Integrand size = 40, antiderivative size = 700 \[ \int \frac {(a+b x)^2 (A+B x)}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 b (7 a B d f h+b (5 A d f h-4 B (d f g+d e h+c f h))) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{15 d^2 f^2 h^2}+\frac {2 b B (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{5 d f h}+\frac {2 \sqrt {-d e+c f} \left (15 a^2 B d^2 f^2 h^2+10 a b d f h (3 A d f h-2 B (d f g+d e h+c f h))-b^2 \left (10 A d f h (d f g+d e h+c f h)-B \left (8 c^2 f^2 h^2+7 c d f h (f g+e h)+d^2 \left (8 f^2 g^2+7 e f g h+8 e^2 h^2\right )\right )\right )\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {g+h x} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{15 d^3 f^{5/2} h^3 \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}-\frac {2 \sqrt {-d e+c f} \left (15 a^2 d^2 f^2 h^2 (B g-A h)+10 a b d f h (3 A d f g h-B (c h (f g-e h)+d g (2 f g+e h)))-b^2 \left (5 A d f h (c h (f g-e h)+d g (2 f g+e h))-B \left (4 c^2 f h^2 (f g-e h)+c d h \left (3 f^2 g^2+e f g h-4 e^2 h^2\right )+d^2 g \left (8 f^2 g^2+3 e f g h+4 e^2 h^2\right )\right )\right )\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{15 d^3 f^{5/2} h^3 \sqrt {e+f x} \sqrt {g+h x}} \]
[Out]
Time = 1.65 (sec) , antiderivative size = 699, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1611, 1614, 1629, 164, 115, 114, 122, 121} \[ \int \frac {(a+b x)^2 (A+B x)}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=-\frac {2 \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right ) \left (15 a^2 d^2 f^2 h^2 (B g-A h)+10 a b d f h (3 A d f g h-B c h (f g-e h)-B d g (e h+2 f g))-\left (b^2 \left (5 A d f h (c h (f g-e h)+d g (e h+2 f g))-B \left (4 c^2 f h^2 (f g-e h)+c d h \left (-4 e^2 h^2+e f g h+3 f^2 g^2\right )+d^2 g \left (4 e^2 h^2+3 e f g h+8 f^2 g^2\right )\right )\right )\right )\right )}{15 d^3 f^{5/2} h^3 \sqrt {e+f x} \sqrt {g+h x}}+\frac {2 \sqrt {g+h x} \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right ) \left (15 a^2 B d^2 f^2 h^2+10 a b d f h (3 A d f h-2 B (c f h+d e h+d f g))-\left (b^2 \left (10 A d f h (c f h+d e h+d f g)-B \left (8 c^2 f^2 h^2+7 c d f h (e h+f g)+d^2 \left (8 e^2 h^2+7 e f g h+8 f^2 g^2\right )\right )\right )\right )\right )}{15 d^3 f^{5/2} h^3 \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}+\frac {2 b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (7 a B d f h+5 A b d f h-4 b B (c f h+d e h+d f g))}{15 d^2 f^2 h^2}+\frac {2 b B (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{5 d f h} \]
[In]
[Out]
Rule 114
Rule 115
Rule 121
Rule 122
Rule 164
Rule 1611
Rule 1614
Rule 1629
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {(a+b x) \left (7 a A d f h+7 (A b+a B) d f h x+7 b B d f h x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{7 d f h} \\ & = \frac {2 b B (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{5 d f h}+\frac {\int \frac {7 d f h \left (5 a^2 A d f h-b B (2 b c e g+a (d e g+c f g+c e h))\right )+7 d f h (5 a (2 A b+a B) d f h-b B (3 b (d e g+c f g+c e h)+2 a (d f g+d e h+c f h))) x+7 b d f h (5 A b d f h+7 a B d f h-4 b B (d f g+d e h+c f h)) x^2}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{35 d^2 f^2 h^2} \\ & = \frac {2 b (5 A b d f h+7 a B d f h-4 b B (d f g+d e h+c f h)) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{15 d^2 f^2 h^2}+\frac {2 b B (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{5 d f h}+\frac {2 \int \frac {\frac {7}{2} d^2 f h \left (15 a^2 A d^2 f^2 h^2-10 a b B d f h (d e g+c f g+c e h)-b^2 \left (5 A d f h (d e g+c f g+c e h)-B \left (4 d^2 e g (f g+e h)+4 c^2 f h (f g+e h)+2 c d \left (2 f^2 g^2+3 e f g h+2 e^2 h^2\right )\right )\right )\right )+\frac {7}{2} d^2 f h \left (15 a^2 B d^2 f^2 h^2+10 a b d f h (3 A d f h-2 B (d f g+d e h+c f h))-b^2 \left (10 A d f h (d f g+d e h+c f h)-B \left (8 c^2 f^2 h^2+7 c d f h (f g+e h)+d^2 \left (8 f^2 g^2+7 e f g h+8 e^2 h^2\right )\right )\right )\right ) x}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{105 d^4 f^3 h^3} \\ & = \frac {2 b (5 A b d f h+7 a B d f h-4 b B (d f g+d e h+c f h)) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{15 d^2 f^2 h^2}+\frac {2 b B (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{5 d f h}-\frac {\left (15 a^2 d^2 f^2 h^2 (B g-A h)+10 a b d f h (3 A d f g h-B c h (f g-e h)-B d g (2 f g+e h))-b^2 \left (5 A d f h (c h (f g-e h)+d g (2 f g+e h))-B \left (4 c^2 f h^2 (f g-e h)+c d h \left (3 f^2 g^2+e f g h-4 e^2 h^2\right )+d^2 g \left (8 f^2 g^2+3 e f g h+4 e^2 h^2\right )\right )\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{15 d^2 f^2 h^3}+\frac {\left (15 a^2 B d^2 f^2 h^2+10 a b d f h (3 A d f h-2 B (d f g+d e h+c f h))-b^2 \left (10 A d f h (d f g+d e h+c f h)-B \left (8 c^2 f^2 h^2+7 c d f h (f g+e h)+d^2 \left (8 f^2 g^2+7 e f g h+8 e^2 h^2\right )\right )\right )\right ) \int \frac {\sqrt {g+h x}}{\sqrt {c+d x} \sqrt {e+f x}} \, dx}{15 d^2 f^2 h^3} \\ & = \frac {2 b (5 A b d f h+7 a B d f h-4 b B (d f g+d e h+c f h)) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{15 d^2 f^2 h^2}+\frac {2 b B (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{5 d f h}-\frac {\left (\left (15 a^2 d^2 f^2 h^2 (B g-A h)+10 a b d f h (3 A d f g h-B c h (f g-e h)-B d g (2 f g+e h))-b^2 \left (5 A d f h (c h (f g-e h)+d g (2 f g+e h))-B \left (4 c^2 f h^2 (f g-e h)+c d h \left (3 f^2 g^2+e f g h-4 e^2 h^2\right )+d^2 g \left (8 f^2 g^2+3 e f g h+4 e^2 h^2\right )\right )\right )\right ) \sqrt {\frac {d (e+f x)}{d e-c f}}\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}} \sqrt {g+h x}} \, dx}{15 d^2 f^2 h^3 \sqrt {e+f x}}+\frac {\left (\left (15 a^2 B d^2 f^2 h^2+10 a b d f h (3 A d f h-2 B (d f g+d e h+c f h))-b^2 \left (10 A d f h (d f g+d e h+c f h)-B \left (8 c^2 f^2 h^2+7 c d f h (f g+e h)+d^2 \left (8 f^2 g^2+7 e f g h+8 e^2 h^2\right )\right )\right )\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {g+h x}\right ) \int \frac {\sqrt {\frac {d g}{d g-c h}+\frac {d h x}{d g-c h}}}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}}} \, dx}{15 d^2 f^2 h^3 \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}} \\ & = \frac {2 b (5 A b d f h+7 a B d f h-4 b B (d f g+d e h+c f h)) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{15 d^2 f^2 h^2}+\frac {2 b B (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{5 d f h}+\frac {2 \sqrt {-d e+c f} \left (15 a^2 B d^2 f^2 h^2+10 a b d f h (3 A d f h-2 B (d f g+d e h+c f h))-b^2 \left (10 A d f h (d f g+d e h+c f h)-B \left (8 c^2 f^2 h^2+7 c d f h (f g+e h)+d^2 \left (8 f^2 g^2+7 e f g h+8 e^2 h^2\right )\right )\right )\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {g+h x} E\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{15 d^3 