Integrand size = 44, antiderivative size = 937 \[ \int \frac {\sqrt {a+b x} \left (A+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {C (a d f h-3 b (d f g+d e h+c f h)) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{4 b d f^2 h^2 \sqrt {c+d x}}+\frac {C \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{2 d f h}-\frac {C \sqrt {d g-c h} \sqrt {f g-e h} (a d f h-3 b (d f g+d e h+c f h)) \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\arcsin \left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{4 b d^2 f^2 h^2 \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}+\frac {C (b e-a f) \sqrt {b g-a h} (a d f h+b (c f h+3 d (f g+e h))) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{4 b^2 d f^2 h^2 \sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}-\frac {\sqrt {-d g+c h} (C (a d f h-3 b (d f g+d e h+c f h)) (a d f h+b (d f g+d e h+c f h))-4 b d f h (2 A b d f h-C (b (d e g+c f g+c e h)+a (d f g+d e h+c f h)))) (a+b x) \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt {\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \operatorname {EllipticPi}\left (-\frac {b (d g-c h)}{(b c-a d) h},\arcsin \left (\frac {\sqrt {b c-a d} \sqrt {g+h x}}{\sqrt {-d g+c h} \sqrt {a+b x}}\right ),\frac {(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{4 b^2 d^2 \sqrt {b c-a d} f^2 h^3 \sqrt {c+d x} \sqrt {e+f x}} \]
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Time = 1.56 (sec) , antiderivative size = 936, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.205, Rules used = {1615, 1616, 1612, 176, 430, 171, 551, 182, 435} \[ \int \frac {\sqrt {a+b x} \left (A+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=-\frac {\sqrt {d g-c h} \sqrt {f g-e h} (a d f h-3 b (d f g+d e h+c f h)) \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\arcsin \left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right ) C}{4 b d^2 f^2 h^2 \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}+\frac {(b e-a f) \sqrt {b g-a h} (b c f h+a d f h+3 b d (f g+e h)) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right ) C}{4 b^2 d f^2 h^2 \sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}+\frac {\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} C}{2 d f h}+\frac {(a d f h-3 b (d f g+d e h+c f h)) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} C}{4 b d f^2 h^2 \sqrt {c+d x}}-\frac {\sqrt {c h-d g} (C (a d f h-3 b (d f g+d e h+c f h)) (a d f h+b (d f g+d e h+c f h))-4 b d f h (2 A b d f h-C (b (d e g+c f g+c e h)+a (d f g+d e h+c f h)))) (a+b x) \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt {\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \operatorname {EllipticPi}\left (-\frac {b (d g-c h)}{(b c-a d) h},\arcsin \left (\frac {\sqrt {b c-a d} \sqrt {g+h x}}{\sqrt {c h-d g} \sqrt {a+b x}}\right ),\frac {(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{4 b^2 d^2 \sqrt {b c-a d} f^2 h^3 \sqrt {c+d x} \sqrt {e+f x}} \]
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Rule 171
Rule 176
Rule 182
Rule 430
Rule 435
Rule 551
Rule 1612
Rule 1615
Rule 1616
Rubi steps \begin{align*} \text {integral}& = \frac {C \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{2 d f h}+\frac {\int \frac {4 a A d f h-C (b c e g+a (d e g+c f g+c e h))+2 (2 A b d f h-C (b (d e g+c f g+c e h)+a (d f g+d e h+c f h))) x+C (a d f h-3 b (d f g+d e h+c f h)) x^2}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{4 d f h} \\ & = \frac {C (a d f h-3 b (d f g+d e h+c f h)) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{4 b d f^2 h^2 \sqrt {c+d x}}+\frac {C \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{2 d f h}+\frac {\int \frac {-C (b d e g+a c f h) (a d f h-3 b (d f g+d e h+c f h))+2 b d f h (4 a A d f h-C (b c e g+a (d e g+c f g+c e h)))-(C (a d f h-3 b (d f g+d e h+c f h)) (a d f h+b (d f g+d e h+c f h))-4 b d f h (2 A b d f h-C (b (d e g+c f g+c e h)+a (d f g+d e h+c f h)))) x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{8 b d^2 f^2 h^2}+\frac {(C (d e-c f) (d g-c h) (a d f h-3 b (d f g+d e h+c f h))) \int \frac {\sqrt {a+b x}}{(c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{8 b d^2 f^2 h^2} \\ & = \frac {C (a d f h-3 b (d f g+d e h+c f