Integrand size = 44, antiderivative size = 1395 \[ \int \frac {(a+b x)^{3/2} \left (A+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {(C (3 a d f h-5 b (d f g+d e h+c f h)) (a d f h-3 b (d f g+d e h+c f h))+8 b d f h (3 A b d f h-C (2 b (d e g+c f g+c e h)+a (d f g+d e h+c f h)))) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{24 b d^2 f^3 h^3 \sqrt {c+d x}}+\frac {C (3 a d f h-5 b (d f g+d e h+c f h)) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{12 d^2 f^2 h^2}+\frac {C (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 d f h}-\frac {\sqrt {d g-c h} \sqrt {f g-e h} (C (3 a d f h-5 b (d f g+d e h+c f h)) (a d f h-3 b (d f g+d e h+c f h))+8 b d f h (3 A b d f h-C (2 b (d e g+c f g+c e h)+a (d f g+d e h+c f h)))) \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 )}{24 b d^3 f^3 h^3 \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}+\frac {(b e-a f) \sqrt {b g-a h} \left (3 a^2 C d^2 f^2 h^2+6 a b C d f h (c f h+2 d (f g+e h))-b^2 \left (24 A d^2 f^2 h^2+C \left (5 c^2 f^2 h^2+4 c d f h (f g+e h)+d^2 \left (15 f^2 g^2+14 e f g h+15 e^2 h^2\right )\right )\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 )}{24 b^2 d^2 f^3 h^3 \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 {\sqrt {-d g+c h} \left (4 b d f h \left (C (b (d e g+c f g+c e h)+a (d f g+d e h+c f h)) (3 a d f h-5 b (d f g+d e h+c f h))+2 d f h \left (3 b^2 c C e g+2 a^2 C (d f g+d e h+c f h)-a b (12 A d f h-5 C (d e g+c f g+c e h))\right )\right )+(a d f h+b (d f g+d e h+c f h)) (C (3 a d f h-5 b (d f g+d e h+c f h)) (a d f h-3 b (d f g+d e h+c f h))+8 b d f h (3 A b d f h-C (2 b (d e g+c f g+c e h)+a (d f g+d e h+c f h))))\right ) (a+b x) \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt {\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \operatorname {EllipticPi}\left (-\frac {b (d g-c h)}{(b c-a d) h},\arcsin \left (\frac {\sqrt {b c-a d} \sqrt {g+h x}}{\sqrt {-d g+c h} \sqrt {a+b x}}\right ),\frac {(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{24 b^2 d^3 \sqrt {b c-a d} f^3 h^4 \sqrt {c+d x} \sqrt {e+f x}} \]
[Out]
Time = 3.98 (sec) , antiderivative size = 1376, normalized size of antiderivative = 0.99, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {1615, 1614, 1616, 1612, 176, 430, 171, 551, 182, 435} \[ \int \frac {(a+b x)^{3/2} \left (A+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {C \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (a+b x)^{3/2}}{3 d f h}-\frac {\sqrt {c h-d g} \left ((a d f h+b (d f g+d e h+c f h)) \left (24 A d^2 f^2 h^2 b^2+15 C (d f g+d e h+c f h)^2 b^2-16 C d f h (d e g+c f g+c e h) b^2-22 a C d f h (d f g+d e h+c f h) b+3 a^2 C d^2 f^2 h^2\right )+4 b d f h \left (C (b (d e g+c f g+c e h)+a (d f g+d e h+c f h)) (3 a d f h-5 b (d f g+d e h+c f h))+2 d f h \left (2 C (d f g+d e h+c f h) a^2-b (12 A d f h-5 C (d e g+c f g+c e h)) a+3 b^2 c C e g\right )\right )\right ) \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt {\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \operatorname {EllipticPi}\left (-\frac {b (d g-c h)}{(b c-a d) h},\arcsin \left (\frac {\sqrt {b c-a d} \sqrt {g+h x}}{\sqrt {c h-d g} \sqrt {a+b x}}\right ),\frac {(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right ) (a+b x)}{24 b^2 d^3 \sqrt {b c-a d} f^3 h^4 \sqrt {c+d x} \sqrt {e+f x}}-\frac {\sqrt {d g-c h} \sqrt {f g-e h} \left (24 A d^2 f^2 h^2 b^2+15 C (d f g+d e h+c f h)^2 b^2-16 C d f h (d e g+c f g+c e h) b^2-22 a C d f h (d f g+d e h+c f h) b+3 a^2 C d^2 f^2 h^2\right ) \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 ) \sqrt {a+b x}}{24 b d^3 f^3 h^3 \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}+\frac {C (3 a d f h-5 b (d f g+d e h+c f h)) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \sqrt {a+b x}}{12 d^2 f^2 h^2}+\frac {\left (24 A b f h d^2+\frac {3 a^2 C f h d^2}{b}-16 b C (d e g+c f g+c e h) d-22 a C (d f g+d e h+c f h) d+\frac {15 b C (d f g+d e h+c f h)^2}{f h}\right ) \sqrt {e+f x} \sqrt {g+h x} \sqrt {a+b x}}{24 d^2 f^2 h^2 \sqrt {c+d x}}+\frac {(b e-a f) \sqrt {b g-a h} \left (-\left (\left (24 A d^2 f^2 h^2+C \left (\left (15 f^2 g^2+14 e f h g+15 e^2 h^2\right ) d^2+4 c f h (f g+e h) d+5 c^2 f^2 h^2\right )\right ) b^2\right )+6 a C d f h (c f h+2 d (f g+e h)) b+3 a^2 C d^2 f^2 h^2\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 )}{24 b^2 d^2 f^3 h^3 \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)}}} \]
[In]
[Out]
Rule 171
Rule 176
Rule 182
Rule 430
Rule 435
Rule 551
Rule 1612
Rule 1614
Rule 1615
Rule 1616
Rubi steps \begin{align*} \text {integral}& = \frac {C (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 d f h}+\frac {\int \frac {\sqrt {a+b x} \left (-3 b c C e g+6 a A d f h-a C (d e g+c f g+c e h)+2 (3 A b d f h-2 b C (d e g+c f g+c e h)-a C (d f g+d e h+c f h)) x+C (3 a d f h-5 b (d f g+d e h+c f h)) x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{6 d f h} \\ & = \frac {C (3 a d f h-5 b (d f g+d e h+c f h)) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{12 d^2 f^2 h^2}+\frac {C (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 d f h}+\frac {\int \frac {-4 a d f h (3 b c C e g-6 a A d f h+a C (d e g+c f g+c e h))-C (b c e g+a (d e g+c f g+c e h)) (3 a d f h-5 b (d f g+d e h+c f h))-2 \left (C (b (d e g+c f g+c e h)+a (d f g+d e h+c f h)) (3 a d f h-5 b (d f g+d e h+c f h))+2 d f h \left (3 b^2 c C e g+2 a^2 C (d f g+d e h+c f h)-a b (12 A d f h-5 C (d e g+c f g+c e h))\right )\right ) x+\left (24 A b^2 d^2 f^2 h^2+3 a^2 C d^2 f^2 h^2-16 b^2 C d f h (d e g+c f g+c e h)-22 a b C d f h (d f g+d e h+c f h)+15 b^2 C (d f g+d e h+c f h)^2\right ) x^2}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{24 d^2 f^2 h^2} \\ & = \frac {\left (24 A b d^2 f h+\frac {3 a^2 C d^2 f h}{b}-16 b C d (d e g+c f g+c e h)-22 a C d (d f g+d e h+c f h)+\frac {15 b C (d f g+d e h+c f h)^2}{f h}\right ) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{24 d^2 f^2 h^2 \sqrt {c+d x}}+\frac {C (3 a d f h-5 b (d f g+d e h+c f h)) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{12 d^2 f^2 h^2}+\frac {C (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 d f h}+\frac {\int \frac {-\left ((b d e g+a c f h) \left (24 A b^2 d^2 f^2 h^2+3 a^2 C d^2 f^2 h^2-16 b^2 C d f h (d e g+c f g+c e h)-22 a b C d f h (d f g+d e h+c f h)+15 b^2 C (d f g+d e h+c f h)^2\right )\right )-2 b d f h (4 a d f h (3 b c C e g-6 a A d f h+a C (d e g+c f g+c e h))+C (b c e g+a (d e g+c f g+c e h)) (3 a d f h-5 b (d f g+d e h+c f h)))-\left ((a d f h+b (d f g+d e h+c f h)) \left (24 A b^2 d^2 f^2 h^2+3 a^2 C d^2 f^2 h^2-16 b^2 C d f h (d e g+c f g+c e h)-22 a b C d f h (d f g+d e h+c f h)+15 b^2 C (d f g+d e h+c f h)^2\right )+4 b d f h \left (C (b (d e g+c f g+c e h)+a (d f g+d e h+c f h)) (3 a d f h-5 b (d f g+d e h+c f h))+2 d f h \left (3 b^2 c C e g+2 a^2 C (d f g+d e h+c f h)-a b (12 A d f h-5 C (d e g+c f g+c e h))\right )\right )\right ) x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{48 b d^3 f^3 h^3}+\frac {\left ((d e-c f) (d g-c h) \left (24 A b^2 d^2 f^2 h^2+3 a^2 C d^2 f^2 h^2-16 b^2 C d f h (d e g+c f g+c e h)-22 a b C d f h (d f g+d e h+c f h)+15 b^2 C (d f g+d e h+c f h)^2\right )\right ) \int \frac {\sqrt {a+b x}}{(c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{48 