Integrand size = 28, antiderivative size = 627 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{d+e x} \, dx=-\frac {\left (512 c^5 d^5+b^5 e^5-128 c^4 d^3 e (11 b d-8 a e)+8 b^3 c e^4 (b d-2 a e)+32 c^3 d e^2 \left (40 b^2 d^2-55 a b d e+16 a^2 e^2\right )-8 b c^2 e^3 \left (49 b^2 d^2-92 a b d e+42 a^2 e^2\right )-2 c e \left (16 c^2 d^2-b^2 e^2-4 c e (4 b d-5 a e)\right ) \left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{256 c^2 e^6}-\frac {\left (64 c^3 d^3-b^3 e^3+4 b c e^2 (14 b d-13 a e)-8 c^2 d e (15 b d-8 a e)-2 c e \left (24 c^2 d^2+b^2 e^2-4 c e (6 b d-5 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{96 c e^4}-\frac {(12 c d-11 b e-10 c e x) \left (a+b x+c x^2\right )^{5/2}}{30 e^2}+\frac {\left (1024 c^6 d^6+b^6 e^6+4 b^4 c e^5 (2 b d-5 a e)-512 c^5 d^4 e (6 b d-5 a e)-320 c^3 e^3 (b d-a e)^2 (4 b d-a e)+640 c^4 d^2 e^2 \left (5 b^2 d^2-8 a b d e+3 a^2 e^2\right )+40 b^2 c^2 e^4 \left (3 b^2 d^2-8 a b d e+6 a^2 e^2\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{512 c^{5/2} e^7}-\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )^{5/2} \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{e^7} \] Output:
-1/256*(512*c^5*d^5+b^5*e^5-128*c^4*d^3*e*(-8*a*e+11*b*d)+8*b^3*c*e^4*(-2* a*e+b*d)+32*c^3*d*e^2*(16*a^2*e^2-55*a*b*d*e+40*b^2*d^2)-8*b*c^2*e^3*(42*a ^2*e^2-92*a*b*d*e+49*b^2*d^2)-2*c*e*(16*c^2*d^2-b^2*e^2-4*c*e*(-5*a*e+4*b* d))*(8*c^2*d^2+b^2*e^2-4*c*e*(-a*e+2*b*d))*x)*(c*x^2+b*x+a)^(1/2)/c^2/e^6- 1/96*(64*c^3*d^3-b^3*e^3+4*b*c*e^2*(-13*a*e+14*b*d)-8*c^2*d*e*(-8*a*e+15*b *d)-2*c*e*(24*c^2*d^2+b^2*e^2-4*c*e*(-5*a*e+6*b*d))*x)*(c*x^2+b*x+a)^(3/2) /c/e^4-1/30*(-10*c*e*x-11*b*e+12*c*d)*(c*x^2+b*x+a)^(5/2)/e^2+1/512*(1024* c^6*d^6+b^6*e^6+4*b^4*c*e^5*(-5*a*e+2*b*d)-512*c^5*d^4*e*(-5*a*e+6*b*d)-32 0*c^3*e^3*(-a*e+b*d)^2*(-a*e+4*b*d)+640*c^4*d^2*e^2*(3*a^2*e^2-8*a*b*d*e+5 *b^2*d^2)+40*b^2*c^2*e^4*(6*a^2*e^2-8*a*b*d*e+3*b^2*d^2))*arctanh(1/2*(2*c *x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(5/2)/e^7-(-b*e+2*c*d)*(a*e^2-b*d*e+c *d^2)^(5/2)*arctanh(1/2*(b*d-2*a*e+(-b*e+2*c*d)*x)/(a*e^2-b*d*e+c*d^2)^(1/ 2)/(c*x^2+b*x+a)^(1/2))/e^7
