Integrand size = 49, antiderivative size = 1090 \[ \int \frac {d e+c f+2 d f x}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {4 d \left (3 a^3 d^2 f^2 h-a^2 b d f (d f g+4 d e h+4 c f h)-b^3 \left (d^2 e^2 g-c d e (f g-e h)+c^2 f (f g+e h)\right )+a b^2 \left (2 c^2 f^2 h+d^2 e (f g+2 e h)+c d f (f g+3 e h)\right )\right ) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {c+d x}}-\frac {2 b (b d e+b c f-2 a d f) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac {4 b \left (3 a^3 d^2 f^2 h-a^2 b d f (d f g+4 d e h+4 c f h)-b^3 \left (d^2 e^2 g-c d e (f g-e h)+c^2 f (f g+e h)\right )+a b^2 \left (2 c^2 f^2 h+d^2 e (f g+2 e h)+c d f (f g+3 e h)\right )\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {a+b x}}-\frac {4 \sqrt {d g-c h} \sqrt {f g-e h} \left (3 a^3 d^2 f^2 h-a^2 b d f (d f g+4 d e h+4 c f h)-b^3 \left (d^2 e^2 g-c d e (f g-e h)+c^2 f (f g+e h)\right )+a b^2 \left (2 c^2 f^2 h+d^2 e (f g+2 e h)+c d f (f g+3 e h)\right )\right ) \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\arcsin \left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}+\frac {2 (d e-c f) \left (3 a^2 d^2 f h-a b d (d f g+3 d e h+2 c f h)+b^2 \left (2 d^2 e g-c d f g+c d e h+c^2 f h\right )\right ) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{3 (b c-a d)^2 (b e-a f) (b g-a h)^{3/2} \sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}} \]
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Time = 2.02 (sec) , antiderivative size = 1090, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1613, 1616, 12, 176, 430, 182, 435} \[ \int \frac {d e+c f+2 d f x}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=-\frac {2 b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (b d e+b c f-2 a d f)}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac {4 \sqrt {d g-c h} \sqrt {f g-e h} \left (3 d^2 f^2 h a^3-b d f (d f g+4 d e h+4 c f h) a^2+b^2 \left (e (f g+2 e h) d^2+c f (f g+3 e h) d+2 c^2 f^2 h\right ) a-b^3 \left (f (f g+e h) c^2-d e (f g-e h) c+d^2 e^2 g\right )\right ) \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\arcsin \left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}+\frac {2 (d e-c f) \left (\left (f h c^2-d f g c+d e h c+2 d^2 e g\right ) b^2-a d (d f g+3 d e h+2 c f h) b+3 a^2 d^2 f h\right ) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{3 (b c-a d)^2 (b e-a f) (b g-a h)^{3/2} \sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}-\frac {4 b \left (3 d^2 f^2 h a^3-b d f (d f g+4 d e h+4 c f h) a^2+b^2 \left (e (f g+2 e h) d^2+c f (f g+3 e h) d+2 c^2 f^2 h\right ) a-b^3 \left (f (f g+e h) c^2-d e (f g-e h) c+d^2 e^2 g\right )\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {a+b x}}+\frac {4 d \left (3 d^2 f^2 h a^3-b d f (d f g+4 d e h+4 c f h) a^2+b^2 \left (e (f g+2 e h) d^2+c f (f g+3 e h) d+2 c^2 f^2 h\right ) a-b^3 \left (f (f g+e h) c^2-d e (f g-e h) c+d^2 e^2 g\right )\right ) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {c+d x}} \]
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Rule 12
Rule 176
Rule 182
Rule 430
Rule 435
Rule 1613
Rule 1616
Rubi steps \begin{align*} \text {integral}& = -\frac {2 b (b d e+b c f-2 a d f) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}+\frac {\int \frac {2 b d f (3 b c e g-a (d e g+c f g+c e h))-(d e+c f) \left (3 a^2 d f h+2 b^2 (d e g+c f g+c e h)-3 a b (d f g+d e h+c f h)\right )+(b d e+b c f-2 a d f) (3 a d f h-b (d f g+d e h+c f h)) x}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{3 (b c-a d) (b e-a f) (b g-a h)} \\ & = -\frac {2 b (b d e+b c f-2 a d f) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac {4 b \left (3 a^3 d^2 f^2 h-a^2 b d f (d f g+4 d e h+4 c f h)-b^3 \left (d^2 e^2 g-c d e (f g-e h)+c^2 f (f g+e h)\right )+a b^2 \left (2 c^2 f^2 h+d^2 e (f g+2 e h)+c d f (f g+3 e h)\right )\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {a+b x}}+\frac {\int \frac {b (b d e+b c f-2 a