b d^3 f^3 h^3} \\ & = \frac {\left (24 A b d^2 f h+\frac {3 a^2 C d^2 f h}{b}-16 b C d (d e g+c f g+c e h)-22 a C d (d f g+d e h+c f h)+\frac {15 b C (d f g+d e h+c f h)^2}{f h}\right ) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{24 d^2 f^2 h^2 \sqrt {c+d x}}+\frac {C (3 a d f h-5 b (d f g+d e h+c f h)) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{12 d^2 f^2 h^2}+\frac {C (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 d f h}+\frac {\left ((b e-a f) (b g-a h) \left (3 a^2 C d^2 f^2 h^2+6 a b C d f h (c f h+2 d (f g+e h))-b^2 \left (24 A d^2 f^2 h^2+C \left (5 c^2 f^2 h^2+4 c d f h (f g+e h)+d^2 \left (15 f^2 g^2+14 e f g h+15 e^2 h^2\right )\right )\right )\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{48 b^2 d^2 f^3 h^3}-\frac {\left ((a d f h+b (d f g+d e h+c f h)) \left (24 A b^2 d^2 f^2 h^2+3 a^2 C d^2 f^2 h^2-16 b^2 C d f h (d e g+c f g+c e h)-22 a b C d f h (d f g+d e h+c f h)+15 b^2 C (d f g+d e h+c f h)^2\right )+4 b d f h \left (C (b (d e g+c f g+c e h)+a (d f g+d e h+c f h)) (3 a d f h-5 b (d f g+d e h+c f h))+2 d f h \left (3 b^2 c C e g+2 a^2 C (d f g+d e h+c f h)-a b (12 A d f h-5 C (d e g+c f g+c e h))\right )\right )\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{48 b^2 d^3 f^3 h^3}-\frac {\left ((d g-c h) \left (24 A b^2 d^2 f^2 h^2+3 a^2 C d^2 f^2 h^2-16 b^2 C d f h (d e g+c f g+c e h)-22 a b C d f h (d f g+d e h+c f h)+15 b^2 C (d f g+d e h+c f h)^2\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 )}{24 b d^3 f^3 h^3 \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}} \\ & = \frac {\left (24 A b d^2 f h+\frac {3 a^2 C d^2 f h}{b}-16 b C d (d e g+c f g+c e h)-22 a C d (d f g+d e h+c f h)+\frac {15 b C (d f g+d e h+c f h)^2}{f h}\right ) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{24 d^2 f^2 h^2 \sqrt {c+d x}}+\frac {C (3 a d f h-5 b (d f g+d e h+c f h)) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{12 d^2 f^2 h^2}+\frac {C (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 d f h}-\frac {\sqrt {d g-c h} \sqrt {f g-e h} \left (24 A b^2 d^2 f^2 h^2+3 a^2 C d^2 f^2 h^2-16 b^2 C d f h (d e g+c f g+c e h)-22 a b C d f h (d f g+d e h+c f h)+15 b^2 C (d f g+d e h+c f h)^2\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 )}{24 b d^3 f^3 h^3 \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}-\frac {\left (\left ((a d f h+b (d f g+d e h+c f h)) \left (24 A b^2 d^2 f^2 h^2+3 a^2 C d^2 f^2 h^2-16 b^2 C d f h (d e g+c f g+c e h)-22 a b C d f h (d f g+d e h+c f h)+15 b^2 C (d f g+d e h+c f h)^2\right )+4 b d f h \left (C (b (d e g+c f g+c e h)+a (d f g+d e h+c f h)) (3 a d f h-5 b (d f g+d e h+c f h))+2 d f h \left (3 b^2 c C e g+2 a^2 C (d f g+d e h+c f h)-a b (12 A d f h-5 C (d e g+c f g+c e h))\right )\right )\right ) (a+b x) \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt {\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}}\right ) \text {Subst}\left (\int \frac {1}{\left (h-b x^2\right ) \sqrt {1+\frac {(b c-a d) x^2}{d g-c h}} \sqrt {1+\frac {(b e-a f) x^2}{f g-e h}}} \, dx,x,\frac {\sqrt {g+h x}}{\sqrt {a+b x}}\right )}{24 b^2 d^3 f^3 h^3 \sqrt {c+d x} \sqrt {e+f x}}+\frac {\left ((b e-a f) (b g-a h) \left (3 a^2 C d^2 f^2 h^2+6 a b C d f h (c f h+2 d (f g+e h))-b^2 \left (24 A d^2 f^2 h^2+C \left (5 c^2 f^2 h^2+4 c d f h (f g+e h)+d^2 \left (15 f^2 g^2+14 e f g h+15 e^2 h^2\right )\right )\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 )}{24 b^2 d^2 f^3 h^3 (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 {\left (24 A b d^2 f h+\frac {3 a^2 C d^2 f h}{b}-16 b C d (d e g+c f g+c e h)-22 a C d (d f g+d e h+c f h)+\frac {15 b