Integrand size = 30, antiderivative size = 322 \[ \int (g+h x) \sqrt {a+b x+c x^2} \left (d+e x+f x^2\right ) \, dx=\frac {\left (32 c^3 d g-7 b^3 f h-8 c^2 (2 b e g+a f g+2 b d h+a e h)+2 b c (6 a f h+5 b (f g+e h))\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{128 c^4}+\frac {f (g+h x)^2 \left (a+b x+c x^2\right )^{3/2}}{5 c h}+\frac {\left (35 b^2 f h^2-16 c^2 \left (3 f g^2-5 h (e g+d h)\right )-2 c h (16 a f h+25 b (f g+e h))-6 c h (6 c f g-10 c e h+7 b f h) x\right ) \left (a+b x+c x^2\right )^{3/2}}{240 c^3 h}-\frac {\left (b^2-4 a c\right ) \left (32 c^3 d g-7 b^3 f h-8 c^2 (2 b e g+a f g+2 b d h+a e h)+2 b c (6 a f h+5 b (f g+e h))\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{9/2}} \]
[Out]
Time = 0.29 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1667, 793, 626, 635, 212} \[ \int (g+h x) \sqrt {a+b x+c x^2} \left (d+e x+f x^2\right ) \, dx=-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-8 c^2 (a e h+a f g+2 b d h+2 b e g)+2 b c (6 a f h+5 b (e h+f g))-7 b^3 f h+32 c^3 d g\right )}{256 c^{9/2}}+\frac {(b+2 c x) \sqrt {a+b x+c x^2} \left (-8 c^2 (a e h+a f g+2 b d h+2 b e g)+2 b c (6 a f h+5 b (e h+f g))-7 b^3 f h+32 c^3 d g\right )}{128 c^4}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (-2 c h (16 a f h+25 b (e h+f g))+35 b^2 f h^2-6 c h x (7 b f h-10 c e h+6 c f g)-\left (c^2 \left (48 f g^2-80 h (d h+e g)\right )\right )\right )}{240 c^3 h}+\frac {f (g+h x)^2 \left (a+b x+c x^2\right )^{3/2}}{5 c h} \]
[In]
[Out]
Rule 212
Rule 626
Rule 635
Rule 793
Rule 1667
Rubi steps \begin{align*} \text {integral}& = \frac {f (g+h x)^2 \left (a+b x+c x^2\right )^{3/2}}{5 c h}+\frac {\int (g+h x) \left (-\frac {1}{2} h (3 b f g-10 c d h+4 a f h)-\frac {1}{2} h (6 c f g-10 c e h+7 b f h) x\right ) \sqrt {a+b x+c x^2} \, dx}{5 c h^2} \\ & = \frac {f (g+h x)^2 \left (a+b x+c x^2\right )^{3/2}}{5 c h}+\frac {\left (35 b^2 f h^2-c^2 \left (48 f g^2-80 h (e g+d h)\right )-2 c h (16 a f h+25 b (f g+e h))-6 c h (6 c f g-10 c e h+7 b f h) x\right ) \left (a+b x+c x^2\right )^{3/2}}{240 c^3 h}+\frac {\left (32 c^3 d g-7 b^3 f h-8 c^2 (2 b e g+a f g+2 b d h+a e h)+2 b c (6 a f h+5 b (f g+e h))\right ) \int \sqrt {a+b x+c x^2} \, dx}{32 c^3} \\ & = \frac {\left (32 c^3 d g-7 b^3 f h-8 c^2 (2 b e g+a f g+2 b d h+a e h)+2 b c (6 a f h+5 b (f g+e h))\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{128 c^4}+\frac {f (g+h x)^2 \left (a+b x+c x^2\right )^{3/2}}{5 c h}+\frac {\left (35 b^2 f h^2-c^2 \left (48 f g^2-80 h (e g+d h)\right )-2 c h (16 a f h+25 b (f g+e h))-6 c h (6 c f g-10 c e h+7 b f h) x\right ) \left (a+b x+c x^2\right )^{3/2}}{240 c^3 h}-\frac {\left (\left (b^2-4 a c\right ) \left (32 c^3 d g-7 b^3 f h-8 c^2 (2 b e g+a f g+2 b d h+a e h)+2 b c (6 a f h+5 b (f g+e h))\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{256 c^4} \\ & = \frac {\left (32 c^3 d g-7 b^3 f h-8 c^2 (2 b e g+a f g+2 b d h+a e h)+2 b c (6 a f h+5 b (f g+e h))\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{128 c^4}+\frac {f (g+h x)^2 \left (a+b x+c x^2\right )^{3/2}}{5 c h}+\frac {\left (35 b^2 f h^2-c^2 \left (48 f g^2-80 h (e g+d h)\right )-2 c h (16 a f h+25 b (f g+e h))-6 c h (6 c f g-10 c e h+7 b f h) x\right ) \left (a+b x+c x^2\right )^{3/2}}{240 c^3 h}-\frac {\left (\left (b^2-4 a c\right ) \left (32 c^3 d g-7 b^3 f h-8 c^2 (2 b e g+a f g+2 b d h+a e h)+2 b c (6 a f h+5 b (f g+e h))\right )\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{128 c^4} \\ & = \frac {\left (32 c^3 d g-7 b^3 f h-8 c^2 (2 b e g+a f g+2 b d h+a e h)+2 b c (6 a f h+5 b (f g+e h))\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{128 c^4}+\frac {f (g+h x)^2 \left (a+b x+c x^2\right )^{3/2}}{5 c h}+\frac {\left (35 b^2 f h^2-c^2 \left (48 f g^2-80 h (e g+d h)\right )-2 c h (16 a f h+25 b (f g+e h))-6 c h (6 c f g-10 c e h+7 b f h) x\right ) \left (a+b x+c x^2\right )^{3/2}}{240 c^3 h}-\frac {\left (b^2-4 a c\right ) \left (32 c^3 d g-7 b^3 f h-8 c^2 (2 b e g+a f g+2 b d h+a e h)+2 b c (6 a f h+5 b (f g+e h))\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{9/2}} \\ \end{align*}
