Integrand size = 29, antiderivative size = 787 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x) (f+g x)^2} \, dx=\frac {\left (8 c^2 d^2+b^2 e^2-2 c e (5 b d-4 a e)-2 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{8 c e (e f-d g)^2}+\frac {3 (4 c f-3 b g-2 c g x) \sqrt {a+b x+c x^2}}{4 g^2 (e f-d g)}-\frac {e \left (8 c^2 f^2+b^2 g^2-2 c g (5 b f-4 a g)-2 c g (2 c f-b g) x\right ) \sqrt {a+b x+c x^2}}{8 c g^2 (e f-d g)^2}+\frac {\left (a+b x+c x^2\right )^{3/2}}{(e f-d g) (f+g x)}-\frac {(2 c d-b e) \left (8 c^2 d^2-b^2 e^2-4 c e (2 b d-3 a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{3/2} e^2 (e f-d g)^2}+\frac {e (2 c f-b g) \left (8 c^2 f^2-b^2 g^2-4 c g (2 b f-3 a g)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{3/2} g^3 (e f-d g)^2}-\frac {3 \left (8 c^2 f^2+b^2 g^2-4 c g (2 b f-a g)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {c} g^3 (e f-d g)}+\frac {\left (c d^2-b d e+a e^2\right )^{3/2} \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{e^2 (e f-d g)^2}+\frac {3 (2 c f-b g) \sqrt {c f^2-b f g+a g^2} \text {arctanh}\left (\frac {b f-2 a g+(2 c f-b g) x}{2 \sqrt {c f^2-b f g+a g^2} \sqrt {a+b x+c x^2}}\right )}{2 g^3 (e f-d g)}-\frac {e \left (c f^2-b f g+a g^2\right )^{3/2} \text {arctanh}\left (\frac {b f-2 a g+(2 c f-b g) x}{2 \sqrt {c f^2-b f g+a g^2} \sqrt {a+b x+c x^2}}\right )}{g^3 (e f-d g)^2} \]
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Time = 0.82 (sec) , antiderivative size = 787, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {974, 748, 828, 857, 635, 212, 738, 746} \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x) (f+g x)^2} \, dx=-\frac {3 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c g (2 b f-a g)+b^2 g^2+8 c^2 f^2\right )}{8 \sqrt {c} g^3 (e f-d g)}-\frac {(2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c e (2 b d-3 a e)-b^2 e^2+8 c^2 d^2\right )}{16 c^{3/2} e^2 (e f-d g)^2}+\frac {e (2 c f-b g) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c g (2 b f-3 a g)-b^2 g^2+8 c^2 f^2\right )}{16 c^{3/2} g^3 (e f-d g)^2}+\frac {\left (a e^2-b d e+c d^2\right )^{3/2} \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{e^2 (e f-d g)^2}-\frac {e \left (a g^2-b f g+c f^2\right )^{3/2} \text {arctanh}\left (\frac {-2 a g+x (2 c f-b g)+b f}{2 \sqrt {a+b x+c x^2} \sqrt {a g^2-b f g+c f^2}}\right )}{g^3 (e f-d g)^2}+\frac {3 (2 c f-b g) \sqrt {a g^2-b f g+c f^2} \text {arctanh}\left (\frac {-2 a g+x (2 c f-b g)+b f}{2 \sqrt {a+b x+c x^2} \sqrt {a g^2-b f g+c f^2}}\right )}{2 g^3 (e f-d g)}+\frac {\sqrt {a+b x+c x^2} \left (-2 c e (5 b d-4 a e)+b^2 e^2-2 c e x (2 c d-b e)+8 c^2 d^2\right )}{8 c e (e f-d g)^2}-\frac {e \sqrt {a+b x+c x^2} \left (-2 c g (5 b f-4 a g)+b^2 g^2-2 c g x (2 c f-b g)+8 c^2 f^2\right )}{8 c g^2 (e f-d g)^2}+\frac {3 \sqrt {a+b x+c x^2} (-3 b g+4 c f-2 c g x)}{4 g^2 (e f-d g)}+\frac {\left (a+b x+c x^2\right )^{3/2}}{(f+g x) (e f-d g)} \]
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Rule 212
Rule 635
Rule 738
Rule 746
Rule 748
Rule 828
Rule 857
Rule 974