f^{5/2} h^3 \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}-\frac {\left (\left (15 a^2 d^2 f^2 h^2 (B g-A h)+10 a b d f h (3 A d f g h-B c h (f g-e h)-B d g (2 f g+e h))-b^2 \left (5 A d f h (c h (f g-e h)+d g (2 f g+e h))-B \left (4 c^2 f h^2 (f g-e h)+c d h \left (3 f^2 g^2+e f g h-4 e^2 h^2\right )+d^2 g \left (8 f^2 g^2+3 e f g h+4 e^2 h^2\right )\right )\right )\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}}\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}} \sqrt {\frac {d g}{d g-c h}+\frac {d h x}{d g-c h}}} \, dx}{15 d^2 f^2 h^3 \sqrt {e+f x} \sqrt {g+h x}} \\ & = \frac {2 b (5 A b d f h+7 a B d f h-4 b B (d f g+d e h+c f h)) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{15 d^2 f^2 h^2}+\frac {2 b B (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{5 d f h}+\frac {2 \sqrt {-d e+c f} \left (15 a^2 B d^2 f^2 h^2+10 a b d f h (3 A d f h-2 B (d f g+d e h+c f h))-b^2 \left (10 A d f h (d f g+d e h+c f h)-B \left (8 c^2 f^2 h^2+7 c d f h (f g+e h)+d^2 \left (8 f^2 g^2+7 e f g h+8 e^2 h^2\right )\right )\right )\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {g+h x} E\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{15 d^3 f^{5/2} h^3 \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}-\frac {2 \sqrt {-d e+c f} \left (15 a^2 d^2 f^2 h^2 (B g-A h)+10 a b d f h (3 A d f g h-B c h (f g-e h)-B d g (2 f g+e h))-b^2 \left (5 A d f h (c h (f g-e h)+d g (2 f g+e h))-B \left (4 c^2 f h^2 (f g-e h)+c d h \left (3 f^2 g^2+e f g h-4 e^2 h^2\right )+d^2 g \left (8 f^2 g^2+3 e f g h+4 e^2 h^2\right )\right )\right )\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{15 d^3 f^{5/2} h^3 \sqrt {e+f x} \sqrt {g+h x}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 28.41 (sec) , antiderivative size = 806, normalized size of antiderivative = 1.15 \[ \int \frac {(a+b x)^2 (A+B x)}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=-\frac {2 \left (-d^2 \sqrt {-c+\frac {d e}{f}} \left (15 a^2 B d^2 f^2 h^2-10 a b d f h (-3 A d f h+2 B (d f g+d e h+c f h))+b^2 \left (-10 A d f h (d f g+d e h+c f h)+B \left (8 c^2 f^2 h^2+7 c d f h (f g+e h)+d^2 \left (8 f^2 g^2+7 e f g h+8 e^2 h^2\right )\right )\right )\right ) (e+f x) (g+h x)+b d^2 \sqrt {-c+\frac {d e}{f}} f h (c+d x) (e+f x) (g+h x) (-5 A b d f h-10 a B d f h+b B (4 c f h+d (4 f g+4 e h-3 f h x)))-i (d e-c f) h \left (15 a^2 B d^2 f^2 h^2-10 a b d f h (-3 A d f h+2 B (d f g+d e h+c f h))+b^2 \left (-10 A d f h (d f g+d e h+c f h)+B \left (8 c^2 f^2 h^2+7 c d f h (f g+e h)+d^2 \left (8 f^2 g^2+7 e f g h+8 e^2 h^2\right )\right )\right )\right ) (c+d x)^{3/2} \sqrt {\frac {d (e+f x)}{f (c+d x)}} \sqrt {\frac {d (g+h x)}{h (c+d x)}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {d e}{f}}}{\sqrt {c+d x}}\right )|\frac {d f g-c f h}{d e h-c f h}\right )-i d h \left (15 a^2 d^2 f^2 (-B e+A f) h^2+10 a b d f h (-3 A d e f h+B c f (-f g+e h)+B d e (f g+2 e h))-b^2 \left (-5 A d f h (c f (-f g+e h)+d e (f g+2 e h))+B \left (4 c^2 f^2 h (-f g+e h)+c d f \left (-4 f^2 g^2+e f g h+3 e^2 h^2\right )+d^2 e \left (4 f^2 g^2+3 e f g h+8 e^2 h^2\right )\right )\right )\right ) (c+d x)^{3/2} \sqrt {\frac {d (e+f x)}{f (c+d x)}} \sqrt {\frac {d (g+h x)}{h (c+d x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {d e}{f}}}{\sqrt {c+d x}}\right ),\frac {d f g-c f h}{d e h-c f h}\right )\right )}{15 d^4 \sqrt {-c+\frac {d e}{f}} f^3 h^3 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \]
