Integrand size = 29, antiderivative size = 1066 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x) (f+g x)^3} \, 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 f-d g)^3}+\frac {3 e (4 c f-3 b g-2 c g x) \sqrt {a+b x+c x^2}}{4 g^2 (e f-d g)^2}-\frac {3 (4 c f-b g+2 c g x) \sqrt {a+b x+c x^2}}{4 g^2 (e f-d g) (f+g x)}-\frac {e^2 \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)^3}+\frac {\left (a+b x+c x^2\right )^{3/2}}{2 (e f-d g) (f+g x)^2}+\frac {e \left (a+b x+c x^2\right )^{3/2}}{(e f-d g)^2 (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 (e f-d g)^3}+\frac {3 \sqrt {c} (2 c f-b g) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 g^3 (e f-d g)}+\frac {e^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 ) \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)^3}-\frac {3 e \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)^2}+\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 (e f-d g)^3}+\frac {3 e (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)^2}-\frac {e^2 \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)^3}-\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 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 )}{8 g^3 (e f-d g) \sqrt {c f^2-b f g+a g^2}} \]
[Out]
Time = 0.99 (sec) , antiderivative size = 1066, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {974, 748, 828, 857, 635, 212, 738, 746, 826} \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x) (f+g x)^3} \, dx=\frac {(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 {c x^2+b x+a}}\right ) e^2}{16 c^{3/2} g^3 (e f-d g)^3}-\frac {\left (c f^2-b g f+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 g f+a g^2} \sqrt {c x^2+b x+a}}\right ) e^2}{g^3 (e f-d g)^3}-\frac {\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 {c x^2+b x+a} e^2}{8 c g^2 (e f-d g)^3}+\frac {\left (c x^2+b x+a\right )^{3/2} e}{(e f-d g)^2 (f+g x)}-\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 {c x^2+b x+a}}\right ) e}{8 \sqrt {c} g^3 (e f-d g)^2}+\frac {3 (2 c f-b g) \sqrt {c f^2-b g f+a g^2} \text {arctanh}\left (\frac {b f-2 a g+(2 c f-b g) x}{2 \sqrt {c f^2-b g f+a g^2} \sqrt {c x^2+b x+a}}\right ) e}{2 g^3 (e f-d g)^2}+\frac {3 (4 c f-3 b g-2 c g x) \sqrt {c x^2+b x+a} e}{4 g^2 (e f-d g)^2}+\frac {\left (c x^2+b x+a\right )^{3/2}}{2 (e f-d g) (f+g x)^2}+\frac {3 \sqrt {c} (2 c f-b g) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right )}{2 g^3 (e f-d g)}-\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 f-2 a g+(2 c f-b g) x}{2 \sqrt {c f^2-b g f+a g^2} \sqrt {c x^2+b x+a}}\right )}{8 g^3 (e f-d g) \sqrt {c f^2-b g f+a g^2}}+\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 {c x^2+b x+a}}{8 c (e f-d g)^3}-\frac {3 (4 c f-b g+2 c g x) \sqrt {c x^2+b x+a}}{4 g^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 {c x^2+b x+a}}\right )}{16 c^{3/2} (e f-d g)^3 e}+\frac {\left (c d^2-b e d+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 e d+a e^2} \sqrt {c x^2+b x+a}}\right )}{(e f-d g)^3 e} \]
[In]
[Out]
Rule 212
Rule 635
Rule 738
Rule 746
Rule 748
Rule 826
Rule 828
Rule 857
Rule 974
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {e^3 \left (a+b x+c x^2\right )^{3/2}}{(e f-d g)^3 (d+e x)}-\frac {g \left (a+b x+c x^2\right )^{3/2}}{(e f-d g) (f+g x)^3}-\frac {e g \left (a+b x+c x^2\right )^{3/2}}{(e f-d g)^2 (f+g x)^2}-\frac {e^2 g \left (a+b x+c x^2\right )^{3/2}}{(e f-d g)^3 (f+g x)}\right ) \, dx \\ & = \frac {e^3 \int \frac {\left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx}{(e f-d g)^3}-\frac {\left (e^2 g\right ) \int \frac {\left (a+b x+c x^2\right )^{3/2}}{f+g x} \, dx}{(e f-d g)^3}-\frac {(e g) \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(f+g x)^2} \, dx}{(e f-d g)^2}-\frac {g \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(f+g x)^3} \, dx}{e f-d g} \\ & = \frac {\left (a+b x+c x^2\right )^{3/2}}{2 (e f-d g) (f+g x)^2}+\frac {e \left (a+b x+c x^2\right )^{3/2}}{(e f-d g)^2 (f+g x)}-\frac {e^2 \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)^3}+\frac {e^2 \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)^3}-\frac {(3 e) \int \frac {(b+2 c 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)^2} \, dx}{4 (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 f-d g)^3}+\frac {3 e (4 c f-3 b g-2 c g x) \sqrt {a+b x+c x^2}}{4 g^2 (e f-d g)^2}-\frac {3 (4 c f-b g+2 c g x) \sqrt {a+b x+c x^2}}{4 g^2 (e f-d g) (f+g x)}-\frac {e^2 \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)^3}+\frac {\left (a+b x+c x^2\right )^{3/2}}{2 (e f-d g) (f+g x)^2}+\frac {e \left (a+b x+c x^2\right )^{3/2}}{(e f-d g)^2 (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 f-d g)^3}-\frac {e^2 \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)^3}+\frac {(3 e) \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)^2}+\frac {3 \int \frac {4 b c f-b^2 g-4 a c g+4 c (2 c f-b g) x}{(f+g x) \sqrt {a+b x+c x^2}} \, dx}{8 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 f-d g)^3}+\frac {3 e (4 c f-3 b g-2 c g x) \sqrt {a+b x+c x^2}}{4 g^2 (e f-d g)^2}-\frac {3 (4 c f-b g+2 c g x) \sqrt {a+b x+c x^2}}{4 g^2 (e f-d g) (f+g x)}-\frac {e^2 \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)^3}+\frac {\left (a+b x+c x^2\right )^{3/2}}{2 (e f-d g) (f+g x)^2}+\frac {e \left (a+b x+c x^2\right )^{3/2}}{(e f-d g)^2 (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 (e f-d g)^3}-\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 (e f-d g)^3}+\frac {(3 c (2 c f-b g)) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{2 g^3 (e f-d g)}+\frac {\left (3 e (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)^2}-\frac {\left (e^2 \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)^3}+\frac {\left (e^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 )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{16 c g^3 (e f-d g)^3}-\frac {\left (3 e \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)^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}{(f+g x) \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 f-d g)^3}+\frac {3 e (4 c f-3 b g-2 c g x) \sqrt {a+b x+c x^2}}{4 g^2 (e f-d g)^2}-\frac {3 (4 c f-b g+2 c g x) \sqrt {a+b x+c x^2}}{4 g^2 (e f-d g) (f+g x)}-\frac {e^2 \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)^3}+\frac {\left (a+b x+c x^2\right )^{3/2}}{2 (e f-d g) (f+g x)^2}+\frac {e \left (a+b x+c x^2\right )^{3/2}}{(e f-d g)^2 (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 (e f-d g)^3}-\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 (e f-d g)^3}+\frac {(3 c (2 c f-b g)) \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 )}{g^3 (e f-d g)}-\frac {\left (3 e (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)^2}+\frac {\left (2 e^2 \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)^3}+\frac {\left (e^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 )\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)^3}-\frac {\left (3 e \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)^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 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 )}{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 f-d g)^3}+\frac {3 e (4 c f-3 b g-2 c g x) \sqrt {a+b x+c x^2}}{4 g^2 (e f-d g)^2}-\frac {3 (4 c f-b g+2 c g x) \sqrt {a+b x+c x^2}}{4 g^2 (e f-d g) (f+g x)}-\frac {e^2 \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)^3}+\frac {\left (a+b x+c x^2\right )^{3/2}}{2 (e f-d g) (f+g x)^2}+\frac {e \left (a+b x+c x^2\right )^{3/2}}{(e f-d g)^2 (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 (e f-d g)^3}+\frac {3 \sqrt {c} (2 c f-b g) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 g^3 (e f-d g)}+\frac {e^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 ) \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)^3}-\frac {3 e \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)^2}+\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 (e f-d g)^3}+\frac {3 e (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)^2}-\frac {e^2 \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)^3}-\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 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 )}{8 g^3 (e f-d g) \sqrt {c f^2-b f g+a g^2}} \\ \end{align*}
Time = 12.29 (sec) , antiderivative size = 1036, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x) (f+g x)^3} \, dx=\frac {1}{4} \left (\frac {2 (a+x (b+c x))^{3/2}}{(e f-d g) (f+g x)^2}+\frac {4 e (a+x (b+c x))^{3/2}}{(e f-d g)^2 (f+g x)}+\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 ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )-2 \sqrt {c} \left (e \sqrt {a+x (b+c x)} \left (-b^2 e^2+4 c^2 d (-2 d+e x)-2 c e (-5 b d+4 a e+b e x)\right )+8 c \left (c d^2+e (-b d+a e)\right )^{3/2} \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 )\right )}{4 c^{3/2} e (e f-d g)^3}-\frac {3 e \left (\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+x (b+c x)}}\right )+2 \sqrt {c} \left (g (-4 c f+3 b g+2 c g x) \sqrt {a+x (b+c x)}+2 (2 c f-b g) \sqrt {c f^2+g (-b f+a g)} \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 )\right )}{2 \sqrt {c} g^3 (e f-d g)^2}+\frac {3 \left (\frac {(-2 c f+b g) (a+x (b+c x))^{3/2}}{f+g x}-\frac {\sqrt {a+x (b+c x)} \left (b^2 g^2+2 c^2 f (2 f-g x)+c g (-5 b f+2 a g+b g x)\right )}{g^2}+\frac {4 \sqrt {c} (2 c f-b g) \left (c f^2+g (-b f+a g)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )+\left (8 c^2 f^2+b^2 g^2+4 c g (-2 b f+a g)\right ) \sqrt {c f^2+g (-b f+a g)} \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 )}{2 g^3}\right )}{(e f-d g) \left (c f^2+g (-b f+a g)\right )}-\frac {e^2 \left ((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+x (b+c x)}}\right )+2 \sqrt {c} \left (g \sqrt {a+x (b+c x)} \left (-b^2 g^2+4 c^2 f (-2 f+g x)-2 c g (-5 b f+4 a g+b g x)\right )+8 c \left (c f^2+g (-b f+a g)\right )^{3/2} \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 )\right )}{4 c^{3/2} g^3 (-e f+d g)^3}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(4241\) vs. \(2(968)=1936\).
Time = 0.96 (sec) , antiderivative size = 4242, normalized size of antiderivative = 3.98
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Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x) (f+g x)^3} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x) (f+g x)^3} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x) (f+g x)^3} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}}{{\left (e x + d\right )} {\left (g x + f\right )}^{3}} \,d x } \]
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Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x) (f+g x)^3} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x) (f+g x)^3} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{{\left (f+g\,x\right )}^3\,\left (d+e\,x\right )} \,d x \]
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