d f) (b c e g-a (d e g+c f g+c e h)) (3 a d f h-b (d f g+d e h+c f h))-a (a d f h-b (d f g+d e h+c f h)) \left (2 b d f (3 b c e g-a (d e g+c f g+c e h))-(d e+c f) \left (3 a^2 d f h+2 b^2 (d e g+c f g+c e h)-3 a b (d f g+d e h+c f h)\right )\right )+2 (a d f h+b (d f g+d e h+c f h)) \left (3 a^3 d^2 f^2 h-a^2 b d f (d f g+4 d e h+4 c f h)-b^3 \left (d^2 e^2 g-c d e (f g-e h)+c^2 f (f g+e h)\right )+a b^2 \left (2 c^2 f^2 h+d^2 e (f g+2 e h)+c d f (f g+3 e h)\right )\right ) x+4 b d f h \left (3 a^3 d^2 f^2 h-a^2 b d f (d f g+4 d e h+4 c f h)-b^3 \left (d^2 e^2 g-c d e (f g-e h)+c^2 f (f g+e h)\right )+a b^2 \left (2 c^2 f^2 h+d^2 e (f g+2 e h)+c d f (f g+3 e h)\right )\right ) x^2}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2} \\ & = \frac {4 d \left (3 a^3 d^2 f^2 h-a^2 b d f (d f g+4 d e h+4 c f h)-b^3 \left (d^2 e^2 g-c d e (f g-e h)+c^2 f (f g+e h)\right )+a b^2 \left (2 c^2 f^2 h+d^2 e (f g+2 e h)+c d f (f g+3 e h)\right )\right ) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {c+d x}}-\frac {2 b (b d e+b c f-2 a d f) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac {4 b \left (3 a^3 d^2 f^2 h-a^2 b d f (d f g+4 d e h+4 c f h)-b^3 \left (d^2 e^2 g-c d e (f g-e h)+c^2 f (f g+e h)\right )+a b^2 \left (2 c^2 f^2 h+d^2 e (f g+2 e h)+c d f (f g+3 e h)\right )\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {a+b x}}+\frac {\int \frac {2 b d f (b e-a f) (d e-c f) h (b g-a h) \left (3 a^2 d^2 f h-a b d (d f g+3 d e h+2 c f h)+b^2 \left (2 d^2 e g-c d f g+c d e h+c^2 f h\right )\right )}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{6 b d (b c-a d)^2 f (b e-a f)^2 h (b g-a h)^2}+\frac {\left (2 (d e-c f) (d g-c h) \left (3 a^3 d^2 f^2 h-a^2 b d f (d f g+4 d e h+4 c f h)-b^3 \left (d^2 e^2 g-c d e (f g-e h)+c^2 f (f g+e h)\right )+a b^2 \left (2 c^2 f^2 h+d^2 e (f g+2 e h)+c d f (f g+3 e h)\right )\right )\right ) \int \frac {\sqrt {a+b x}}{(c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2} \\ & = \frac {4 d \left (3 a^3 d^2 f^2 h-a^2 b d f (d f g+4 d e h+4 c f h)-b^3 \left (d^2 e^2 g-c d e (f g-e h)+c^2 f (f g+e h)\right )+a b^2 \left (2 c^2 f^2 h+d^2 e (f g+2 e h)+c d f (f g+3 e h)\right )\right ) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {c+d x}}-\frac {2 b (b d e+b c f-2 a d f) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac {4 b \left (3 a^3 d^2 f^2 h-a^2 b d f (d f g+4 d e h+4 c f h)-b^3 \left (d^2 e^2 g-c d e (f g-e h)+c^2 f (f g+e h)\right )+a b^2 \left (2 c^2 f^2 h+d^2 e (f g+2 e h)+c d f (f g+3 e h)\right )\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {a+b x}}+\frac {\left ((d e-c f) \left (3 a^2 d^2 f h-a b d (d f g+3 d e h+2 c f h)+b^2 \left (2 d^2 e g-c d f g+c d e h+c^2 f h\right )\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{3 (b c-a d)^2 (b e-a f) (b g-a h)}-\frac {\left (4 (d g-c h) \left (3 a^3 d^2 f^2 h-a^2 b d f (d f g+4 d e h+4 c f h)-b^3 \left (d^2 e^2 g-c d e (f g-e h)+c^2 f (f g+e h)\right )+a b^2 \left (2 c^2 f^2 h+d^2 e (f g+2 e h)+c d f (f g+3 e h)\right )\right ) \sqrt {a+b x} \sqrt {\frac {(-d e+c f) (g+h x)}{(f g-e h) (c+d x)}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {(-b c+a d) x^2}{b e-a f}}}{\sqrt {1-\frac {(d g-c h) x^2}{f g-e h}}} \, dx,x,\frac {\sqrt {e+f x}}{\sqrt {c+d x}}\right )}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}} \\ & = \frac {4 d \left (3 a^3 d^2 f^2 h-a^2 b d f (d f g+4 d e h+4 c f h)-b^3 \left (d^2 e^2 g-c d e (f g-e h)+c^2 f (f g+e h)\right )+a b^2 \left (2 c^2 f^2 h+d^2 e (f g+2 e h)+c d f (f g+3 e h)\right )\right ) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {c+d x}}-\frac {2 b (b d e+b c f-2 a d f) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac {4 b \left (3 a^3 d^2 f^2 h-a^2 b d f (d f g+4 d e h+4 c f h)-b^3 \left (d^2 e^2 g-c d e (f g-e h)+c^2 f (f g+e h)\right )+a b^2 \left (2 c^2 f^2 h+d^2 e (f g+2 e h)+c d f (f g+3 e h)\right )\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {a+b x}}-\frac {4 \sqrt {d g-c h} \sqrt {f g-e h} \left (3 a^3 d^2 f^2 h-a^2 b d f (d f g+4 d e h+4 c f h)-b^3 \left (d^2 e^2 g-c d e (f g-e h)+c^2 f (f g+e h)\right )+a b^2 \left (2 c^2 f^2 h+d^2 e (f g+2 e h)+c d f (f g+3 e h)\right )\right ) \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\sin ^{-1}\left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}+\frac {\left (2 (d e-c f) \left (3 a^2 d^2 f h-a b d (d f g+3 d e h+2 c f h)+b^2 \left (2 d^2 e g-c d f g+c d e h+c^2 f h\right )\right ) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {(b c-a d) x^2}{d e-c f}} \sqrt {1-\frac {(b g-a h) x^2}{f g-e h}}} \, dx,x,\frac {\sqrt {e+f x}}{\sqrt {a+b x}}\right )}{3 (b c-a d)^2 (b e-a f) (b g-a h) (f g-e h) \sqrt {c+d x} \sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}}} \\ & = \frac {4 d \left (3 a^3 d^2 f^2 h-a^2 b d f (d f g+4 d e h+4 c f h)-b^3 \left (d^2 e^2 g-c d e (f g-e h)+c^2 f (f g+e h)\right )+a b^2 \left (2 c^2 f^2 h+d^2 e (f g+2 e h)+c d f (f g+3 e h)\right )\right ) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {c+d x}}-\frac {2 b (b d e+b c f-2 a d f) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac {4 b \left (3 a^3 d^2 f^2 h-a^2 b d f (d f g+4 d e h+4 c f h)-b^3 \left (d^2 e^2 g-c d e (f g-e h)+c^2 f (f g+e h)\right )+a b^2 \left (2 c^2 f^2 h+d^2 e (f g+2 e h)+c d f (f g+3 e h)\right )\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {a+b x}}-\frac {4 \sqrt {d g-c h} \sqrt {f g-e h} \left (3 a^3 d^2 f^2 h-a^2 b d f (d f g+4 d e h+4 c f h)-b^3 \left (d^2 e^2 g-c d e (f g-e h)+c^2 f (f g+e h)\right )+a b^2 \left (2 c^2 f^2 h+d^2 e (f g+2 e h)+c d f (f g+3 e h)\right )\right ) \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\sin ^{-1}\left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}+\frac {2 (d e-c f) \left (3 a^2 d^2 f h-a b d (d f g+3 d e h+2 c f h)+b^2 \left (2 d^2 e g-c d f g+c d e h+c^2 f h\right )\right ) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} F\left (\sin ^{-1}\left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right )|-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{3 (b c-a d)^2 (b e-a f) (b g-a h)^{3/2} \sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(10790\) vs. \(2(1090)=2180\).
Time = 38.04 (sec) , antiderivative size = 10790, normalized size of antiderivative = 9.90 \[ \int \frac {d e+c f+2 d f x}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Result too large to show} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(3570\) vs. \(2(1018)=2036\).
Time = 9.42 (sec) , antiderivative size = 3571, normalized size of antiderivative = 3.28
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(3571\) |
default | \(\text {Expression too large to display}\) | \(87910\) |
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\[ \int \frac {d e+c f+2 d f x}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {2 \, d f x + d e + c f}{{\left (b x + a\right )}^{\frac {5}{2}} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
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Timed out. \[ \int \frac {d e+c f+2 d f x}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Timed out} \]
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\[ \int \frac {d e+c f+2 d f x}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {2 \, d f x + d e + c f}{{\left (b x + a\right )}^{\frac {5}{2}} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
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\[ \int \frac {d e+c f+2 d f x}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {2 \, d f x + d e + c f}{{\left (b x + a\right )}^{\frac {5}{2}} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
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Timed out. \[ \int \frac {d e+c f+2 d f x}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Hanged} \]
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