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {e^2 \left (a+b x+c x^2\right )^{3/2}}{(e f-d g)^2 (d+e x)}-\frac {g \left (a+b x+c x^2\right )^{3/2}}{(e f-d g) (f+g x)^2}-\frac {e g \left (a+b x+c x^2\right )^{3/2}}{(e f-d g)^2 (f+g x)}\right ) \, dx \\ & = \frac {e^2 \int \frac {\left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx}{(e f-d g)^2}-\frac {(e g) \int \frac {\left (a+b x+c x^2\right )^{3/2}}{f+g x} \, dx}{(e f-d g)^2}-\frac {g \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(f+g x)^2} \, dx}{e f-d g} \\ & = \frac {\left (a+b x+c x^2\right )^{3/2}}{(e f-d g) (f+g x)}-\frac {e \int \frac {(b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{d+e x} \, dx}{2 (e f-d g)^2}+\frac {e \int \frac {(b f-2 a g+(2 c f-b g) x) \sqrt {a+b x+c x^2}}{f+g x} \, dx}{2 (e f-d g)^2}-\frac {3 \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{f+g x} \, dx}{2 (e f-d g)} \\ & = \frac {\left (8 c^2 d^2+b^2 e^2-2 c e (5 b d-4 a e)-2 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{8 c e (e f-d g)^2}+\frac {3 (4 c f-3 b g-2 c g x) \sqrt {a+b x+c x^2}}{4 g^2 (e f-d g)}-\frac {e \left (8 c^2 f^2+b^2 g^2-2 c g (5 b f-4 a g)-2 c g (2 c f-b g) x\right ) \sqrt {a+b x+c x^2}}{8 c g^2 (e f-d g)^2}+\frac {\left (a+b x+c x^2\right )^{3/2}}{(e f-d g) (f+g x)}+\frac {\int \frac {\frac {1}{2} \left (4 c e (b d-2 a e)^2-d (2 c d-b e) \left (4 b c d-b^2 e-4 a c e\right )\right )-\frac {1}{2} (2 c d-b e) \left (8 c^2 d^2-b^2 e^2-4 c e (2 b d-3 a e)\right ) x}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{8 c e (e f-d g)^2}-\frac {e \int \frac {\frac {1}{2} \left (4 c g (b f-2 a g)^2-f (2 c f-b g) \left (4 b c f-b^2 g-4 a c g\right )\right )-\frac {1}{2} (2 c f-b g) \left (8 c^2 f^2-b^2 g^2-4 c g (2 b f-3 a g)\right ) x}{(f+g x) \sqrt {a+b x+c x^2}} \, dx}{8 c g^2 (e f-d g)^2}+\frac {3 \int \frac {c \left (3 b^2 f g+4 a c f g-4 b \left (c f^2+a g^2\right )\right )-c \left (8 c^2 f^2+b^2 g^2-4 c g (2 b f-a g)\right ) x}{(f+g x) \sqrt {a+b x+c x^2}} \, dx}{8 c g^2 (e f-d g)} \\ & = \frac {\left (8 c^2 d^2+b^2 e^2-2 c e (5 b d-4 a e)-2 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{8 c e (e f-d g)^2}+\frac {3 (4 c f-3 b g-2 c g x) \sqrt {a+b x+c x^2}}{4 g^2 (e f-d g)}-\frac {e \left (8 c^2 f^2+b^2 g^2-2 c g (5 b f-4 a g)-2 c g (2 c f-b g) x\right ) \sqrt {a+b x+c x^2}}{8 c g^2 (e f-d g)^2}+\frac {\left (a+b x+c x^2\right )^{3/2}}{(e f-d g) (f+g x)}+\frac {\left (c d^2-b d e+a e^2\right )^2 \int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{e^2 (e f-d g)^2}-\frac {\left ((2 c d-b e) \left (8 c^2 d^2-b^2 e^2-4 c e (2 b d-3 a e)\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{16 c e^2 (e f-d g)^2}+\frac {\left (3 (2 c f-b g) \left (c f^2-b f g+a g^2\right )\right ) \int \frac {1}{(f+g x) \sqrt {a+b x+c x^2}} \, dx}{2 g^3 (e f-d g)}-\frac {\left (e \left (c f^2-b f g+a g^2\right )^2\right ) \int \frac {1}{(f+g x) \sqrt {a+b x+c x^2}} \, dx}{g^3 (e f-d g)^2}+\frac {\left (e (2 c f-b g) \left (8 c^2 f^2-b^2 g^2-4 c g (2 b f-3 a g)\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{16 c g^3 (e f-d g)^2}-\frac {\left (3 \left (8 c^2 f^2+b^2 g^2-4 c g (2 b f-a g)\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{8 g^3 (e f-d g)} \\ & = \frac {\left (8 c^2 d^2+b^2 e^2-2 c e (5 b d-4 a e)-2 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{8 