Integrand size = 28, antiderivative size = 417 \[ \int \frac {x^2 \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=-\frac {\left (b \left (80 c^2 d-3 b^2 f+12 a c f\right )+2 c \left (16 c^2 d-3 b^2 f+12 a c f\right ) x\right ) \sqrt {a+b x+c x^2}}{64 c^2 f^2}-\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c f}-\frac {\left (128 c^4 d^2+192 a c^3 d f+3 b^4 f^2-24 a b^2 c f^2+48 c^2 f \left (b^2 d+a^2 f\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 c^{5/2} f^3}+\frac {\sqrt {d} \left (c d-b \sqrt {d} \sqrt {f}+a f\right )^{3/2} \text {arctanh}\left (\frac {b \sqrt {d}-2 a \sqrt {f}+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x}{2 \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 f^3}+\frac {\sqrt {d} \left (c d+b \sqrt {d} \sqrt {f}+a f\right )^{3/2} \text {arctanh}\left (\frac {b \sqrt {d}+2 a \sqrt {f}+\left (2 c \sqrt {d}+b \sqrt {f}\right ) x}{2 \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 f^3} \]
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Time = 0.65 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1085, 1084, 1092, 635, 212, 1047, 738} \[ \int \frac {x^2 \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=-\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (48 c^2 f \left (a^2 f+b^2 d\right )-24 a b^2 c f^2+192 a c^3 d f+3 b^4 f^2+128 c^4 d^2\right )}{128 c^{5/2} f^3}+\frac {\sqrt {d} \left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right )^{3/2} \text {arctanh}\left (\frac {-2 a \sqrt {f}+x \left (2 c \sqrt {d}-b \sqrt {f}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )}{2 f^3}+\frac {\sqrt {d} \left (a f+b \sqrt {d} \sqrt {f}+c d\right )^{3/2} \text {arctanh}\left (\frac {2 a \sqrt {f}+x \left (b \sqrt {f}+2 c \sqrt {d}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )}{2 f^3}-\frac {\sqrt {a+b x+c x^2} \left (2 c x \left (12 a c f-3 b^2 f+16 c^2 d\right )+b \left (12 a c f-3 b^2 f+80 c^2 d\right )\right )}{64 c^2 f^2}-\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c f} \]
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Rule 212
Rule 635
Rule 738
Rule 1047
Rule 1084
Rule 1085
Rule 1092
Rubi steps \begin{align*} \text {integral}& = -\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c f}-\frac {\int \frac {\sqrt {a+b x+c x^2} \left (-\frac {3}{4} \left (3 b^2+4 a c\right ) d f-12 b c d f x-\frac {3}{4} f \left (16 c^2 d-3 \left (b^2-4 a c\right ) f\right ) x^2\right )}{d-f x^2} \, dx}{12 c f^2} \\ & = -\frac {\left (b \left (80 c^2 d-3 b^2 f+12 a c f\right )+2 c \left (16 c^2 d-3 b^2 f+12 a c f\right ) x\right ) \sqrt {a+b x+c x^2}}{64 c^2 f^2}-\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c f}+\frac {\int \frac {-\frac {3}{16} d f^2 \left (3 b^4 f-8 b^2 c (10 c d+3 a f)-16 a c^2 (4 c d+5 a f)\right )+48 b c^2 d f^2 (c d+a f) x+\frac {3}{16} f^2 \left (128 c^4 d^2+192 a c^3 d f+3 b^4 f^2-24 a b^2 c f^2+48 c^2 f \left (b^2 d+a^2 f\right )\right ) x^2}{\sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx}{24 c^2 f^4} \\ & = -\frac {\left (b \left (80 c^2 d-3 b^2 f+12 a c f\right )+2 c \left (16 c^2 d-3 b^2 f+12 a c f\right ) x\right ) \sqrt {a+b x+c x^2}}{64 c^2 f^2}-\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c f}-\frac {\int \frac {\frac {3}{16} d f^3 \left (3 b^4 f-8 b^2 c (10 c d+3 a f)-16 a c^2 (4 c d+5 a f)\right )-\frac {3}{16} d f^2 \left (128 c^4 d^2+192 a c^3 d f+3 b^4 f^2-24 a b^2 c f^2+48 c^2 f \left (b^2 d+a^2 f\right )\right )-48 b c^2 d f^3 (c d+a f) x}{\sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx}{24 c^2 f^5}-\frac {\left (128 c^4 d^2+192 a c^3 d f+3 b^4 f^2-24 a b^2 c f^2+48 c^2 f \left (b^2 d+a^2 f\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{128 c^2 f^3} \\ & = -\frac {\left (b \left (80 c^2 d-3 b^2 f+12 a c f\right )+2 c \left (16 c^2 d-3 b^2 f+12 a c f\right ) x\right ) \sqrt {a+b x+c x^2}}{64 c^2 f^2}-\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c f}-\frac {\left (\sqrt {d} \left (c d-b \sqrt {d} \sqrt {f}+a f\right )^2\right ) \int \frac {1}{\left (-\sqrt {d} \sqrt {f}-f x\right ) \sqrt {a+b x+c x^2}} \, dx}{2 f^{5/2}}+\frac {\left (\sqrt {d} \left (c d+b \sqrt {d} \sqrt {f}+a f\right )^2\right ) \int \frac {1}{\left (\sqrt {d} \sqrt {f}-f x\right ) \sqrt {a+b x+c x^2}} \, dx}{2 f^{5/2}}-\frac {\left (128 c^4 d^2+192 a c^3 d f+3 b^4 f^2-24 a b^2 c f^2+48 c^2 f \left (b^2 d+a^2 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 )}{64 c^2 f^3} \\ & = -\frac {\left (b \left (80 c^2 d-3 b^2 f+12 a c f\right )+2 c \left (16 c^2 d-3 b^2 f+12 a c f\right ) x\right ) \sqrt {a+b x+c x^2}}{64 c^2 f^2}-\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c f}-\frac {\left (128 c^4 d^2+192 a c^3 d f+3 b^4 f^2-24 a b^2 c f^2+48 c^2 f \left (b^2 d+a^2 