Integrand size = 44, antiderivative size = 360 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^9} \, dx=-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{13 e^2 (2 c d-b e) (d+e x)^9}-\frac {2 (8 c e f+18 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{143 e^2 (2 c d-b e)^2 (d+e x)^8}-\frac {4 c (8 c e f+18 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{429 e^2 (2 c d-b e)^3 (d+e x)^7}-\frac {16 c^2 (8 c e f+18 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3003 e^2 (2 c d-b e)^4 (d+e x)^6}-\frac {32 c^3 (8 c e f+18 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{15015 e^2 (2 c d-b e)^5 (d+e x)^5} \] Output:
-2/13*(-d*g+e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(5/2)/e^2/(-b*e+2*c*d)/( e*x+d)^9-2/143*(-13*b*e*g+18*c*d*g+8*c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^ 2)^(5/2)/e^2/(-b*e+2*c*d)^2/(e*x+d)^8-4/429*c*(-13*b*e*g+18*c*d*g+8*c*e*f) *(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(5/2)/e^2/(-b*e+2*c*d)^3/(e*x+d)^7-16/30 03*c^2*(-13*b*e*g+18*c*d*g+8*c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(5/2) /e^2/(-b*e+2*c*d)^4/(e*x+d)^6-32/15015*c^3*(-13*b*e*g+18*c*d*g+8*c*e*f)*(d *(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(5/2)/e^2/(-b*e+2*c*d)^5/(e*x+d)^5
Time = 0.69 (sec) , antiderivative size = 349, normalized size of antiderivative = 0.97 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^9} \, dx=\frac {2 (-c d+b e+c e x)^2 \sqrt {(d+e x) (-b e+c (d-e x))} \left (105 b^4 e^4 (11 e f+2 d g+13 e g x)-70 b^3 c e^3 \left (25 d^2 g+e^2 x (12 f+13 g x)+2 d e (72 f+85 g x)\right )+20 b^2 c^2 e^2 \left (271 d^3 g+2 e^3 x^2 (14 f+13 g x)+d e^2 x (308 f+323 g x)+2 d^2 e (833 f+977 g x)\right )+16 c^4 \left (213 d^5 g+8 e^5 f x^4+18 d e^4 x^3 (4 f+g x)+2 d^2 e^3 x^2 (154 f+81 g x)+3 d^3 e^2 x (284 f+231 g x)+d^4 e (1763 f+1917 g x)\right )-8 b c^3 e \left (911 d^4 g+2 e^4 x^3 (20 f+13 g x)+4 d e^3 x^2 (100 f+81 g x)+d^2 e^2 x (1940 f+1901 g x)+d^3 e (6200 f+7134 g x)\right )\right )}{15015 e^2 (-2 c d+b e)^5 (d+e x)^7} \] Input:
Integrate[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2))/(d + e*x )^9,x]
Output:
(2*(-(c*d) + b*e + c*e*x)^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(105*b^ 4*e^4*(11*e*f + 2*d*g + 13*e*g*x) - 70*b^3*c*e^3*(25*d^2*g + e^2*x*(12*f + 13*g*x) + 2*d*e*(72*f + 85*g*x)) + 20*b^2*c^2*e^2*(271*d^3*g + 2*e^3*x^2* (14*f + 13*g*x) + d*e^2*x*(308*f + 323*g*x) + 2*d^2*e*(833*f + 977*g*x)) + 16*c^4*(213*d^5*g + 8*e^5*f*x^4 + 18*d*e^4*x^3*(4*f + g*x) + 2*d^2*e^3*x^ 2*(154*f + 81*g*x) + 3*d^3*e^2*x*(284*f + 231*g*x) + d^4*e*(1763*f + 1917* g*x)) - 8*b*c^3*e*(911*d^4*g + 2*e^4*x^3*(20*f + 13*g*x) + 4*d*e^3*x^2*(10 0*f + 81*g*x) + d^2*e^2*x*(1940*f + 1901*g*x) + d^3*e*(6200*f + 7134*g*x)) ))/(15015*e^2*(-2*c*d + b*e)^5*(d + e*x)^7)
Time = 1.01 (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.