h)) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{4 b d f^2 h^2 \sqrt {c+d x}}+\frac {C \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{2 d f h}+\frac {(C (b e-a f) (b g-a h) (b c f h+a d f h+3 b d (f g+e h))) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{8 b^2 d f^2 h^2}-\frac {(C (a d f h-3 b (d f g+d e h+c f h)) (a d f h+b (d f g+d e h+c f h))-4 b d f h (2 A b d f h-C (b (d e g+c f g+c e h)+a (d f g+d e h+c f h)))) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{8 b^2 d^2 f^2 h^2}-\frac {\left (C (d g-c h) (a d f h-3 b (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)}}\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 )}{4 b d^2 f^2 h^2 \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}} \\ & = \frac {C (a d f h-3 b (d f g+d e h+c f h)) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{4 b d f^2 h^2 \sqrt {c+d x}}+\frac {C \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{2 d f h}-\frac {C \sqrt {d g-c h} \sqrt {f g-e h} (a d f h-3 b (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 (\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 )}{4 b d^2 f^2 h^2 \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}-\frac {\left ((C (a d f h-3 b (d f g+d e h+c f h)) (a d f h+b (d f g+d e h+c f h))-4 b d f h (2 A b d f h-C (b (d e g+c f g+c e h)+a (d f g+d e h+c f h)))) (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 )}{4 b^2 d^2 f^2 h^2 \sqrt {c+d x} \sqrt {e+f x}}+\frac {\left (C (b e-a f) (b g-a h) (b c f h+a d f h+3 b d (f g+e h)) \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 )}{4 b^2 d f^2 h^2 (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 {C (a d f h-3 b (d f g+d e h+c f h)) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{4 b d f^2 h^2 \sqrt {c+d x}}+\frac {C \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{2 d f h}-\frac {C \sqrt {d g-c h} \sqrt {f g-e h} (a d f h-3 b (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 (\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 )}{4 b d^2 f^2 h^2 \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}+\frac {C (b e-a f) \sqrt {b g-a h} (b c f h+a d f h+3 b d (f g+e h)) \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 )}{4 b^2 d f^2 h^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 {\sqrt {-d g+c h} (C (a d f h-3 b (d f g+d e h+c f h)) (a d f h+b (d f g+d e h+c f h))-4 b d f h (2 A b d f h-C (b (d e g+c f g+c e h)+a (d f g+d e h+c f h)))) (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 )}{4 b^2 d^2 \sqrt {b c-a d} f^2 h^3 \sqrt {c+d x} \sqrt {e+f x}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(16972\) vs. \(2(937)=1874\).
Time = 36.21 (sec) , antiderivative size = 16972, normalized size of antiderivative = 18.11 \[ \int \frac {\sqrt {a+b x} \left (A+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Result too large to show} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1793\) vs. \(2(854)=1708\).
Time = 5.23 (sec) , antiderivative size = 1794, normalized size of antiderivative = 1.91
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(1794\) |
default | \(\text {Expression too large to display}\) | \(43214\) |
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Timed out. \[ \int \frac {\sqrt {a+b x} \left (A+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt {a+b x} \left (A+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {\left (A + C x^{2}\right ) \sqrt {a + b x}}{\sqrt {c + d x} \sqrt {e + f x} \sqrt {g + h x}}\, dx \]
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\[ \int \frac {\sqrt {a+b x} \left (A+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {{\left (C x^{2} + A\right )} \sqrt {b x + a}}{\sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
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\[ \int \frac {\sqrt {a+b x} \left (A+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {{\left (C x^{2} + A\right )} \sqrt {b x + a}}{\sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {a+b x} \left (A+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {\left (C\,x^2+A\right )\,\sqrt {a+b\,x}}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,\sqrt {c+d\,x}} \,d x \]
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