C (d f g+d e h+c f h)^2}{f h}\right ) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{24 d^2 f^2 h^2 \sqrt {c+d x}}+\frac {C (3 a d f h-5 b (d f g+d e h+c f h)) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{12 d^2 f^2 h^2}+\frac {C (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 d f h}-\frac {\sqrt {d g-c h} \sqrt {f g-e h} \left (24 A b^2 d^2 f^2 h^2+3 a^2 C d^2 f^2 h^2-16 b^2 C d f h (d e g+c f g+c e h)-22 a b C d f h (d f g+d e h+c f h)+15 b^2 C (d f g+d e h+c f h)^2\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 )}{24 b d^3 f^3 h^3 \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}+\frac {(b e-a f) \sqrt {b g-a h} \left (3 a^2 C d^2 f^2 h^2+6 a b C d f h (c f h+2 d (f g+e h))-b^2 \left (24 A d^2 f^2 h^2+C \left (5 c^2 f^2 h^2+4 c d f h (f g+e h)+d^2 \left (15 f^2 g^2+14 e f g h+15 e^2 h^2\right )\right )\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 )}{24 b^2 d^2 f^3 h^3 \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 {\sqrt {-d g+c h} \left ((a d f h+b (d f g+d e h+c f h)) \left (24 A b^2 d^2 f^2 h^2+3 a^2 C d^2 f^2 h^2-16 b^2 C d f h (d e g+c f g+c e h)-22 a b C d f h (d f g+d e h+c f h)+15 b^2 C (d f g+d e h+c f h)^2\right )+4 b d f h \left (C (b (d e g+c f g+c e h)+a (d f g+d e h+c f h)) (3 a d f h-5 b (d f g+d e h+c f h))+2 d f h \left (3 b^2 c C e g+2 a^2 C (d f g+d e h+c f h)-a b (12 A d f h-5 C (d e g+c f g+c e h))\right )\right )\right ) (a+b x) \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt {\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \Pi \left (-\frac {b (d g-c h)}{(b c-a d) h};\sin ^{-1}\left (\frac {\sqrt {b c-a d} \sqrt {g+h x}}{\sqrt {-d g+c h} \sqrt {a+b x}}\right )|\frac {(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{24 b^2 d^3 \sqrt {b c-a d} f^3 h^4 \sqrt {c+d x} \sqrt {e+f x}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(39032\) vs. \(2(1395)=2790\).
Time = 40.08 (sec) , antiderivative size = 39032, normalized size of antiderivative = 27.98 \[ \int \frac {(a+b x)^{3/2} \left (A+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Result too large to show} \]
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Time = 6.74 (sec) , antiderivative size = 2228, normalized size of antiderivative = 1.60
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(2228\) |
default | \(\text {Expression too large to display}\) | \(92114\) |
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Timed out. \[ \int \frac {(a+b x)^{3/2} \left (A+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+b x)^{3/2} \left (A+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {\left (A + C x^{2}\right ) \left (a + b x\right )^{\frac {3}{2}}}{\sqrt {c + d x} \sqrt {e + f x} \sqrt {g + h x}}\, dx \]
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\[ \int \frac {(a+b x)^{3/2} \left (A+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {{\left (C x^{2} + A\right )} {\left (b x + a\right )}^{\frac {3}{2}}}{\sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
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\[ \int \frac {(a+b x)^{3/2} \left (A+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {{\left (C x^{2} + A\right )} {\left (b x + a\right )}^{\frac {3}{2}}}{\sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
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Timed out. \[ \int \frac {(a+b x)^{3/2} \left (A+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {\left (C\,x^2+A\right )\,{\left (a+b\,x\right )}^{3/2}}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,\sqrt {c+d\,x}} \,d x \]
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