| \(\displaystyle \int \frac {(f+g x) \left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^9} \, dx\) |
| \(\Big \downarrow \) 1220 |
| \(\displaystyle \frac {(-13 b e g+18 c d g+8 c e f) \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}{(d+e x)^8}dx}{13 e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{13 e^2 (d+e x)^9 (2 c d-b e)}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {(-13 b e g+18 c d g+8 c e f) \left (\frac {6 c \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}{(d+e x)^7}dx}{11 (2 c d-b e)}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{11 e (d+e x)^8 (2 c d-b e)}\right )}{13 e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{13 e^2 (d+e x)^9 (2 c d-b e)}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {(-13 b e g+18 c d g+8 c e f) \left (\frac {6 c \left (\frac {4 c \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}{(d+e x)^6}dx}{9 (2 c d-b e)}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{9 e (d+e x)^7 (2 c d-b e)}\right )}{11 (2 c d-b e)}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{11 e (d+e x)^8 (2 c d-b e)}\right )}{13 e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{13 e^2 (d+e x)^9 (2 c d-b e)}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {(-13 b e g+18 c d g+8 c e f) \left (\frac {6 c \left (\frac {4 c \left (\frac {2 c \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}{(d+e x)^5}dx}{7 (2 c d-b e)}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 e (d+e x)^6 (2 c d-b e)}\right )}{9 (2 c d-b e)}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{9 e (d+e x)^7 (2 c d-b e)}\right )}{11 (2 c d-b e)}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{11 e (d+e x)^8 (2 c d-b e)}\right )}{13 e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{13 e^2 (d+e x)^9 (2 c d-b e)}\) |
| \(\Big \downarrow \) 1123 |
| \(\displaystyle \frac {\left (\frac {6 c \left (\frac {4 c \left (-\frac {4 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{35 e (d+e x)^5 (2 c d-b e)^2}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 e (d+e x)^6 (2 c d-b e)}\right )}{9 (2 c d-b e)}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{9 e (d+e x)^7 (2 c d-b e)}\right )}{11 (2 c d-b e)}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{11 e (d+e x)^8 (2 c d-b e)}\right ) (-13 b e g+18 c d g+8 c e f)}{13 e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{13 e^2 (d+e x)^9 (2 c d-b e)}\) |
Input:
Int[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2))/(d + e*x)^9,x]
Output:
(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(13*e^2*(2*c* d - b*e)*(d + e*x)^9) + ((8*c*e*f + 18*c*d*g - 13*b*e*g)*((-2*(d*(c*d - b* e) - b*e^2*x - c*e^2*x^2)^(5/2))/(11*e*(2*c*d - b*e)*(d + e*x)^8) + (6*c*( (-2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(9*e*(2*c*d - b*e)*(d + e *x)^7) + (4*c*((-2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(7*e*(2*c* d - b*e)*(d + e*x)^6) - (4*c*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/ (35*e*(2*c*d - b*e)^2*(d + e*x)^5)))/(9*(2*c*d - b*e))))/(11*(2*c*d - b*e) )))/(13*e*(2*c*d - b*e))
