Integrand size = 28, antiderivative size = 287 \[ \int \frac {x^3}{\sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx=-\frac {\sqrt {a+b x+c x^2}}{c f}+\frac {b \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 c^{3/2} f}-\frac {d \text {arctanh}\left (\frac {b \sqrt {d}-2 a \sqrt {f}+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x}{2 \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 f^{3/2} \sqrt {c d-b \sqrt {d} \sqrt {f}+a f}}+\frac {d \text {arctanh}\left (\frac {b \sqrt {d}+2 a \sqrt {f}+\left (2 c \sqrt {d}+b \sqrt {f}\right ) x}{2 \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 f^{3/2} \sqrt {c d+b \sqrt {d} \sqrt {f}+a f}} \]
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Time = 0.41 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {6857, 654, 635, 212, 1047, 738} \[ \int \frac {x^3}{\sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx=\frac {b \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 c^{3/2} f}-\frac {d \text {arctanh}\left (\frac {-2 a \sqrt {f}+x \left (2 c \sqrt {d}-b \sqrt {f}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )}{2 f^{3/2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}+\frac {d \text {arctanh}\left (\frac {2 a \sqrt {f}+x \left (b \sqrt {f}+2 c \sqrt {d}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )}{2 f^{3/2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}-\frac {\sqrt {a+b x+c x^2}}{c f} \]
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Rule 212
Rule 635
Rule 654
Rule 738
Rule 1047
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {x}{f \sqrt {a+b x+c x^2}}+\frac {d x}{f \sqrt {a+b x+c x^2} \left (d-f x^2\right )}\right ) \, dx \\ & = -\frac {\int \frac {x}{\sqrt {a+b x+c x^2}} \, dx}{f}+\frac {d \int \frac {x}{\sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx}{f} \\ & = -\frac {\sqrt {a+b x+c x^2}}{c f}+\frac {b \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{2 c f}+\frac {d \int \frac {1}{\left (-\sqrt {d} \sqrt {f}-f x\right ) \sqrt {a+b x+c x^2}} \, dx}{2 f}+\frac {d \int \frac {1}{\left (\sqrt {d} \sqrt {f}-f x\right ) \sqrt {a+b x+c x^2}} \, dx}{2 f} \\ & = -\frac {\sqrt {a+b x+c x^2}}{c f}+\frac {b \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 )}{c f}-\frac {d \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 )}{f}-\frac {d \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 )}{f} \\ & = -\frac {\sqrt {a+b x+c x^2}}{c f}+\frac {b \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 c^{3/2} f}-\frac {d \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 f^{3/2} \sqrt {c d-b \sqrt {d} \sqrt {f}+a f}}+\frac {d \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 f^{3/2} \sqrt {c d+b \sqrt {d} \sqrt {f}+a f}} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.35 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.73 \[ \int \frac {x^3}{\sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx=-\frac {\frac {2 \sqrt {a+x (b+c x)}}{c}+\frac {b \log \left (c f \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )\right )}{c^{3/2}}+d \text {RootSum}\left [b^2 d-a^2 f-4 b \sqrt {c} d \text {$\#$1}+4 c d \text {$\#$1}^2+2 a f \text {$\#$1}^2-f \text {$\#$1}^4\&,\frac {a \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-\log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-b \sqrt {c} d+2 c d \text {$\#$1}+a f \text {$\#$1}-f \text {$\#$1}^3}\&\right ]}{2 f} \]
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Time = 0.74 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.43
method | result | size |
default | \(-\frac {\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}}{f}+\frac {d \ln \left (\frac {\frac {-2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x +\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+f a +c d}{f}}}{x +\frac {\sqrt {d f}}{f}}\right )}{2 f^{2} \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}}+\frac {d \ln \left (\frac {\frac {2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x -\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+f a +c d}{f}}}{x -\frac {\sqrt {d f}}{f}}\right )}{2 f^{2} \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}}\) | \(409\) |
risch | \(-\frac {\sqrt {c \,x^{2}+b x +a}}{c f}+\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}} f}+\frac {d \ln \left (\frac {\frac {-2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x +\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+f a +c d}{f}}}{x +\frac {\sqrt {d f}}{f}}\right )}{2 f^{2} \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}}+\frac {d \ln \left (\frac {\frac {2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x -\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+f a +c d}{f}}}{x -\frac {\sqrt {d f}}{f}}\right )}{2 f^{2} \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}}\) | \(410\) |
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Timed out. \[ \int \frac {x^3}{\sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {x^3}{\sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx=- \int \frac {x^{3}}{- d \sqrt {a + b x + c x^{2}} + f x^{2} \sqrt {a + b x + c x^{2}}}\, dx \]
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Exception generated. \[ \int \frac {x^3}{\sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx=\text {Exception raised: ValueError} \]
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Exception generated. \[ \int \frac {x^3}{\sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^3}{\sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx=\int \frac {x^3}{\left (d-f\,x^2\right )\,\sqrt {c\,x^2+b\,x+a}} \,d x \]
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