Integrand size = 27, antiderivative size = 891 \[ \int \frac {\left (d+e x+f x^2\right )^3}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \left (3 a b^4 c e f^2-a b^5 f^3+a b^3 c f \left (5 a f^2-3 c \left (e^2+d f\right )\right )-b c^2 \left (c^3 d^3+5 a^3 f^3+3 a c^2 d \left (e^2+d f\right )-9 a^2 c f \left (e^2+d f\right )\right )-a b^2 c^2 e \left (12 a f^2-c \left (e^2+6 d f\right )\right )+2 a c^3 e \left (3 c^2 d^2+3 a^2 f^2-a c \left (e^2+6 d f\right )\right )-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) \left (c^4 d^2-b c^3 d e+b^2 c^2 e^2-3 a c^3 e^2+b^2 c^2 d f-2 a c^3 d f-2 b^3 c e f+7 a b c^2 e f+b^4 f^2-4 a b^2 c f^2+a^2 c^2 f^2\right ) x\right )}{3 c^5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (3 b^6 c e f^2-b^7 f^3+3 b^5 c f \left (6 a f^2-c \left (e^2+d f\right )\right )-3 b^3 c^2 \left (29 a^2 f^3+c^2 d \left (e^2+d f\right )-10 a c f \left (e^2+d f\right )\right )-4 b c^3 \left (2 c^3 d^3-29 a^3 f^3+3 a c^2 d \left (e^2+d f\right )+24 a^2 c f \left (e^2+d f\right )\right )-24 a^2 c^4 e \left (6 a f^2-c \left (e^2+6 d f\right )\right )-b^4 c^2 e \left (42 a f^2-c \left (e^2+6 d f\right )\right )+6 b^2 c^3 e \left (2 c^2 d^2+28 a^2 f^2-a c \left (e^2+6 d f\right )\right )-c \left (16 c^6 d^3-10 b^6 f^3+3 b^4 c f^2 (7 b e+26 a f)-24 c^5 d \left (b d e-a \left (e^2+d f\right )\right )-6 b^2 c^2 f \left (25 a b e f+27 a^2 f^2+2 b^2 \left (e^2+d f\right )\right )+6 c^4 \left (b^2 d \left (e^2+d f\right )-16 a^2 f \left (e^2+d f\right )-2 a b e \left (e^2+6 d f\right )\right )+c^3 \left (240 a^2 b e f^2+56 a^3 f^3+84 a b^2 f \left (e^2+d f\right )+b^3 \left (e^3+6 d e f\right )\right )\right ) x\right )}{3 c^5 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}+\frac {f^2 (12 c e-11 b f) \sqrt {a+b x+c x^2}}{4 c^4}+\frac {f^3 x \sqrt {a+b x+c x^2}}{2 c^3}+\frac {f \left (35 b^2 f^2-20 c f (3 b e+a f)+24 c^2 \left (e^2+d f\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{9/2}} \]
2/3*(3*a*b^4*c*e*f^2-a*b^5*f^3+a*b^3*c*f*(5*a*f^2-3*c*(d*f+e^2))-b*c^2*(c^ 3*d^3+5*a^3*f^3+3*a*c^2*d*(d*f+e^2)-9*a^2*c*f*(d*f+e^2))-a*b^2*c^2*e*(12*a *f^2-c*(6*d*f+e^2))+2*a*c^3*e*(3*c^2*d^2+3*a^2*f^2-a*c*(6*d*f+e^2))-(-2*a* c*f+b^2*f-b*c*e+2*c^2*d)*(a^2*c^2*f^2-4*a*b^2*c*f^2+7*a*b*c^2*e*f-2*a*c^3* d*f-3*a*c^3*e^2+b^4*f^2-2*b^3*c*e*f+b^2*c^2*d*f+b^2*c^2*e^2-b*c^3*d*e+c^4* d^2)*x)/c^5/(-4*a*c+b^2)/(c*x^2+b*x+a)^(3/2)+1/8*f*(35*b^2*f^2-20*c*f*(a*f +3*b*e)+24*c^2*(d*f+e^2))*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2 ))/c^(9/2)-2/3*(3*b^6*c*e*f^2-b^7*f^3+3*b^5*c*f*(6*a*f^2-c*(d*f+e^2))-3*b^ 3*c^2*(29*a^2*f^3+c^2*d*(d*f+e^2)-10*a*c*f*(d*f+e^2))-4*b*c^3*(2*c^3*d^3-2 9*a^3*f^3+3*a*c^2*d*(d*f+e^2)+24*a^2*c*f*(d*f+e^2))-24*a^2*c^4*e*(6*a*f^2- c*(6*d*f+e^2))-b^4*c^2*e*(42*a*f^2-c*(6*d*f+e^2))+6*b^2*c^3*e*(2*c^2*d^2+2 8*a^2*f^2-a*c*(6*d*f+e^2))-c*(16*c^6*d^3-10*b^6*f^3+3*b^4*c*f^2*(26*a*f+7* b*e)-24*c^5*d*(b*d*e-a*(d*f+e^2))-6*b^2*c^2*f*(25*a*b*e*f+27*a^2*f^2+2*b^2 *(d*f+e^2))+6*c^4*(b^2*d*(d*f+e^2)-16*a^2*f*(d*f+e^2)-2*a*b*e*(6*d*f+e^2)) +c^3*(240*a^2*b*e*f^2+56*a^3*f^3+84*a*b^2*f*(d*f+e^2)+b^3*(6*d*e*f+e^3)))* x)/c^5/(-4*a*c+b^2)^2/(c*x^2+b*x+a)^(1/2)+1/4*f^2*(-11*b*f+12*c*e)*(c*x^2+ b*x+a)^(1/2)/c^4+1/2*f^3*x*(c*x^2+b*x+a)^(1/2)/c^3
