Optimal. Leaf size=1226 \[ -\frac {\left (128 c^4 f g^7-32 c^3 f g^5 h (11 b g-10 a h)+8 c^2 g h^2 \left (38 b^2 f g^4+2 a^2 h^2 \left (13 f g^2+3 d h^2\right )-a b g h \left (65 f g^2+3 d h^2\right )\right )-2 c h^3 \left (8 a^3 h^3 (2 f g-3 e h)-2 a b^2 g^2 h (34 f g+3 e h)+4 a^2 b h^2 \left (5 f g^2+6 e g h+3 d h^2\right )+b^3 \left (35 f g^4-3 d g^2 h^2\right )\right )-b h^4 (b g-2 a h) \left (16 a^2 f h^2-2 a b h (10 f g+3 e h)+b^2 \left (7 f g^2+3 h (e g+d h)\right )\right )+h \left (128 c f \left (c g^2-h (b g-a h)\right )^3+(2 c g-b h) \left (32 c^3 f g^5-8 c^2 g h \left (10 b f g^3-11 a f g^2 h+3 a d h^3\right )+2 c h^2 \left (4 a^2 h^2 (10 f g-3 e h)-6 a b h \left (11 f g^2-e g h-d h^2\right )+b^2 \left (29 f g^3+3 d g h^2\right )\right )-b h^3 \left (16 a^2 f h^2-2 a b h (10 f g+3 e h)+b^2 \left (7 f g^2+3 h (e g+d h)\right )\right )\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{128 h^5 \left (c g^2-b g h+a h^2\right )^3 (g+h x)^2}-\frac {\left (16 c^2 f g^5-2 c g h \left (13 b f g^3-10 a f g^2 h+3 b d g h^2-6 a d h^3\right )-h^2 \left (4 a^2 h^2 (2 f g-3 e h)-b^2 g \left (7 f g^2+3 h (e g+d h)\right )+2 a b h \left (f g^2+3 h (2 e g+d h)\right )\right )+h \left (4 c^2 \left (7 f g^4-3 d g^2 h^2\right )+2 c g h \left (2 a h (14 f g-3 e h)-b \left (28 f g^2-3 e g h-6 d h^2\right )\right )+h^2 \left (16 a^2 f h^2-2 a b h (22 f g-3 e h)+b^2 \left (25 f g^2-3 h (e g+d h)\right )\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 h^3 \left (c g^2-b g h+a h^2\right )^2 (g+h x)^4}-\frac {\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{5 h \left (c g^2-b g h+a h^2\right ) (g+h x)^5}+\frac {c^{3/2} f \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{h^6}-\frac {\left (256 c^5 f g^7-896 c^4 f g^5 h (b g-a h)+32 c^3 g h^2 \left (35 b^2 f g^4-70 a b f g^3 h+a^2 h^2 \left (35 f g^2-3 d h^2\right )\right )-16 c^2 h^3 \left (35 b^3 f g^4-6 a^3 h^3 (6 f g-e h)+3 a^2 b h^2 \left (35 f g^2-e g h-d h^2\right )-3 a b^2 g h \left (35 f g^2+d h^2\right )\right )+b^3 h^5 \left (16 a^2 f h^2-2 a b h (10 f g+3 e h)+b^2 \left (7 f g^2+3 h (e g+d h)\right )\right )-2 b c h^4 \left (96 a^3 f h^3-24 a^2 b h^2 (8 f g+e h)-b^3 \left (35 f g^3-3 d g h^2\right )+4 a b^2 h \left (35 f g^2+3 h (e g+d h)\right )\right )\right ) \tanh ^{-1}\left (\frac {b g-2 a h+(2 c g-b h) x}{2 \sqrt {c g^2-b g h+a h^2} \sqrt {a+b x+c x^2}}\right )}{256 h^6 \left (c g^2-b g h+a h^2\right )^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 2.46, antiderivative size = 1223, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1664, 824, 857,