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 = 18.31 (sec) , antiderivative size = 564, normalized size of antiderivative = 1.57
| method | result | size |
| gosper | \(-\frac {2 \left (c e x +b e -c d \right ) \left (-208 b \,c^{3} e^{5} g \,x^{4}+288 c^{4} d \,e^{4} g \,x^{4}+128 c^{4} e^{5} f \,x^{4}+520 b^{2} c^{2} e^{5} g \,x^{3}-2592 b \,c^{3} d \,e^{4} g \,x^{3}-320 b \,c^{3} e^{5} f \,x^{3}+2592 c^{4} d^{2} e^{3} g \,x^{3}+1152 c^{4} d \,e^{4} f \,x^{3}-910 b^{3} c \,e^{5} g \,x^{2}+6460 b^{2} c^{2} d \,e^{4} g \,x^{2}+560 b^{2} c^{2} e^{5} f \,x^{2}-15208 b \,c^{3} d^{2} e^{3} g \,x^{2}-3200 b \,c^{3} d \,e^{4} f \,x^{2}+11088 c^{4} d^{3} e^{2} g \,x^{2}+4928 c^{4} d^{2} e^{3} f \,x^{2}+1365 b^{4} e^{5} g x -11900 b^{3} c d \,e^{4} g x -840 b^{3} c \,e^{5} f x +39080 b^{2} c^{2} d^{2} e^{3} g x +6160 b^{2} c^{2} d \,e^{4} f x -57072 b \,c^{3} d^{3} e^{2} g x -15520 b \,c^{3} d^{2} e^{3} f x +30672 c^{4} d^{4} e g x +13632 c^{4} d^{3} e^{2} f x +210 b^{4} d \,e^{4} g +1155 b^{4} e^{5} f -1750 b^{3} c \,d^{2} e^{3} g -10080 b^{3} c d \,e^{4} f +5420 b^{2} c^{2} d^{3} e^{2} g +33320 b^{2} c^{2} d^{2} e^{3} f -7288 b \,c^{3} d^{4} e g -49600 b \,c^{3} d^{3} e^{2} f +3408 c^{4} d^{5} g +28208 c^{4} d^{4} e f \right ) \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {3}{2}}}{15015 \left (e x +d \right )^{8} e^{2} \left (b^{5} e^{5}-10 b^{4} c d \,e^{4}+40 b^{3} d^{2} e^{3} c^{2}-80 b^{2} c^{3} d^{3} e^{2}+80 b \,c^{4} d^{4} e -32 d^{5} c^{5}\right )}\) | \(564\) |
| orering | \(-\frac {2 \left (c e x +b e -c d \right ) \left (-208 b \,c^{3} e^{5} g \,x^{4}+288 c^{4} d \,e^{4} g \,x^{4}+128 c^{4} e^{5} f \,x^{4}+520 b^{2} c^{2} e^{5} g \,x^{3}-2592 b \,c^{3} d \,e^{4} g \,x^{3}-320 b \,c^{3} e^{5} f \,x^{3}+2592 c^{4} d^{2} e^{3} g \,x^{3}+1152 c^{4} d \,e^{4} f \,x^{3}-910 b^{3} c \,e^{5} g \,x^{2}+6460 b^{2} c^{2} d \,e^{4} g \,x^{2}+560 b^{2} c^{2} e^{5} f \,x^{2}-15208 b \,c^{3} d^{2} e^{3} g \,x^{2}-3200 b \,c^{3} d \,e^{4} f \,x^{2}+11088 c^{4} d^{3} e^{2} g \,x^{2}+4928 c^{4} d^{2} e^{3} f \,x^{2}+1365 b^{4} e^{5} g x -11900 b^{3} c d \,e^{4} g x -840 b^{3} c \,e^{5} f x +39080 b^{2} c^{2} d^{2} e^{3} g x +6160 b^{2} c^{2} d \,e^{4} f x -57072 b \,c^{3} d^{3} e^{2} g x -15520 b \,c^{3} d^{2} e^{3} f x +30672 c^{4} d^{4} e g x +13632 c^{4} d^{3} e^{2} f x +210 b^{4} d \,e^{4} g +1155 b^{4} e^{5} f -1750 b^{3} c \,d^{2} e^{3} g -10080 b^{3} c d \,e^{4} f +5420 b^{2} c^{2} d^{3} e^{2} g +33320 b^{2} c^{2} d^{2} e^{3} f -7288 b \,c^{3} d^{4} e g -49600 b \,c^{3} d^{3} e^{2} f +3408 c^{4} d^{5} g +28208 c^{4} d^{4} e f \right ) \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {3}{2}}}{15015 \left (e x +d \right )^{8} e^{2} \left (b^{5} e^{5}-10 b^{4} c d \,e^{4}+40 b^{3} d^{2} e^{3} c^{2}-80 b^{2} c^{3} d^{3} e^{2}+80 b \,c^{4} d^{4} e -32 d^{5} c^{5}\right )}\) | \(564\) |