Time = 3.46 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.07 \[ \int (g+h x) \sqrt {a+b x+c x^2} \left (d+e x+f x^2\right ) \, dx=\frac {\sqrt {c} \sqrt {a+x (b+c x)} \left (-105 b^4 f h+10 b^3 c (15 f g+15 e h+7 f h x)-4 b^2 c \left (-115 a f h+c \left (60 e g+60 d h+25 f g x+25 e h x+14 f h x^2\right )\right )+8 b c^2 (20 c d (3 g+h x)-a (65 f g+65 e h+29 f h x)+2 c x (5 e (2 g+h x)+f x (5 g+3 h x)))+16 c^2 \left (-16 a^2 f h+a c (40 d h+5 e (8 g+3 h x)+f x (15 g+8 h x))+2 c^2 x (10 d (3 g+2 h x)+x (5 e (4 g+3 h x)+3 f x (5 g+4 h x)))\right )\right )+15 \left (b^2-4 a c\right ) \left (-32 c^3 d g+7 b^3 f h+8 c^2 (2 b e g+a f g+2 b d h+a e h)-2 b c (6 a f h+5 b (f g+e h))\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )}{1920 c^{9/2}} \]
[In]
[Out]
Time = 0.77 (sec) , antiderivative size = 488, normalized size of antiderivative = 1.52
method | result | size |
risch | \(-\frac {\left (-384 h f \,c^{4} x^{4}-48 b \,c^{3} f h \,x^{3}-480 c^{4} e h \,x^{3}-480 c^{4} f g \,x^{3}-128 a \,c^{3} f h \,x^{2}+56 b^{2} c^{2} f h \,x^{2}-80 b \,c^{3} e h \,x^{2}-80 b \,c^{3} f g \,x^{2}-640 c^{4} d h \,x^{2}-640 c^{4} e g \,x^{2}+232 a b \,c^{2} f h x -240 a \,c^{3} e h x -240 a \,c^{3} f g x -70 b^{3} c f h x +100 b^{2} c^{2} e h x +100 b^{2} c^{2} f g x -160 b \,c^{3} d h x -160 b \,c^{3} e g x -960 c^{4} d g x +256 a^{2} c^{2} f h -460 a \,b^{2} c f h +520 a b \,c^{2} e h +520 a b \,c^{2} f g -640 a \,c^{3} d h -640 a \,c^{3} e g +105 b^{4} f h -150 b^{3} c e h -150 b^{3} c f g +240 b^{2} c^{2} d h +240 b^{2} c^{2} e g -480 b \,c^{3} d g \right ) \sqrt {c \,x^{2}+b x +a}}{1920 c^{4}}+\frac {\left (48 a^{2} b \,c^{2} f h -32 a^{2} c^{3} e h -32 a^{2} c^{3} f g -40 a \,b^{3} c f h +48 a \,b^{2} c^{2} e h +48 a \,b^{2} c^{2} f g -64 a b \,c^{3} d h -64 a b \,c^{3} e g +128 a \,c^{4} d g +7 b^{5} f h -10 b^{4} c e h -10 b^{4} c f g +16 b^{3} c^{2} d h +16 b^{3} c^{2} e g -32 b^{2} c^{3} d g \right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{256 c^{\frac {9}{2}}}\) | \(488\) |
default | \(d g \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )+h f \left (\frac {x^{2} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{5 c}-\frac {7 b \left (\frac {x \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{4 c}-\frac {5 b \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{3 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{2 c}\right )}{8 c}-\frac {a \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{4 c}\right )}{10 c}-\frac {2 a \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{3 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{2 c}\right )}{5 c}\right )+\left (e h +f g \right ) \left (\frac {x \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{4 c}-\frac {5 b \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{3 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{2 c}\right )}{8 c}-\frac {a \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{4 c}\right )+\left (d h +e g \right ) \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{3 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{2 c}\right )\) | \(663\) |
[In]
[Out]
none
Time = 0.46 (sec) , antiderivative size = 1009, normalized size of antiderivative = 3.13 \[ \int (g+h x) \sqrt {a+b x+c x^2} \left (d+e x+f x^2\right ) \, dx=\text {Too large to display} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 993 vs. \(2 (332) = 664\).
