Integrand size = 28, antiderivative size = 614 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^3 \left (d-f x^2\right )} \, dx=-\frac {3 (b-2 c x) \sqrt {a+b x+c x^2}}{4 d x}+\frac {f \left (b^2+8 a c+2 b c x\right ) \sqrt {a+b x+c x^2}}{8 c d^2}-\frac {\left (8 c^2 d+b^2 f+8 a c f+2 b c f x\right ) \sqrt {a+b x+c x^2}}{8 c d^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{2 d x^2}-\frac {3 \left (b^2+4 a c\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {a} d}-\frac {a^{3/2} f \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{d^2}+\frac {3 b \sqrt {c} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 d}-\frac {b \left (b^2-12 a c\right ) f \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{3/2} d^2}-\frac {b \left (24 c^2 d-b^2 f+12 a c f\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{3/2} d^2}-\frac {\left (c d-b \sqrt {d} \sqrt {f}+a f\right )^{3/2} \text {arctanh}\left (\frac {b \sqrt {d}-2 a \sqrt {f}+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x}{2 \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 d^2 \sqrt {f}}+\frac {\left (c d+b \sqrt {d} \sqrt {f}+a f\right )^{3/2} \text {arctanh}\left (\frac {b \sqrt {d}+2 a \sqrt {f}+\left (2 c \sqrt {d}+b \sqrt {f}\right ) x}{2 \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 d^2 \sqrt {f}} \]
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Time = 0.93 (sec) , antiderivative size = 614, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {6857, 746, 826, 857, 635, 212, 738, 748, 828, 1035, 1084, 1092, 1047} \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^3 \left (d-f x^2\right )} \, dx=-\frac {a^{3/2} f \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{d^2}-\frac {b f \left (b^2-12 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{3/2} d^2}-\frac {b \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (12 a c f+b^2 (-f)+24 c^2 d\right )}{16 c^{3/2} d^2}-\frac {3 \left (4 a c+b^2\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {a} d}-\frac {\left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right )^{3/2} \text {arctanh}\left (\frac {-2 a \sqrt {f}+x \left (2 c \sqrt {d}-b \sqrt {f}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )}{2 d^2 \sqrt {f}}+\frac {\left (a f+b \sqrt {d} \sqrt {f}+c d\right )^{3/2} \text {arctanh}\left (\frac {2 a \sqrt {f}+x \left (b \sqrt {f}+2 c \sqrt {d}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )}{2 d^2 \sqrt {f}}+\frac {3 b \sqrt {c} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 d}-\frac {\sqrt {a+b x+c x^2} \left (8 a c f+b^2 f+2 b c f x+8 c^2 d\right )}{8 c d^2}+\frac {f \left (8 a c+b^2+2 b c x\right ) \sqrt {a+b x+c x^2}}{8 c d^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{2 d x^2}-\frac {3 (b-2 c x) \sqrt {a+b x+c x^2}}{4 d x} \]
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Rule 212
Rule 635
Rule 738
Rule 746
Rule 748
Rule 826
Rule 828
Rule 857
Rule 1035
Rule 1047
Rule 1084
Rule 1092
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (a+b x+c x^2\right )^{3/2}}{d x^3}+\frac {f \left (a+b x+c x^2\right )^{3/2}}{d^2 x}+\frac {f^2 x \left (a+b x+c x^2\right )^{3/2}}{d^2 \left (d-f x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^3} \, dx}{d}+\frac {f \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x} \, dx}{d^2}+\frac {f^2 \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx}{d^2} \\ & = -\frac {\left (a+b x+c x^2\right )^{3/2}}{2 d x^2}+\frac {3 \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{x^2} \, dx}{4 d}+\frac {f \int \frac {\sqrt {a+b x+c x^2} \left (\frac {3 b d}{2}+3 (c d+a f) x+\frac {3}{2} b f x^2\right )}{d-f x^2} \, dx}{3 d^2}-\frac {f \int \frac {(-2 a-b x) \sqrt {a+b x+c x^2}}{x} \, dx}{2 d^2} \\ & = -\frac {3 (b-2 c x) \sqrt {a+b x+c x^2}}{4 d x}+\frac {f \left (b^2+8 a c+2 b c x\right ) \sqrt {a+b x+c x^2}}{8 c d^2}-\frac {\left (8 c^2 d+b^2 f+8 a c f+2 b c f x\right ) \sqrt {a+b x+c x^2}}{8 c d^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{2 d x^2}-\frac {3 \int \frac {-b^2-4 a c-4 b c x}{x \sqrt {a+b x+c x^2}} \, dx}{8 d}-\frac {\int \frac {-\frac {3}{8} b d f \left (8 c^2 d+b^2 f+20 a c f\right )-6 c f \left (b^2 d f+(c d+a f)^2\right ) x-\frac {3}{8} b f^2 \left (24 c^2 d-b^2 f+12 a c f\right ) x^2}{\sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx}{6 c d^2 f}+\frac {f \int \frac {8 a^2 c-\frac {1}{2} b \left (b^2-12 a c\right ) x}{x \sqrt {a+b x+c x^2}} \, dx}{8 c d^2} \\ & = -\frac {3 (b-2 c x) \sqrt {a+b x+c x^2}}{4 d x}+\frac {f \left (b^2+8 a c+2 b c x\right ) \sqrt {a+b x+c x^2}}{8 c d^2}-\frac {\left (8 c^2 d+b^2 f+8 a c f+2 b c f x\right ) \sqrt {a+b x+c x^2}}{8 c d^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{2 d x^2}+\frac {(3 b c) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{2 d}+\frac {\left (3 \left (b^2+4 a c\right )\right ) \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx}{8 d}+\frac {\int \frac {\frac {3}{8} b d f^2 \left (24 c^2 d-b^2 f+12 a c f\right )+\frac {3}{8} b d f^2 \left (8 c^2 d+b^2 f+20 a c f\right )+6 c f^2 \left (b^2 d f+(c d+a f)^2\right ) x}{\sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx}{6 c d^2 f^2}+\frac {\left (a^2 f\right ) \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx}{d^2}-\frac {\left (b \left (b^2-12 a c\right ) f\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{16 c d^2}-\frac {\left (b \left (24 c^2 d-b^2 f+12 a c f\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{16 c d^2} \\ & = -\frac {3 (b-2 c x) \sqrt {a+b x+c x^2}}{4 d x}+\frac {f \left (b^2+8 a c+2 b c x\right ) \sqrt {a+b x+c x^2}}{8 c d^2}-\frac {\left (8 c^2 d+b^2 f+8 a c f+2 b c f x\right ) \sqrt {a+b x+c x^2}}{8 c d^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{2 d x^2}+\frac {(3 b c) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{d}-\frac {\left (3 \left (b^2+4 a c\right )\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 d}-\frac {\left (2 a^2 f\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 )}{d^2}-\frac {\left (b \left (b^2-12 a c\right ) f\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{8 c d^2}+\frac {\left (c d-b \sqrt {d} \sqrt {f}+a f\right )^2 \int \frac {1}{\left (-\sqrt {d} \sqrt {f}-f x\right ) \sqrt {a+b x+c x^2}} \, dx}{2 d^2}+\frac {\left (c d+b \sqrt {d} \sqrt {f}+a f\right )^2 \int \frac {1}{\left (\sqrt {d} \sqrt {f}-f x\right ) \sqrt {a+b x+c x^2}} \, dx}{2 d^2}-\frac {\left (b \left (24 c^2 d-b^2 f+12 a c f\right )\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{8 c d^2} \\ & = -\frac {3 (b-2 c x) \sqrt {a+b x+c x^2}}{4 d x}+\frac {f \left (b^2+8 a c+2 b c x\right ) \sqrt {a+b x+c x^2}}{8 c d^2}-\frac {\left (8 c^2 d+b^2 f+8 a c f+2 b c f x\right ) \sqrt {a+b x+c x^2}}{8 c d^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{2 d x^2}-\frac {3 \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 \sqrt {a} d}-\frac {a^{3/2} f \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{d^2}+\frac {3 b \sqrt {c} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 d}-\frac {b \left (b^2-12 a c\right ) f \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{3/2} d^2}-\frac {b \left (24 c^2 d-b^2 f+12 a c f\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{3/2} d^2}-\frac {\left (c d-b \sqrt {d} \sqrt {f}+a f\right )^2 \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 (c d+b \sqrt {d} \sqrt {f}+a f\right )^2 \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 {3 (b-2 c x) \sqrt {a+b x+c x^2}}{4 d x}+\frac {f \left (b^2+8 a c+2 b c x\right ) \sqrt {a+b x+c x^2}}{8 c d^2}-\frac {\left (8 c^2 d+b^2 f+8 a c f+2 b c f x\right ) \sqrt {a+b x+c x^2}}{8 c d^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{2 d x^2}-\frac {3 \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 \sqrt {a} d}-\frac {a^{3/2} f \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{d^2}+\frac {3 b \sqrt {c} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 d}-\frac {b \left (b^2-12 a c\right ) f \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{3/2} d^2}-\frac {b \left (24 c^2 