| default | \(\frac {g \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{11 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{8}}+\frac {6 c \,e^{2} \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{9 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{7}}+\frac {4 c \,e^{2} \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{7 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{6}}-\frac {4 c \,e^{2} \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{35 \left (-b \,e^{2}+2 d e c \right )^{2} \left (x +\frac {d}{e}\right )^{5}}\right )}{9 \left (-b \,e^{2}+2 d e c \right )}\right )}{11 \left (-b \,e^{2}+2 d e c \right )}\right )}{e^{9}}-\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 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{13 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{9}}+\frac {8 c \,e^{2} \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{11 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{8}}+\frac {6 c \,e^{2} \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{9 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{7}}+\frac {4 c \,e^{2} \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{7 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{6}}-\frac {4 c \,e^{2} \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{35 \left (-b \,e^{2}+2 d e c \right )^{2} \left (x +\frac {d}{e}\right )^{5}}\right )}{9 \left (-b \,e^{2}+2 d e c \right )}\right )}{11 \left (-b \,e^{2}+2 d e c \right )}\right )}{13 \left (-b \,e^{2}+2 d e c \right )}\right )}{e^{10}}\) | \(702\) |
| trager | \(\frac {2 \left (-208 b \,c^{5} e^{7} g \,x^{6}+288 c^{6} d \,e^{6} g \,x^{6}+128 c^{6} e^{7} f \,x^{6}+104 b^{2} c^{4} e^{7} g \,x^{5}-1600 b \,c^{5} d \,e^{6} g \,x^{5}-64 b \,c^{5} e^{7} f \,x^{5}+2016 c^{6} d^{2} e^{5} g \,x^{5}+896 c^{6} d \,e^{6} f \,x^{5}-78 b^{3} c^{3} e^{7} g \,x^{4}+940 b^{2} c^{4} d \,e^{6} g \,x^{4}+48 b^{2} c^{4} e^{7} f \,x^{4}-5624 b \,c^{5} d^{2} e^{5} g \,x^{4}-512 b \,c^{5} d \,e^{6} f \,x^{4}+6192 c^{6} d^{3} e^{4} g \,x^{4}+2752 c^{6} d^{2} e^{5} f \,x^{4}+65 b^{4} c^{2} e^{7} g \,x^{3}-792 b^{3} c^{3} d \,e^{6} g \,x^{3}-40 b^{3} c^{3} e^{7} f \,x^{3}+4040 b^{2} c^{4} d^{2} e^{5} g \,x^{3}+432 b^{2} c^{4} d \,e^{6} f \,x^{3}-12256 b \,c^{5} d^{3} e^{4} g \,x^{3}-1888 b \,c^{5} d^{2} e^{5} f \,x^{3}+11088 c^{6} d^{4} e^{3} g \,x^{3}+4928 c^{6} d^{3} e^{4} f \,x^{3}+1820 b^{5} c \,e^{7} g \,x^{2}-18040 b^{4} c^{2} d \,e^{6} g \,x^{2}+35 b^{4} c^{2} e^{7} f \,x^{2}+71172 b^{3} c^{3} d^{2} e^{5} g \,x^{2}-400 b^{3} c^{3} d \,e^{6} f \,x^{2}-138920 b^{2} c^{4} d^{3} e^{4} g \,x^{2}+1848 b^{2} c^{4} d^{2} e^{5} f \,x^{2}+130816 b \,c^{5} d^{4} e^{3} g \,x^{2}-4352 b \,c^{5} d^{3} e^{4} f \,x^{2}-46848 c^{6} d^{5} e^{2} g \,x^{2}+5872 c^{6} d^{4} e^{3} f \,x^{2}+1365 b^{6} e^{7} g x -14210 b^{5} c d \,e^{6} g x +1470 b^{5} c \,e^{7} f x +60325 b^{4} c^{2} d^{2} e^{5} g x -14630 b^{4} c^{2} d \,e^{6} f x -132792 b^{3} c^{3} d^{3} e^{4} g x +58120 b^{3} c^{3} d^{2} e^{5} f x +158480 b^{2} c^{4} d^{4} e^{3} g