Time = 11.98 (sec) , antiderivative size = 872, normalized size of antiderivative = 0.98 \[ \int \frac {\left (d+e x+f x^2\right )^3}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {-105 b^7 f^3 x^2-10 b^6 f^2 x (21 a f+2 c x (-9 e+7 f x))+6 b^4 c f \left (5 a^2 f (6 e+53 f x)-6 a c x \left (4 e^2+4 d f+30 e f x-31 f^2 x^2\right )+c^2 x^3 \left (-16 e^2-16 d f+6 e f x+f^2 x^2\right )\right )-3 b^5 f \left (35 a^2 f^2-10 a c f x (12 e+23 f x)+c^2 x^2 \left (24 e^2+24 d f-80 e f x+7 f^2 x^2\right )\right )-48 b c^2 \left (27 a^4 f^3-4 c^4 d^2 x^2 (d-e x)+a^2 c^2 \left (-4 d^2 f+4 e^3 x-64 e f^2 x^3+7 f^3 x^4-4 d e (e-6 f x)\right )-2 a c^3 \left (d^3-e^3 x^3+3 d e x^2 (e-2 f x)+3 d^2 x (-e+f x)\right )-2 a^3 c f \left (5 e^2+39 e f x+f \left (5 d-14 f x^2\right )\right )\right )-8 b^3 c \left (-95 a^3 f^3+c^3 \left (d^3-e^3 x^3+9 d^2 x (e-f x)-3 d e x^2 (3 e+2 f x)\right )-3 a c^2 f x^2 \left (18 e^2-74 e f x+f \left (18 d+7 f x^2\right )\right )+3 a^2 c f \left (3 e^2+105 e f x+f \left (3 d+29 f x^2\right )\right )\right )+32 c^3 \left (4 c^4 d^3 x^3+3 a^4 f^2 (16 e+5 f x)+6 a c^3 d x \left (d^2+e^2 x^2+d f x^2\right )-2 a^3 c \left (2 e^3+9 e^2 f x+f^2 x \left (9 d-10 f x^2\right )+12 e f \left (d-3 f x^2\right )\right )-3 a^2 c^2 \left (2 d^2 e+4 d f x^2 (3 e+2 f x)+x^2 \left (2 e^3+8 e^2 f x-6 e f^2 x^2-f^3 x^3\right )\right )\right )-48 b^2 c^2 \left (a^3 f^2 (25 e+63 f x)-c^3 d x \left (d^2+e^2 x^2+d x (-6 e+f x)\right )+a^2 c f x \left (-21 e^2-12 e f x+7 f \left (-3 d+7 f x^2\right )\right )+a c^2 \left (d^2 (e-6 f x)-2 d x \left (3 e^2-3 e f x+7 f^2 x^2\right )+x^2 \left (e^3-14 e^2 f x+6 e f^2 x^2+f^3 x^3\right )\right )\right )}{12 c^4 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}}+\frac {f \left (35 b^2 f^2-20 c f (3 b e+a f)+24 c^2 \left (e^2+d f\right )\right ) \log \left (b+2 c x+2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{8 c^{9/2}} \]
(-105*b^7*f^3*x^2 - 10*b^6*f^2*x*(21*a*f + 2*c*x*(-9*e + 7*f*x)) + 6*b^4*c *f*(5*a^2*f*(6*e + 53*f*x) - 6*a*c*x*(4*e^2 + 4*d*f + 30*e*f*x - 31*f^2*x^ 2) + c^2*x^3*(-16*e^2 - 16*d*f + 6*e*f*x + f^2*x^2)) - 3*b^5*f*(35*a^2*f^2 - 10*a*c*f*x*(12*e + 23*f*x) + c^2*x^2*(24*e^2 + 24*d*f - 80*e*f*x + 7*f^ 2*x^2)) - 48*b*c^2*(27*a^4*f^3 - 4*c^4*d^2*x^2*(d - e*x) + a^2*c^2*(-4*d^2 *f + 4*e^3*x - 64*e*f^2*x^3 + 7*f^3*x^4 - 4*d*e*(e - 6*f*x)) - 2*a*c^3*(d^ 3 - e^3*x^3 + 3*d*e*x^2*(e - 2*f*x) + 3*d^2*x*(-e + f*x)) - 2*a^3*c*f*(5*e ^2 + 39*e*f*x + f*(5*d - 14*f*x^2))) - 8*b^3*c*(-95*a^3*f^3 + c^3*(d^3 - e ^3*x^3 + 9*d^2*x*(e - f*x) - 3*d*e*x^2*(3*e + 2*f*x)) - 3*a*c^2*f*x^2*(18* e^2 - 74*e*f*x + f*(18*d + 7*f*x^2)) + 3*a^2*c*f*(3*e^2 + 105*e*f*x + f*(3 *d + 29*f*x^2))) + 32*c^3*(4*c^4*d^3*x^3 + 3*a^4*f^2*(16*e + 5*f*x) + 6*a* c^3*d*x*(d^2 + e^2*x^2 + d*f*x^2) - 2*a^3*c*(2*e^3 + 9*e^2*f*x + f^2*x*(9* d - 10*f*x^2) + 12*e*f*(d - 3*f*x^2)) - 3*a^2*c^2*(2*d^2*e + 4*d*f*x^2*(3* e + 2*f*x) + x^2*(2*e^3 + 8*e^2*f*x - 6*e*f^2*x^2 - f^3*x^3))) - 48*b^2*c^ 2*(a^3*f^2*(25*e + 63*f*x) - c^3*d*x*(d^2 + e^2*x^2 + d*x*(-6*e + f*x)) + a^2*c*f*x*(-21*e^2 - 12*e*f*x + 7*f*(-3*d + 7*f*x^2)) + a*c^2*(d^2*(e - 6* f*x) - 2*d*x*(3*e^2 - 3*e*f*x + 7*f^2*x^2) + x^2*(e^3 - 14*e^2*f*x + 6*e*f ^2*x^2 + f^3*x^3))))/(12*c^4*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^(3/2)) + (f *(35*b^2*f^2 - 20*c*f*(3*b*e + a*f) + 24*c^2*(e^2 + d*f))*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(8*c^(9/2))
Time = 2.38 (sec) , antiderivative size = 937, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2191, 27, 2191, 27, 2192, 27, 1160, 1092, 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.