Time = 1.04 (sec) , antiderivative size = 993, normalized size of antiderivative = 3.08 \[ \int (g+h x) \sqrt {a+b x+c x^2} \left (d+e x+f x^2\right ) \, dx=\begin {cases} \sqrt {a + b x + c x^{2}} \left (\frac {f h x^{4}}{5} + \frac {x^{3} \left (\frac {b f h}{10} + c e h + c f g\right )}{4 c} + \frac {x^{2} \left (\frac {a f h}{5} + b e h + b f g - \frac {7 b \left (\frac {b f h}{10} + c e h + c f g\right )}{8 c} + c d h + c e g\right )}{3 c} + \frac {x \left (a e h + a f g - \frac {3 a \left (\frac {b f h}{10} + c e h + c f g\right )}{4 c} + b d h + b e g - \frac {5 b \left (\frac {a f h}{5} + b e h + b f g - \frac {7 b \left (\frac {b f h}{10} + c e h + c f g\right )}{8 c} + c d h + c e g\right )}{6 c} + c d g\right )}{2 c} + \frac {a d h + a e g - \frac {2 a \left (\frac {a f h}{5} + b e h + b f g - \frac {7 b \left (\frac {b f h}{10} + c e h + c f g\right )}{8 c} + c d h + c e g\right )}{3 c} + b d g - \frac {3 b \left (a e h + a f g - \frac {3 a \left (\frac {b f h}{10} + c e h + c f g\right )}{4 c} + b d h + b e g - \frac {5 b \left (\frac {a f h}{5} + b e h + b f g - \frac {7 b \left (\frac {b f h}{10} + c e h + c f g\right )}{8 c} + c d h + c e g\right )}{6 c} + c d g\right )}{4 c}}{c}\right ) + \left (a d g - \frac {a \left (a e h + a f g - \frac {3 a \left (\frac {b f h}{10} + c e h + c f g\right )}{4 c} + b d h + b e g - \frac {5 b \left (\frac {a f h}{5} + b e h + b f g - \frac {7 b \left (\frac {b f h}{10} + c e h + c f g\right )}{8 c} + c d h + c e g\right )}{6 c} + c d g\right )}{2 c} - \frac {b \left (a d h + a e g - \frac {2 a \left (\frac {a f h}{5} + b e h + b f g - \frac {7 b \left (\frac {b f h}{10} + c e h + c f g\right )}{8 c} + c d h + c e g\right )}{3 c} + b d g - \frac {3 b \left (a e h + a f g - \frac {3 a \left (\frac {b f h}{10} + c e h + c f g\right )}{4 c} + b d h + b e g - \frac {5 b \left (\frac {a f h}{5} + b e h + b f g - \frac {7 b \left (\frac {b f h}{10} + c e h + c f g\right )}{8 c} + c d h + c e g\right )}{6 c} + c d g\right )}{4 c}\right )}{2 c}\right ) \left (\begin {cases} \frac {\log {\left (b + 2 \sqrt {c} \sqrt {a + b x + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: a - \frac {b^{2}}{4 c} \neq 0 \\\frac {\left (\frac {b}{2 c} + x\right ) \log {\left (\frac {b}{2 c} + x \right )}}{\sqrt {c \left (\frac {b}{2 c} + x\right )^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: c \neq 0 \\\frac {2 \left (\frac {f h \left (a + b x\right )^{\frac {9}{2}}}{9 b^{3}} + \frac {\left (a + b x\right )^{\frac {7}{2}} \left (- 3 a f h + b e h + b f g\right )}{7 b^{3}} + \frac {\left (a + b x\right )^{\frac {5}{2}} \cdot \left (3 a^{2} f h - 2 a b e h - 2 a b f g + b^{2} d h + b^{2} e g\right )}{5 b^{3}} + \frac {\left (a + b x\right )^{\frac {3}{2}} \left (- a^{3} f h + a^{2} b e h + a^{2} b f g - a b^{2} d h - a b^{2} e g + b^{3} d g\right )}{3 b^{3}}\right )}{b} & \text {for}\: b \neq 0 \\\sqrt {a} \left (d g x + \frac {f h x^{4}}{4} + \frac {x^{3} \left (e h + f g\right )}{3} + \frac {x^{2} \left (d h + e g\right )}{2}\right ) & \text {otherwise} \end {cases} \]
[In]
[Out]