x -115008 b^{2} c^{4} d^{3} e^{4} f x -97024 b \,c^{5} d^{5} e^{2} g x +112832 b \,c^{5} d^{4} e^{3} f x +23856 c^{6} d^{6} e g x -42784 c^{6} d^{5} e^{2} f x +210 b^{6} d \,e^{6} g +1155 b^{6} e^{7} f -2170 b^{5} c \,d^{2} e^{5} g -12390 b^{5} c d \,e^{6} f +9130 b^{4} c^{2} d^{3} e^{4} g +54635 b^{4} c^{2} d^{2} e^{5} f -19878 b^{3} c^{3} d^{4} e^{3} g -126320 b^{3} c^{3} d^{3} e^{4} f +23404 b^{2} c^{4} d^{5} e^{2} g +160728 b^{2} c^{4} d^{4} e^{3} f -14104 b \,c^{5} d^{6} e g -106016 b \,c^{5} d^{5} e^{2} f +3408 c^{6} d^{7} g +28208 c^{6} d^{6} e f \right ) \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}{15015 \left (b^{5} e^{5}-10 b^{4} c d \,e^{4}+40 b^{3} d^{2} e^{3} c^{2}-80 b^{2} c^{3} d^{3} e^{2}+80 b \,c^{4} d^{4} e -32 d^{5} c^{5}\right ) e^{2} \left (e x +d \right )^{7}}\) | \(996\) |
Input:
int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^9,x,method=_RET URNVERBOSE)
Output:
-2/15015*(c*e*x+b*e-c*d)*(-208*b*c^3*e^5*g*x^4+288*c^4*d*e^4*g*x^4+128*c^4 *e^5*f*x^4+520*b^2*c^2*e^5*g*x^3-2592*b*c^3*d*e^4*g*x^3-320*b*c^3*e^5*f*x^ 3+2592*c^4*d^2*e^3*g*x^3+1152*c^4*d*e^4*f*x^3-910*b^3*c*e^5*g*x^2+6460*b^2 *c^2*d*e^4*g*x^2+560*b^2*c^2*e^5*f*x^2-15208*b*c^3*d^2*e^3*g*x^2-3200*b*c^ 3*d*e^4*f*x^2+11088*c^4*d^3*e^2*g*x^2+4928*c^4*d^2*e^3*f*x^2+1365*b^4*e^5* g*x-11900*b^3*c*d*e^4*g*x-840*b^3*c*e^5*f*x+39080*b^2*c^2*d^2*e^3*g*x+6160 *b^2*c^2*d*e^4*f*x-57072*b*c^3*d^3*e^2*g*x-15520*b*c^3*d^2*e^3*f*x+30672*c ^4*d^4*e*g*x+13632*c^4*d^3*e^2*f*x+210*b^4*d*e^4*g+1155*b^4*e^5*f-1750*b^3 *c*d^2*e^3*g-10080*b^3*c*d*e^4*f+5420*b^2*c^2*d^3*e^2*g+33320*b^2*c^2*d^2* e^3*f-7288*b*c^3*d^4*e*g-49600*b*c^3*d^3*e^2*f+3408*c^4*d^5*g+28208*c^4*d^ 4*e*f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^8/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-32*c^5*d^5)
Timed out. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^9} \, dx=\text {Timed out} \] Input:
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^9,x, algo rithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^9} \, dx=\text {Timed out} \] Input:
integrate((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(3/2)/(e*x+d)**9,x )
Output:
Timed out
Exception generated. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^9} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^9,x, algo rithm="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(b*e-2*c*d>0)', see `assume?` for more deta
Timed out. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^9} \, dx=\text {Timed out} \] Input:
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^9,x, algo rithm="giac")
Output:
Timed out
Time = 53.01 (sec) , antiderivative size = 33375, normalized size of antiderivative = 92.71 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^9} \, dx=\text {Too large to display} \] Input:
int(((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2))/(d + e*x)^9,x)
Output:
(((d*((d*((128*c^7*(5*b*e*g - 8*c*d*g + c*e*f))/(135135*(b*e - 2*c*d)^7) - (128*c^8*d*g)/(135135*(b*e - 2*c*d)^7)))/e - (4352*c^8*d^2*g + 1376*b^2*c ^6*e^2*g - 1024*c^8*d*e*f + 640*b*c^7*e^2*f - 4864*b*c^7*d*e*g)/(135135*e* (b*e - 2*c*d)^7)))/e + (32*b*c^5*(17*b^2*e^2*g + 68*c^2*d^2*g + 9*b*c*e^2* f - 16*c^2*d*e*f - 68*b*c*d*e*g))/(135135*e*(b*e - 2*c*d)^7))*(c*d^2 - c*e ^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((d*((d*((64*c^7*(15*b*e*g - 26*c*d*g + 2*c*e*f))/(135135*(b*e - 2*c*d)^7) - (128*c^8*d*g)/(135135*(b* e - 2*c*d)^7)))/e - (9472*c^8*d^2*g + 2816*b^2*c^6*e^2*g - 1664*c^8*d*e*f + 960*b*c^7*e^2*f - 10304*b*c^7*d*e*g)/(135135*e*(b*e - 2*c*d)^7)))/e + (3 2*b*c^5*(37*b^2*e^2*g + 148*c^2*d^2*g + 14*b*c*e^2*f - 26*c^2*d*e*f - 148* b*c*d*e*g))/(135135*e*(b*e - 2*c*d)^7))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2 *x)^(1/2))/(d + e*x) - (((d*((d*((64*c^7*(17*b*e*g - 30*c*d*g + 2*c*e*f))/ (135135*(b*e - 2*c*d)^7) - (128*c^8*d*g)/(135135*(b*e - 2*c*d)^7)))/e - (1 1520*c^8*d^2*g + 3392*b^2*c^6*e^2*g - 1920*c^8*d*e*f + 1088*b*c^7*e^2*f - 12480*b*c^7*d*e*g)/(135135*e*(b*e - 2*c*d)^7)))/e + (32*b*c^5*(45*b^2*e^2* g + 180*c^2*d^2*g + 16*b*c*e^2*f - 30*c^2*d*e*f - 180*b*c*d*e*g))/(135135* e*(b*e - 2*c*d)^7))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((d*((d*((64*c^7*(19*b*e*g - 34*c*d*g + 2*c*e*f))/(135135*(b*e - 2*c*d )^7) - (128*c^8*d*g)/(135135*(b*e - 2*c*d)^7)))/e - (13568*c^8*d^2*g + 396 8*b^2*c^6*e^2*g - 2176*c^8*d*e*f + 1216*b*c^7*e^2*f - 14656*b*c^7*d*e*g...
Time = 8.29 (sec) , antiderivative size = 4808, normalized size of antiderivative = 13.36 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^9} \, dx =\text {Too large to display} \] Input:
int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^9,x)
Output:
(2*i*(210*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**6*d*e**6*g + 1155*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**6*e**7*f + 1365*sqrt(d + e*x )*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**6*e **7*g*x - 2170*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**5*c*d**2*e**5*g - 12390*sqrt(d + e*x)*sqrt(b*e - 2 *c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**5*c*d*e**6*f - 14 210*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**5*c*d*e**6*g*x + 1470*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**5*c*e**7*f*x + 1820*sqrt(d + e *x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**5 *c*e**7*g*x**2 + 9130*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d) *sqrt( - b*e + c*d - c*e*x)*b**4*c**2*d**3*e**4*g + 54635*sqrt(d + e*x)*sq rt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**4*c**2* d**2*e**5*f + 60325*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*s qrt( - b*e + c*d - c*e*x)*b**4*c**2*d**2*e**5*g*x - 14630*sqrt(d + e*x)*sq rt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**4*c**2* d*e**6*f*x - 18040*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sq rt( - b*e + c*d - c*e*x)*b**4*c**2*d*e**6*g*x**2 + 35*sqrt(d + e*x)*sqrt(b *e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**4*c**2*e...