| \(\displaystyle \int \frac {\left (d+e x+f x^2\right )^3}{\left (a+b x+c x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 2191 |
| \(\displaystyle \frac {2 \left (-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right ) \left (a^2 c^2 f^2-4 a b^2 c f^2+7 a b c^2 e f-2 a c^3 d f-3 a c^3 e^2+b^4 f^2-2 b^3 c e f+b^2 c^2 d f+b^2 c^2 e^2-b c^3 d e+c^4 d^2\right )+2 a c^3 e \left (3 a^2 f^2-a c \left (6 d f+e^2\right )+3 c^2 d^2\right )-b c^2 \left (5 a^3 f^3-9 a^2 c f \left (d f+e^2\right )+3 a c^2 d \left (d f+e^2\right )+c^3 d^3\right )-a b^5 f^3+3 a b^4 c e f^2+a b^3 c f \left (5 a f^2-3 c \left (d f+e^2\right )\right )-a b^2 c^2 e \left (12 a f^2-c \left (6 d f+e^2\right )\right )\right )}{3 c^5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \int \frac {3 \left (4 a-\frac {b^2}{c}\right ) f^3 x^4-\frac {3 \left (b^2-4 a c\right ) f^2 (3 c e-b f) x^3}{c^2}-\frac {3 \left (b^2-4 a c\right ) f \left (3 \left (e^2+d f\right ) c^2-f (3 b e+a f) c+b^2 f^2\right ) x^2}{c^3}+\frac {3 \left (b^2-4 a c\right ) \left (-\left (\left (e^3+6 d f e\right ) c^3\right )+3 f \left (a e f+b \left (e^2+d f\right )\right ) c^2-b f^2 (3 b e+2 a f) c+b^3 f^3\right ) x}{c^4}+\frac {f^3 b^6-3 c f^2 (b e+a f) b^4+3 c^2 f \left (\left (e^2+d f\right ) b^2+2 a e f b-a^2 f^2\right ) b^2+8 c^6 d^3+3 c^4 \left (b^2 d-4 a^2 f\right ) \left (e^2+d f\right )-12 c^5 d \left (b d e-a \left (e^2+d f\right )\right )+c^3 \left (-\left (\left (e^3+6 d f e\right ) b^3\right )-3 a f \left (e^2+d f\right ) b^2+12 a^2 e f^2 b+4 a^3 f^3\right )}{c^5}}{2 \left (c x^2+b x+a\right )^{3/2}}dx}{3 \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right ) \left (a^2 c^2 f^2-4 a b^2 c f^2+7 a b c^2 e f-2 a c^3 d f-3 a c^3 e^2+b^4 f^2-2 b^3 c e f+b^2 c^2 d f+b^2 c^2 e^2-b c^3 d e+c^4 d^2\right )+2 a c^3 e \left (3 a^2 f^2-a c \left (6 d f+e^2\right )+3 c^2 d^2\right )-b c^2 \left (5 a^3 f^3-9 a^2 c f \left (d f+e^2\right )+3 a c^2 d \left (d f+e^2\right )+c^3 d^3\right )-a b^5 f^3+3 a b^4 c e f^2+a b^3 c f \left (5 a f^2-3 c \left (d f+e^2\right )\right )-a b^2 c^2 e \left (12 a f^2-c \left (6 d f+e^2\right )\right )\right )}{3 c^5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {\int \frac {3 \left (4 a-\frac {b^2}{c}\right ) f^3 x^4-\frac {3 \left (b^2-4 a c\right ) f^2 (3 c e-b f) x^3}{c^2}-\frac {3 \left (b^2-4 a c\right ) f \left (3 \left (e^2+d f\right ) c^2-f (3 b e+a f) c+b^2 f^2\right ) x^2}{c^3}+\frac {3 \left (b^2-4 a c\right ) \left (-\left (\left (e^3+6 d f e\right ) c^3\right )+3 f \left (a e f+b \left (e^2+d f\right )\right ) c^2-b f^2 (3 b e+2 a f) c+b^3 f^3\right ) x}{c^4}+\frac {f^3 b^6-3 c f^2 (b e+a f) b^4+3 c^2 f \left (\left (e^2+d f\right ) b^2+2 a e f b-a^2 f^2\right ) b^2+8 c^6 d^3+3 c^4 \left (b^2 d-4 a^2 f\right ) \left (e^2+d f\right )-12 c^5 d \left (b d e-a \left (e^2+d f\right )\right )+c^3 \left (-\left (\left (e^3+6 d f e\right ) b^3\right )-3 a f \left (e^2+d f\right ) b^2+12 a^2 e f^2 b+4 a^3 f^3\right )}{c^5}}{\left (c x^2+b x+a\right )^{3/2}}dx}{3 \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 2191 |
| \(\displaystyle \frac {2 \left (-a f^3 b^5+3 a c e f^2 b^4+a c f \left (5 a f^2-3 c \left (e^2+d f\right )\right ) b^3-a c^2 e \left (12 a f^2-c \left (e^2+6 d f\right )\right ) b^2-c^2 \left (c^3 d^3+3 a c^2 \left (e^2+d f\right ) d+5 a^3 f^3-9 a^2 c f \left (e^2+d f\right )\right ) b+2 a c^3 e \left (3 c^2 d^2+3 a^2 f^2-a c \left (e^2+6 d f\right )\right )-\left (f b^2-c e b+2 c^2 d-2 a c f\right ) \left (f^2 b^4-2 c e f b^3+c^2 e^2 b^2-4 a c f^2 b^2+c^2 d f b^2-c^3 d e b+7 a c^2 e f b+c^4 d^2-3 a c^3 e^2+a^2 c^2 f^2-2 a c^3 d f\right ) x\right )}{3 c^5 \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{3/2}}-\frac {\frac {2 \left (-f^3 b^7+3 c e f^2 b^6+3 c f \left (6 a f^2-c \left (e^2+d f\right )\right ) b^5-c^2 e \left (42 a f^2-c \left (e^2+6 d f\right )\right ) b^4-3 c^2 \left (29 a^2 f^3-10 a c \left (e^2+d f\right ) f+c^2 d \left (e^2+d f\right )\right ) b^3+6 c^3 e \left (2 c^2 d^2+28 a^2 f^2-a c \left (e^2+6 d f\right )\right ) b^2-4 c^3 \left (2 c^3 d^3+3 a c^2 \left (e^2+d f\right ) d-29 a^3 f^3+24 a^2 c f \left (e^2+d f\right )\right ) b-24 a^2 c^4 e \left (6 a f^2-c \left (e^2+6 d f\right )\right )-c \left (-10 f^3 b^6+3 c f^2 (7 b e+26 a f) b^4-6 c^2 f \left (2 \left (e^2+d f\right ) b^2+25 a e f b+27 a^2 f^2\right ) b^2+16 c^6 d^3-24 c^5 d \left (b d e-a \left (e^2+d f\right )\right )+6 c^4 \left (-16 f \left (e^2+d f\right ) a^2-2 b e \left (e^2+6 d f\right ) a+b^2 d \left (e^2+d f\right )\right )+c^3 \left (\left (e^3+6 d f e\right ) b^3+84 a f \left (e^2+d f\right ) b^2+240 a^2 e f^2 b+56 a^3 f^3\right )\right ) x\right )}{c^5 \left (b^2-4 a c\right ) \sqrt {c x^2+b x+a}}-\frac {2 \int \frac {3 \left (\frac {\left (b^2-4 a c\right )^2 x^2 f^3}{c^2}+\frac {\left (b^2-4 a c\right )^2 (3 c e-2 b f) x f^2}{c^3}+\frac {\left (b^2-4 a c\right )^2 \left (3 \left (e^2+d f\right ) c^2-2 f (3 b e+a f) c+3 b^2 f^2\right ) f}{c^4}\right )}{2 \sqrt {c x^2+b x+a}}dx}{b^2-4 a c}}{3 \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (-a f^3 b^5+3 a c e f^2 b^4+a c f \left (5 a f^2-3 c \left (e^2+d f\right )\right ) b^3-a c^2 e \left (12 a f^2-c \left (e^2+6 d f\right )\right ) b^2-c^2 \left (c^3 d^3+3 a c^2 \left (e^2+d f\right ) d+5 a^3 f^3-9 a^2 c f \left (e^2+d f\right )\right ) b+2 a c^3 e \left (3 c^2 d^2+3 a^2 f^2-a c \left (e^2+6 d f\right )\right )-\left (f b^2-c e b+2 c^2 d-2 a c f\right ) \left (f^2 b^4-2 c e f b^3+c^2 e^2 b^2-4 a c f^2 b^2+c^2 d f b^2-c^3 d e b+7 a c^2 e f b+c^4 d^2-3 a c^3 e^2+a^2 c^2 f^2-2 a c^3 d f\right ) x\right )}{3 c^5 \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{3/2}}-\frac {\frac {2 \left (-f^3 b^7+3 c e f^2 b^6+3 c f \left (6 a f^2-c \left (e^2+d f\right )\right ) b^5-c^2 e \left (42 a f^2-c \left (e^2+6 d f\right )\right ) b^4-3 c^2 \left (29 a^2 f^3-10 a c \left (e^2+d f\right ) f+c^2 d \left (e^2+d f\right )\right ) b^3+6 c^3 e \left (2 c^2 d^2+28 a^2 f^2-a c \left (e^2+6 d f\right )\right ) b^2-4 c^3 \left (2 c^3 d^3+3 a c^2 \left (e^2+d f\right ) d-29 a^3 f^3+24 a^2 c f \left (e^2+d f\right )\right ) b-24 a^2 c^4 e \left (6 a f^2-c \left (e^2+6 d f\right )\right )-c \left (-10 f^3 b^6+3 c f^2 (7 b e+26 a f) b^4-6 c^2 f \left (2 \left (e^2+d f\right ) b^2+25 a e f b+27 a^2 f^2\right ) b^2+16 c^6 d^3-24 c^5 d \left (b d e-a \left (e^2+d f\right )\right )+6 c^4 \left (-16 f \left (e^2+d f\right ) a^2-2 b e \left (e^2+6 d f\right ) a+b^2 d \left (e^2+d f\right )\right )+c^3 \left (\left (e^3+6 d f e\right ) b^3+84 a f \left (e^2+d f\right ) b^2+240 a^2 e f^2 b+56 a^3 f^3\right )\right ) x\right )}{c^5 \left (b^2-4 a c\right ) \sqrt {c x^2+b x+a}}-\frac {3 \int \frac {\frac {\left (b^2-4 a c\right )^2 x^2 f^3}{c^2}+\frac {\left (b^2-4 a c\right )^2 (3 c e-2 b f) x f^2}{c^3}+\frac {\left (b^2-4 a c\right )^2 \left (3 \left (e^2+d f\right ) c^2-2 f (3 b e+a f) c+3 b^2 f^2\right ) f}{c^4}}{\sqrt {c x^2+b x+a}}dx}{b^2-4 a c}}{3 \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {2 \left (-a f^3 b^5+3 a c e f^2 b^4+a c f \left (5 a f^2-3 c \left (e^2+d f\right )\right ) b^3-a c^2 e \left (12 a f^2-c \left (e^2+6 d f\right )\right ) b^2-c^2 \left (c^3 d^3+3 a c^2 \left (e^2+d f\right ) d+5 a^3 f^3-9 a^2 c f \left (e^2+d f\right )\right ) b+2 a c^3 e \left (3 c^2 d^2+3 a^2 f^2-a c \left (e^2+6 d f\right )\right )-\left (f b^2-c e b+2 c^2 d-2 a c f\right ) \left (f^2 b^4-2 c e f b^3+c^2 e^2 b^2-4 a c f^2 b^2+c^2 d f b^2-c^3 d e b+7 a c^2 e f b+c^4 d^2-3 a c^3 e^2+a^2 c^2 f^2-2 a c^3 d f\right ) x\right )}{3 c^5 \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{3/2}}-\frac {\frac {2 \left (-f^3 b^7+3 c e f^2 b^6+3 c f \left (6 a f^2-c \left (e^2+d f\right )\right ) b^5-c^2 e \left (42 a f^2-c \left (e^2+6 d f\right )\right ) b^4-3 c^2 \left (29 a^2 f^3-10 a c \left (e^2+d f\right ) f+c^2 d \left (e^2+d f\right )\right ) b^3+6 c^3 e \left (2 c^2 d^2+28 a^2 f^2-a c \left (e^2+6 d f\right )\right ) b^2-4 c^3 \left (2 c^3 d^3+3 a c^2 \left (e^2+d f\right ) d-29 a^3 f^3+24 a^2 c f \left (e^2+d f\right )\right ) b-24 a^2 c^4 e \left (6 a f^2-c \left (e^2+6 d f\right )\right )-c \left (-10 f^3 b^6+3 c f^2 (7 b e+26 a f) b^4-6 c^2 f \left (2 \left (e^2+d f\right ) b^2+25 a e f b+27 a^2 f^2\right ) b^2+16 c^6 d^3-24 c^5 d \left (b d e-a \left (e^2+d f\right )\right )+6 c^4 \left (-16 f \left (e^2+d f\right ) a^2-2 b e \left (e^2+6 d f\right ) a+b^2 d \left (e^2+d f\right )\right )+c^3 \left (\left (e^3+6 d f