Integrand size = 44, antiderivative size = 1070 \[ \int \frac {A+C x^2}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=-\frac {4 d \left (A b^3 (d e g+c f g+c e h)+a^3 C (d f g+d e h+c f h)+a^2 b (3 A d f h-2 C (d e g+c f g+c e h))-a b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h))\right ) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {c+d x}}-\frac {2 \left (A b^2+a^2 C\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}+\frac {4 b \left (A b^3 (d e g+c f g+c e h)+a^3 C (d f g+d e h+c f h)+a^2 b (3 A d f h-2 C (d e g+c f g+c e h))-a b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h))\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {a+b x}}+\frac {4 \sqrt {d g-c h} \sqrt {f g-e h} \left (A b^3 (d e g+c f g+c e h)+a^3 C (d f g+d e h+c f h)+a^2 b (3 A d f h-2 C (d e g+c f g+c e h))-a b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h))\right ) \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\arcsin \left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}-\frac {2 \left (3 a b \left (c^2 C+A d^2\right ) (f g+e h)-b^2 \left (2 A d^2 e g+A c d (f g+e h)+c^2 (3 C e g-A f h)\right )-a^2 \left (3 A d^2 f h-C \left (d^2 e g-c d f g-c d e h-2 c^2 f h\right )\right )\right ) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{3 (b c-a d)^2 (b e-a f) (b g-a h)^{3/2} \sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}} \]
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Time = 2.32 (sec) , antiderivative size = 1070, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1619, 1613, 1616, 12, 176, 430, 182, 435} \[ \int \frac {A+C x^2}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=-\frac {2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (C a^2+A b^2\right )}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}+\frac {4 \sqrt {d g-c h} \sqrt {f g-e h} \left (C (d f g+d e h+c f h) a^3+b (3 A d f h-2 C (d e g+c f g+c e h)) a^2-b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h)) a+A b^3 (d e g+c f g+c e h)\right ) \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\arcsin \left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}-\frac {2 \left (-\left (\left (3 A d^2 f h-C \left (-2 f h c^2-d f g c-d e h c+d^2 e g\right )\right ) a^2\right )+3 b \left (C c^2+A d^2\right ) (f g+e h) a-b^2 \left ((3 C e g-A f h) c^2+A d (f g+e h) c+2 A d^2 e g\right )\right ) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{3 (b c-a d)^2 (b e-a f) (b g-a h)^{3/2} \sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}+\frac {4 b \left (C (d f g+d e h+c f h) a^3+b (3 A d f h-2 C (d e g+c f g+c e h)) a^2-b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h)) a+A b^3 (d e g+c f g+c e h)\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {a+b x}}-\frac {4 d \left (C (d f g+d e h+c f h) a^3+b (3 A d f h-2 C (d e g+c f g+c e h)) a^2-b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h)) a+A b^3 (d e g+c f g+c e h)\right ) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {c+d x}} \]
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Rule 12
Rule 176
Rule 182
Rule 430
Rule 435
Rule 1613
Rule 1616
Rule 1619
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (A b^2+a^2 C\right ) \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 A b^2 (d e g+c f g+c e h)-3 a b (c C e g-A d f g-A d e h-A c f h)-a^2 (3 A d f h-C (d e g+c f g+c e h))+\left (2 a^2 C (d f g+d e h+c f h)+b^2 (3 c C e g-A d f g-A d e h-A c f h)+3 a b (A d f h-C (d e g+c f g+c e h))\right ) 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 \left (A b^2+a^2 C\right ) \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 (A b^3 (d e g+c f g+c e h)+a^3 C (d f g+d e h+c f h)+a^2 b (3 A d f h-2 C (d e g+c f g+c e h))-a b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h))\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 c e g-a (d e g+c f g+c e h)) \left (2 a^2 C (d f g+d e h+c f h)+b^2 (3 c C e g-A d f g-A d e h-A c f h)+3 a b (A d f h-C (d e g+c f g+c e h))\right )+a (a d f h-b (d f g+d e h+c f h)) \left (2 A b^2 (d e g+c f g+c e h)+3 a b (c C e g-A d f g-A d e h-A c f h)+a^2 (3 A d f h-C (d e g+c f g+c e h))\right )-2 (a d f h+b (d f g+d e h+c f h)) \left (A b^3 (d e g+c f g+c e h)+a^3 C (d f g+d e h+c f h)+a^2 b (3 A d f h-2 C (d e g+c f g+c e h))-a b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h))\right ) x-4 b d f h \left (A b^3 (d e g+c f g+c e h)+a^3 C (d f g+d e h+c f h)+a^2 b (3 A d f h-2 C (d e g+c f g+c e h))-a b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h))\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 (A b^3 (d e g+c f g+c e h)+a^3 C (d f g+d e h+c f h)+a^2 b (3 A d f h-2 C (d e g+c f g+c e h))-a b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h))\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 \left (A b^2+a^2 C\right ) \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 (A b^3 (d e g+c f g+c e h)+a^3 C (d f g+d e h+c f h)+a^2 b (3 A d f h-2 C (d e g+c f g+c e h))-a b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h))\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) h (b g-a h) \left (3 a b \left (c^2 C+A d^2\right ) (f g+e h)-b^2 \left (2 A d^2 e g+A c d (f g+e h)+c^2 (3 C e g-A f h)\right )-a^2 \left (3 A d^2 f h-C \left (d^2 e g-c d f g-c d e h-2 c^2 f h\right )\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 (A b^3 (d e g+c f g+c e h)+a^3 C (d f g+d e h+c f h)+a^2 b (3 A d f h-2 C (d e g+c f g+c e h))-a b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h))\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 (A b^3 (d e g+c f g+c e h)+a^3 C (d f g+d e h+c f h)+a^2 b (3 A d f h-2 C (d e g+c f g+c e h))-a b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h))\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 \left (A b^2+a^2 C\right ) \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 (A b^3 (d e g+c f g+c e h)+a^3 C (d f g+d e h+c f h)+a^2 b (3 A d f h-2 C (d e g+c f g+c e h))-a b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h))\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 (3 a b \left (c^2 C+A d^2\right ) (f g+e h)-b^2 \left (2 A d^2 e g+A c d (f g+e h)+c^2 (3 C e g-A f h)\right )-a^2 \left (3 A d^2 f h-C \left (d^2 e g-2 c^2 f h-c d (f g+e 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 (A b^3 (d e g+c f g+c e h)+a^3 C (d f g+d e h+c f h)+a^2 b (3 A d f h-2 C (d e g+c f g+c e h))-a b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h))\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 (A b^3 (d e g+c f g+c e h)+a^3 C (d f g+d e h+c f h)+a^2 b (3 A d f h-2 C (d e g+c f g+c e h))-a b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h))\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 \left (A b^2+a^2 C\right ) \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 (A b^3 (d e g+c f g+c e h)+a^3 C (d f g+d e h+c f h)+a^2 b (3 A d f h-2 C (d e g+c f g+c e h))-a b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h))\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 (A b^3 (d e g+c f g+c e h)+a^3 C (d f g+d e h+c f h)+a^2 b (3 A d f h-2 C (d e g+c f g+c e h))-a b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h))\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 \left (3 a b \left (c^2 C+A d^2\right ) (f g+e h)-b^2 \left (2 A d^2 e g+A c d (f g+e h)+c^2 (3 C e g-A f h)\right )-a^2 \left (3 A d^2 f h-C \left (d^2 e g-2 c^2 f h-c d (f g+e h)\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 )}{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 (A b^3 (d e g+c f g+c e h)+a^3 C (d f g+d e h+c f h)+a^2 b (3 A d f h-2 C (d e g+c f g+c e h))-a b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h))\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 \left (A b^2+a^2 C\right ) \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 (A b^3 (d e g+c f g+c e h)+a^3 C (d f g+d e h+c f h)+a^2 b (3 A d f h-2 C (d e g+c f g+c e h))-a b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h))\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 (A b^3 (d e g+c f g+c e h)+a^3 C (d f g+d e h+c f h)+a^2 b (3 A d f h-2 C (d e g+c f g+c e h))-a b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h))\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 \left (3 a b \left (c^2 C+A d^2\right ) (f g+e h)-b^2 \left (2 A d^2 e g+A c d (f g+e h)+c^2 (3 C e g-A f h)\right )-a^2 \left (3 A d^2 f h-C \left (d^2 e g-c d f g-c d e h-2 c^2 f h\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 )}{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. \(11363\) vs. \(2(1070)=2140\).
Time = 40.54 (sec) , antiderivative size = 11363, normalized size of antiderivative = 10.62 \[ \int \frac {A+C x^2}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Result too large to show} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(3341\) vs. \(2(998)=1996\).
Time = 10.35 (sec) , antiderivative size = 3342, normalized size of antiderivative = 3.12
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(3342\) |
default | \(\text {Expression too large to display}\) | \(106972\) |
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\[ \int \frac {A+C x^2}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {C x^{2} + A}{{\left (b x + a\right )}^{\frac {5}{2}} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
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Timed out. \[ \int \frac {A+C x^2}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Timed out} \]
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\[ \int \frac {A+C x^2}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {C x^{2} + A}{{\left (b x + a\right )}^{\frac {5}{2}} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
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\[ \int \frac {A+C x^2}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {C x^{2} + A}{{\left (b x + a\right )}^{\frac {5}{2}} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
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Timed out. \[ \int \frac {A+C x^2}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {C\,x^2+A}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,{\left (a+b\,x\right )}^{5/2}\,\sqrt {c+d\,x}} \,d x \]
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