635, 212, 738} \begin {gather*} -\frac {\left (f g^2-h (e g-d h)\right ) \left (c x^2+b x+a\right )^{5/2}}{5 h \left (c g^2-b h g+a h^2\right ) (g+h x)^5}-\frac {\left (16 c^2 f g^5-2 c h \left (13 b f g^3-10 a f h g^2+3 b d h^2 g-6 a d h^3\right ) g-h^2 \left (-g \left (7 f g^2+3 h (e g+d h)\right ) b^2+2 a h \left (f g^2+3 h (2 e g+d h)\right ) b+4 a^2 h^2 (2 f g-3 e h)\right )+h^2 \left (16 a^2 f h^3+4 a c g (14 f g-3 e h) h+b^2 \left (25 f g^2-3 h (e g+d h)\right ) h+c^2 \left (\frac {28 f g^4}{h}-12 d g^2 h\right )-b \left (56 c f g^3-6 c h (e g+2 d h) g+2 a h^2 (22 f g-3 e h)\right )\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{48 h^3 \left (c g^2-b h g+a h^2\right )^2 (g+h x)^4}-\frac {\left (\frac {128 c^4 f g^7}{h}-32 c^3 f (11 b g-10 a h) g^5+8 c^2 h \left (38 b^2 f g^4-a b h \left (65 f g^2+3 d h^2\right ) g+2 a^2 h^2 \left (13 f g^2+3 d h^2\right )\right ) g-b h^3 (b g-2 a h) \left (\left (7 f g^2+3 h (e g+d h)\right ) b^2-2 a h (10 f g+3 e h) b+16 a^2 f h^2\right )-2 c h^2 \left (\left (35 f g^4-3 d g^2 h^2\right ) b^3-2 a g^2 h (34 f g+3 e h) b^2+4 a^2 h^2 \left (5 f g^2+3 h (2 e g+d h)\right ) b+8 a^3 h^3 (2 f g-3 e h)\right )+\left (128 c f \left (c g^2-h (b g-a h)\right )^3+(2 c g-b h) \left (32 c^3 f g^5-8 c^2 h \left (10 b f g^3-11 a f h g^2+3 a d h^3\right ) g+2 c h^2 \left (\left (29 f g^3+3 d h^2 g\right ) b^2-6 a h \left (11 f g^2-e h g-d h^2\right ) b+4 a^2 h^2 (10 f g-3 e h)\right )-b h^3 \left (\left (7 f g^2+3 h (e g+d h)\right ) b^2-2 a h (10 f g+3 e h) b+16 a^2 f h^2\right )\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{128 h^4 \left (c g^2-b h g+a h^2\right )^3 (g+h x)^2}+\frac {c^{3/2} f \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right )}{h^6}-\frac {\left (256 c^5 f g^7-896 c^4 f h (b g-a h) g^5+32 c^3 h^2 \left (35 b^2 f g^4-70 a b f h g^3+a^2 h^2 \left (35 f g^2-3 d h^2\right )\right ) g-16 c^2 h^3 \left (35 b^3 f g^4-3 a b^2 h \left (35 f g^2+d h^2\right ) g-6 a^3 h^3 (6 f g-e h)+3 a^2 b h^2 \left (35 f g^2-e h g-d h^2\right )\right )+b^3 h^5 \left (\left (7 f g^2+3 h (e g+d h)\right ) b^2-2 a h (10 f g+3 e h) b+16 a^2 f h^2\right )-2 b c h^4 \left (-\left (\left (35 f g^3-3 d g h^2\right ) b^3\right )+4 a h \left (35 f g^2+3 h (e g+d h)\right ) b^2-24 a^2 h^2 (8 f g+e h) b+96 a^3 f h^3\right )\right ) \tanh ^{-1}\left (\frac {b g-2 a h+(2 c g-b h) x}{2 \sqrt {c g^2-b h g+a h^2} \sqrt {c x^2+b x+a}}\right )}{256 h^6 \left (c g^2-b h g+a h^2\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 635
Rule 738
Rule 824
Rule 857
Rule 1664
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{(g+h x)^6} \, dx &=-\frac {\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{5 h \left (c g^2-b g h+a h^2\right ) (g+h x)^5}-\frac {\int \frac {\left (-\frac {5}{2} \left (2 c d g-b e g-2 a f g+\frac {b f g^2}{h}-b d h+2 a e h\right )+5 f \left (b g-\frac {c g^2}{h}-a h\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{(g+h x)^5} \, dx}{5 \left (c g^2-b g h+a h^2\right )}\\ &=-\frac {\left (16 c^2 f g^5-2 c g h \left (13 b