Integrand size = 28, antiderivative size = 463 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^2 \left (d-f x^2\right )} \, dx=\frac {3 (3 b+2 c x) \sqrt {a+b x+c x^2}}{4 d}-\frac {(5 b+2 c x) \sqrt {a+b x+c x^2}}{4 d}-\frac {\left (a+b x+c x^2\right )^{3/2}}{d x}-\frac {3 \sqrt {a} b \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 d}+\frac {3 \left (b^2+4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {c} d}-\frac {\left (8 c^2 d+3 b^2 f+12 a c f\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {c} d f}+\frac {\left (c d-b \sqrt {d} \sqrt {f}+a f\right )^{3/2} \text {arctanh}\left (\frac {b \sqrt {d}-2 a \sqrt {f}+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x}{2 \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 d^{3/2} f}+\frac {\left (c d+b \sqrt {d} \sqrt {f}+a f\right )^{3/2} \text {arctanh}\left (\frac {b \sqrt {d}+2 a \sqrt {f}+\left (2 c \sqrt {d}+b \sqrt {f}\right ) x}{2 \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 d^{3/2} f} \]
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Time = 0.79 (sec) , antiderivative size = 463, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {6857, 746, 828, 857, 635, 212, 738, 992, 1092, 1047} \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^2 \left (d-f x^2\right )} \, dx=-\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (12 a c f+3 b^2 f+8 c^2 d\right )}{8 \sqrt {c} d f}+\frac {3 \left (4 a c+b^2\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {c} d}+\frac {\left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right )^{3/2} \text {arctanh}\left (\frac {-2 a \sqrt {f}+x \left (2 c \sqrt {d}-b \sqrt {f}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )}{2 d^{3/2} f}+\frac {\left (a f+b \sqrt {d} \sqrt {f}+c d\right )^{3/2} \text {arctanh}\left (\frac {2 a \sqrt {f}+x \left (b \sqrt {f}+2 c \sqrt {d}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )}{2 d^{3/2} f}-\frac {3 \sqrt {a} b \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 d}-\frac {\left (a+b x+c x^2\right )^{3/2}}{d x}+\frac {3 (3 b+2 c x) \sqrt {a+b x+c x^2}}{4 d}-\frac {(5 b+2 c x) \sqrt {a+b x+c x^2}}{4 d} \]
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Rule 212
Rule 635
Rule 738
Rule 746
Rule 828
Rule 857
Rule 992
Rule 1047
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^2}+\frac {f \left (a+b x+c x^2\right )^{3/2}}{d \left (d-f x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^2} \, dx}{d}+\frac {f \int \frac {\left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx}{d} \\ & = -\frac {(5 b+2 c x) \sqrt {a+b x+c x^2}}{4 d}-\frac {\left (a+b x+c x^2\right )^{3/2}}{d x}+\frac {\int \frac {\frac {1}{4} \left (5 b^2 d+4 a (c d+2 a f)\right )+4 b (c d+a f) x+\frac {1}{4} \left (8 c^2 d+3 b^2 f+12 a c f\right ) x^2}{\sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx}{2 d}+\frac {3 \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{x} \, dx}{2 d} \\ & = \frac {3 (3 b+2 c x) \sqrt {a+b x+c x^2}}{4 d}-\frac {(5 b+2 c x) \sqrt {a+b x+c x^2}}{4 d}-\frac {\left (a+b x+c x^2\right )^{3/2}}{d x}-\frac {3 \int \frac {-4 a b c-c \left (b^2+4 a c\right ) x}{x \sqrt {a+b x+c x^2}} \, dx}{8 c d}-\frac {\int \frac {-\frac {1}{4} d \left (8 c^2 d+3 b^2 f+12 a c f\right )-\frac {1}{4} f \left (5 b^2 d+4 a (c d+2 a f)\right )-4 b f (c d+a f) x}{\sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx}{2 d f}-\frac {\left (8 c^2 d+3 b^2 f+12 a c f\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{8 d f} \\ & = \frac {3 (3 b+2 c x) \sqrt {a+b x+c x^2}}{4 d}-\frac {(5 b+2 c x) \sqrt {a+b x+c x^2}}{4 d}-\frac {\left (a+b x+c x^2\right )^{3/2}}{d x}+\frac {(3 a b) \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx}{2 d}+\frac {\left (3 \left (b^2+4 a c\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{8 d}-\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^{3/2} \sqrt {f}}+\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^{3/2} \sqrt {f}}-\frac {\left (8 c^2 d+3 b^2 f+12 a c 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 )}{4 d f} \\ & = \frac {3 (3 b+2 c x) \sqrt {a+b x+c x^2}}{4 d}-\frac {(5 b+2 c x) \sqrt {a+b x+c x^2}}{4 d}-\frac {\left (a+b x+c x^2\right )^{3/2}}{d x}-\frac {\left (8 c^2 d+3 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 )}{8 \sqrt {c} d f}-\frac {(3 a b) \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}+\frac {\left (3 \left (b^2+4 a c\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 )}{4 d}+\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^{3/2} \sqrt {f}}-\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^{3/2} \sqrt {f}} \\ & = \frac {3 (3 b+2 c x) \sqrt {a+b x+c x^2}}{4 d}-\frac {(5 b+2 c x) \sqrt {a+b x+c x^2}}{4 d}-\frac {\left (a+b x+c x^2\right )^{3/2}}{d x}-\frac {3 \sqrt {a} b \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 d}+\frac {3 \left (b^2+4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {c} d}-\frac {\left (8 c^2 d+3 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 )}{8 \sqrt {c} d 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^{3/2} 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^{3/2} f} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.65 (sec) , antiderivative size = 541, normalized size of antiderivative = 1.17 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^2 \left (d-f x^2\right )} \, dx=\frac {-2 a f \sqrt {a+x (b+c x)}+6 \sqrt {a} b f x \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )+2 c^{3/2} d x \log \left (f \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )\right )-x \text {RootSum}\left [b^2 d-a^2 f-4 b \sqrt {c} d \text {$\#$1}+4 c d \text {$\#$1}^2+2 a f \text {$\#$1}^2-f \text {$\#$1}^4\&,\frac {b c^2 d^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )+b^3 d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-a^2 b f^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-2 c^{5/2} d^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-2 b^2 \sqrt {c} d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-4 a c^{3/2} d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-2 a^2 \sqrt {c} f^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}+2 b c d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+2 a b f^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{b \sqrt {c} d-2 c d \text {$\#$1}-a f \text {$\#$1}+f \text {$\#$1}^3}\&\right ]}{2 d f x} \]
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Time = 0.85 (sec) , antiderivative size = 585, normalized size of antiderivative = 1.26
method | result | size |
risch | \(-\frac {a \sqrt {c \,x^{2}+b x +a}}{d x}-\frac {\frac {2 c^{\frac {3}{2}} d \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{f}+3 b \sqrt {a}\, \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )-\frac {\left (-\sqrt {d f}\, a^{2} f^{2}-2 \sqrt {d f}\, a c d f -\sqrt {d f}\, b^{2} d f -\sqrt {d f}\, c^{2} d^{2}+2 a b d \,f^{2}+2 b c \,d^{2} f \right ) \ln \left (\frac {\frac {-2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x +\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+f a +c d}{f}}}{x +\frac {\sqrt {d f}}{f}}\right )}{d \,f^{2} \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}}-\frac {\left (\sqrt {d f}\, a^{2} f^{2}+2 \sqrt {d f}\, a c d f +\sqrt {d f}\, b^{2} d f +\sqrt {d f}\, c^{2} d^{2}+2 a b d \,f^{2}+2 b c \,d^{2} f \right ) \ln \left (\frac {\frac {2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x -\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+f a +c d}{f}}}{x -\frac {\sqrt {d f}}{f}}\right )}{f^{2} d \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}}}{2 d}\) | \(585\) |
default | \(\text {Expression too large to display}\) | \(1787\) |
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Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^2 \left (d-f x^2\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^2 \left (d-f x^2\right )} \, dx=- \int \frac {a \sqrt {a + b x + c x^{2}}}{- d x^{2} + f x^{4}}\, dx - \int \frac {b x \sqrt {a + b x + c x^{2}}}{- d x^{2} + f x^{4}}\, dx - \int \frac {c x^{2} \sqrt {a + b x + c x^{2}}}{- d x^{2} + f x^{4}}\, dx \]
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\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^2 \left (d-f x^2\right )} \, dx=\int { -\frac {{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}}{{\left (f x^{2} - d\right )} x^{2}} \,d x } \]
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Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^2 \left (d-f x^2\right )} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^2 \left (d-f x^2\right )} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{x^2\,\left (d-f\,x^2\right )} \,d x \]
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