Integrand size = 44, antiderivative size = 285 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^8} \, dx=-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{11 e^2 (2 c d-b e) (d+e x)^8}-\frac {2 (6 c e f+16 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{99 e^2 (2 c d-b e)^2 (d+e x)^7}-\frac {8 c (6 c e f+16 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{693 e^2 (2 c d-b e)^3 (d+e x)^6}-\frac {16 c^2 (6 c e f+16 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3465 e^2 (2 c d-b e)^4 (d+e x)^5} \]
-2/11*(-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)^8-2/99*(-11*b*e*g+16*c*d*g+6*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)^7-8/693*c*(-11*b*e*g+16*c*d*g+6*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)^6-16/346 5*c^2*(-11*b*e*g+16*c*d*g+6*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)^5
Time = 0.30 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.87 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^8} \, dx=-\frac {2 (-c d+b e+c e x)^2 \sqrt {(d+e x) (-b e+c (d-e x))} \left (-35 b^3 e^3 (9 e f+2 d g+11 e g x)+8 c^3 \left (61 d^4 g+6 e^4 f x^3+16 d e^3 x^2 (3 f+g x)+8 d^3 e (57 f+61 g x)+d^2 e^2 x (183 f+128 g x)\right )+10 b^2 c e^2 \left (43 d^2 g+e^2 x (21 f+22 g x)+d e (210 f+254 g x)\right )-4 b c^2 e \left (212 d^3 g+2 e^3 x^2 (15 f+11 g x)+2 d e^2 x (135 f+128 g x)+d^2 e (1185 f+1391 g x)\right )\right )}{3465 e^2 (-2 c d+b e)^4 (d+e x)^6} \]
(-2*(-(c*d) + b*e + c*e*x)^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-35*b ^3*e^3*(9*e*f + 2*d*g + 11*e*g*x) + 8*c^3*(61*d^4*g + 6*e^4*f*x^3 + 16*d*e ^3*x^2*(3*f + g*x) + 8*d^3*e*(57*f + 61*g*x) + d^2*e^2*x*(183*f + 128*g*x) ) + 10*b^2*c*e^2*(43*d^2*g + e^2*x*(21*f + 22*g*x) + d*e*(210*f + 254*g*x) ) - 4*b*c^2*e*(212*d^3*g + 2*e^3*x^2*(15*f + 11*g*x) + 2*d*e^2*x*(135*f + 128*g*x) + d^2*e*(1185*f + 1391*g*x))))/(3465*e^2*(-2*c*d + b*e)^4*(d + e* x)^6)
Time = 0.51 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1220, 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.
| \(\displaystyle \int \frac {(f+g x) \left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^8} \, dx\) |
| \(\Big \downarrow \) 1220 |
| \(\displaystyle \frac {(-11 b e g+16 c d g+6 c e f) \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}{(d+e x)^7}dx}{11 e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{11 e^2 (d+e x)^8 (2 c d-b e)}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {(-11 b e g+16 c d g+6 c e f) \left (\frac {4 c \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}{(d+e x)^6}dx}{9 (2 c d-b e)}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{9 e (d+e x)^7 (2 c d-b e)}\right )}{11 e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{11 e^2 (d+e x)^8 (2 c d-b e)}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {(-11 b e g+16 c d g+6 c e f) \left (\frac {4 c \left (\frac {2 c \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}{(d+e x)^5}dx}{7 (2 c d-b e)}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 e (d+e x)^6 (2 c d-b e)}\right )}{9 (2 c d-b e)}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{9 e (d+e x)^7 (2 c d-b e)}\right )}{11 e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{11 e^2 (d+e x)^8 (2 c d-b e)}\) |
| \(\Big \downarrow \) 1123 |
| \(\displaystyle \frac {\left (\frac {4 c \left (-\frac {4 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{35 e (d+e x)^5 (2 c d-b e)^2}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 e (d+e x)^6 (2 c d-b e)}\right )}{9 (2 c d-b e)}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{9 e (d+e x)^7 (2 c d-b e)}\right ) (-11 b e g+16 c d g+6 c e f)}{11 e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{11 e^2 (d+e x)^8 (2 c d-b e)}\) |