c e (e f-d g)^2}+\frac {3 (4 c f-3 b g-2 c g x) \sqrt {a+b x+c x^2}}{4 g^2 (e f-d g)}-\frac {e \left (8 c^2 f^2+b^2 g^2-2 c g (5 b f-4 a g)-2 c g (2 c f-b g) x\right ) \sqrt {a+b x+c x^2}}{8 c g^2 (e f-d g)^2}+\frac {\left (a+b x+c x^2\right )^{3/2}}{(e f-d g) (f+g x)}-\frac {\left (2 \left (c d^2-b d e+a e^2\right )^2\right ) \text {Subst}\left (\int \frac {1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac {-b d+2 a e-(2 c d-b e) x}{\sqrt {a+b x+c x^2}}\right )}{e^2 (e f-d g)^2}-\frac {\left ((2 c d-b e) \left (8 c^2 d^2-b^2 e^2-4 c e (2 b d-3 a e)\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 e^2 (e f-d g)^2}-\frac {\left (3 (2 c f-b g) \left (c f^2-b f g+a g^2\right )\right ) \text {Subst}\left (\int \frac {1}{4 c f^2-4 b f g+4 a g^2-x^2} \, dx,x,\frac {-b f+2 a g-(2 c f-b g) x}{\sqrt {a+b x+c x^2}}\right )}{g^3 (e f-d g)}+\frac {\left (2 e \left (c f^2-b f g+a g^2\right )^2\right ) \text {Subst}\left (\int \frac {1}{4 c f^2-4 b f g+4 a g^2-x^2} \, dx,x,\frac {-b f+2 a g-(2 c f-b g) x}{\sqrt {a+b x+c x^2}}\right )}{g^3 (e f-d g)^2}+\frac {\left (e (2 c f-b g) \left (8 c^2 f^2-b^2 g^2-4 c g (2 b f-3 a g)\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 g^3 (e f-d g)^2}-\frac {\left (3 \left (8 c^2 f^2+b^2 g^2-4 c g (2 b f-a g)\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 g^3 (e f-d g)} \\ & = \frac {\left (8 c^2 d^2+b^2 e^2-2 c e (5 b d-4 a e)-2 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{8 c e (e f-d g)^2}+\frac {3 (4 c f-3 b g-2 c g x) \sqrt {a+b x+c x^2}}{4 g^2 (e f-d g)}-\frac {e \left (8 c^2 f^2+b^2 g^2-2 c g (5 b f-4 a g)-2 c g (2 c f-b g) x\right ) \sqrt {a+b x+c x^2}}{8 c g^2 (e f-d g)^2}+\frac {\left (a+b x+c x^2\right )^{3/2}}{(e f-d g) (f+g x)}-\frac {(2 c d-b e) \left (8 c^2 d^2-b^2 e^2-4 c e (2 b d-3 a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{3/2} e^2 (e f-d g)^2}+\frac {e (2 c f-b g) \left (8 c^2 f^2-b^2 g^2-4 c g (2 b f-3 a g)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{3/2} g^3 (e f-d g)^2}-\frac {3 \left (8 c^2 f^2+b^2 g^2-4 c g (2 b f-a g)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {c} g^3 (e f-d g)}+\frac {\left (c d^2-b d e+a e^2\right )^{3/2} \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{e^2 (e f-d g)^2}+\frac {3 (2 c f-b g) \sqrt {c f^2-b f g+a g^2} \tanh ^{-1}\left (\frac {b f-2 a g+(2 c f-b g) x}{2 \sqrt {c f^2-b f g+a g^2} \sqrt {a+b x+c x^2}}\right )}{2 g^3 (e f-d g)}-\frac {e \left (c f^2-b f g+a g^2\right )^{3/2} \tanh ^{-1}\left (\frac {b f-2 a g+(2 c f-b g) x}{2 \sqrt {c f^2-b f g+a g^2} \sqrt {a+b x+c x^2}}\right )}{g^3 (e f-d g)^2} \\ \end{align*}
Time = 10.93 (sec) , antiderivative size = 357, normalized size of antiderivative = 0.45 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x) (f+g x)^2} \, dx=\frac {-\sqrt {c} (e f-d g)^2 (4 c e f+2 c d g-3 b e g) (f+g x) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )-2 \left (c d^2+e (-b d+a e)\right )^{3/2} g^3 (f+g x) \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )+e \left (2 g (-e f+d g) \sqrt {a+x (b+c x)} (e g (b f-a g)+c d g (f+g x)-c e f (2 f+g x))-e \sqrt {c f^2+g (-b f+a g)} (2 c f (2 e f-3 d g)+g (-b e f+3 b d g-2 a e g)) (f+g x) \text {arctanh}\left (\frac {-b f+2 a g-2 c f x+b g x}{2 \sqrt {c f^2+g (-b f+a g)} \sqrt {a+x (b+c x)}}\right )\right )}{2 e^2 g^3 (e f-d g)^2 (f+g x)} \]
