Integrand size = 29, antiderivative size = 507 \[ \int \frac {\left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{(g+h x)^6} \, dx=-\frac {c \left (8 c^3 f g^7+20 a c^2 f g^5 h^2-a^3 h^6 (2 f g-3 e h)+a^2 c g h^4 \left (13 f g^2+3 d h^2\right )+h \left (12 c^3 f g^6+8 a^3 f h^6+a^2 c g h^4 (34 f g-3 e h)+a c^2 g^2 h^2 \left (35 f g^2-3 d h^2\right )\right ) x\right ) \sqrt {a+c x^2}}{8 h^5 \left (c g^2+a h^2\right )^3 (g+h x)^2}-\frac {\left (4 c^2 f g^5-a^2 h^4 (2 f g-3 e h)+a c g h^2 \left (5 f g^2+3 d h^2\right )+h \left (4 a^2 f h^4+a c g h^2 (14 f g-3 e h)+c^2 \left (7 f g^4-3 d g^2 h^2\right )\right ) x\right ) \left (a+c x^2\right )^{3/2}}{12 h^3 \left (c g^2+a h^2\right )^2 (g+h x)^4}-\frac {\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{5 h \left (c g^2+a h^2\right ) (g+h x)^5}+\frac {c^{3/2} f \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{h^6}+\frac {c^2 \left (8 c^3 f g^7+28 a c^2 f g^5 h^2+3 a^3 h^6 (6 f g-e h)+a^2 c g h^4 \left (35 f g^2-3 d h^2\right )\right ) \text {arctanh}\left (\frac {a h-c g x}{\sqrt {c g^2+a h^2} \sqrt {a+c x^2}}\right )}{8 h^6 \left (c g^2+a h^2\right )^{7/2}} \]
-1/12*(4*c^2*f*g^5-a^2*h^4*(-3*e*h+2*f*g)+a*c*g*h^2*(3*d*h^2+5*f*g^2)+h*(4 *a^2*f*h^4+a*c*g*h^2*(-3*e*h+14*f*g)+c^2*(-3*d*g^2*h^2+7*f*g^4))*x)*(c*x^2 +a)^(3/2)/h^3/(a*h^2+c*g^2)^2/(h*x+g)^4-1/5*(d*h^2-e*g*h+f*g^2)*(c*x^2+a)^ (5/2)/h/(a*h^2+c*g^2)/(h*x+g)^5+c^(3/2)*f*arctanh(x*c^(1/2)/(c*x^2+a)^(1/2 ))/h^6+1/8*c^2*(8*c^3*f*g^7+28*a*c^2*f*g^5*h^2+3*a^3*h^6*(-e*h+6*f*g)+a^2* c*g*h^4*(-3*d*h^2+35*f*g^2))*arctanh((-c*g*x+a*h)/(a*h^2+c*g^2)^(1/2)/(c*x ^2+a)^(1/2))/h^6/(a*h^2+c*g^2)^(7/2)-1/8*c*(8*c^3*f*g^7+20*a*c^2*f*g^5*h^2 -a^3*h^6*(-3*e*h+2*f*g)+a^2*c*g*h^4*(3*d*h^2+13*f*g^2)+h*(12*c^3*f*g^6+8*a ^3*f*h^6+a^2*c*g*h^4*(-3*e*h+34*f*g)+a*c^2*g^2*h^2*(-3*d*h^2+35*f*g^2))*x) *(c*x^2+a)^(1/2)/h^5/(a*h^2+c*g^2)^3/(h*x+g)^2
Time = 11.29 (sec) , antiderivative size = 639, normalized size of antiderivative = 1.26 \[ \int \frac {\left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{(g+h x)^6} \, dx=\frac {-\frac {h \sqrt {a+c x^2} \left (24 \left (c g^2+a h^2\right )^4 \left (f g^2+h (-e g+d h)\right )-6 \left (c g^2+a h^2\right )^3 \left (21 c f g^3+c g h (-16 e g+11 d h)-5 a h^2 (-2 f g+e h)\right ) (g+h x)+2 \left (c g^2+a h^2\right )^2 \left (20 a^2 f h^4+c^2 \left (137 f g^4+9 g^2 h (-8 e g+3 d h)\right )+a c h^2 \left (154 f g^2+3 h (-23 e g+8 d h)\right )\right ) (g+h x)^2-c \left (c g^2+a h^2\right ) \left (5 a^2 h^4 (58 f g-15 e h)+c^2 \left (326 f g^5+6 g^3 h (-16 e g+d h)\right )+a c g h^2 \left (631 f g^2+3 h (-62 e g+7 d h)\right )\right ) (g+h