Integrand size = 28, antiderivative size = 341 \[ \int \frac {x^3}{\left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx=-\frac {2 (2 a+b x)}{\left (b^2-4 a c\right ) f \sqrt {a+b x+c x^2}}-\frac {2 d \left (a \left (2 c^2 d-b^2 f+2 a c f\right )+b c (c d-a f) x\right )}{\left (b^2-4 a c\right ) f \left (b^2 d f-(c d+a f)^2\right ) \sqrt {a+b x+c x^2}}-\frac {d \text {arctanh}\left (\frac {b \sqrt {d}-2 a \sqrt {f}+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x}{2 \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {f} \left (c d-b \sqrt {d} \sqrt {f}+a f\right )^{3/2}}+\frac {d \text {arctanh}\left (\frac {b \sqrt {d}+2 a \sqrt {f}+\left (2 c \sqrt {d}+b \sqrt {f}\right ) x}{2 \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {f} \left (c d+b \sqrt {d} \sqrt {f}+a f\right )^{3/2}} \]
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Time = 0.63 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {6857, 650, 1032, 1047, 738, 212} \[ \int \frac {x^3}{\left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx=-\frac {d \text {arctanh}\left (\frac {-2 a \sqrt {f}+x \left (2 c \sqrt {d}-b \sqrt {f}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )}{2 \sqrt {f} \left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right )^{3/2}}+\frac {d \text {arctanh}\left (\frac {2 a \sqrt {f}+x \left (b \sqrt {f}+2 c \sqrt {d}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )}{2 \sqrt {f} \left (a f+b \sqrt {d} \sqrt {f}+c d\right )^{3/2}}-\frac {2 d \left (a \left (2 a c f+b^2 (-f)+2 c^2 d\right )+b c x (c d-a f)\right )}{f \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (b^2 d f-(a f+c d)^2\right )}-\frac {2 (2 a+b x)}{f \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}} \]
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Rule 212
Rule 650
Rule 738
Rule 1032
Rule 1047
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {x}{f \left (a+b x+c x^2\right )^{3/2}}+\frac {d x}{f \left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )}\right ) \, dx \\ & = -\frac {\int \frac {x}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{f}+\frac {d \int \frac {x}{\left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx}{f} \\ & = -\frac {2 (2 a+b x)}{\left (b^2-4 a c\right ) f \sqrt {a+b x+c x^2}}-\frac {2 d \left (a \left (2 c^2 d-b^2 f+2 a c f\right )+b c (c d-a f) x\right )}{\left (b^2-4 a c\right ) f \left (b^2 d f-(c d+a f)^2\right ) \sqrt {a+b x+c x^2}}+\frac {(2 d) \int \frac {\frac {1}{2} b \left (b^2-4 a c\right ) d f-\frac {1}{2} \left (b^2-4 a c\right ) f (c d+a f) x}{\sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx}{\left (b^2-4 a c\right ) f \left (b^2 d f-(c d+a f)^2\right )} \\ & = -\frac {2 (2 a+b x)}{\left (b^2-4 a c\right ) f \sqrt {a+b x+c x^2}}-\frac {2 d \left (a \left (2 c^2 d-b^2 f+2 a c f\right )+b c (c d-a f) x\right )}{\left (b^2-4 a c\right ) f \left (b^2 d f-(c d+a f)^2\right ) \sqrt {a+b x+c x^2}}+\frac {d \int \frac {1}{\left (-\sqrt {d} \sqrt {f}-f x\right ) \sqrt {a+b x+c x^2}} \, dx}{2 \left (c d-b \sqrt {d} \sqrt {f}+a f\right )}+\frac {d \int \frac {1}{\left (\sqrt {d} \sqrt {f}-f x\right ) \sqrt {a+b x+c x^2}} \, dx}{2 \left (c d+b \sqrt {d} \sqrt {f}+a f\right )} \\ & = -\frac {2 (2 a+b x)}{\left (b^2-4 a c\right ) f \sqrt {a+b x+c x^2}}-\frac {2 d \left (a \left (2 c^2 d-b^2 f+2 a c f\right )+b c (c d-a f) x\right )}{\left (b^2-4 a c\right ) f \left (b^2 d f-(c d+a f)^2\right ) \sqrt {a+b x+c x^2}}-\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 )}{c d-b \sqrt {d} \sqrt {f}+a 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 )}{c d+b \sqrt {d} \sqrt {f}+a f} \\ & = -\frac {2 (2 a+b x)}{\left (b^2-4 a c\right ) f \sqrt {a+b x+c x^2}}-\frac {2 d \left (a \left (2 c^2 d-b^2 f+2 a c f\right )+b c (c d-a f) x\right )}{\left (b^2-4 a c\right ) f \left (b^2 d f-(c d+a f)^2\right ) \sqrt {a+b x+c x^2}}-\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 \sqrt {f} \left (c d-b \sqrt {d} \sqrt {f}+a f\right )^{3/2}}+\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 \sqrt {f} \left (c d+b \sqrt {d} \sqrt {f}+a f\right )^{3/2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.74 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.20 \[ \int \frac {x^3}{\left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx=\frac {8 a^3 f-4 b^3 d x-4 a b d (b-3 c x)+4 a^2 (2 c d+b f x)-\left (b^2-4 a c\right ) d \sqrt {a+x (b+c x)} \text {RootSum}\left [b^2 d-a^2 f-4 b \sqrt {c} d \text {$\#$1}+4 c d \text {$\#$1}^2+2 a f \text {$\#$1}^2-f \text {$\#$1}^4\&,\frac {b^2 d \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )+a c d \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )+a^2 f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-2 b \sqrt {c} d \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-c d \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2-a f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{b \sqrt {c} d-2 c d \text {$\#$1}-a f \text {$\#$1}+f \text {$\#$1}^3}\&\right ]}{2 \left (b^2-4 a c\right ) \left (-c^2 d^2-2 a c d f+f \left (b^2 d-a^2 f\right )\right ) \sqrt {a+x (b+c x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(959\) vs. \(2(281)=562\).
Time = 0.87 (sec) , antiderivative size = 960, normalized size of antiderivative = 2.82
method | result | size |
default | \(-\frac {-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}}{f}-\frac {d \left (\frac {f}{\left (-b \sqrt {d f}+f a +c d \right ) \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}}}-\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (2 c \left (x +\frac {\sqrt {d f}}{f}\right )+\frac {-2 c \sqrt {d f}+b f}{f}\right )}{\left (-b \sqrt {d f}+f a +c d \right ) \left (\frac {4 c \left (-b \sqrt {d f}+f a +c d \right )}{f}-\frac {\left (-2 c \sqrt {d f}+b f \right )^{2}}{f^{2}}\right ) \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}}}-\frac {f \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 )}{\left (-b \sqrt {d f}+f a +c d \right ) \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}}\right )}{2 f^{2}}-\frac {d \left (\frac {f}{\left (b \sqrt {d f}+f a +c d \right ) \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}}}-\frac {\left (2 c \sqrt {d f}+b f \right ) \left (2 c \left (x -\frac {\sqrt {d f}}{f}\right )+\frac {2 c \sqrt {d f}+b f}{f}\right )}{\left (b \sqrt {d f}+f a +c d \right ) \left (\frac {4 c \left (b \sqrt {d f}+f a +c d \right )}{f}-\frac {\left (2 c \sqrt {d f}+b f \right )^{2}}{f^{2}}\right ) \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}}}-\frac {f \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 )}{\left (b \sqrt {d f}+f a +c d \right ) \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}}\right )}{2 f^{2}}\) | \(960\) |
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Leaf count of result is larger than twice the leaf count of optimal. 17339 vs. \(2 (281) = 562\).
Time = 22.37 (sec) , antiderivative size = 17339, normalized size of antiderivative = 50.85 \[ \int \frac {x^3}{\left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {x^3}{\left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {x^3}{\left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx=\text {Exception raised: ValueError} \]
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Exception generated. \[ \int \frac {x^3}{\left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^3}{\left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx=\int \frac {x^3}{\left (d-f\,x^2\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \]
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