \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-a^2 f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-2 b \sqrt {c} d \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}+c d \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+a f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{b \sqrt {c} d-2 c d \text {$\#$1}-a f \text {$\#$1}+f \text {$\#$1}^3}\&\right ]}{4 d^2} \]
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Time = 0.96 (sec) , antiderivative size = 479, normalized size of antiderivative = 1.36
method | result | size |
risch | \(-\frac {\left (b x +2 a \right ) \sqrt {c \,x^{2}+b x +a}}{4 a d \,x^{2}}-\frac {-\frac {\left (-8 a^{2} f -4 a c d +b^{2} d \right ) \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{d \sqrt {a}}-\frac {4 a \left (b \sqrt {d f}+f a +c d \right ) \ln \left (\frac {\frac {2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x -\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+f a +c d}{f}}}{x -\frac {\sqrt {d f}}{f}}\right )}{d \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}}+\frac {4 a \left (b \sqrt {d f}-f a -c d \right ) \ln \left (\frac {\frac {-2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x +\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+f a +c d}{f}}}{x +\frac {\sqrt {d f}}{f}}\right )}{d \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}}}{8 a d}\) | \(479\) |
default | \(\text {Expression too large to display}\) | \(1143\) |
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Leaf count of result is larger than twice the leaf count of optimal. 738 vs. \(2 (275) = 550\).
Time = 81.78 (sec) , antiderivative size = 1485, normalized size of antiderivative = 4.21 \[ \int \frac {\sqrt {a+b x+c x^2}}{x^3 \left (d-f x^2\right )} \, dx=\text {Too large to display} \]
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\[ \int \frac {\sqrt {a+b x+c x^2}}{x^3 \left (d-f x^2\right )} \, dx=- \int \frac {\sqrt {a + b x + c x^{2}}}{- d x^{3} + f x^{5}}\, dx \]
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\[ \int \frac {\sqrt {a+b x+c x^2}}{x^3 \left (d-f x^2\right )} \, dx=\int { -\frac {\sqrt {c x^{2} + b x + a}}{{\left (f x^{2} - d\right )} x^{3}} \,d x } \]
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Exception generated. \[ \int \frac {\sqrt {a+b x+c x^2}}{x^3 \left (d-f x^2\right )} \, dx=\text {Exception raised: AttributeError} \]
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Timed out. \[ \int \frac {\sqrt {a+b x+c x^2}}{x^3 \left (d-f x^2\right )} \, dx=\int \frac {\sqrt {c\,x^2+b\,x+a}}{x^3\,\left (d-f\,x^2\right )} \,d x \]
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