x)^3+c \left (160 a^3 f h^6+c^3 \left (274 f g^6-6 g^4 h (4 e g+d h)\right )+3 a^2 c h^4 \left (238 f g^2+h (-33 e g+8 d h)\right )+3 a c^2 g^2 h^2 \left (261 f g^2-h (26 e g+9 d h)\right )\right ) (g+h x)^4\right )}{\left (c g^2+a h^2\right )^3 (g+h x)^5}-\frac {15 c^2 \left (8 c^3 f g^7+28 a c^2 f g^5 h^2-3 a^3 h^6 (-6 f g+e h)+a^2 c g h^4 \left (35 f g^2-3 d h^2\right )\right ) \log (g+h x)}{\left (c g^2+a h^2\right )^{7/2}}+120 c^{3/2} f \log \left (c x+\sqrt {c} \sqrt {a+c x^2}\right )+\frac {15 c^2 \left (8 c^3 f g^7+28 a c^2 f g^5 h^2-3 a^3 h^6 (-6 f g+e h)+a^2 c g h^4 \left (35 f g^2-3 d h^2\right )\right ) \log \left (a h-c g x+\sqrt {c g^2+a h^2} \sqrt {a+c x^2}\right )}{\left (c g^2+a h^2\right )^{7/2}}}{120 h^6} \]
(-((h*Sqrt[a + c*x^2]*(24*(c*g^2 + a*h^2)^4*(f*g^2 + h*(-(e*g) + d*h)) - 6 *(c*g^2 + a*h^2)^3*(21*c*f*g^3 + c*g*h*(-16*e*g + 11*d*h) - 5*a*h^2*(-2*f* g + e*h))*(g + h*x) + 2*(c*g^2 + a*h^2)^2*(20*a^2*f*h^4 + c^2*(137*f*g^4 + 9*g^2*h*(-8*e*g + 3*d*h)) + a*c*h^2*(154*f*g^2 + 3*h*(-23*e*g + 8*d*h)))* (g + h*x)^2 - c*(c*g^2 + a*h^2)*(5*a^2*h^4*(58*f*g - 15*e*h) + c^2*(326*f* g^5 + 6*g^3*h*(-16*e*g + d*h)) + a*c*g*h^2*(631*f*g^2 + 3*h*(-62*e*g + 7*d *h)))*(g + h*x)^3 + c*(160*a^3*f*h^6 + c^3*(274*f*g^6 - 6*g^4*h*(4*e*g + d *h)) + 3*a^2*c*h^4*(238*f*g^2 + h*(-33*e*g + 8*d*h)) + 3*a*c^2*g^2*h^2*(26 1*f*g^2 - h*(26*e*g + 9*d*h)))*(g + h*x)^4))/((c*g^2 + a*h^2)^3*(g + h*x)^ 5)) - (15*c^2*(8*c^3*f*g^7 + 28*a*c^2*f*g^5*h^2 - 3*a^3*h^6*(-6*f*g + e*h) + a^2*c*g*h^4*(35*f*g^2 - 3*d*h^2))*Log[g + h*x])/(c*g^2 + a*h^2)^(7/2) + 120*c^(3/2)*f*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]] + (15*c^2*(8*c^3*f*g^7 + 28*a*c^2*f*g^5*h^2 - 3*a^3*h^6*(-6*f*g + e*h) + a^2*c*g*h^4*(35*f*g^2 - 3 *d*h^2))*Log[a*h - c*g*x + Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2]])/(c*g^2 + a*h^2)^(7/2))/(120*h^6)
Time = 0.92 (sec) , antiderivative size = 574, normalized size of antiderivative = 1.13, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {2182, 27, 680, 27, 680, 27, 719, 224, 219, 488, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{(g+h x)^6} \, dx\) |
| \(\Big \downarrow \) 2182 |
| \(\displaystyle -\frac {\int -\frac {5 \left (c d g-a f g+a e h+f \left (\frac {c g^2}{h}+a h\right ) x\right ) \left (c x^2+a\right )^{3/2}}{(g+h x)^5}dx}{5 \left (a h^2+c g^2\right )}-\frac {\left (a+c x^2\right )^{5/2} \left (d h^2-e g h+f g^2\right )}{5 h (g+h x)^5 \left (a h^2+c g^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\left (c d g-a f g+a e h+f \left (\frac {c g^2}{h}+a h\right ) x\right ) \left (c x^2+a\right )^{3/2}}{(g+h x)^5}dx}{a h^2+c g^2}-\frac {\left (a+c x^2\right )^{5/2} \left (d h^2-e g h+f g^2\right )}{5 h (g+h x)^5 \left (a h^2+c g^2\right )}\) |