d-b^2 f+12 a c f\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{3/2} d^2}-\frac {\left (c d-b \sqrt {d} \sqrt {f}+a f\right )^{3/2} \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 \sqrt {f}}+\frac {\left (c d+b \sqrt {d} \sqrt {f}+a f\right )^{3/2} \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 \sqrt {f}} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.98 (sec) , antiderivative size = 587, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^3 \left (d-f x^2\right )} \, dx=-\frac {\frac {d (2 a+5 b x) \sqrt {a+x (b+c x)}}{x^2}+\frac {\left (3 b^2 d+4 a (3 c d+2 a f)\right ) \text {arctanh}\left (\frac {-\sqrt {c} x+\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{\sqrt {a}}+2 \text {RootSum}\left [b^2 d-a^2 f-4 b \sqrt {c} d \text {$\#$1}+4 c d \text {$\#$1}^2+2 a f \text {$\#$1}^2-f \text {$\#$1}^4\&,\frac {2 b^2 c d^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-a c^2 d^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )+a b^2 d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-2 a^2 c d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-a^3 f^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-4 b c^{3/2} d^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-4 a b \sqrt {c} d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}+c^2 d^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+b^2 d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+2 a c d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+a^2 f^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{b \sqrt {c} d-2 c d \text {$\#$1}-a f \text {$\#$1}+f \text {$\#$1}^3}\&\right ]}{4 d^2} \]
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Time = 0.92 (sec) , antiderivative size = 549, normalized size of antiderivative = 0.89
method | result | size |
risch | \(-\frac {\sqrt {c \,x^{2}+b x +a}\, \left (5 b x +2 a \right )}{4 d \,x^{2}}-\frac {-\frac {\left (-8 a^{2} f -12 a c d -3 b^{2} d \right ) \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{d \sqrt {a}}-\frac {\left (-8 \sqrt {d f}\, a b f -8 \sqrt {d f}\, b c d +4 a^{2} f^{2}+8 a c d f +4 b^{2} d f +4 c^{2} d^{2}\right ) \ln \left (\frac {\frac {-2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x +\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+f a +c d}{f}}}{x +\frac {\sqrt {d f}}{f}}\right )}{d f \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}}-\frac {\left (8 \sqrt {d f}\, a b f +8 \sqrt {d f}\, b c d +4 a^{2} f^{2}+8 a c d f +4 b^{2} d f +4 c^{2} d^{2}\right ) \ln \left (\frac {\frac {2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x -\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+f a +c d}{f}}}{x -\frac {\sqrt {d f}}{f}}\right )}{d f \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}}}{8 d}\) | \(549\) |
default | \(\text {Expression too large to display}\) | \(2139\) |
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Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^3 \left (d-f x^2\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^3 \left (d-f x^2\right )} \, dx=- \int \frac {a \sqrt {a + b x + c x^{2}}}{- d x^{3} + f x^{5}}\, dx - \int \frac {b x \sqrt {a + b x + c x^{2}}}{- d x^{3} + f x^{5}}\, dx - \int \frac {c x^{2} \sqrt {a + b x + c x^{2}}}{- d x^{3} + f x^{5}}\, dx \]
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\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^3 \left (d-f x^2\right )} \, dx=\int { -\frac {{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}}{{\left (f x^{2} - d\right )} x^{3}} \,d x } \]
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Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^3 \left (d-f x^2\right )} \, dx=\text {Exception raised: AttributeError} \]
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Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^3 \left (d-f x^2\right )} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{x^3\,\left (d-f\,x^2\right )} \,d x \]
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