Time = 11.79 (sec) , antiderivative size = 607, normalized size of antiderivative = 0.97 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{d+e x} \, dx=\frac {(-12 c d+11 b e+10 c e x) (a+x (b+c x))^{5/2}}{30 e^2}+\frac {(a+x (b+c x))^{3/2} \left (b^3 e^3+16 c^3 d^2 (-4 d+3 e x)+2 b c e^2 (-28 b d+26 a e+b e x)+8 c^2 e (3 b d (5 d-2 e x)+a e (-8 d+5 e x))\right )}{96 c e^4}+\frac {-2 c e \sqrt {a+x (b+c x)} \left (b^5 e^5+256 c^5 d^4 (2 d-e x)+2 b^3 c e^4 (4 b d-8 a e+b e x)-8 b c^2 e^3 \left (42 a^2 e^2+b^2 d (49 d-2 e x)+4 a b e (-23 d+e x)\right )-64 c^4 d^2 e (2 b d (11 d-4 e x)+a e (-16 d+7 e x))+16 c^3 e^2 \left (b^2 d^2 (80 d-17 e x)+2 a^2 e^2 (16 d-5 e x)+2 a b d e (-55 d+14 e x)\right )\right )+\sqrt {c} \left (1024 c^6 d^6+b^6 e^6+4 b^4 c e^5 (2 b d-5 a e)-512 c^5 d^4 e (6 b d-5 a e)+320 c^3 e^3 (b d-a e)^2 (-4 b d+a e)+640 c^4 d^2 e^2 \left (5 b^2 d^2-8 a b d e+3 a^2 e^2\right )+40 b^2 c^2 e^4 \left (3 b^2 d^2-8 a b d e+6 a^2 e^2\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )+512 c^3 (2 c d-b e) \left (c d^2+e (-b d+a e)\right )^{5/2} \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{512 c^3 e^7} \] Input:
Integrate[((b + 2*c*x)*(a + b*x + c*x^2)^(5/2))/(d + e*x),x]
Output:
((-12*c*d + 11*b*e + 10*c*e*x)*(a + x*(b + c*x))^(5/2))/(30*e^2) + ((a + x *(b + c*x))^(3/2)*(b^3*e^3 + 16*c^3*d^2*(-4*d + 3*e*x) + 2*b*c*e^2*(-28*b* d + 26*a*e + b*e*x) + 8*c^2*e*(3*b*d*(5*d - 2*e*x) + a*e*(-8*d + 5*e*x)))) /(96*c*e^4) + (-2*c*e*Sqrt[a + x*(b + c*x)]*(b^5*e^5 + 256*c^5*d^4*(2*d - e*x) + 2*b^3*c*e^4*(4*b*d - 8*a*e + b*e*x) - 8*b*c^2*e^3*(42*a^2*e^2 + b^2 *d*(49*d - 2*e*x) + 4*a*b*e*(-23*d + e*x)) - 64*c^4*d^2*e*(2*b*d*(11*d - 4 *e*x) + a*e*(-16*d + 7*e*x)) + 16*c^3*e^2*(b^2*d^2*(80*d - 17*e*x) + 2*a^2 *e^2*(16*d - 5*e*x) + 2*a*b*d*e*(-55*d + 14*e*x))) + Sqrt[c]*(1024*c^6*d^6 + b^6*e^6 + 4*b^4*c*e^5*(2*b*d - 5*a*e) - 512*c^5*d^4*e*(6*b*d - 5*a*e) + 320*c^3*e^3*(b*d - a*e)^2*(-4*b*d + a*e) + 640*c^4*d^2*e^2*(5*b^2*d^2 - 8 *a*b*d*e + 3*a^2*e^2) + 40*b^2*c^2*e^4*(3*b^2*d^2 - 8*a*b*d*e + 6*a^2*e^2) )*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] + 512*c^3*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^(5/2)*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(512*c ^3*e^7)
Time = 2.63 (sec) , antiderivative size = 658, normalized size of antiderivative = 1.05, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {1231, 27, 1231, 27, 1231, 27, 1269, 1092, 219, 1154, 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.