| \(\Big \downarrow \) 680 |
| \(\displaystyle \frac {-\frac {\int \frac {2 c \left (3 a h \left (a (2 f g-e h) h^2+c \left (f g^3-d g h^2\right )\right )-4 f \left (c g^2+a h^2\right )^2 x\right ) \sqrt {c x^2+a}}{h (g+h x)^3}dx}{8 h^2 \left (a h^2+c g^2\right )}-\frac {\left (a+c x^2\right )^{3/2} \left (x \left (4 a^2 f h^4+a c g h^2 (14 f g-3 e h)+c^2 \left (7 f g^4-3 d g^2 h^2\right )\right )-a^2 h^3 (2 f g-3 e h)+a c g h \left (3 d h^2+5 f g^2\right )+\frac {4 c^2 f g^5}{h}\right )}{12 h^2 (g+h x)^4 \left (a h^2+c g^2\right )}}{a h^2+c g^2}-\frac {\left (a+c x^2\right )^{5/2} \left (d h^2-e g h+f g^2\right )}{5 h (g+h x)^5 \left (a h^2+c g^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {c \int \frac {\left (3 a h \left (a (2 f g-e h) h^2+c \left (f g^3-d g h^2\right )\right )-4 f \left (c g^2+a h^2\right )^2 x\right ) \sqrt {c x^2+a}}{(g+h x)^3}dx}{4 h^3 \left (a h^2+c g^2\right )}-\frac {\left (a+c x^2\right )^{3/2} \left (x \left (4 a^2 f h^4+a c g h^2 (14 f g-3 e h)+c^2 \left (7 f g^4-3 d g^2 h^2\right )\right )-a^2 h^3 (2 f g-3 e h)+a c g h \left (3 d h^2+5 f g^2\right )+\frac {4 c^2 f g^5}{h}\right )}{12 h^2 (g+h x)^4 \left (a h^2+c g^2\right )}}{a h^2+c g^2}-\frac {\left (a+c x^2\right )^{5/2} \left (d h^2-e g h+f g^2\right )}{5 h (g+h x)^5 \left (a h^2+c g^2\right )}\) |
| \(\Big \downarrow \) 680 |
| \(\displaystyle \frac {-\frac {c \left (\frac {\sqrt {a+c x^2} \left (-a^3 h^6 (2 f g-3 e h)+a^2 c g h^4 \left (3 d h^2+13 f g^2\right )+h x \left (8 a^3 f h^6+a^2 c g h^4 (34 f g-3 e h)+a c^2 g^2 h^2 \left (35 f g^2-3 d h^2\right )+12 c^3 f g^6\right )+20 a c^2 f g^5 h^2+8 c^3 f g^7\right )}{2 h^2 (g+h x)^2 \left (a h^2+c g^2\right )}-\frac {\int -\frac {2 c \left (a h \left (4 c^2 f g^5+a c h^2 \left (11 f g^2-3 d h^2\right ) g+a^2 h^4 (10 f g-3 e h)\right )-8 f \left (c g^2+a h^2\right )^3 x\right )}{(g+h x) \sqrt {c x^2+a}}dx}{4 h^2 \left (a h^2+c g^2\right )}\right )}{4 h^3 \left (a h^2+c g^2\right )}-\frac {\left (a+c x^2\right )^{3/2} \left (x \left (4 a^2 f h^4+a c g h^2 (14 f g-3 e h)+c^2 \left (7 f g^4-3 d g^2 h^2\right )\right )-a^2 h^3 (2 f g-3 e h)+a c g h \left (3 d h^2+5 f g^2\right )+\frac {4 c^2 f g^5}{h}\right )}{12 h^2 (g+h x)^4 \left (a h^2+c g^2\right )}}{a h^2+c g^2}-\frac {\left (a+c x^2\right )^{5/2} \left (d h^2-e g h+f g^2\right )}{5 h (g+h x)^5 \left (a h^2+c g^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {c \left (\frac {c \int \frac {a h \left (4 c^2 f g^5+a c h^2 \left (11 f g^2-3 d h^2\right ) g+a^2 