[In]
[Out]
Time = 2.79 (sec) , antiderivative size = 866, normalized size of antiderivative = 1.24
method | result | size |
elliptic | \(\frac {\sqrt {\left (d x +c \right ) \left (f x +e \right ) \left (h x +g \right )}\, \left (\frac {2 B \,b^{2} x \sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}}{5 d f h}+\frac {2 \left (b^{2} A +2 a b B -\frac {2 B \,b^{2} \left (2 c f h +2 d e h +2 d f g \right )}{5 d f h}\right ) \sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}}{3 d f h}+\frac {2 \left (a^{2} A -\frac {2 B \,b^{2} c e g}{5 d f h}-\frac {2 \left (b^{2} A +2 a b B -\frac {2 B \,b^{2} \left (2 c f h +2 d e h +2 d f g \right )}{5 d f h}\right ) \left (\frac {1}{2} c e h +\frac {1}{2} c f g +\frac {1}{2} d e g \right )}{3 d f h}\right ) \left (\frac {g}{h}-\frac {e}{f}\right ) \sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {g}{h}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {g}{h}+\frac {e}{f}}}\, F\left (\sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}, \sqrt {\frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {c}{d}}}\right )}{\sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}}+\frac {2 \left (2 a b A +a^{2} B -\frac {2 B \,b^{2} \left (\frac {3}{2} c e h +\frac {3}{2} c f g +\frac {3}{2} d e g \right )}{5 d f h}-\frac {2 \left (b^{2} A +2 a b B -\frac {2 B \,b^{2} \left (2 c f h +2 d e h +2 d f g \right )}{5 d f h}\right ) \left (c f h +d e h +d f g \right )}{3 d f h}\right ) \left (\frac {g}{h}-\frac {e}{f}\right ) \sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {g}{h}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {g}{h}+\frac {e}{f}}}\, \left (\left (-\frac {g}{h}+\frac {c}{d}\right ) E\left (\sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}, \sqrt {\frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {c}{d}}}\right )-\frac {c F\left (\sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}, \sqrt {\frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {c}{d}}}\right )}{d}\right )}{\sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}}\right )}{\sqrt {d x +c}\, \sqrt {f x +e}\, \sqrt {h x +g}}\) | \(866\) |
default | \(\text {Expression too large to display}\) | \(8121\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.15 (sec) , antiderivative size = 1236, normalized size of antiderivative = 1.77 \[ \int \frac {(a+b x)^2 (A+B x)}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Too large to display} \]
[In]
[Out]
\[ \int \frac {(a+b x)^2 (A+B x)}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {\left (A + B x\right ) \left (a + b x\right )^{2}}{\sqrt {c + d x} \sqrt {e + f x} \sqrt {g + h x}}\, dx \]
[In]
[Out]
\[ \int \frac {(a+b x)^2 (A+B x)}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {{\left (B x + A\right )} {\left (b x + a\right )}^{2}}{\sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
[In]
[Out]
\[ \int \frac {(a+b x)^2 (A+B x)}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {{\left (B x + A\right )} {\left (b x + a\right )}^{2}}{\sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(a+b x)^2 (A+B x)}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (a+b\,x\right )}^2}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,\sqrt {c+d\,x}} \,d x \]
[In]
[Out]