| \(\displaystyle \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{d+e x} \, dx\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle -\frac {\int \frac {c \left (11 d e b^2-12 \left (c d^2+a e^2\right ) b+4 a c d e-\left (24 c^2 d^2+b^2 e^2-4 c e (6 b d-5 a e)\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{d+e x}dx}{12 c e^2}-\frac {\left (a+b x+c x^2\right )^{5/2} (-11 b e+12 c d-10 c e x)}{30 e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {\left (11 d e b^2-12 \left (c d^2+a e^2\right ) b+4 a c d e-\left (24 c^2 d^2+b^2 e^2-4 c e (6 b d-5 a e)\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{d+e x}dx}{12 e^2}-\frac {\left (a+b x+c x^2\right )^{5/2} (-11 b e+12 c d-10 c e x)}{30 e^2}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle -\frac {\frac {\left (a+b x+c x^2\right )^{3/2} \left (-2 c e x \left (-4 c e (6 b d-5 a e)+b^2 e^2+24 c^2 d^2\right )-8 c^2 d e (15 b d-8 a e)+4 b c e^2 (14 b d-13 a e)-b^3 e^3+64 c^3 d^3\right )}{8 c e^2}-\frac {\int \frac {\left (d \left (-3 e b^2+8 c d b-4 a c e\right ) \left (24 c^2 d^2+b^2 e^2-4 c e (6 b d-5 a e)\right )+8 c e (b d-2 a e) \left (11 d e b^2-12 \left (c d^2+a e^2\right ) b+4 a c d e\right )+3 \left (16 c^2 d^2-b^2 e^2-4 c e (4 b d-5 a e)\right ) \left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right ) x\right ) \sqrt {c x^2+b x+a}}{2 (d+e x)}dx}{8 c e^2}}{12 e^2}-\frac {\left (a+b x+c x^2\right )^{5/2} (-11 b e+12 c d-10 c e x)}{30 e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\left (a+b x+c x^2\right )^{3/2} \left (-2 c e x \left (-4 c e (6 b d-5 a e)+b^2 e^2+24 c^2 d^2\right )-8 c^2 d e (15 b d-8 a e)+4 b c e^2 (14 b d-13 a e)-b^3 e^3+64 c^3 d^3\right )}{8 c e^2}-\frac {\int \frac {\left (d \left (-3 e b^2+8 c d b-4 a c e\right ) \left (24 c^2 d^2+b^2 e^2-4 c e (6 b d-5 a e)\right )+8 c e (b d-2 a e) \left (11 d e b^2-12 \left (c d^2+a e^2\right ) b+4 a c d e\right )+3 \left (16 c^2 d^2-b^2 e^2-4 c e (4 b d-5 a e)\right ) \left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right ) x\right ) \sqrt {c x^2+b x+a}}{d+e x}dx}{16 c e^2}}{12 e^2}-\frac {\left (a+b x+c x^2\right )^{5/2} (-11 b e+12 c d-10 c e x)}{30 e^2}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle -\frac {\frac {\left (a+b x+c x^2\right )^{3/2} \left (-2 c e x \left (-4 c e (6 b d-5 a e)+b^2 e^2+24 c^2 d^2\right )-8 c^2 d e (15 b d-8 a e)+4 b c e^2 (14 b d-13 a e)-b^3 e^3+64 c^3 d^3\right )}{8 c e^2}-\frac {-\frac {\int -\frac {3 d \left (-e b^2+4 c d b-4 a c e\right ) \left (16 c^2 d^2-b^2 e^2-4 c e (4 b d-5 a e)\right ) \left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right )-4 c e (b d-2 a e) \left (d \left (-3 e b^2+8 c d b-4 a c e\right ) \left (24 c^2 d^2+b^2 e^2-4 c e (6 b d-5 a e)\right )+8 c e (b d-2 a e) \left (11 d e b^2-12 \left (c d^2+a e^2\right ) b+4 a c d e\right )\right )+3 \left (1024 c^6 d^6-512 c^5 e (6 b d-5 a e) d^4+640 c^4 e^2 \left (5 b^2 d^2-8 a b e d+3 a^2 e^2\right ) d^2+b^6 e^6+4 b^4 c e^5 (2 b d-5 a e)-320 c^3 e^3 (b d-a e)^2 (4 b d-a e)+40 b^2 c^2 e^4 \left (3 b^2 d^2-8 a b e d+6 a^2 e^2\right )\right ) x}{2 (d+e x) \sqrt {c x^2+b x+a}}dx}{4 c e^2}-\frac {3 \sqrt {a+b x+c x^2} \left (32 c^3 d e^2 \left (16 a^2 e^2-55 a b d e+40 b^2 d^2\right )-8 b c^2 e^3 \left (42 a^2 e^2-92 a b d e+49 b^2 d^2\right )+8 b^3 c e^4 (b d-2 a e)-2 c e x \left (-4 c e (4 b d-5 a e)-b^2 e^2+16 c^2 d^2\right ) \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )-128 c^4 d^3 e (11 b d-8 a e)+b^5 e^5+512 c^5 d^5\right )}{4 c e^2}}{16 c e^2}}{12 e^2}-\frac {\left (a+b x+c x^2\right )^{5/2} (-11 b e+12 c d-10 c e x)}{30 e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\left (a+b x+c x^2\right )^{3/2} \left (-2 c e x \left (-4 c e (6 b d-5 a e)+b^2 e^2+24 c^2 d^2\right )-8 c^2 d e (15 b d-8 a e)+4 b c e^2 (14 b d-13 a e)-b^3 e^3+64 c^3 d^3\right )}{8 c e^2}-\frac {\frac {\int \frac {3 d \left (-e b^2+4 c d b-4 a c e\right ) \left (16 c^2 d^2-b^2 e^2-4 c e (4 b d-5 a e)\right ) \left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right )-4 c e (b d-2 a e) \left (d \left (-3 e b^2+8 c d b-4 a c e\right ) \left (24 c^2 d^2+b^2 e^2-4 c e (6 b d-5 a e)\right )+8 c e (b d-2 a e) \left (11 d e b^2-12 \left (c d^2+a e^2\right ) b+4 a c d e\right )\right )+3 \left (1024 c^6 d^6-512 c^5 e (6 b d-5 a e) d^4+640 c^4 e^2 \left (5 b^2 d^2-8 a b e d+3 a^2 e^2\right ) d^2+b^6 e^6+4 b^4 c e^5 (2 b d-5 a e)-320 c^3 e^3 (b d-a e)^2 (4 b d-a e)+40 b^2 c^2 e^4 \left (3 b^2 d^2-8 a b e d+6 a^2 e^2\right )\right ) x}{(d+e x) \sqrt {c x^2+b x+a}}dx}{8 c e^2}-\frac {3 \sqrt {a+b x+c x^2} \left (32 c^3 d e^2 \left (16 a^2 e^2-55 a b d e+40 b^2 d^2\right )-8 b c^2 e^3 \left (42 a^2 e^2-92 a b d e+49 b^2 d^2\right )+8 b^3 c e^4 (b d-2 a e)-2 c e x \left (-4 c e (4 b d-5 a e)-b^2 e^2+16 c^2 d^2\right ) \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )-128 c^4 d^3 e (11 b d-8 a e)+b^5 e^5+512 c^5 d^5\right )}{4 c e^2}}{16 c e^2}}{12 e^2}-\frac {\left (a+b x+c x^2\right )^{5/2} (-11 b e+12 c d-10 c e x)}{30 e^2}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle -\frac {\frac {\left (a+b x+c x^2\right )^{3/2} \left (-2 c e x \left (-4 c e (6 b d-5 a e)+b^2 e^2+24 c^2 d^2\right )-8 c^2 d e (15 b d-8 a e)+4 b c e^2 (14 b d-13 a e)-b^3 e^3+64 c^3 d^3\right )}{8 c e^2}-\frac {\frac {\frac {3 \left (640 c^4 d^2 e^2 \left (3 a^2 e^2-8 a b d e+5 b^2 d^2\right )+40 b^2 c^2 e^4 \left (6 a^2 e^2-8 a b d e+3 b^2 d^2\right )+4 b^4 c e^5 (2 b d-5 a e)-512 c^5 d^4 e (6 b d-5 a e)-320 c^3 e^3 (b d-a e)^2 (4 b d-a e)+b^6 e^6+1024 c^6 d^6\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{e}-\frac {1536 c^2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3 \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{8 c e^2}-\frac {3 \sqrt {a+b x+c x^2} \left (32 c^3 d e^2 \left (16 a^2 e^2-55 a b d e+40 b^2 d^2\right )-8 b c^2 e^3 \left (42 a^2 e^2-92 a b d e+49 b^2 d^2\right )+8 b^3 c