h^4 (10 f g-3 e h)\right )-8 f \left (c g^2+a h^2\right )^3 x}{(g+h x) \sqrt {c x^2+a}}dx}{2 h^2 \left (a h^2+c g^2\right )}+\frac {\sqrt {a+c x^2} \left (-a^3 h^6 (2 f g-3 e h)+a^2 c g h^4 \left (3 d h^2+13 f g^2\right )+h x \left (8 a^3 f h^6+a^2 c g h^4 (34 f g-3 e h)+a c^2 g^2 h^2 \left (35 f g^2-3 d h^2\right )+12 c^3 f g^6\right )+20 a c^2 f g^5 h^2+8 c^3 f g^7\right )}{2 h^2 (g+h x)^2 \left (a h^2+c g^2\right )}\right )}{4 h^3 \left (a h^2+c g^2\right )}-\frac {\left (a+c x^2\right )^{3/2} \left (x \left (4 a^2 f h^4+a c g h^2 (14 f g-3 e h)+c^2 \left (7 f g^4-3 d g^2 h^2\right )\right )-a^2 h^3 (2 f g-3 e h)+a c g h \left (3 d h^2+5 f g^2\right )+\frac {4 c^2 f g^5}{h}\right )}{12 h^2 (g+h x)^4 \left (a h^2+c g^2\right )}}{a h^2+c g^2}-\frac {\left (a+c x^2\right )^{5/2} \left (d h^2-e g h+f g^2\right )}{5 h (g+h x)^5 \left (a h^2+c g^2\right )}\) |
| \(\Big \downarrow \) 719 |
| \(\displaystyle \frac {-\frac {c \left (\frac {c \left (\frac {\left (3 a^3 h^6 (6 f g-e h)+a^2 c g h^4 \left (35 f g^2-3 d h^2\right )+28 a c^2 f g^5 h^2+8 c^3 f g^7\right ) \int \frac {1}{(g+h x) \sqrt {c x^2+a}}dx}{h}-\frac {8 f \left (a h^2+c g^2\right )^3 \int \frac {1}{\sqrt {c x^2+a}}dx}{h}\right )}{2 h^2 \left (a h^2+c g^2\right )}+\frac {\sqrt {a+c x^2} \left (-a^3 h^6 (2 f g-3 e h)+a^2 c g h^4 \left (3 d h^2+13 f g^2\right )+h x \left (8 a^3 f h^6+a^2 c g h^4 (34 f g-3 e h)+a c^2 g^2 h^2 \left (35 f g^2-3 d h^2\right )+12 c^3 f g^6\right )+20 a c^2 f g^5 h^2+8 c^3 f g^7\right )}{2 h^2 (g+h x)^2 \left (a h^2+c g^2\right )}\right )}{4 h^3 \left (a h^2+c g^2\right )}-\frac {\left (a+c x^2\right )^{3/2} \left (x \left (4 a^2 f h^4+a c g h^2 (14 f g-3 e h)+c^2 \left (7 f g^4-3 d g^2 h^2\right )\right )-a^2 h^3 (2 f g-3 e h)+a c g h \left (3 d h^2+5 f g^2\right )+\frac {4 c^2 f g^5}{h}\right )}{12 h^2 (g+h x)^4 \left (a h^2+c g^2\right )}}{a h^2+c g^2}-\frac {\left (a+c x^2\right )^{5/2} \left (d h^2-e g h+f g^2\right )}{5 h (g+h x)^5 \left (a h^2+c g^2\right )}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {-\frac {c \left (\frac {c \left (\frac {\left (3 a^3 h^6 (6 f g-e h)+a^2 c g h^4 \left (35 f g^2-3 d h^2\right )+28 a c^2 f g^5 h^2+8 c^3 f g^7\right ) \int \frac {1}{(g+h x) \sqrt {c x^2+a}}dx}{h}-\frac {8 f \left (a h^2+c g^2\right )^3 \int \frac {1}{1-\frac {c x^2}{c x^2+a}}d\frac {x}{\sqrt {c x^2+a}}}{h}\right )}{2 h^2 \left (a h^2+c g^2\right )}+\frac {\sqrt {a+c x^2} \left (-a^3 h^6 (2 f g-3 e h)+a^2 c g h^4 \left (3 d h^2+13 f g^2\right )+h x \left (8 a^3 f h^6+a^2 c g h^4 (34 f g-3 e h)+a c^2 g^2 h^2 \left (35 f g^2-3 d h^2\right )+12 c^3 f g^6\right )+20 a