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Time = 0.92 (sec) , antiderivative size = 900, normalized size of antiderivative = 1.14
method | result | size |
risch | \(\frac {\sqrt {c \,x^{2}+b x +a}\, c}{g^{2} e}+\frac {\frac {\sqrt {c}\, \left (3 b e g -2 c d g -4 c e f \right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{e g}+\frac {2 e \left (a^{2} g^{4}-2 a b f \,g^{3}+2 a c \,f^{2} g^{2}+b^{2} f^{2} g^{2}-2 b c \,f^{3} g +c^{2} f^{4}\right ) \left (-\frac {g^{2} \sqrt {\left (x +\frac {f}{g}\right )^{2} c +\frac {\left (b g -2 c f \right ) \left (x +\frac {f}{g}\right )}{g}+\frac {a \,g^{2}-b f g +c \,f^{2}}{g^{2}}}}{\left (a \,g^{2}-b f g +c \,f^{2}\right ) \left (x +\frac {f}{g}\right )}+\frac {\left (b g -2 c f \right ) g \ln \left (\frac {\frac {2 a \,g^{2}-2 b f g +2 c \,f^{2}}{g^{2}}+\frac {\left (b g -2 c f \right ) \left (x +\frac {f}{g}\right )}{g}+2 \sqrt {\frac {a \,g^{2}-b f g +c \,f^{2}}{g^{2}}}\, \sqrt {\left (x +\frac {f}{g}\right )^{2} c +\frac {\left (b g -2 c f \right ) \left (x +\frac {f}{g}\right )}{g}+\frac {a \,g^{2}-b f g +c \,f^{2}}{g^{2}}}}{x +\frac {f}{g}}\right )}{2 \left (a \,g^{2}-b f g +c \,f^{2}\right ) \sqrt {\frac {a \,g^{2}-b f g +c \,f^{2}}{g^{2}}}}\right )}{g^{3} \left (d g -e f \right )}-\frac {2 g^{2} \left (a^{2} e^{4}-2 a b d \,e^{3}+2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}-2 b c \,d^{3} e +c^{2} d^{4}\right ) \ln \left (\frac {\frac {2 e^{2} a -2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \left (d g -e f \right )^{2} \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}+\frac {2 e \left (a^{2} e \,g^{4}-2 a b d \,g^{4}+4 a c d f \,g^{3}-2 a c e \,f^{2} g^{2}+2 b^{2} d f \,g^{3}-b^{2} e \,f^{2} g^{2}-6 b c d \,f^{2} g^{2}+4 b c e \,f^{3} g +4 c^{2} d \,f^{3} g -3 c^{2} e \,f^{4}\right ) \ln \left (\frac {\frac {2 a \,g^{2}-2 b f g +2 c \,f^{2}}{g^{2}}+\frac {\left (b g -2 c f \right ) \left (x +\frac {f}{g}\right )}{g}+2 \sqrt {\frac {a \,g^{2}-b f g +c \,f^{2}}{g^{2}}}\, \sqrt {\left (x +\frac {f}{g}\right )^{2} c +\frac {\left (b g -2 c f \right ) \left (x +\frac {f}{g}\right )}{g}+\frac {a \,g^{2}-b f g +c \,f^{2}}{g^{2}}}}{x +\frac {f}{g}}\right )}{g^{2} \left (d g -e f \right )^{2} \sqrt {\frac {a \,g^{2}-b f g +c \,f^{2}}{g^{2}}}}}{2 g^{2} e}\) | \(900\) |
default | \(\text {Expression too large to display}\) | \(2372\) |
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Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x) (f+g x)^2} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x) (f+g x)^2} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x) (f+g x)^2} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}}{{\left (e x + d\right )} {\left (g x + f\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x) (f+g x)^2} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x) (f+g x)^2} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{{\left (f+g\,x\right )}^2\,\left (d+e\,x\right )} \,d x \]
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