Exception generated. \[ \int (g+h x) \sqrt {a+b x+c x^2} \left (d+e x+f x^2\right ) \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 478, normalized size of antiderivative = 1.48 \[ \int (g+h x) \sqrt {a+b x+c x^2} \left (d+e x+f x^2\right ) \, dx=\frac {1}{1920} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, f h x + \frac {10 \, c^{4} f g + 10 \, c^{4} e h + b c^{3} f h}{c^{4}}\right )} x + \frac {80 \, c^{4} e g + 10 \, b c^{3} f g + 80 \, c^{4} d h + 10 \, b c^{3} e h - 7 \, b^{2} c^{2} f h + 16 \, a c^{3} f h}{c^{4}}\right )} x + \frac {480 \, c^{4} d g + 80 \, b c^{3} e g - 50 \, b^{2} c^{2} f g + 120 \, a c^{3} f g + 80 \, b c^{3} d h - 50 \, b^{2} c^{2} e h + 120 \, a c^{3} e h + 35 \, b^{3} c f h - 116 \, a b c^{2} f h}{c^{4}}\right )} x + \frac {480 \, b c^{3} d g - 240 \, b^{2} c^{2} e g + 640 \, a c^{3} e g + 150 \, b^{3} c f g - 520 \, a b c^{2} f g - 240 \, b^{2} c^{2} d h + 640 \, a c^{3} d h + 150 \, b^{3} c e h - 520 \, a b c^{2} e h - 105 \, b^{4} f h + 460 \, a b^{2} c f h - 256 \, a^{2} c^{2} f h}{c^{4}}\right )} + \frac {{\left (32 \, b^{2} c^{3} d g - 128 \, a c^{4} d g - 16 \, b^{3} c^{2} e g + 64 \, a b c^{3} e g + 10 \, b^{4} c f g - 48 \, a b^{2} c^{2} f g + 32 \, a^{2} c^{3} f g - 16 \, b^{3} c^{2} d h + 64 \, a b c^{3} d h + 10 \, b^{4} c e h - 48 \, a b^{2} c^{2} e h + 32 \, a^{2} c^{3} e h - 7 \, b^{5} f h + 40 \, a b^{3} c f h - 48 \, a^{2} b c^{2} f h\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{256 \, c^{\frac {9}{2}}} \]
[In]
[Out]
Time = 15.10 (sec) , antiderivative size = 877, normalized size of antiderivative = 2.72 \[ \int (g+h x) \sqrt {a+b x+c x^2} \left (d+e x+f x^2\right ) \, dx=d\,g\,\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,x^2+b\,x+a}-\frac {2\,a\,f\,h\,\left (\frac {\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}+\frac {\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}\right )}{5\,c}-\frac {5\,b\,e\,h\,\left (\frac {\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}+\frac {\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}\right )}{8\,c}-\frac {5\,b\,f\,g\,\left (\frac {\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}+\frac {\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}\right )}{8\,c}+\frac {d\,h\,\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}+\frac {e\,g\,\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}+\frac {e\,h\,x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{4\,c}+\frac {f\,g\,x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{4\,c}+\frac {7\,b\,f\,h\,\left (\frac {5\,b\,\left (\frac {\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}+\frac {\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}\right )}{8\,c}-\frac {x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{4\,c}+\frac {a\,\left (\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,x^2+b\,x+a}+\frac {\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}\right )}{4\,c}\right )}{10\,c}+\frac {f\,h\,x^2\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{5\,c}-\frac {a\,e\,h\,\left (\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,x^2+b\,x+a}+\frac {\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}\right )}{4\,c}-\frac {a\,f\,g\,\left (\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,x^2+b\,x+a}+\frac {\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}\right )}{4\,c}+\frac {d\,g\,\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}+\frac {d\,h\,\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}+\frac {e\,g\,\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}} \]
[In]
[Out]