e\right ) b^3+84 a f \left (e^2+d f\right ) b^2+240 a^2 e f^2 b+56 a^3 f^3\right )\right ) x\right )}{c^5 \left (b^2-4 a c\right ) \sqrt {c x^2+b x+a}}-\frac {3 \left (\frac {\left (b^2-4 a c\right )^2 x \sqrt {c x^2+b x+a} f^3}{2 c^3}+\frac {\int \frac {\left (b^2-4 a c\right )^2 f \left (2 \left (6 \left (e^2+d f\right ) c^2-f (12 b e+5 a f) c+6 b^2 f^2\right )+c f (12 c e-11 b f) x\right )}{2 c^3 \sqrt {c x^2+b x+a}}dx}{2 c}\right )}{b^2-4 a c}}{3 \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (-a f^3 b^5+3 a c e f^2 b^4+a c f \left (5 a f^2-3 c \left (e^2+d f\right )\right ) b^3-a c^2 e \left (12 a f^2-c \left (e^2+6 d f\right )\right ) b^2-c^2 \left (c^3 d^3+3 a c^2 \left (e^2+d f\right ) d+5 a^3 f^3-9 a^2 c f \left (e^2+d f\right )\right ) b+2 a c^3 e \left (3 c^2 d^2+3 a^2 f^2-a c \left (e^2+6 d f\right )\right )-\left (f b^2-c e b+2 c^2 d-2 a c f\right ) \left (f^2 b^4-2 c e f b^3+c^2 e^2 b^2-4 a c f^2 b^2+c^2 d f b^2-c^3 d e b+7 a c^2 e f b+c^4 d^2-3 a c^3 e^2+a^2 c^2 f^2-2 a c^3 d f\right ) x\right )}{3 c^5 \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{3/2}}-\frac {\frac {2 \left (-f^3 b^7+3 c e f^2 b^6+3 c f \left (6 a f^2-c \left (e^2+d f\right )\right ) b^5-c^2 e \left (42 a f^2-c \left (e^2+6 d f\right )\right ) b^4-3 c^2 \left (29 a^2 f^3-10 a c \left (e^2+d f\right ) f+c^2 d \left (e^2+d f\right )\right ) b^3+6 c^3 e \left (2 c^2 d^2+28 a^2 f^2-a c \left (e^2+6 d f\right )\right ) b^2-4 c^3 \left (2 c^3 d^3+3 a c^2 \left (e^2+d f\right ) d-29 a^3 f^3+24 a^2 c f \left (e^2+d f\right )\right ) b-24 a^2 c^4 e \left (6 a f^2-c \left (e^2+6 d f\right )\right )-c \left (-10 f^3 b^6+3 c f^2 (7 b e+26 a f) b^4-6 c^2 f \left (2 \left (e^2+d f\right ) b^2+25 a e f b+27 a^2 f^2\right ) b^2+16 c^6 d^3-24 c^5 d \left (b d e-a \left (e^2+d f\right )\right )+6 c^4 \left (-16 f \left (e^2+d f\right ) a^2-2 b e \left (e^2+6 d f\right ) a+b^2 d \left (e^2+d f\right )\right )+c^3 \left (\left (e^3+6 d f e\right ) b^3+84 a f \left (e^2+d f\right ) b^2+240 a^2 e f^2 b+56 a^3 f^3\right )\right ) x\right )}{c^5 \left (b^2-4 a c\right ) \sqrt {c x^2+b x+a}}-\frac {3 \left (\frac {\left (b^2-4 a c\right )^2 x \sqrt {c x^2+b x+a} f^3}{2 c^3}+\frac {\left (b^2-4 a c\right )^2 \int \frac {2 \left (6 \left (e^2+d f\right ) c^2-f (12 b e+5 a f) c+6 b^2 f^2\right )+c f (12 c e-11 b f) x}{\sqrt {c x^2+b x+a}}dx f}{4 c^4}\right )}{b^2-4 a c}}{3 \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle \frac {2 \left (-a f^3 b^5+3 a c e f^2 b^4+a c f \left (5 a f^2-3 c \left (e^2+d f\right )\right ) b^3-a c^2 e \left (12 a f^2-c \left (e^2+6 d f\right )\right ) b^2-c^2 \left (c^3 d^3+3 a c^2 \left (e^2+d f\right ) d+5 a^3 f^3-9 a^2 c f \left (e^2+d f\right )\right ) b+2 a c^3 e \left (3 c^2 d^2+3 a^2 f^2-a c \left (e^2+6 d f\right )\right )-\left (f b^2-c e b+2 c^2 d-2 a c f\right ) \left (f^2 b^4-2 c e f b^3+c^2 e^2 b^2-4 a c f^2 b^2+c^2 d f b^2-c^3 d e b+7 a c^2 e f b+c^4 d^2-3 a c^3 e^2+a^2 c^2 f^2-2 a c^3 d f\right ) x\right )}{3 c^5 \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{3/2}}-\frac {\frac {2 \left (-f^3 b^7+3 c e f^2 b^6+3 c f \left (6 a f^2-c \left (e^2+d f\right )\right ) b^5-c^2 e \left (42 a f^2-c \left (e^2+6 d f\right )\right ) b^4-3 c^2 \left (29 a^2 f^3-10 a c \left (e^2+d f\right ) f+c^2 d \left (e^2+d f\right )\right ) b^3+6 c^3 e \left (2 c^2 d^2+28 a^2 f^2-a c \left (e^2+6 d f\right )\right ) b^2-4 c^3 \left (2 c^3 d^3+3 a c^2 \left (e^2+d f\right ) d-29 a^3 f^3+24 a^2 c f \left (e^2+d f\right )\right ) b-24 a^2 c^4 e \left (6 a f^2-c \left (e^2+6 d f\right )\right )-c \left (-10 f^3 b^6+3 c f^2 (7 b e+26 a f) b^4-6 c^2 f \left (2 \left (e^2+d f\right ) b^2+25 a e f b+27 a^2 f^2\right ) b^2+16 c^6 d^3-24 c^5 d \left (b d e-a \left (e^2+d f\right )\right )+6 c^4 \left (-16 f \left (e^2+d f\right ) a^2-2 b e \left (e^2+6 d f\right ) a+b^2 d \left (e^2+d f\right )\right )+c^3 \left (\left (e^3+6 d f e\right ) b^3+84 a f \left (e^2+d f\right ) b^2+240 a^2 e f^2 b+56 a^3 f^3\right )\right ) x\right )}{c^5 \left (b^2-4 a c\right ) \sqrt {c x^2+b x+a}}-\frac {3 \left (\frac {\left (b^2-4 a c\right )^2 x \sqrt {c x^2+b x+a} f^3}{2 c^3}+\frac {\left (b^2-4 a c\right )^2 \left (f \sqrt {c x^2+b x+a} (12 c e-11 b f)+\frac {1}{2} \left (24 \left (e^2+d f\right ) c^2-20 f (3 b e+a f) c+35 b^2 f^2\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx\right ) f}{4 c^4}\right )}{b^2-4 a c}}{3 \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {2 \left (-a f^3 b^5+3 a c e f^2 b^4+a c f \left (5 a f^2-3 c \left (e^2+d f\right )\right ) b^3-a c^2 e \left (12 a f^2-c \left (e^2+6 d f\right )\right ) b^2-c^2 \left (c^3 d^3+3 a c^2 \left (e^2+d f\right ) d+5 a^3 f^3-9 a^2 c f \left (e^2+d f\right )\right ) b+2 a c^3 e \left (3 c^2 d^2+3 a^2 f^2-a c \left (e^2+6 d f\right )\right )-\left (f b^2-c e b+2 c^2 d-2 a c f\right ) \left (f^2 b^4-2 c e f b^3+c^2 e^2 b^2-4 a c f^2 b^2+c^2 d f b^2-c^3 d e b+7 a c^2 e f b+c^4 d^2-3 a c^3 e^2+a^2 c^2 f^2-2 a c^3 d f\right ) x\right )}{3 c^5 \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{3/2}}-\frac {\frac {2 \left (-f^3 b^7+3 c e f^2 b^6+3 c f \left (6 a f^2-c \left (e^2+d f\right )\right ) b^5-c^2 e \left (42 a f^2-c \left (e^2+6 d f\right )\right ) b^4-3 c^2 \left (29 a^2 f^3-10 a c \left (e^2+d f\right ) f+c^2 d \left (e^2+d f\right )\right ) b^3+6 c^3 e \left (2 c^2 d^2+28 a^2 f^2-a c \left (e^2+6 d f\right )\right ) b^2-4 c^3 \left (2 c^3 d^3+3 a c^2 \left (e^2+d f\right ) d-29 a^3 f^3+24 a^2 c f \left (e^2+d f\right )\right ) b-24 a^2 c^4 e \left (6 a f^2-c \left (e^2+6 d f\right )\right )-c \left (-10 f^3 b^6+3 c f^2 (7 b e+26 a f) b^4-6 c^2 f \left (2 \left (e^2+d f\right ) b^2+25 a e f b+27 a^2 f^2\right ) b^2+16 c^6 d^3-24 c^5 d \left (b d e-a \left (e^2+d f\right )\right )+6 c^4 \left (-16 f \left (e^2+d f\right ) a^2-2 b e \left (e^2+6 d f\right ) a+b^2 d \left (e^2+d f\right )\right )+c^3 \left (\left (e^3+6 d f e\right ) b^3+84 a f \left (e^2+d f\right ) b^2+240 a^2 e f^2 b+56 a^3 f^3\right )\right ) x\right )}{c^5 \left (b^2-4 a c\right ) \sqrt {c x^2+b x+a}}-\frac {3 \left (\frac {\left (b^2-4 a c\right )^2 x \sqrt {c x^2+b x+a} f^3}{2 c^3}+\frac {\left (b^2-4 a c\right )^2 \left (f \sqrt {c x^2+b x+a} (12 c e-11 b f)+\left (24 \left (e^2+d f\right ) c^2-20 f (3 b e+a f) c+35 b^2 f^2\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}\right ) f}{4 c^4}\right )}{b^2-4 a c}}{3 \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {2 \left (-a f^3 b^5+3 a c e f^2 b^4+a c f \left (5 a f^2-3 c \left (e^2+d f\right )\right ) b^3-a c^2 e \left (12 a f^2-c \left (e^2+6 d f\right )\right ) b^2-c^2 \left (c^3 d^3+3 a c^2 \left (e^2+d f\right ) d+5 a^3 f^3-9 a^2 c f \left (e^2+d f\right )\right ) b+2 a c^3 e \left (3 c^2 d^2+3 a^2 f^2-a c \left (e^2+6 d f\right )\right )-\left (f b^2-c e b+2 c^2 d-2 a c f\right ) \left (f^2 b^4-2 c e f b^3+c^2 e^2 b^2-4 a c f^2 b^2+c^2 d f b^2-c^3 d e b+7 a c^2 e f b+c^4 d^2-3 a c^3 e^2+a^2 c^2 f^2-2 a c^3 d f\right ) x\right )}{3 c^5 \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{3/2}}-\frac {\frac {2 \left (-f^3 b^7+3 c e f^2 b^6+3 c f \left (6 a f^2-c \left (e^2+d f\right )\right ) b^5-c^2 e \left (42 a f^2-c \left (e^2+6 d f\right )\right ) b^4-3 c^2 \left (29 a^2 f^3-10 a c \left (e^2+d f\right ) f+c^2 d \left (e^2+d f\right )\right ) b^3+6 c^3 e \left (2 c^2 d^2+28 a^2 f^2-a c \left (e^2+6 d f\right )\right ) b^2-4 c^3 \left (2 c^3 d^3+3 a c^2 \left (e^2+d f\right ) d-29 a^3 f^3+24 a^2 c f \left (e^2+d f\right )\right ) b-24 a^2 c^4 e \left (6 a f^2-c \left (e^2+6 d f\right )\right )-c \left (-10 f^3 b^6+3 c f^2 (7 b e+26 a f) b^4-6 c^2 f \left (2 \left (e^2+d f\right ) b^2+25 a e f b+27 a^2 f^2\right ) b^2+16 c^6 d^3-24 c^5 d \left (b d e-a \left (e^2+d f\right )\right )+6 c^4 \left (-16 f \left (e^2+d f\right ) a^2-2 b e \left (e^2+6 d f\right ) a+b^2 d \left (e^2+d f\right )\right )+c^3 \left (\left (e^3+6 d f e\right ) b^3+84 a f \left (e^2+d f\right ) b^2+240 a^2 e f^2 b+56 a^3 f^3\right )\right ) x\right )}{c^5 \left (b^2-4 a c\right ) \sqrt {c x^2+b x+a}}-\frac {3 \left (\frac {\left (b^2-4 a c\right )^2 x \sqrt {c x^2+b x+a} f^3}{2 c^3}+\frac {\left (b^2-4 a c\right )^2 \left (f \sqrt {c x^2+b x+a} (12 c e-11 b f)+\frac {\left (24 \left (e^2+d f\right ) c^2-20 f (3 b e+a f) c+35 b^2 f^2\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right )}{2 \sqrt {c}}\right ) f}{4 c^4}\right )}{b^2-4 a c}}{3 \left (b^2-4 a c\right )}\) |
