Integrand size = 44, antiderivative size = 360 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{11}} \, dx=-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 e^2 (2 c d-b e) (d+e x)^{11}}-\frac {2 (8 c e f+22 c d g-15 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{195 e^2 (2 c d-b e)^2 (d+e x)^{10}}-\frac {4 c (8 c e f+22 c d g-15 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{715 e^2 (2 c d-b e)^3 (d+e x)^9}-\frac {16 c^2 (8 c e f+22 c d g-15 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{6435 e^2 (2 c d-b e)^4 (d+e x)^8}-\frac {32 c^3 (8 c e f+22 c d g-15 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{45045 e^2 (2 c d-b e)^5 (d+e x)^7} \]
-2/15*(-d*g+e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(7/2)/e^2/(-b*e+2*c*d)/( e*x+d)^11-2/195*(-15*b*e*g+22*c*d*g+8*c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x ^2)^(7/2)/e^2/(-b*e+2*c*d)^2/(e*x+d)^10-4/715*c*(-15*b*e*g+22*c*d*g+8*c*e* f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(7/2)/e^2/(-b*e+2*c*d)^3/(e*x+d)^9-16/ 6435*c^2*(-15*b*e*g+22*c*d*g+8*c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(7/ 2)/e^2/(-b*e+2*c*d)^4/(e*x+d)^8-32/45045*c^3*(-15*b*e*g+22*c*d*g+8*c*e*f)* (d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(7/2)/e^2/(-b*e+2*c*d)^5/(e*x+d)^7
Time = 0.41 (sec) , antiderivative size = 351, normalized size of antiderivative = 0.98 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{11}} \, dx=-\frac {2 (-c d+b e+c e x)^3 \sqrt {(d+e x) (-b e+c (d-e x))} \left (231 b^4 e^4 (13 e f+2 d g+15 e g x)+84 b^2 c^2 e^2 \left (133 d^3 g+2 e^3 x^2 (6 f+5 g x)+3 d e^2 x (52 f+51 g x)+6 d^2 e (167 f+189 g x)\right )-42 b^3 c e^3 \left (89 d^2 g+e^2 x (44 f+45 g x)+d e (616 f+706 g x)\right )+16 c^4 \left (407 d^5 g+8 e^5 f x^4+22 d e^4 x^3 (4 f+g x)+11 d^3 e^2 x (148 f+117 g x)+2 d^2 e^3 x^2 (234 f+121 g x)+d^4 e (4243 f+4477 g x)\right )-8 b c^3 e \left (1801 d^4 g+2 e^4 x^3 (28 f+15 g x)+4 d e^3 x^2 (168 f+121 g x)+3 d^2 e^2 x (1316 f+1201 g x)+2 d^3 e (7672 f+8481 g x)\right )\right )}{45045 e^2 (-2 c d+b e)^5 (d+e x)^8} \]
(-2*(-(c*d) + b*e + c*e*x)^3*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(231*b ^4*e^4*(13*e*f + 2*d*g + 15*e*g*x) + 84*b^2*c^2*e^2*(133*d^3*g + 2*e^3*x^2 *(6*f + 5*g*x) + 3*d*e^2*x*(52*f + 51*g*x) + 6*d^2*e*(167*f + 189*g*x)) - 42*b^3*c*e^3*(89*d^2*g + e^2*x*(44*f + 45*g*x) + d*e*(616*f + 706*g*x)) + 16*c^4*(407*d^5*g + 8*e^5*f*x^4 + 22*d*e^4*x^3*(4*f + g*x) + 11*d^3*e^2*x* (148*f + 117*g*x) + 2*d^2*e^3*x^2*(234*f + 121*g*x) + d^4*e*(4243*f + 4477 *g*x)) - 8*b*c^3*e*(1801*d^4*g + 2*e^4*x^3*(28*f + 15*g*x) + 4*d*e^3*x^2*( 168*f + 121*g*x) + 3*d^2*e^2*x*(1316*f + 1201*g*x) + 2*d^3*e*(7672*f + 848 1*g*x))))/(45045*e^2*(-2*c*d + b*e)^5*(d + e*x)^8)
Time = 0.59 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {1220, 1129, 1129, 1129, 1123}
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.