e^4 (b d-2 a e)-2 c e x \left (-4 c e (4 b d-5 a e)-b^2 e^2+16 c^2 d^2\right ) \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )-128 c^4 d^3 e (11 b d-8 a e)+b^5 e^5+512 c^5 d^5\right )}{4 c e^2}}{16 c e^2}}{12 e^2}-\frac {\left (a+b x+c x^2\right )^{5/2} (-11 b e+12 c d-10 c e x)}{30 e^2}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle -\frac {\frac {\left (a+b x+c x^2\right )^{3/2} \left (-2 c e x \left (-4 c e (6 b d-5 a e)+b^2 e^2+24 c^2 d^2\right )-8 c^2 d e (15 b d-8 a e)+4 b c e^2 (14 b d-13 a e)-b^3 e^3+64 c^3 d^3\right )}{8 c e^2}-\frac {\frac {\frac {6 \left (640 c^4 d^2 e^2 \left (3 a^2 e^2-8 a b d e+5 b^2 d^2\right )+40 b^2 c^2 e^4 \left (6 a^2 e^2-8 a b d e+3 b^2 d^2\right )+4 b^4 c e^5 (2 b d-5 a e)-512 c^5 d^4 e (6 b d-5 a e)-320 c^3 e^3 (b d-a e)^2 (4 b d-a e)+b^6 e^6+1024 c^6 d^6\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{e}-\frac {1536 c^2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3 \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{8 c e^2}-\frac {3 \sqrt {a+b x+c x^2} \left (32 c^3 d e^2 \left (16 a^2 e^2-55 a b d e+40 b^2 d^2\right )-8 b c^2 e^3 \left (42 a^2 e^2-92 a b d e+49 b^2 d^2\right )+8 b^3 c e^4 (b d-2 a e)-2 c e x \left (-4 c e (4 b d-5 a e)-b^2 e^2+16 c^2 d^2\right ) \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )-128 c^4 d^3 e (11 b d-8 a e)+b^5 e^5+512 c^5 d^5\right )}{4 c e^2}}{16 c e^2}}{12 e^2}-\frac {\left (a+b x+c x^2\right )^{5/2} (-11 b e+12 c d-10 c e x)}{30 e^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\frac {\left (a+b x+c x^2\right )^{3/2} \left (-2 c e x \left (-4 c e (6 b d-5 a e)+b^2 e^2+24 c^2 d^2\right )-8 c^2 d e (15 b d-8 a e)+4 b c e^2 (14 b d-13 a e)-b^3 e^3+64 c^3 d^3\right )}{8 c e^2}-\frac {\frac {\frac {3 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (640 c^4 d^2 e^2 \left (3 a^2 e^2-8 a b d e+5 b^2 d^2\right )+40 b^2 c^2 e^4 \left (6 a^2 e^2-8 a b d e+3 b^2 d^2\right )+4 b^4 c e^5 (2 b d-5 a e)-512 c^5 d^4 e (6 b d-5 a e)-320 c^3 e^3 (b d-a e)^2 (4 b d-a e)+b^6 e^6+1024 c^6 d^6\right )}{\sqrt {c} e}-\frac {1536 c^2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3 \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{8 c e^2}-\frac {3 \sqrt {a+b x+c x^2} \left (32 c^3 d e^2 \left (16 a^2 e^2-55 a b d e+40 b^2 d^2\right )-8 b c^2 e^3 \left (42 a^2 e^2-92 a b d e+49 b^2 d^2\right )+8 b^3 c e^4 (b d-2 a e)-2 c e x \left (-4 c e (4 b d-5 a e)-b^2 e^2+16 c^2 d^2\right ) \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )-128 c^4 d^3 e (11 b d-8 a e)+b^5 e^5+512 c^5 d^5\right )}{4 c e^2}}{16 c e^2}}{12 e^2}-\frac {\left (a+b x+c x^2\right )^{5/2} (-11 b e+12 c d-10 c e x)}{30 e^2}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle -\frac {\frac {\left (a+b x+c x^2\right )^{3/2} \left (-2 c e x \left (-4 c e (6 b d-5 a e)+b^2 e^2+24 c^2 d^2\right )-8 c^2 d e (15 b d-8 a e)+4 b c e^2 (14 b d-13 a e)-b^3 e^3+64 c^3 d^3\right )}{8 c e^2}-\frac {\frac {\frac {3072 c^2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3 \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{e}+\frac {3 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (640 c^4 d^2 e^2 \left (3 a^2 e^2-8 a b d e+5 b^2 d^2\right )+40 b^2 c^2 e^4 \left (6 a^2 e^2-8 a b d e+3 b^2 d^2\right )+4 b^4 c e^5 (2 b d-5 a e)-512 c^5 d^4 e (6 b d-5 a e)-320 c^3 e^3 (b d-a e)^2 (4 b d-a e)+b^6 e^6+1024 c^6 d^6\right )}{\sqrt {c} e}}{8 c e^2}-\frac {3 \sqrt {a+b x+c x^2} \left (32 c^3 d e^2 \left (16 a^2 e^2-55 a b d e+40 b^2 d^2\right )-8 b c^2 e^3 \left (42 a^2 e^2-92 a b d e+49 b^2 d^2\right )+8 b^3 c e^4 (b d-2 a e)-2 c e x \left (-4 c e (4 b d-5 a e)-b^2 e^2+16 c^2 d^2\right ) \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )-128 c^4 d^3 e (11 b d-8 a e)+b^5 e^5+512 c^5 d^5\right )}{4 c e^2}}{16 c e^2}}{12 e^2}-\frac {\left (a+b x+c x^2\right )^{5/2} (-11 b e+12 c d-10 c e x)}{30 e^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\frac {\left (a+b x+c x^2\right )^{3/2} \left (-2 c e x \left (-4 c e (6 b d-5 a e)+b^2 e^2+24 c^2 d^2\right )-8 c^2 d e (15 b d-8 a e)+4 b c e^2 (14 b d-13 a e)-b^3 e^3+64 c^3 d^3\right )}{8 c e^2}-\frac {\frac {\frac {3 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (640 c^4 d^2 e^2 \left (3 a^2 e^2-8 a b d e+5 b^2 d^2\right )+40 b^2 c^2 e^4 \left (6 a^2 e^2-8 a b d e+3 b^2 d^2\right )+4 b^4 c e^5 (2 b d-5 a e)-512 c^5 d^4 e (6 b d-5 a e)-320 c^3 e^3 (b d-a e)^2 (4 b d-a e)+b^6 e^6+1024 c^6 d^6\right )}{\sqrt {c} e}-\frac {1536 c^2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^{5/2} \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{e}}{8 c e^2}-\frac {3 \sqrt {a+b x+c x^2} \left (32 c^3 d e^2 \left (16 a^2 e^2-55 a b d e+40 b^2 d^2\right )-8 b c^2 e^3 \left (42 a^2 e^2-92 a b d e+49 b^2 d^2\right )+8 b^3 c e^4 (b d-2 a e)-2 c e x \left (-4 c e (4 b d-5 a e)-b^2 e^2+16 c^2 d^2\right ) \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )-128 c^4 d^3 e (11 b d-8 a e)+b^5 e^5+512 c^5 d^5\right )}{4 c e^2}}{16 c e^2}}{12 e^2}-\frac {\left (a+b x+c x^2\right )^{5/2} (-11 b e+12 c d-10 c e x)}{30 e^2}\) |
Input:
Int[((b + 2*c*x)*(a + b*x + c*x^2)^(5/2))/(d + e*x),x]
Output:
-1/30*((12*c*d - 11*b*e - 10*c*e*x)*(a + b*x + c*x^2)^(5/2))/e^2 - (((64*c ^3*d^3 - b^3*e^3 + 4*b*c*e^2*(14*b*d - 13*a*e) - 8*c^2*d*e*(15*b*d - 8*a*e ) - 2*c*e*(24*c^2*d^2 + b^2*e^2 - 4*c*e*(6*b*d - 5*a*e))*x)*(a + b*x + c*x ^2)^(3/2))/(8*c*e^2) - ((-3*(512*c^5*d^5 + b^5*e^5 - 128*c^4*d^3*e*(11*b*d - 8*a*e) + 8*b^3*c*e^4*(b*d - 2*a*e) + 32*c^3*d*e^2*(40*b^2*d^2 - 55*a*b* d*e + 16*a^2*e^2) - 8*b*c^2*e^3*(49*b^2*d^2 - 92*a*b*d*e + 42*a^2*e^2) - 2 *c*e*(16*c^2*d^2 - b^2*e^2 - 4*c*e*(4*b*d - 5*a*e))*(8*c^2*d^2 + b^2*e^2 - 4*c*e*(2*b*d - a*e))*x)*Sqrt[a + b*x + c*x^2])/(4*c*e^2) + ((3*(1024*c^6* d^6 + b^6*e^6 + 4*b^4*c*e^5*(2*b*d - 5*a*e) - 512*c^5*d^4*e*(6*b*d - 5*a*e ) - 320*c^3*e^3*(b*d - a*e)^2*(4*b*d - a*e) + 640*c^4*d^2*e^2*(5*b^2*d^2 - 8*a*b*d*e + 3*a^2*e^2) + 40*b^2*c^2*e^4*(3*b^2*d^2 - 8*a*b*d*e + 6*a^2*e^ 2))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(Sqrt[c]*e) - (1536*c^2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^(5/2)*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])] )/e)/(8*c*e^2))/(16*c*e^2))/(12*e^2)