c^2 f g^5 h^2+8 c^3 f g^7\right )}{2 h^2 (g+h x)^2 \left (a h^2+c g^2\right )}\right )}{4 h^3 \left (a h^2+c g^2\right )}-\frac {\left (a+c x^2\right )^{3/2} \left (x \left (4 a^2 f h^4+a c g h^2 (14 f g-3 e h)+c^2 \left (7 f g^4-3 d g^2 h^2\right )\right )-a^2 h^3 (2 f g-3 e h)+a c g h \left (3 d h^2+5 f g^2\right )+\frac {4 c^2 f g^5}{h}\right )}{12 h^2 (g+h x)^4 \left (a h^2+c g^2\right )}}{a h^2+c g^2}-\frac {\left (a+c x^2\right )^{5/2} \left (d h^2-e g h+f g^2\right )}{5 h (g+h x)^5 \left (a h^2+c g^2\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {-\frac {c \left (\frac {c \left (\frac {\left (3 a^3 h^6 (6 f g-e h)+a^2 c g h^4 \left (35 f g^2-3 d h^2\right )+28 a c^2 f g^5 h^2+8 c^3 f g^7\right ) \int \frac {1}{(g+h x) \sqrt {c x^2+a}}dx}{h}-\frac {8 f \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (a h^2+c g^2\right )^3}{\sqrt {c} h}\right )}{2 h^2 \left (a h^2+c g^2\right )}+\frac {\sqrt {a+c x^2} \left (-a^3 h^6 (2 f g-3 e h)+a^2 c g h^4 \left (3 d h^2+13 f g^2\right )+h x \left (8 a^3 f h^6+a^2 c g h^4 (34 f g-3 e h)+a c^2 g^2 h^2 \left (35 f g^2-3 d h^2\right )+12 c^3 f g^6\right )+20 a c^2 f g^5 h^2+8 c^3 f g^7\right )}{2 h^2 (g+h x)^2 \left (a h^2+c g^2\right )}\right )}{4 h^3 \left (a h^2+c g^2\right )}-\frac {\left (a+c x^2\right )^{3/2} \left (x \left (4 a^2 f h^4+a c g h^2 (14 f g-3 e h)+c^2 \left (7 f g^4-3 d g^2 h^2\right )\right )-a^2 h^3 (2 f g-3 e h)+a c g h \left (3 d h^2+5 f g^2\right )+\frac {4 c^2 f g^5}{h}\right )}{12 h^2 (g+h x)^4 \left (a h^2+c g^2\right )}}{a h^2+c g^2}-\frac {\left (a+c x^2\right )^{5/2} \left (d h^2-e g h+f g^2\right )}{5 h (g+h x)^5 \left (a h^2+c g^2\right )}\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle \frac {-\frac {c \left (\frac {c \left (-\frac {\left (3 a^3 h^6 (6 f g-e h)+a^2 c g h^4 \left (35 f g^2-3 d h^2\right )+28 a c^2 f g^5 h^2+8 c^3 f g^7\right ) \int \frac {1}{c g^2+a h^2-\frac {(a h-c g x)^2}{c x^2+a}}d\frac {a h-c g x}{\sqrt {c x^2+a}}}{h}-\frac {8 f \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (a h^2+c g^2\right )^3}{\sqrt {c} h}\right )}{2 h^2 \left (a h^2+c g^2\right )}+\frac {\sqrt {a+c x^2} \left (-a^3 h^6 (2 f g-3 e h)+a^2 c g h^4 \left (3 d h^2+13 f g^2\right )+h x \left (8 a^3 f h^6+a^2 c g h^4 (34 f g-3 e h)+a c^2 g^2 h^2 \left (35 f g^2-3 d h^2\right )+12 c^3 f g^6\right )+20 a c^2 f g^5 h^2+8 c^3 f g^7\right )}{2 h^2 (g+h x)^2 \left (a h^2+c g^2\right )}\right )}{4 h^3 \left (a h^2+c g^2\right )}-\frac {\left (a+c x^2\right )^{3/2} \left (x \left (4 a^2 f h^4+a c g h^2 (14 f g-3 e h)+c^2 \left (7 f g^4-3 d g^2 h^2\right )\right )-a^2 h^3 (2 f g-3 e h)+a c g