f g^3-10 a f g^2 h+3 b d g h^2-6 a d h^3\right )-h^2 \left (4 a^2 h^2 (2 f g-3 e h)-b^2 g \left (7 f g^2+3 h (e g+d h)\right )+2 a b h \left (f g^2+3 h (2 e g+d h)\right )\right )+h^2 \left (16 a^2 f h^3+4 a c g h (14 f g-3 e h)+c^2 \left (\frac {28 f g^4}{h}-12 d g^2 h\right )+b^2 h \left (25 f g^2-3 h (e g+d h)\right )-b \left (56 c f g^3-6 c g h (e g+2 d h)+2 a h^2 (22 f g-3 e h)\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 h^3 \left (c g^2-b g h+a h^2\right )^2 (g+h x)^4}-\frac {\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{5 h \left (c g^2-b g h+a h^2\right ) (g+h x)^5}+\frac {\int \frac {\left (\frac {5 \left (b^3 h^2 \left (7 f g^2+3 h (e g+d h)\right )-24 a c h \left (a h^2 (2 f g-e h)+c \left (f g^3-d g h^2\right )\right )-2 b^2 \left (a h^3 (10 f g+3 e h)+c \left (13 f g^3 h+3 d g h^3\right )\right )+4 b \left (4 c^2 f g^4+4 a^2 f h^4+a c h^2 \left (17 f g^2-3 h (e g+d h)\right )\right )\right )}{4 h}+\frac {40 c f \left (c g^2-b g h+a h^2\right )^2 x}{h}\right ) \sqrt {a+b x+c x^2}}{(g+h x)^3} \, dx}{40 h^2 \left (c g^2-b g h+a h^2\right )^2}\\ &=-\frac {\left (\frac {128 c^4 f g^7}{h}-32 c^3 f g^5 (11 b g-10 a h)+8 c^2 g h \left (38 b^2 f g^4+2 a^2 h^2 \left (13 f g^2+3 d h^2\right )-a b g h \left (65 f g^2+3 d h^2\right )\right )-b h^3 (b g-2 a h) \left (16 a^2 f h^2-2 a b h (10 f g+3 e h)+b^2 \left (7 f g^2+3 h (e g+d h)\right )\right )-2 c h^2 \left (8 a^3 h^3 (2 f g-3 e h)-2 a b^2 g^2 h (34 f g+3 e h)+b^3 \left (35 f g^4-3 d g^2 h^2\right )+4 a^2 b h^2 \left (5 f g^2+3 h (2 e g+d h)\right )\right )+\left (128 c f \left (c g^2-h (b g-a h)\right )^3+(2 c g-b h) \left (32 c^3 f g^5-8 c^2 g h \left (10 b f g^3-11 a f g^2 h+3 a d h^3\right )+2 c h^2 \left (4 a^2 h^2 (10 f g-3 e h)-6 a b h \left (11 f g^2-e g h-d h^2\right )+b^2 \left (29 f g^3+3 d g h^2\right )\right )-b h^3 \left (16 a^2 f h^2-2 a b h (10 f g+3 e h)+b^2 \left (7 f g^2+3 h (e g+d h)\right )\right )\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{128 h^4 \left (c g^2-b g h+a h^2\right )^3 (g+h x)^2}-\frac {\left (16 c^2 f g^5-2 c g h \left (13 b f g^3-10 a f g^2 h+3 b d g h^2-6 a d h^3\right )-h^2 \left (4 a^2 h^2 (2 f g-3 e h)-b^2 g \left (7 f g^2+3 h (e g+d h)\right )+2 a b h \left (f g^2+3 h (2 e g+d h)\right )\right )+h^2 \left (16 a^2 f h^3+4 a c g h (14 f g-3 e h)+c^2 \left (\frac {28 f g^4}{h}-12 d g^2 h\right )+b^2 h \left (25 f g^2-3 h (e g+d h)\right )-b \left (56 c f g^3-6 c g h (e g+2 d h)+2 a h^2 (22 f g-3 e h)\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 h^3 \left (c g^2-b g h+a h^2\right )^2 (g+h x)^4}-\frac {\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{5 h \left (c g^2-b g h+a h^2\right ) (g+h x)^5}-\frac {\int \frac {\frac {5 \left (b^4 \left (70 c f g^3 h^3-6 c d g h^5-20 a f g h^5-6 a e h^6\right )+b^5 h^4 \left (7 