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[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ (c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)) Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c ^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x ] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || !R ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Integer Q[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Time = 2.02 (sec) , antiderivative size = 998, normalized size of antiderivative = 1.59
| method | result | size |
| risch | \(\frac {\left (1280 c^{5} e^{5} x^{5}+3968 b \,c^{4} e^{5} x^{4}-1536 c^{5} d \,e^{4} x^{4}+4160 a \,c^{4} e^{5} x^{3}+4176 b^{2} c^{3} e^{5} x^{3}-4992 b \,c^{4} d \,e^{4} x^{3}+1920 c^{5} d^{2} e^{3} x^{3}+9056 a b \,c^{3} e^{5} x^{2}-5632 a \,c^{4} d \,e^{4} x^{2}+1528 b^{3} c^{2} e^{5} x^{2}-5696 b^{2} c^{3} d \,e^{4} x^{2}+6720 b \,c^{4} d^{2} e^{3} x^{2}-2560 c^{5} d^{3} e^{2} x^{2}+5280 a^{2} c^{3} e^{5} x +5456 a \,b^{2} c^{2} e^{5} x -14272 a b \,c^{3} d \,e^{4} x +8640 a \,c^{4} d^{2} e^{3} x +10 b^{4} c \,e^{5} x -2480 b^{3} c^{2} d \,e^{4} x +8880 b^{2} c^{3} d^{2} e^{3} x -10240 b \,c^{4} d^{3} e^{2} x +3840 c^{5} d^{4} e x +8528 a^{2} b \,c^{2} e^{5}-11776 a^{2} c^{3} d \,e^{4}+280 a \,b^{3} c \,e^{5}-13280 a \,b^{2} c^{2} d \,e^{4}+31200 a b \,c^{3} d^{2} e^{3}-17920 a \,c^{4} d^{3} e^{2}-15 b^{5} e^{5}-120 b^{4} c d \,e^{4}+5880 b^{3} d^{2} e^{3} c^{2}-19200 b^{2} c^{3} d^{3} e^{2}+21120 b \,c^{4} d^{4} e -7680 d^{5} c^{5}\right ) \sqrt {c \,x^{2}+b x +a}}{3840 c^{2} e^{6}}+\frac {\frac {\left (320 a^{3} c^{3} e^{6}+240 a^{2} b^{2} c^{2} e^{6}-1920 a^{2} b \,c^{3} d \,e^{5}+1920 a^{2} c^{4} d^{2} e^{4}-20 a \,b^{4} c \,e^{6}-320 a \,b^{3} c^{2} d \,e^{5}+2880 a \,b^{2} c^{3} d^{2} e^{4}-5120 a b \,c^{4} d^{3} e^{3}+2560 a \,c^{5} d^{4} e^{2}+b^{6} e^{6}+8 b^{5} c d \,e^{5}+120 b^{4} c^{2} d^{2} e^{4}-1280 b^{3} c^{3} d^{3} e^{3}+3200 b^{2} c^{4} d^{4} e^{2}-3072 b \,c^{5} d^{5} e +1024 c^{6} d^{6}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{e \sqrt {c}}-\frac {512 \left (a^{3} b \,e^{7}-2 d \,e^{6} c \,a^{3}-3 a^{2} b^{2} d \,e^{6}+9 a^{2} b c \,d^{2} e^{5}-6 d^{3} e^{4} a^{2} c^{2}+3 a \,b^{3} d^{2} e^{5}-12 a \,b^{2} c \,d^{3} e^{4}+15 a b \,c^{2} d^{4} e^{3}-6 d^{5} e^{2} a \,c^{3}-b^{4} d^{3} e^{4}+5 b^{3} c \,d^{4} e^{3}-9 b^{2} c^{2} d^{5} e^{2}+7 b \,c^{3} d^{6} e -2 d^{7} c^{4}\right ) c^{2} \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}}{512 e^{6} c^{2}}\) | \(998\) |
| default | \(\text {Expression too large to display}\) | \(1203\) |
Input:
int((2*c*x+b)*(c*x^2+b*x+a)^(5/2)/(e*x+d),x,method=_RETURNVERBOSE)
Output:
1/3840/c^2*(1280*c^5*e^5*x^5+3968*b*c^4*e^5*x^4-1536*c^5*d*e^4*x^4+4160*a* c^4*e^5*x^3+4176*b^2*c^3*e^5*x^3-4992*b*c^4*d*e^4*x^3+1920*c^5*d^2*e^3*x^3 +9056*a*b*c^3*e^5*x^2-5632*a*c^4*d*e^4*x^2+1528*b^3*c^2*e^5*x^2-5696*b^2*c ^3*d*e^4*x^2+6720*b*c^4*d^2*e^3*x^2-2560*c^5*d^3*e^2*x^2+5280*a^2*c^3*e^5* x+5456*a*b^2*c^2*e^5*x-14272*a*b*c^3*d*e^4*x+8640*a*c^4*d^2*e^3*x+10*b^4*c *e^5*x-2480*b^3*c^2*d*e^4*x+8880*b^2*c^3*d^2*e^3*x-10240*b*c^4*d^3*e^2*x+3 840*c^5*d^4*e*x+8528*a^2*b*c^2*e^5-11776*a^2*c^3*d*e^4+280*a*b^3*c*e^5-132 80*a*b^2*c^2*d*e^4+31200*a*b*c^3*d^2*e^3-17920*a*c^4*d^3*e^2-15*b^5*e^5-12 0*b^4*c*d*e^4+5880*b^3*c^2*d^2*e^3-19200*b^2*c^3*d^3*e^2+21120*b*c^4*d^4*e -7680*c^5*d^5)*(c*x^2+b*x+a)^(1/2)/e^6+1/512/e^6/c^2*((320*a^3*c^3*e^6+240 *a^2*b^2*c^2*e^6-1920*a^2*b*c^3*d*e^5+1920*a^2*c^4*d^2*e^4-20*a*b^4*c*e^6- 320*a*b^3*c^2*d*e^5+2880*a*b^2*c^3*d^2*e^4-5120*a*b*c^4*d^3*e^3+2560*a*c^5 *d^4*e^2+b^6*e^6+8*b^5*c*d*e^5+120*b^4*c^2*d^2*e^4-1280*b^3*c^3*d^3*e^3+32 00*b^2*c^4*d^4*e^2-3072*b*c^5*d^5*e+1024*c^6*d^6)/e*ln((1/2*b+c*x)/c^(1/2) +(c*x^2+b*x+a)^(1/2))/c^(1/2)-512*(a^3*b*e^7-2*a^3*c*d*e^6-3*a^2*b^2*d*e^6 +9*a^2*b*c*d^2*e^5-6*a^2*c^2*d^3*e^4+3*a*b^3*d^2*e^5-12*a*b^2*c*d^3*e^4+15 *a*b*c^2*d^4*e^3-6*a*c^3*d^5*e^2-b^4*d^3*e^4+5*b^3*c*d^4*e^3-9*b^2*c^2*d^5 *e^2+7*b*c^3*d^6*e-2*c^4*d^7)*c^2/e^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln(( 2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2 )^(1/2)*(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/...
Timed out. \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{d+e x} \, dx=\text {Timed out} \] Input:
integrate((2*c*x+b)*(c*x^2+b*x+a)^(5/2)/(e*x+d),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{d+e x} \, dx=\int \frac {\left (b + 2 c x\right ) \left (a + b x + c x^{2}\right )^{\frac {5}{2}}}{d + e x}\, dx \] Input:
integrate((2*c*x+b)*(c*x**2+b*x+a)**(5/2)/(e*x+d),x)
Output:
Integral((b + 2*c*x)*(a + b*x + c*x**2)**(5/2)/(d + e*x), x)
Exception generated. \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{d+e x} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((2*c*x+b)*(c*x^2+b*x+a)^(5/2)/(e*x+d),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Exception generated. \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{d+e x} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((2*c*x+b)*(c*x^2+b*x+a)^(5/2)/(e*x+d),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{d+e x} \, dx=\int \frac {\left (b+2\,c\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{5/2}}{d+e\,x} \,d x \] Input:
int(((b + 2*c*x)*(a + b*x + c*x^2)^(5/2))/(d + e*x),x)
Output:
int(((b + 2*c*x)*(a + b*x + c*x^2)^(5/2))/(d + e*x), x)
\[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{d+e x} \, dx=\int \frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{e x +d}d x \] Input:
int((2*c*x+b)*(c*x^2+b*x+a)^(5/2)/(e*x+d),x)
Output:
int((2*c*x+b)*(c*x^2+b*x+a)^(5/2)/(e*x+d),x)