h \left (3 d h^2+5 f g^2\right )+\frac {4 c^2 f g^5}{h}\right )}{12 h^2 (g+h x)^4 \left (a h^2+c g^2\right )}}{a h^2+c g^2}-\frac {\left (a+c x^2\right )^{5/2} \left (d h^2-e g h+f g^2\right )}{5 h (g+h x)^5 \left (a h^2+c g^2\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {-\frac {\left (a+c x^2\right )^{3/2} \left (x \left (4 a^2 f h^4+a c g h^2 (14 f g-3 e h)+c^2 \left (7 f g^4-3 d g^2 h^2\right )\right )-a^2 h^3 (2 f g-3 e h)+a c g h \left (3 d h^2+5 f g^2\right )+\frac {4 c^2 f g^5}{h}\right )}{12 h^2 (g+h x)^4 \left (a h^2+c g^2\right )}-\frac {c \left (\frac {c \left (-\frac {\text {arctanh}\left (\frac {a h-c g x}{\sqrt {a+c x^2} \sqrt {a h^2+c g^2}}\right ) \left (3 a^3 h^6 (6 f g-e h)+a^2 c g h^4 \left (35 f g^2-3 d h^2\right )+28 a c^2 f g^5 h^2+8 c^3 f g^7\right )}{h \sqrt {a h^2+c g^2}}-\frac {8 f \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (a h^2+c g^2\right )^3}{\sqrt {c} h}\right )}{2 h^2 \left (a h^2+c g^2\right )}+\frac {\sqrt {a+c x^2} \left (-a^3 h^6 (2 f g-3 e h)+a^2 c g h^4 \left (3 d h^2+13 f g^2\right )+h x \left (8 a^3 f h^6+a^2 c g h^4 (34 f g-3 e h)+a c^2 g^2 h^2 \left (35 f g^2-3 d h^2\right )+12 c^3 f g^6\right )+20 a c^2 f g^5 h^2+8 c^3 f g^7\right )}{2 h^2 (g+h x)^2 \left (a h^2+c g^2\right )}\right )}{4 h^3 \left (a h^2+c g^2\right )}}{a h^2+c g^2}-\frac {\left (a+c x^2\right )^{5/2} \left (d h^2-e g h+f g^2\right )}{5 h (g+h x)^5 \left (a h^2+c g^2\right )}\) |
-1/5*((f*g^2 - e*g*h + d*h^2)*(a + c*x^2)^(5/2))/(h*(c*g^2 + a*h^2)*(g + h *x)^5) + (-1/12*(((4*c^2*f*g^5)/h - a^2*h^3*(2*f*g - 3*e*h) + a*c*g*h*(5*f *g^2 + 3*d*h^2) + (4*a^2*f*h^4 + a*c*g*h^2*(14*f*g - 3*e*h) + c^2*(7*f*g^4 - 3*d*g^2*h^2))*x)*(a + c*x^2)^(3/2))/(h^2*(c*g^2 + a*h^2)*(g + h*x)^4) - (c*(((8*c^3*f*g^7 + 20*a*c^2*f*g^5*h^2 - a^3*h^6*(2*f*g - 3*e*h) + a^2*c* g*h^4*(13*f*g^2 + 3*d*h^2) + h*(12*c^3*f*g^6 + 8*a^3*f*h^6 + a^2*c*g*h^4*( 34*f*g - 3*e*h) + a*c^2*g^2*h^2*(35*f*g^2 - 3*d*h^2))*x)*Sqrt[a + c*x^2])/ (2*h^2*(c*g^2 + a*h^2)*(g + h*x)^2) + (c*((-8*f*(c*g^2 + a*h^2)^3*ArcTanh[ (Sqrt[c]*x)/Sqrt[a + c*x^2]])/(Sqrt[c]*h) - ((8*c^3*f*g^7 + 28*a*c^2*f*g^5 *h^2 + 3*a^3*h^6*(6*f*g - e*h) + a^2*c*g*h^4*(35*f*g^2 - 3*d*h^2))*ArcTanh [(a*h - c*g*x)/(Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2])])/(h*Sqrt[c*g^2 + a*h ^2])))/(2*h^2*(c*g^2 + a*h^2))))/(4*h^3*(c*g^2 + a*h^2)))/(c*g^2 + a*h^2)