(2*(3*a*b^4*c*e*f^2 - a*b^5*f^3 + a*b^3*c*f*(5*a*f^2 - 3*c*(e^2 + d*f)) - b*c^2*(c^3*d^3 + 5*a^3*f^3 + 3*a*c^2*d*(e^2 + d*f) - 9*a^2*c*f*(e^2 + d*f) ) - a*b^2*c^2*e*(12*a*f^2 - c*(e^2 + 6*d*f)) + 2*a*c^3*e*(3*c^2*d^2 + 3*a^ 2*f^2 - a*c*(e^2 + 6*d*f)) - (2*c^2*d - b*c*e + b^2*f - 2*a*c*f)*(c^4*d^2 - b*c^3*d*e + b^2*c^2*e^2 - 3*a*c^3*e^2 + b^2*c^2*d*f - 2*a*c^3*d*f - 2*b^ 3*c*e*f + 7*a*b*c^2*e*f + b^4*f^2 - 4*a*b^2*c*f^2 + a^2*c^2*f^2)*x))/(3*c^ 5*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(3/2)) - ((2*(3*b^6*c*e*f^2 - b^7*f^3 + 3*b^5*c*f*(6*a*f^2 - c*(e^2 + d*f)) - 3*b^3*c^2*(29*a^2*f^3 + c^2*d*(e^2 + d*f) - 10*a*c*f*(e^2 + d*f)) - 4*b*c^3*(2*c^3*d^3 - 29*a^3*f^3 + 3*a*c^2* d*(e^2 + d*f) + 24*a^2*c*f*(e^2 + d*f)) - 24*a^2*c^4*e*(6*a*f^2 - c*(e^2 + 6*d*f)) - b^4*c^2*e*(42*a*f^2 - c*(e^2 + 6*d*f)) + 6*b^2*c^3*e*(2*c^2*d^2 + 28*a^2*f^2 - a*c*(e^2 + 6*d*f)) - c*(16*c^6*d^3 - 10*b^6*f^3 + 3*b^4*c* f^2*(7*b*e + 26*a*f) - 24*c^5*d*(b*d*e - a*(e^2 + d*f)) - 6*b^2*c^2*f*(25* a*b*e*f + 27*a^2*f^2 + 2*b^2*(e^2 + d*f)) + 6*c^4*(b^2*d*(e^2 + d*f) - 16* a^2*f*(e^2 + d*f) - 2*a*b*e*(e^2 + 6*d*f)) + c^3*(240*a^2*b*e*f^2 + 56*a^3 *f^3 + 84*a*b^2*f*(e^2 + d*f) + b^3*(e^3 + 6*d*e*f)))*x))/(c^5*(b^2 - 4*a* c)*Sqrt[a + b*x + c*x^2]) - (3*(((b^2 - 4*a*c)^2*f^3*x*Sqrt[a + b*x + c*x^ 2])/(2*c^3) + ((b^2 - 4*a*c)^2*f*(f*(12*c*e - 11*b*f)*Sqrt[a + b*x + c*x^2 ] + ((35*b^2*f^2 - 20*c*f*(3*b*e + a*f) + 24*c^2*(e^2 + d*f))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[c])))/(4*c^4)))/(b^...
3.2.18.3.1 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_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b *e)/(2*c) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[P q, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 1]}, Simp[(b*f - 2*a*g + (2*c*f - b*g)*x)*((a + b*x + c*x^2)^ (p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)) Int [(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (2*p + 3)* (2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^ 2 - 4*a*c, 0] && LtQ[p, -1]
Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(q + 2*p + 1))), x] + Simp[1/(c*(q + 2*p + 1)) Int[(a + b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b *e*(q + p)*x^(q - 1) - c*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c , p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && !LeQ[p, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(3250\) vs. \(2(865)=1730\).
Time = 1.57 (sec) , antiderivative size = 3251, normalized size of antiderivative = 3.65
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(3251\) |
| risch | \(\text {Expression too large to display}\) | \(19191\) |
d^3*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2 *c*x+b)/(c*x^2+b*x+a)^(1/2))+f^3*(1/2*x^5/c/(c*x^2+b*x+a)^(3/2)-7/4*b/c*(x ^4/c/(c*x^2+b*x+a)^(3/2)-5/2*b/c*(-1/3*x^3/c/(c*x^2+b*x+a)^(3/2)-1/2*b/c*( -x^2/c/(c*x^2+b*x+a)^(3/2)+1/2*b/c*(-1/2*x/c/(c*x^2+b*x+a)^(3/2)-1/4*b/c*( -1/3/c/(c*x^2+b*x+a)^(3/2)-1/2*b/c*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a )^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2)))+1/2*a/c*(2/3* (2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/( c*x^2+b*x+a)^(1/2)))+2*a/c*(-1/3/c/(c*x^2+b*x+a)^(3/2)-1/2*b/c*(2/3*(2*c*x +b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+ b*x+a)^(1/2))))+1/c*(-x/c/(c*x^2+b*x+a)^(1/2)-1/2*b/c*(-1/c/(c*x^2+b*x+a)^ (1/2)-b/c*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2))+1/c^(3/2)*ln((1/2*b+c *x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))-4*a/c*(-x^2/c/(c*x^2+b*x+a)^(3/2)+1/2*b /c*(-1/2*x/c/(c*x^2+b*x+a)^(3/2)-1/4*b/c*(-1/3/c/(c*x^2+b*x+a)^(3/2)-1/2*b /c*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2* c*x+b)/(c*x^2+b*x+a)^(1/2)))+1/2*a/c*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x +a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2)))+2*a/c*(-1/3 /c/(c*x^2+b*x+a)^(3/2)-1/2*b/c*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3 /2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2)))))-5/2*a/c*(-1/3*x ^3/c/(c*x^2+b*x+a)^(3/2)-1/2*b/c*(-x^2/c/(c*x^2+b*x+a)^(3/2)+1/2*b/c*(-1/2 *x/c/(c*x^2+b*x+a)^(3/2)-1/4*b/c*(-1/3/c/(c*x^2+b*x+a)^(3/2)-1/2*b/c*(2...
Leaf count of result is larger than twice the leaf count of optimal. 1996 vs. \(2 (865) = 1730\).