f g^2+3 h (e g+d h)\right )-16 b c \left (8 c^3 f g^6+44 a c^2 f g^4 h^2+12 a^3 f h^6+3 a^2 c h^4 \left (19 f g^2-e g h-d h^2\right )\right )+16 b^2 c h \left (22 c^2 f g^5+3 a^2 h^4 (8 f g+e h)+3 a c g h^2 \left (19 f g^2+d h^2\right )\right )+32 a c^2 h \left (4 c^2 f g^5+a^2 h^4 (10 f g-3 e h)+a c \left (11 f g^3 h^2-3 d g h^4\right )\right )-8 b^3 \left (38 c^2 f g^4 h^2-2 a^2 f h^6+a c h^4 \left (35 f g^2+3 h (e g+d h)\right )\right )\right )}{8 h}-\frac {160 c^2 f \left (c g^2-b g h+a h^2\right )^3 x}{h}}{(g+h x) \sqrt {a+b x+c x^2}} \, dx}{160 h^4 \left (c g^2-b g h+a h^2\right )^3}\\ &=-\frac {\left (\frac {128 c^4 f g^7}{h}-32 c^3 f g^5 (11 b g-10 a h)+8 c^2 g h \left (38 b^2 f g^4+2 a^2 h^2 \left (13 f g^2+3 d h^2\right )-a b g h \left (65 f g^2+3 d h^2\right )\right )-b h^3 (b g-2 a h) \left (16 a^2 f h^2-2 a b h (10 f g+3 e h)+b^2 \left (7 f g^2+3 h (e g+d h)\right )\right )-2 c h^2 \left (8 a^3 h^3 (2 f g-3 e h)-2 a b^2 g^2 h (34 f g+3 e h)+b^3 \left (35 f g^4-3 d g^2 h^2\right )+4 a^2 b h^2 \left (5 f g^2+3 h (2 e g+d h)\right )\right )+\left (128 c f \left (c g^2-h (b g-a h)\right )^3+(2 c g-b h) \left (32 c^3 f g^5-8 c^2 g h \left (10 b f g^3-11 a f g^2 h+3 a d h^3\right )+2 c h^2 \left (4 a^2 h^2 (10 f g-3 e h)-6 a b h \left (11 f g^2-e g h-d h^2\right )+b^2 \left (29 f g^3+3 d g h^2\right )\right )-b h^3 \left (16 a^2 f h^2-2 a b h (10 f g+3 e h)+b^2 \left (7 f g^2+3 h (e g+d h)\right )\right )\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{128 h^4 \left (c g^2-b g h+a h^2\right )^3 (g+h x)^2}-\frac {\left (16 c^2 f g^5-2 c g h \left (13 b f g^3-10 a f g^2 h+3 b d g h^2-6 a d h^3\right )-h^2 \left (4 a^2 h^2 (2 f g-3 e h)-b^2 g \left (7 f g^2+3 h (e g+d h)\right )+2 a b h \left (f g^2+3 h (2 e g+d h)\right )\right )+h^2 \left (16 a^2 f h^3+4 a c g h (14 f g-3 e h)+c^2 \left (\frac {28 f g^4}{h}-12 d g^2 h\right )+b^2 h \left (25 f g^2-3 h (e g+d h)\right )-b \left (56 c f g^3-6 c g h (e g+2 d h)+2 a h^2 (22 f g-3 e h)\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 h^3 \left (c g^2-b g h+a h^2\right )^2 (g+h x)^4}-\frac {\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{5 h \left (c g^2-b g h+a h^2\right ) (g+h x)^5}+\frac {\left (c^2 f\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{h^6}-\frac {\left (256 c^5 f g^7-896 c^4 f g^5 h (b g-a h)+32 c^3 g h^2 \left (35 b^2 f g^4-70 a b f g^3 h+a^2 h^2 \left (35 f g^2-3 d h^2\right )\right )-16 c^2 h^3 \left (35 b^3 f g^4-6 a^3 h^3 (6 f g-e h)+3 a^2 b h^2 \left (35 f g^2-e g h-d h^2\right )-3 a b^2 g h \left (35 f g^2+d h^2\right )\right )+b^3 h^5 \left (16 a^2 f h^2-2 a b h (10 f g+3 e h)+b^2 \left (7 f g^2+3 h (e g+d h)\right )\right )-2 b c h^4 \left (96 a^3 f h^3-24 a^2 b h^2 (8 f g+e h)-b^3 \left (35 f g^3-3 d g h^2\right )+4 a b^2 h \left (35 f g^2+3 h (e g+d h)\right )\right )\right ) \int \frac {1}{(g+h x) \sqrt {a+b x+c x^2}} \, dx}{256 h^6 \left (c g^2-b g h+a h^2\right )^3}\\ &=-\frac {\left (\frac {128 c^4 f g^7}{h}-32 c^3 f g^5 (11 b g-10 a h)+8 c^2 g h \left (38 b^2 f g^4+2 a^2 h^2 \left (13 f g^2+3 d h^2\right )-a b g h \left (65 f g^2+3 d h^2\right )\right )-b h^3 (b g-2 a h) \left (16 a^2 f h^2-2 a b h (10 f g+3 e h)+b^2 \left (7 f g^2+3 h (e g+d h)\right )\right )-2 c h^2 \left (8 a^3 h^3 (2 f g-3 e h)-2 a b^2 g^2 h (34 f g+3 e h)+b^3 \left (35 f g^4-3 d g^2 h^2\right )+4 a^2 b h^2 \left (5 f g^2+3 h (2 e g+d h)\right )\right )+\left (128 c f \left (c g^2-h (b g-a h)\right )^3+(2 c g-b h) \left (32 c^3 f g^5-8 c^2 g h \left (10 b f g^3-11 a f g^2 h+3 a d h^3\right )+2 c h^2 \left (4 a^2 h^2 (10 f g-3 e h)-6 a b h \left (11 f g^2-e g h-d h^2\right )+b^2 \left (29 f g^3+3 d g h^2\right )\right )-b h^3 \left (16 a^2 f h^2-2 a b h (10 f g+3 e h)+b^2 \left (7 f g^2+3 h (e g+d h)\right )\right )\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{128 h^4 \left (c g^2-b g h+a h^2\right )^3 (g+h x)^2}-\frac {\left (16 c^2 f g^5-2 c g h \left (13 b f g^3-10 a f g^2 h+3 b d g h^2-6 a d h^3\right )-h^2 \left (4 a^2 h^2 (2 f g-3 e h)-b^2 g \left (7 f g^2+3 h (e g+d h)\right )+2 a b h \left (f g^2+3 h (2 e g+d h)\right )\right )+h^2 \left (16 a^2 f h^3+4 a c g h (14 f g-3 e h)+c^2 \left (\frac {28 f g^4}{h}-12 d g^2 h\right )+b^2 h \left (25 f g^2-3 h (e g+d h)\right )-b \left (56 c f g^3-6 c g h (e g+2 d h)+2 a h^2 (22 f g-3 e h)\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 h^3 \left (c g^2-b g h+a h^2\right )^2 (g+h x)^4}-\frac {\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{5 h \left (c g^2-b g h+a h^2\right ) (g+h x)^5}+\frac {\left (2 c^2 f\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 )}{h^6}+\frac {\left (256 c^5 f g^7-896 c^4 f g^5 h (b g-a h)+32 c^3 g h^2 \left (35 b^2 f g^4-70 a b f g^3 h+a^2 h^2 \left (35 f g^2-3 d h^2\right )\right )-16 c^2 h^3 \left (35 b^3 f g^4-6 a^3 h^3 (6 f g-e h)+3 a^2 b h^2 \left (35 f g^2-e g h-d h^2\right )-3 a b^2 g h \left (35 f g^2+d h^2\right )\right )+b^3 h^5 \left (16 a^2 f h^2-2 a b h (10 f g+3 e h)+b^2 \left (7 f g^2+3 h (e g+d h)\right )\right )-2 b c h^4 \left (96 a^3 f h^3-24 a^2 b h^2 (8 f g+e h)-b^3 \left (35 f g^3-3 d g h^2\right )+4 a b^2 h \left (35 f g^2+3 h (e g+d h)\right )\right )\right ) \text {Subst}\left (\int \frac {1}{4 c g^2-4 b g h+4 a h^2-x^2} \, dx,x,\frac {-b g+2 a h-(2 c g-b h) x}{\sqrt {a+b x+c x^2}}\right )}{128 h^6 \left (c g^2-b g h+a h^2\right )^3}\\ &=-\frac {\left (\frac {128 c^4 f g^7}{h}-32 c^3 f g^5 (11 b g-10 a h)+8 c^2 g h \left (38 b^2 f g^4+2 a^2 h^2 \left (13 f g^2+3 d h^2\right )-a b g h \left (65 f g^2+3 d h^2\right )\right )-b h^3 (b g-2 a h) \left (16 a^2 f h^2-2 a b h (10 f g+3 e h)+b^2 \left (7 f g^2+3 h (e g+d h)\right )\right )-2 c h^2 \left (8 a^3 h^3 (2 f g-3 e h)-2 a b^2 g^2 h (34 f g+3 e h)+b^3 \left (35 f g^4-3 d g^2 h^2\right )+4 a^2 b h^2 \left (5 f g^2+3 h (2 e g+d h)\right )\right )+\left (128 c f \left (c g^2-h (b g-a h)\right )^3+(2 c g-b h) \left (32 c^3 f g^5-8 c^2 g h \left (10 b f g^3-11 a f g^2 h+3 a d h^3\right )+2 c h^2 \left (4 a^2 h^2 (10 f g-3 e h)-6 a b h \left (11 f g^2-e g h-d h^2\right )+b^2 \left (29 f g^3+3 d g h^2\right )\right )-b h^3 \left (16 a^2 f h^2-2 a b h (10 f g+3 e h)+b^2 \left (7 f g^2+3 h (e g+d h)\right )\right )\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{128 h^4 \left (c g^2-b g h+a h^2\right )^3 (g+h x)^2}-\frac {\left (16 c^2 f g^5-2 c g h \left (13 b f g^3-10 a f g^2 h+3 b d g h^2-6 a d h^3\right )-h^2 \left (4 a^2 h^2 (2 f g-3 e h)-b^2 g \left (7 f g^2+3 h (e g+d h)\right )+2 a b h \left (f g^2+3 h (2 e g+d h)\right )\right )+h^2 \left (16 a^2 f h^3+4 a c g h (14 f g-3 e h)+c^2 \left (\frac {28 f g^4}{h}-12 d g^2 h\right )+b^2 h \left (25 f g^2-3 h (e g+d h)\right )-b \left (56 c f g^3-6 c g h (e g+2 d h)+2 a h^2 (22 f g-3 e h)\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 h^3 \left (c g^2-b g h+a h^2\right )^2 (g+h x)^4}-\frac {\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{5 h \left (c g^2-b g h+a h^2\right ) (g+h x)^5}+\frac {c^{3/2} f \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{h^6}-\frac {\left (256 c^5 f g^7-896 c^4 f g^5 h (b g-a h)+32 c^3 g h^2 \left (35 b^2 f g^4-70 a b f g^3 h+a^2 h^2 \left (35 f g^2-3 d h^2\right )\right )-16 c^2 h^3 \left (35 b^3 f g^4-6 a^3 h^3 (6 f g-e h)+3 a^2 b h^2 \left (35 f g^2-e g h-d h^2\right )-3 a b^2 g h \left (35 f g^2+d h^2\right )\right )+b^3 h^5 \left (16 a^2 f h^2-2 a b h (10 f g+3 e h)+b^2 \left (7 f g^2+3 h (e g+d h)\right )\right )-2 b c h^4 \left (96 a^3 f h^3-24 a^2 b h^2 (8 f g+e h)-b^3 \left (35 f g^3-3 d g h^2\right )+4 a b^2 h \left (35 f g^2+3 h (e g+d h)\right )\right )\right ) \tanh ^{-1}\left (\frac {b g-2 a h+(2 c g-b h) x}{2 \sqrt {c g^2-b g h+a h^2} \sqrt {a+b x+c x^2}}\right )}{256 h^6 \left (c g^2-b g h+a h^2\right )^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 16.74, size = 2229, normalized size = 1.82 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(13371\) vs.
\(2(1196)=2392\).
time = 0.14, size = 13372, normalized size = 10.91
method | result | size |
default | \(\text {Expression too large to display}\) | \(13372\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x + c x^{2}\right )^{\frac {3}{2}} \left (d + e x + f x^{2}\right )}{\left (g + h x\right )^{6}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}\,\left (f\,x^2+e\,x+d\right )}{{\left (g+h\,x\right )}^6} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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