| \(\displaystyle \int \frac {(f+g x) \left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{5/2}}{(d+e x)^{11}} \, dx\) |
| \(\Big \downarrow \) 1220 |
| \(\displaystyle \frac {(-15 b e g+22 c d g+8 c e f) \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}}{(d+e x)^{10}}dx}{15 e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 e^2 (d+e x)^{11} (2 c d-b e)}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {(-15 b e g+22 c d g+8 c e f) \left (\frac {6 c \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}}{(d+e x)^9}dx}{13 (2 c d-b e)}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 e (d+e x)^{10} (2 c d-b e)}\right )}{15 e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 e^2 (d+e x)^{11} (2 c d-b e)}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {(-15 b e g+22 c d g+8 c e f) \left (\frac {6 c \left (\frac {4 c \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}}{(d+e x)^8}dx}{11 (2 c d-b e)}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 e (d+e x)^9 (2 c d-b e)}\right )}{13 (2 c d-b e)}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 e (d+e x)^{10} (2 c d-b e)}\right )}{15 e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 e^2 (d+e x)^{11} (2 c d-b e)}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {(-15 b e g+22 c d g+8 c e f) \left (\frac {6 c \left (\frac {4 c \left (\frac {2 c \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}}{(d+e x)^7}dx}{9 (2 c d-b e)}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 e (d+e x)^8 (2 c d-b e)}\right )}{11 (2 c d-b e)}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 e (d+e x)^9 (2 c d-b e)}\right )}{13 (2 c d-b e)}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 e (d+e x)^{10} (2 c d-b e)}\right )}{15 e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 e^2 (d+e x)^{11} (2 c d-b e)}\) |
| \(\Big \downarrow \) 1123 |
| \(\displaystyle \frac {\left (\frac {6 c \left (\frac {4 c \left (-\frac {4 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{63 e (d+e x)^7 (2 c d-b e)^2}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 e (d+e x)^8 (2 c d-b e)}\right )}{11 (2 c d-b e)}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 e (d+e x)^9 (2 c d-b e)}\right )}{13 (2 c d-b e)}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 e (d+e x)^{10} (2 c d-b e)}\right ) (-15 b e g+22 c d g+8 c e f)}{15 e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 e^2 (d+e x)^{11} (2 c d-b e)}\) |
(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(15*e^2*(2*c* d - b*e)*(d + e*x)^11) + ((8*c*e*f + 22*c*d*g - 15*b*e*g)*((-2*(d*(c*d - b *e) - b*e^2*x - c*e^2*x^2)^(7/2))/(13*e*(2*c*d - b*e)*(d + e*x)^10) + (6*c *((-2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(11*e*(2*c*d - b*e)*(d + e*x)^9) + (4*c*((-2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(9*e*(2 *c*d - b*e)*(d + e*x)^8) - (4*c*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2 ))/(63*e*(2*c*d - b*e)^2*(d + e*x)^7)))/(11*(2*c*d - b*e))))/(13*(2*c*d - b*e))))/(15*e*(2*c*d - b*e))
3.23.8.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(2*c*d - b *e))), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[m + 2*p + 2, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(-e)*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((m + p + 1)*(2* c*d - b*e))), x] + Simp[c*(Simplify[m + 2*p + 2]/((m + p + 1)*(2*c*d - b*e) )) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d , e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[Simplify[m + 2*p + 2], 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d*g - e*f)*(d + e*x)^m*((a + b*x + c*x ^2)^(p + 1)/((2*c*d - b*e)*(m + p + 1))), x] + Simp[(m*(g*(c*d - b*e) + c*e *f) + e*(p + 1)*(2*c*f - b*g))/(e*(2*c*d - b*e)*(m + p + 1)) Int[(d + e*x )^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ((LtQ[m, -1] && !IGtQ[m + p + 1, 0 ]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0 ]
Time = 20.11 (sec) , antiderivative size = 564, normalized size of antiderivative = 1.57
| method | result | size |
| gosper | \(-\frac {2 \left (x c e +b e -c d \right ) \left (-240 b \,c^{3} e^{5} g \,x^{4}+352 c^{4} d \,e^{4} g \,x^{4}+128 c^{4} e^{5} f \,x^{4}+840 b^{2} c^{2} e^{5} g \,x^{3}-3872 b \,c^{3} d \,e^{4} g \,x^{3}-448 b \,c^{3} e^{5} f \,x^{3}+3872 c^{4} d^{2} e^{3} g \,x^{3}+1408 c^{4} d \,e^{4} f \,x^{3}-1890 b^{3} c \,e^{5} g \,x^{2}+12852 b^{2} c^{2} d \,e^{4} g \,x^{2}+1008 b^{2} c^{2} e^{5} f \,x^{2}-28824 b \,c^{3} d^{2} e^{3} g \,x^{2}-5376 b \,c^{3} d \,e^{4} f \,x^{2}+20592 c^{4} d^{3} e^{2} g \,x^{2}+7488 c^{4} d^{2} e^{3} f \,x^{2}+3465 b^{4} e^{5} g x -29652 b^{3} c d \,e^{4} g x -1848 b^{3} c \,e^{5} f x +95256 b^{2} c^{2} d^{2} e^{3} g x +13104 b^{2} c^{2} d \,e^{4} f x -135696 b \,c^{3} d^{3} e^{2} g x -31584 b \,c^{3} d^{2} e^{3} f x +71632 c^{4} d^{4} e g x +26048 c^{4} d^{3} e^{2} f x +462 b^{4} d \,e^{4} g +3003 b^{4} e^{5} f -3738 b^{3} c \,d^{2} e^{3} g -25872 b^{3} c d \,e^{4} f +11172 b^{2} c^{2} d^{3} e^{2} g +84168 b^{2} c^{2} d^{2} e^{3} f -14408 b \,c^{3} d^{4} e g -122752 b \,c^{3} d^{3} e^{2} f +6512 c^{4} d^{5} g +67888 c^{4} d^{4} e f \right ) \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {5}{2}}}{45045 \left (e x +d \right )^{10} e^{2} \left (b^{5} e^{5}-10 b^{4} c d \,e^{4}+40 b^{3} c^{2} d^{2} e^{3}-80 b^{2} c^{3} d^{3} e^{2}+80 b \,c^{4} d^{4} e -32 c^{5} d^{5}\right )}\) | \(564\) |
| default | \(\frac {g \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{13 \left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )^{10}}+\frac {6 c \,e^{2} \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{11 \left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )^{9}}+\frac {4 c \,e^{2} \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{9 \left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )^{8}}-\frac {4 c \,e^{2} \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{63 \left (-b \,e^{2}+2 c d e \right )^{2} \left (x +\frac {d}{e}\right )^{7}}\right )}{11 \left (-b \,e^{2}+2 c d e \right )}\right )}{13 \left (-b \,e^{2}+2 c d e \right )}\right )}{e^{11}}+\frac {\left (-d g +e f \right ) \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{15 \left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )^{11}}+\frac {8 c \,e^{2} \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{13 \left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )^{10}}+\frac {6 c \,e^{2} \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{11 \left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )^{9}}+\frac {4 c \,e^{2} \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{9 \left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )^{8}}-\frac {4 c \,e^{2} \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{63 \left (-b \,e^{2}+2 c d e \right )^{2} \left (x +\frac {d}{e}\right )^{7}}\right )}{11 \left (-b \,e^{2}+2 c d e \right )}\right )}{13 \left (-b \,e^{2}+2 c d e \right )}\right )}{15 \left (-b \,e^{2}+2 c d e \right )}\right )}{e^{12}}\) | \(701\) |
| trager | \(\text {Expression too large to display}\) | \(1272\) |
-2/45045*(c*e*x+b*e-c*d)*(-240*b*c^3*e^5*g*x^4+352*c^4*d*e^4*g*x^4+128*c^4 *e^5*f*x^4+840*b^2*c^2*e^5*g*x^3-3872*b*c^3*d*e^4*g*x^3-448*b*c^3*e^5*f*x^ 3+3872*c^4*d^2*e^3*g*x^3+1408*c^4*d*e^4*f*x^3-1890*b^3*c*e^5*g*x^2+12852*b ^2*c^2*d*e^4*g*x^2+1008*b^2*c^2*e^5*f*x^2-28824*b*c^3*d^2*e^3*g*x^2-5376*b *c^3*d*e^4*f*x^2+20592*c^4*d^3*e^2*g*x^2+7488*c^4*d^2*e^3*f*x^2+3465*b^4*e ^5*g*x-29652*b^3*c*d*e^4*g*x-1848*b^3*c*e^5*f*x+95256*b^2*c^2*d^2*e^3*g*x+ 13104*b^2*c^2*d*e^4*f*x-135696*b*c^3*d^3*e^2*g*x-31584*b*c^3*d^2*e^3*f*x+7 1632*c^4*d^4*e*g*x+26048*c^4*d^3*e^2*f*x+462*b^4*d*e^4*g+3003*b^4*e^5*f-37 38*b^3*c*d^2*e^3*g-25872*b^3*c*d*e^4*f+11172*b^2*c^2*d^3*e^2*g+84168*b^2*c ^2*d^2*e^3*f-14408*b*c^3*d^4*e*g-122752*b*c^3*d^3*e^2*f+6512*c^4*d^5*g+678 88*c^4*d^4*e*f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^10/e^2/(b^5 *e^5-10*b^4*c*d*e^4+40*b^3*c^2*d^2*e^3-80*b^2*c^3*d^3*e^2+80*b*c^4*d^4*e-3 2*c^5*d^5)
Timed out. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{11}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{11}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{11}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(b*e-2*c*d>0)', see `assume?` for more deta
Timed out. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{11}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{11}} \, dx=\text {Hanged} \]