Time = 4.62 (sec) , antiderivative size = 3995, normalized size of antiderivative = 4.48 \[ \int \frac {\left (d+e x+f x^2\right )^3}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Too large to display} \]
[-1/48*(3*((24*(b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*e^2*f + 5*(7*b^6*c^2 - 60*a*b^4*c^3 + 144*a^2*b^2*c^4 - 64*a^3*c^5)*f^3 + 12*(2*(b^4*c^4 - 8*a*b ^2*c^5 + 16*a^2*c^6)*d - 5*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*e)*f^2)* x^4 + 24*(a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)*e^2*f + 5*(7*a^2*b^6 - 60*a^3*b^4*c + 144*a^4*b^2*c^2 - 64*a^5*c^3)*f^3 + 2*(24*(b^5*c^3 - 8*a*b ^3*c^4 + 16*a^2*b*c^5)*e^2*f + 5*(7*b^7*c - 60*a*b^5*c^2 + 144*a^2*b^3*c^3 - 64*a^3*b*c^4)*f^3 + 12*(2*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*d - 5* (b^6*c^2 - 8*a*b^4*c^3 + 16*a^2*b^2*c^4)*e)*f^2)*x^3 + 12*(2*(a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)*d - 5*(a^2*b^5*c - 8*a^3*b^3*c^2 + 16*a^4*b* c^3)*e)*f^2 + (24*(b^6*c^2 - 6*a*b^4*c^3 + 32*a^3*c^5)*e^2*f + 5*(7*b^8 - 46*a*b^6*c + 24*a^2*b^4*c^2 + 224*a^3*b^2*c^3 - 128*a^4*c^4)*f^3 + 12*(2*( b^6*c^2 - 6*a*b^4*c^3 + 32*a^3*c^5)*d - 5*(b^7*c - 6*a*b^5*c^2 + 32*a^3*b* c^4)*e)*f^2)*x^2 + 2*(24*(a*b^5*c^2 - 8*a^2*b^3*c^3 + 16*a^3*b*c^4)*e^2*f + 5*(7*a*b^7 - 60*a^2*b^5*c + 144*a^3*b^3*c^2 - 64*a^4*b*c^3)*f^3 + 12*(2* (a*b^5*c^2 - 8*a^2*b^3*c^3 + 16*a^3*b*c^4)*d - 5*(a*b^6*c - 8*a^2*b^4*c^2 + 16*a^3*b^2*c^3)*e)*f^2)*x)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 + 4*sq rt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) - 4*(192*a^2*b*c^5*d*e^2 - 128*a^3*c^5*e^3 + 6*(b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*f^3*x^5 + 3*(12 *(b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*e*f^2 - 7*(b^5*c^3 - 8*a*b^3*c^4 + 1 6*a^2*b*c^5)*f^3)*x^4 - 8*(b^3*c^5 - 12*a*b*c^6)*d^3 - 48*(a*b^2*c^5 + ...
Timed out. \[ \int \frac {\left (d+e x+f x^2\right )^3}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\left (d+e x+f x^2\right )^3}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Time = 0.32 (sec) , antiderivative size = 1392, normalized size of antiderivative = 1.56 \[ \int \frac {\left (d+e x+f x^2\right )^3}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Too large to display} \]
1/12*((((3*(2*(b^4*c^3*f^3 - 8*a*b^2*c^4*f^3 + 16*a^2*c^5*f^3)*x/(b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6) + (12*b^4*c^3*e*f^2 - 96*a*b^2*c^4*e*f^2 + 192 *a^2*c^5*e*f^2 - 7*b^5*c^2*f^3 + 56*a*b^3*c^3*f^3 - 112*a^2*b*c^4*f^3)/(b^ 4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6))*x + 4*(32*c^7*d^3 - 48*b*c^6*d^2*e + 12 *b^2*c^5*d*e^2 + 48*a*c^6*d*e^2 + 2*b^3*c^4*e^3 - 24*a*b*c^5*e^3 + 12*b^2* c^5*d^2*f + 48*a*c^6*d^2*f + 12*b^3*c^4*d*e*f - 144*a*b*c^5*d*e*f - 24*b^4 *c^3*e^2*f + 168*a*b^2*c^4*e^2*f - 192*a^2*c^5*e^2*f - 24*b^4*c^3*d*f^2 + 168*a*b^2*c^4*d*f^2 - 192*a^2*c^5*d*f^2 + 60*b^5*c^2*e*f^2 - 444*a*b^3*c^3 *e*f^2 + 768*a^2*b*c^4*e*f^2 - 35*b^6*c*f^3 + 279*a*b^4*c^2*f^3 - 588*a^2* b^2*c^3*f^3 + 160*a^3*c^4*f^3)/(b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6))*x + 3 *(64*b*c^6*d^3 - 96*b^2*c^5*d^2*e + 24*b^3*c^4*d*e^2 + 96*a*b*c^5*d*e^2 - 16*a*b^2*c^4*e^3 - 64*a^2*c^5*e^3 + 24*b^3*c^4*d^2*f + 96*a*b*c^5*d^2*f - 96*a*b^2*c^4*d*e*f - 384*a^2*c^5*d*e*f - 24*b^5*c^2*e^2*f + 144*a*b^3*c^3* e^2*f - 24*b^5*c^2*d*f^2 + 144*a*b^3*c^3*d*f^2 + 60*b^6*c*e*f^2 - 360*a*b^ 4*c^2*e*f^2 + 192*a^2*b^2*c^3*e*f^2 + 768*a^3*c^4*e*f^2 - 35*b^7*f^3 + 230 *a*b^5*c*f^3 - 232*a^2*b^3*c^2*f^3 - 448*a^3*b*c^3*f^3)/(b^4*c^4 - 8*a*b^2 *c^5 + 16*a^2*c^6))*x + 6*(8*b^2*c^5*d^3 + 32*a*c^6*d^3 - 12*b^3*c^4*d^2*e - 48*a*b*c^5*d^2*e + 48*a*b^2*c^4*d*e^2 - 32*a^2*b*c^4*e^3 + 48*a*b^2*c^4 *d^2*f - 192*a^2*b*c^4*d*e*f - 24*a*b^4*c^2*e^2*f + 168*a^2*b^2*c^3*e^2*f - 96*a^3*c^4*e^2*f - 24*a*b^4*c^2*d*f^2 + 168*a^2*b^2*c^3*d*f^2 - 96*a^...
Timed out. \[ \int \frac {\left (d+e x+f x^2\right )^3}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {{\left (f\,x^2+e\,x+d\right )}^3}{{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \]