Integrand size = 42, antiderivative size = 1081 \[ \int \frac {A+B x}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 d \left (3 a^3 B d f h+b^3 (3 B c e g-2 A (d e g+c f g+c e h))-a b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h))-a^2 b (6 A d f h+B (d f g+d e h+c f h))\right ) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {c+d x}}-\frac {2 b (A b-a B) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac {2 b \left (3 a^3 B d f h+b^3 (3 B c e g-2 A (d e g+c f g+c e h))-a b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h))-a^2 b (6 A d f h+B (d f g+d e h+c f h))\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {a+b x}}-\frac {2 \sqrt {d g-c h} \sqrt {f g-e h} \left (3 a^3 B d f h+b^3 (3 B c e g-2 A (d e g+c f g+c e h))-a b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h))-a^2 b (6 A d f h+B (d f g+d e h+c f h))\right ) \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\arcsin \left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}-\frac {2 \left (3 a^2 d (B c-A d) f h+b^2 \left (3 B c d e g-A \left (2 d^2 e g-c^2 f h+c d (f g+e h)\right )\right )+a b \left (3 A d^2 (f g+e h)-B \left (d^2 e g+c^2 f h+2 c d (f g+e h)\right )\right )\right ) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{3 (b c-a d)^2 (b e-a f) (b g-a h)^{3/2} \sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}} \]
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Time = 2.25 (sec) , antiderivative size = 1080, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1613, 1616, 12, 176, 430, 182, 435} \[ \int \frac {A+B x}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=-\frac {2 b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (A b-a B)}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac {2 \sqrt {d g-c h} \sqrt {f g-e h} \left (3 B d f h a^3-b (6 A d f h+B (d f g+d e h+c f h)) a^2-b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h)) a+b^3 (3 B c e g-2 A (d e g+c f g+c e h))\right ) \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\arcsin \left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}-\frac {2 \left (3 d (B c-A d) f h a^2+b \left (3 A d^2 (f g+e h)-B \left (f h c^2+2 d (f g+e h) c+d^2 e g\right )\right ) a+b^2 \left (A f h c^2+3 B d e g c-A d (f g+e h) c-2 A d^2 e g\right )\right ) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{3 (b c-a d)^2 (b e-a f) (b g-a h)^{3/2} \sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}-\frac {2 b \left (3 B d f h a^3-b (6 A d f h+B (d f g+d e h+c f h)) a^2-b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h)) a+b^3 (3 B c e g-2 A (d e g+c f g+c e h))\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {a+b x}}+\frac {2 d \left (3 B d f h a^3-b (6 A d f h+B (d f g+d e h+c f h)) a^2-b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h)) a+b^3 (3 B c e g-2 A (d e g+c f g+c e h))\right ) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {c+d x}} \]
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Rule 12
Rule 176
Rule 182
Rule 430
Rule 435
Rule 1613
Rule 1616
Rubi steps \begin{align*} \text {integral}& = -\frac {2 b (A b-a B) \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 {-3 a^2 A d f h+b^2 (3 B c e g-2 A (d e g+c f g+c e h))-a b (B (d e g+c f g+c e h)-3 A (d f g+d e h+c f h))+(A b-a B) (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 (A b-a B) \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 {2 b \left (3 a^3 B d f h+b^3 (3 B c e g-2 A (d e g+c f g+c e h))-a b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h))-a^2 b (6 A d f h+B (d f g+d e h+c 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 (A b-a B) (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 (3 a^2 A d f h-b^2 (3 B c e g-2 A (d e g+c f g+c e h))+a b (B (d e g+c f g+c e h)-3 A (d f g+d e h+c f h))\right )+(a d f h+b (d f g+d e h+c f h)) \left (3 a^3 B d f h+b^3 (3 B c e g-2 A (d e g+c f g+c e h))-a b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h))-a^2 b (6 A d f h+B (d f g+d e h+c f h))\right ) x+2 b d f h \left (3 a^3 B d f h+b^3 (3 B c e g-2 A (d e g+c f g+c e h))-a b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h))-a^2 b (6 A d f h+B (d f g+d e h+c 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 {2 d \left (3 a^3 B d f h+b^3 (3 B c e g-2 A (d e g+c f g+c e h))-a b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h))-a^2 b (6 A d f h+B (d f g+d e h+c 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 b (A b-a B) \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 {2 b \left (3 a^3 B d f h+b^3 (3 B c e g-2 A (d e g+c f g+c e h))-a b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h))-a^2 b (6 A d f h+B (d f g+d e h+c 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^2 d (B c-A d) f h+b^2 \left (3 B c d e g-2 A d^2 e g+A c^2 f h-A c d (f g+e h)\right )+a b \left (3 A d^2 (f g+e h)-B \left (d^2 e g+c^2 f h+2 c d (f g+e 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 ((d e-c f) (d g-c h) \left (3 a^3 B d f h+b^3 (3 B c e g-2 A (d e g+c f g+c e h))-a b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h))-a^2 b (6 A d f h+B (d f g+d e h+c 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 {2 d \left (3 a^3 B d f h+b^3 (3 B c e g-2 A (d e g+c f g+c e h))-a b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h))-a^2 b (6 A d f h+B (d f g+d e h+c 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 b (A b-a B) \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 {2 b \left (3 a^3 B d f h+b^3 (3 B c e g-2 A (d e g+c f g+c e h))-a b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h))-a^2 b (6 A d f h+B (d f g+d e h+c 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^2 d (B c-A d) f h+b^2 \left (3 B c d e g-2 A d^2 e g+A c^2 f h-A c d (f g+e h)\right )+a b \left (3 A d^2 (f g+e h)-B \left (d^2 e g+c^2 f h+2 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 (2 (d g-c h) \left (3 a^3 B d f h+b^3 (3 B c e g-2 A (d e g+c f g+c e h))-a b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h))-a^2 b (6 A d f h+B (d f g+d e h+c 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 {2 d \left (3 a^3 B d f h+b^3 (3 B c e g-2 A (d e g+c f g+c e h))-a b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h))-a^2 b (6 A d f h+B (d f g+d e h+c 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 b (A b-a B) \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 {2 b \left (3 a^3 B d f h+b^3 (3 B c e g-2 A (d e g+c f g+c e h))-a b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h))-a^2 b (6 A d f h+B (d f g+d e h+c 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 {2 \sqrt {d g-c h} \sqrt {f g-e h} \left (3 a^3 B d f h+b^3 (3 B c e g-2 A (d e g+c f g+c e h))-a b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h))-a^2 b (6 A d f h+B (d f g+d e h+c 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^2 d (B c-A d) f h+b^2 \left (3 B c d e g-2 A d^2 e g+A c^2 f h-A c d (f g+e h)\right )+a b \left (3 A d^2 (f g+e h)-B \left (d^2 e g+c^2 f h+2 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 {2 d \left (3 a^3 B d f h+b^3 (3 B c e g-2 A (d e g+c f g+c e h))-a b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h))-a^2 b (6 A d f h+B (d f g+d e h+c 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 b (A b-a B) \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 {2 b \left (3 a^3 B d f h+b^3 (3 B c e g-2 A (d e g+c f g+c e h))-a b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h))-a^2 b (6 A d f h+B (d f g+d e h+c 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 {2 \sqrt {d g-c h} \sqrt {f g-e h} \left (3 a^3 B d f h+b^3 (3 B c e g-2 A (d e g+c f g+c e h))-a b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h))-a^2 b (6 A d f h+B (d f g+d e h+c 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^2 d (B c-A d) f h+b^2 \left (3 B c d e g-2 A d^2 e g+A c^2 f h-A c d (f g+e h)\right )+a b \left (3 A d^2 (f g+e h)-B \left (d^2 e g+c^2 f h+2 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} 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. \(10828\) vs. \(2(1081)=2162\).
Time = 39.61 (sec) , antiderivative size = 10828, normalized size of antiderivative = 10.02 \[ \int \frac {A+B x}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Result too large to show} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(3388\) vs. \(2(1009)=2018\).
Time = 10.18 (sec) , antiderivative size = 3389, normalized size of antiderivative = 3.14
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(3389\) |
default | \(\text {Expression too large to display}\) | \(104801\) |
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\[ \int \frac {A+B x}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {B x + A}{{\left (b x + a\right )}^{\frac {5}{2}} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
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Timed out. \[ \int \frac {A+B x}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Timed out} \]
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\[ \int \frac {A+B x}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {B x + A}{{\left (b x + a\right )}^{\frac {5}{2}} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
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\[ \int \frac {A+B x}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {B x + A}{{\left (b x + a\right )}^{\frac {5}{2}} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
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Timed out. \[ \int \frac {A+B x}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {A+B\,x}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,{\left (a+b\,x\right )}^{5/2}\,\sqrt {c+d\,x}} \,d x \]
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