Integrand size = 26, antiderivative size = 349 \[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=-\frac {\left (8 c^2 d+b^2 f+8 a c f+2 b c f x\right ) \sqrt {a+b x+c x^2}}{8 c f^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 f}-\frac {b \left (24 c^2 d-b^2 f+12 a c f\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{3/2} f^2}-\frac {\left (c d-b \sqrt {d} \sqrt {f}+a f\right )^{3/2} \text {arctanh}\left (\frac {b \sqrt {d}-2 a \sqrt {f}+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x}{2 \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 f^{5/2}}+\frac {\left (c d+b \sqrt {d} \sqrt {f}+a f\right )^{3/2} \text {arctanh}\left (\frac {b \sqrt {d}+2 a \sqrt {f}+\left (2 c \sqrt {d}+b \sqrt {f}\right ) x}{2 \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 f^{5/2}} \]
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Time = 0.33 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {1035, 1084, 1092, 635, 212, 1047, 738} \[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=-\frac {b \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (12 a c f+b^2 (-f)+24 c^2 d\right )}{16 c^{3/2} f^2}-\frac {\left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right )^{3/2} \text {arctanh}\left (\frac {-2 a \sqrt {f}+x \left (2 c \sqrt {d}-b \sqrt {f}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )}{2 f^{5/2}}+\frac {\left (a f+b \sqrt {d} \sqrt {f}+c d\right )^{3/2} \text {arctanh}\left (\frac {2 a \sqrt {f}+x \left (b \sqrt {f}+2 c \sqrt {d}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )}{2 f^{5/2}}-\frac {\sqrt {a+b x+c x^2} \left (8 a c f+b^2 f+2 b c f x+8 c^2 d\right )}{8 c f^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 f} \]
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Rule 212
Rule 635
Rule 738
Rule 1035
Rule 1047
Rule 1084
Rule 1092
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a+b x+c x^2\right )^{3/2}}{3 f}+\frac {\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 f} \\ & = -\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 f^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 f}-\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 f^3} \\ & = -\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 f^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 f}+\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 f^4}-\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 f^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 f^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 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 f^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 f^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 f^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 f^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 f}-\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} f^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 )}{f^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 )}{f^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 f^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 f}-\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} f^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 f^{5/2}}+\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 f^{5/2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.98 (sec) , antiderivative size = 619, normalized size of antiderivative = 1.77 \[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=\frac {-2 \sqrt {c} \sqrt {a+x (b+c x)} \left (3 b^2 f+2 c f (16 a+7 b x)+8 c^2 \left (3 d+f x^2\right )\right )+3 b \left (-24 c^2 d+b^2 f-12 a c f\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )-24 c^{3/2} \text {RootSum}\left [b^2 d-a^2 f-4 b \sqrt {c} d \text {$\#$1}+4 c d \text {$\#$1}^2+2 a f \text {$\#$1}^2-f \text {$\#$1}^4\&,\frac {2 b^2 c d^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-a c^2 d^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )+a b^2 d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-2 a^2 c d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-a^3 f^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-4 b c^{3/2} d^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-4 a b \sqrt {c} d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}+c^2 d^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+b^2 d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+2 a c d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+a^2 f^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{b \sqrt {c} d-2 c d \text {$\#$1}-a f \text {$\#$1}+f \text {$\#$1}^3}\&\right ]}{48 c^{3/2} f^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(601\) vs. \(2(275)=550\).
Time = 0.73 (sec) , antiderivative size = 602, normalized size of antiderivative = 1.72
method | result | size |
risch | \(-\frac {\left (8 f \,c^{2} x^{2}+14 b c f x +32 a c f +3 b^{2} f +24 c^{2} d \right ) \sqrt {c \,x^{2}+b x +a}}{24 c \,f^{2}}-\frac {\frac {b \left (12 a c f -b^{2} f +24 c^{2} d \right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{\sqrt {c}}-\frac {8 c \left (\sqrt {d f}\, a^{2} f^{2}+2 \sqrt {d f}\, a c d f +\sqrt {d f}\, b^{2} d f +\sqrt {d f}\, c^{2} d^{2}+2 a b d \,f^{2}+2 b c \,d^{2} f \right ) \ln \left (\frac {\frac {2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x -\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+f a +c d}{f}}}{x -\frac {\sqrt {d f}}{f}}\right )}{\sqrt {d f}\, f \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}}-\frac {8 c \left (\sqrt {d f}\, a^{2} f^{2}+2 \sqrt {d f}\, a c d f +\sqrt {d f}\, b^{2} d f +\sqrt {d f}\, c^{2} d^{2}-2 a b d \,f^{2}-2 b c \,d^{2} f \right ) \ln \left (\frac {\frac {-2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x +\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+f a +c d}{f}}}{x +\frac {\sqrt {d f}}{f}}\right )}{\sqrt {d f}\, f \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}}}{16 f^{2} c}\) | \(602\) |
default | \(\text {Expression too large to display}\) | \(1473\) |
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Timed out. \[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=\text {Timed out} \]
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\[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=- \int \frac {a x \sqrt {a + b x + c x^{2}}}{- d + f x^{2}}\, dx - \int \frac {b x^{2} \sqrt {a + b x + c x^{2}}}{- d + f x^{2}}\, dx - \int \frac {c x^{3} \sqrt {a + b x + c x^{2}}}{- d + f x^{2}}\, dx \]
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Exception generated. \[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=\text {Exception raised: ValueError} \]
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Exception generated. \[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=\int \frac {x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{d-f\,x^2} \,d x \]
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