f\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 c^{5/2} f^3}+\frac {\left (\sqrt {d} \left (c d-b \sqrt {d} \sqrt {f}+a f\right )^2\right ) \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^{5/2}}-\frac {\left (\sqrt {d} \left (c d+b \sqrt {d} \sqrt {f}+a f\right )^2\right ) \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^{5/2}} \\ & = -\frac {\left (b \left (80 c^2 d-3 b^2 f+12 a c f\right )+2 c \left (16 c^2 d-3 b^2 f+12 a c f\right ) x\right ) \sqrt {a+b x+c x^2}}{64 c^2 f^2}-\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c f}-\frac {\left (128 c^4 d^2+192 a c^3 d f+3 b^4 f^2-24 a b^2 c f^2+48 c^2 f \left (b^2 d+a^2 f\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 c^{5/2} f^3}+\frac {\sqrt {d} \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^3}+\frac {\sqrt {d} \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^3} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 1.77 (sec) , antiderivative size = 608, normalized size of antiderivative = 1.46 \[ \int \frac {x^2 \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=\frac {-2 \sqrt {c} f \sqrt {a+x (b+c x)} \left (-3 b^3 f+2 b^2 c f x+8 c^2 x \left (4 c d+5 a f+2 c f x^2\right )+4 b c \left (20 c d+5 a f+6 c f x^2\right )\right )+\left (128 c^4 d^2+192 a c^3 d f+3 b^4 f^2-24 a b^2 c f^2+48 c^2 f \left (b^2 d+a^2 f\right )\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )-64 c^{5/2} d \text {RootSum}\left [b^2 d-a^2 f-4 b \sqrt {c} d \text {$\#$1}+4 c d \text {$\#$1}^2+2 a f \text {$\#$1}^2-f \text {$\#$1}^4\&,\frac {b c^2 d^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )+b^3 d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-a^2 b f^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-2 c^{5/2} d^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-2 b^2 \sqrt {c} d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-4 a c^{3/2} d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-2 a^2 \sqrt {c} f^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}+2 b c d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+2 a b f^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{b \sqrt {c} d-2 c d \text {$\#$1}-a f \text {$\#$1}+f \text {$\#$1}^3}\&\right ]}{128 c^{5/2} f^3} \]
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Time = 0.76 (sec) , antiderivative size = 648, normalized size of antiderivative = 1.55
method | result | size |
risch | \(-\frac {\left (16 f \,c^{3} x^{3}+24 b \,c^{2} f \,x^{2}+40 a \,c^{2} f x +2 b^{2} c f x +32 c^{3} d x +20 a b c f -3 b^{3} f +80 b d \,c^{2}\right ) \sqrt {c \,x^{2}+b x +a}}{64 c^{2} f^{2}}-\frac {\frac {\left (48 a^{2} c^{2} f^{2}-24 a \,b^{2} c \,f^{2}+192 a \,c^{3} d f +3 b^{4} f^{2}+48 b^{2} c^{2} d f +128 c^{4} d^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{f \sqrt {c}}-\frac {64 c^{2} d \left (2 \sqrt {d f}\, a b f +2 \sqrt {d f}\, b c d -a^{2} f^{2}-2 a c d f -b^{2} d f -c^{2} d^{2}\right ) \ln \left (\frac {\frac {-2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x +\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+f a +c d}{f}}}{x +\frac {\sqrt {d f}}{f}}\right )}{\sqrt {d f}\, f \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}}-\frac {64 c^{2} d \left (2 \sqrt {d f}\, a b f +2 \sqrt {d f}\, b c d +a^{2} f^{2}+2 a c d f +b^{2} d f +c^{2} d^{2}\right ) \ln \left (\frac {\frac {2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x -\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+f a +c d}{f}}}{x -\frac {\sqrt {d f}}{f}}\right )}{\sqrt {d f}\, f \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}}}{128 c^{2} f^{2}}\) | \(648\) |
default | \(\text {Expression too large to display}\) | \(1593\) |
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Timed out. \[ \int \frac {x^2 \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=\text {Timed out} \]
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\[ \int \frac {x^2 \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=- \int \frac {a x^{2} \sqrt {a + b x + c x^{2}}}{- d + f x^{2}}\, dx - \int \frac {b x^{3} \sqrt {a + b x + c x^{2}}}{- d + f x^{2}}\, dx - \int \frac {c x^{4} \sqrt {a + b x + c x^{2}}}{- d + f x^{2}}\, dx \]
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Exception generated. \[ \int \frac {x^2 \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=\text {Exception raised: ValueError} \]
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Exception generated. \[ \int \frac {x^2 \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^2 \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=\int \frac {x^2\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{d-f\,x^2} \,d x \]
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