3.1.97.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ [{a, b, c, d}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*((a + c*x^2)^p/(e^2*(m + 1)*(m + 2)*(c*d^2 + a*e^2)))*((d*g - e*f*(m + 2))*(c*d^2 + a*e^2) - 2*c*d^2*p*(e* f - d*g) - e*(g*(m + 1)*(c*d^2 + a*e^2) + 2*c*d*p*(e*f - d*g))*x), x] - Sim p[p/(e^2*(m + 1)*(m + 2)*(c*d^2 + a*e^2)) Int[(d + e*x)^(m + 2)*(a + c*x^ 2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) - c*(2*c*d*(d*g*(2*p + 1) - e*f *(m + 2*p + 2)) - 2*a*e^2*g*(m + 1))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] && !ILtQ[m + 2*p + 3 , 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[e*R*(d + e*x)^(m + 1)*((a + b*x^2)^(p + 1)/((m + 1)*(b* d^2 + a*e^2))), x] + Simp[1/((m + 1)*(b*d^2 + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[(m + 1)*(b*d^2 + a*e^2)*Qx + b*d*R*(m + 1) - b *e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, d, e, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] && LtQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(10597\) vs. \(2(481)=962\).
Time = 0.75 (sec) , antiderivative size = 10598, normalized size of antiderivative = 20.90
Timed out. \[ \int \frac {\left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{(g+h x)^6} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{(g+h x)^6} \, dx=\int \frac {\left (a + c x^{2}\right )^{\frac {3}{2}} \left (d + e x + f x^{2}\right )}{\left (g + h x\right )^{6}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 6650 vs. \(2 (482) = 964\).
Time = 0.49 (sec) , antiderivative size = 6650, normalized size of antiderivative = 13.12 \[ \int \frac {\left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{(g+h x)^6} \, dx=\text {Too large to display} \]
3/8*sqrt(c*x^2 + a)*c^5*f*g^7/(c^4*g^8*h^5 + 4*a*c^3*g^6*h^7 + 6*a^2*c^2*g ^4*h^9 + 4*a^3*c*g^2*h^11 + a^4*h^13) - 3/8*sqrt(c*x^2 + a)*c^5*f*g^6*x/(c ^4*g^8*h^4 + 4*a*c^3*g^6*h^6 + 6*a^2*c^2*g^4*h^8 + 4*a^3*c*g^2*h^10 + a^4* h^12) - 3/8*sqrt(c*x^2 + a)*c^5*e*g^6/(c^4*g^8*h^4 + 4*a*c^3*g^6*h^6 + 6*a ^2*c^2*g^4*h^8 + 4*a^3*c*g^2*h^10 + a^4*h^12) + 1/8*(c*x^2 + a)^(3/2)*c^4* f*g^6/(c^4*g^8*h^4*x + 4*a*c^3*g^6*h^6*x + 6*a^2*c^2*g^4*h^8*x + 4*a^3*c*g ^2*h^10*x + a^4*h^12*x + c^4*g^9*h^3 + 4*a*c^3*g^7*h^5 + 6*a^2*c^2*g^5*h^7 + 4*a^3*c*g^3*h^9 + a^4*g*h^11) + 3/8*sqrt(c*x^2 + a)*c^5*e*g^5*x/(c^4*g^ 8*h^3 + 4*a*c^3*g^6*h^5 + 6*a^2*c^2*g^4*h^7 + 4*a^3*c*g^2*h^9 + a^4*h^11) + 3/8*sqrt(c*x^2 + a)*c^5*d*g^5/(c^4*g^8*h^3 + 4*a*c^3*g^6*h^5 + 6*a^2*c^2 *g^4*h^7 + 4*a^3*c*g^2*h^9 + a^4*h^11) - 1/8*(c*x^2 + a)^(3/2)*c^4*e*g^5/( c^4*g^8*h^3*x + 4*a*c^3*g^6*h^5*x + 6*a^2*c^2*g^4*h^7*x + 4*a^3*c*g^2*h^9* x + a^4*h^11*x + c^4*g^9*h^2 + 4*a*c^3*g^7*h^4 + 6*a^2*c^2*g^5*h^6 + 4*a^3 *c*g^3*h^8 + a^4*g*h^10) - 1/8*(c*x^2 + a)^(5/2)*c^3*f*g^5/(c^4*g^8*h^3*x^ 2 + 4*a*c^3*g^6*h^5*x^2 + 6*a^2*c^2*g^4*h^7*x^2 + 4*a^3*c*g^2*h^9*x^2 + a^ 4*h^11*x^2 + 2*c^4*g^9*h^2*x + 8*a*c^3*g^7*h^4*x + 12*a^2*c^2*g^5*h^6*x + 8*a^3*c*g^3*h^8*x + 2*a^4*g*h^10*x + c^4*g^10*h + 4*a*c^3*g^8*h^3 + 6*a^2* c^2*g^6*h^5 + 4*a^3*c*g^4*h^7 + a^4*g^2*h^9) + 1/8*(c*x^2 + a)^(3/2)*c^4*f *g^5/(c^4*g^8*h^3 + 4*a*c^3*g^6*h^5 + 6*a^2*c^2*g^4*h^7 + 4*a^3*c*g^2*h^9 + a^4*h^11) - 3/8*sqrt(c*x^2 + a)*c^5*d*g^4*x/(c^4*g^8*h^2 + 4*a*c^3*g^...
Leaf count of result is larger than twice the leaf count of optimal. 4363 vs. \(2 (482) = 964\).
Time = 0.55 (sec) , antiderivative size = 4363, normalized size of antiderivative = 8.61 \[ \int \frac {\left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{(g+h x)^6} \, dx=\text {Too large to display} \]
-1/4*(8*c^5*f*g^7 + 28*a*c^4*f*g^5*h^2 + 35*a^2*c^3*f*g^3*h^4 - 3*a^2*c^3* d*g*h^6 + 18*a^3*c^2*f*g*h^6 - 3*a^3*c^2*e*h^7)*arctan(-((sqrt(c)*x - sqrt (c*x^2 + a))*h + sqrt(c)*g)/sqrt(-c*g^2 - a*h^2))/((c^3*g^6*h^6 + 3*a*c^2* g^4*h^8 + 3*a^2*c*g^2*h^10 + a^3*h^12)*sqrt(-c*g^2 - a*h^2)) - c^(3/2)*f*l og(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/h^6 - 1/60*(600*(sqrt(c)*x - sqrt(c* x^2 + a))^9*c^5*f*g^7*h^4 - 120*(sqrt(c)*x - sqrt(c*x^2 + a))^9*c^5*e*g^6* h^5 + 1740*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a*c^4*f*g^5*h^6 - 360*(sqrt(c)* x - sqrt(c*x^2 + a))^9*a*c^4*e*g^4*h^7 + 1635*(sqrt(c)*x - sqrt(c*x^2 + a) )^9*a^2*c^3*f*g^3*h^8 - 360*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^2*c^3*e*g^2* h^9 + 45*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^2*c^3*d*g*h^10 + 450*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^3*c^2*f*g*h^10 - 75*(sqrt(c)*x - sqrt(c*x^2 + a))^ 9*a^3*c^2*e*h^11 + 3600*(sqrt(c)*x - sqrt(c*x^2 + a))^8*c^(11/2)*f*g^8*h^3 - 480*(sqrt(c)*x - sqrt(c*x^2 + a))^8*c^(11/2)*e*g^7*h^4 - 120*(sqrt(c)*x - sqrt(c*x^2 + a))^8*c^(11/2)*d*g^6*h^5 + 10020*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a*c^(9/2)*f*g^6*h^5 - 1440*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a*c^(9/2 )*e*g^5*h^6 - 360*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a*c^(9/2)*d*g^4*h^7 + 85 95*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^2*c^(7/2)*f*g^4*h^7 - 1440*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^2*c^(7/2)*e*g^3*h^8 + 45*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^2*c^(7/2)*d*g^2*h^9 + 1530*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^3*c^ (5/2)*f*g^2*h^9 - 75*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^3*c^(5/2)*e*g*h^...
Timed out. \[ \int \frac {\left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{(g+h x)^6} \, dx=\int \frac {{\left (c\,x^2+a\right )}^{3/2}\,\left (f\,x^2+e\,x+d\right )}{{\left (g+h\,x\right )}^6} \,d x \]