(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(11*e^2*(2*c* d - b*e)*(d + e*x)^8) + ((6*c*e*f + 16*c*d*g - 11*b*e*g)*((-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*e*(2*c*d - b*e))
3.22.93.3.1 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 = 5.83 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.34
| method | result | size |
| gosper | \(-\frac {2 \left (x c e +b e -c d \right ) \left (88 b \,c^{2} e^{4} g \,x^{3}-128 c^{3} d \,e^{3} g \,x^{3}-48 c^{3} e^{4} f \,x^{3}-220 b^{2} c \,e^{4} g \,x^{2}+1024 b \,c^{2} d \,e^{3} g \,x^{2}+120 b \,c^{2} e^{4} f \,x^{2}-1024 c^{3} d^{2} e^{2} g \,x^{2}-384 c^{3} d \,e^{3} f \,x^{2}+385 b^{3} e^{4} g x -2540 b^{2} c d \,e^{3} g x -210 b^{2} c \,e^{4} f x +5564 b \,c^{2} d^{2} e^{2} g x +1080 b \,c^{2} d \,e^{3} f x -3904 c^{3} d^{3} e g x -1464 c^{3} d^{2} e^{2} f x +70 b^{3} d \,e^{3} g +315 b^{3} e^{4} f -430 b^{2} c \,d^{2} e^{2} g -2100 b^{2} c d \,e^{3} f +848 b \,c^{2} d^{3} e g +4740 b \,c^{2} d^{2} e^{2} f -488 c^{3} d^{4} g -3648 c^{3} d^{3} e f \right ) \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {3}{2}}}{3465 \left (e x +d \right )^{7} e^{2} \left (b^{4} e^{4}-8 b^{3} c d \,e^{3}+24 b^{2} c^{2} d^{2} e^{2}-32 b \,c^{3} d^{3} e +16 c^{4} d^{4}\right )}\) | \(382\) |
| default | \(\frac {g \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{9 \left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )^{7}}+\frac {4 c \,e^{2} \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{7 \left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )^{6}}-\frac {4 c \,e^{2} \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{35 \left (-b \,e^{2}+2 c d e \right )^{2} \left (x +\frac {d}{e}\right )^{5}}\right )}{9 \left (-b \,e^{2}+2 c d e \right )}\right )}{e^{8}}+\frac {\left (-d g +e f \right ) \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{11 \left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )^{8}}+\frac {6 c \,e^{2} \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{9 \left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )^{7}}+\frac {4 c \,e^{2} \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{7 \left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )^{6}}-\frac {4 c \,e^{2} \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{35 \left (-b \,e^{2}+2 c d e \right )^{2} \left (x +\frac {d}{e}\right )^{5}}\right )}{9 \left (-b \,e^{2}+2 c d e \right )}\right )}{11 \left (-b \,e^{2}+2 c d e \right )}\right )}{e^{9}}\) | \(533\) |
| trager | \(\frac {2 \left (88 b \,c^{4} e^{6} g \,x^{5}-128 c^{5} d \,e^{5} g \,x^{5}-48 c^{5} e^{6} f \,x^{5}-44 b^{2} c^{3} e^{6} g \,x^{4}+592 b \,c^{4} d \,e^{5} g \,x^{4}+24 b \,c^{4} e^{6} f \,x^{4}-768 c^{5} d^{2} e^{4} g \,x^{4}-288 c^{5} d \,e^{5} f \,x^{4}+33 b^{3} c^{2} e^{6} g \,x^{3}-356 b^{2} c^{3} d \,e^{5} g \,x^{3}-18 b^{2} c^{3} e^{6} f \,x^{3}+1812 b \,c^{4} d^{2} e^{4} g \,x^{3}+168 b \,c^{4} d \,e^{5} f \,x^{3}-1984 c^{5} d^{3} e^{3} g \,x^{3}-744 c^{5} d^{2} e^{4} f \,x^{3}+550 b^{4} c \,e^{6} g \,x^{2}-4316 b^{3} c^{2} d \,e^{5} g \,x^{2}+15 b^{3} c^{2} e^{6} f \,x^{2}+12486 b^{2} c^{3} d^{2} e^{4} g \,x^{2}-144 b^{2} c^{3} d \,e^{5} f \,x^{2}-15016 b \,c^{4} d^{3} e^{3} g \,x^{2}+540 b \,c^{4} d^{2} e^{4} f \,x^{2}+6296 c^{5} d^{4} e^{2} g \,x^{2}-1104 c^{5} d^{3} e^{3} f \,x^{2}+385 b^{5} e^{6} g x -3170 b^{4} c d \,e^{5} g x +420 b^{4} c \,e^{6} f x +10029 b^{3} c^{2} d^{2} e^{4} g x -3330 b^{3} c^{2} d \,e^{5} f x -15016 b^{2} c^{3} d^{3} e^{3} g x +9846 b^{2} c^{3} d^{2} e^{4} f x +10700 b \,c^{4} d^{4} e^{2} g x -12768 b \,c^{4} d^{3} e^{3} f x -2928 c^{5} d^{5} e g x +5832 c^{5} d^{4} e^{2} f x +70 b^{5} d \,e^{5} g +315 b^{5} e^{6} f -570 b^{4} c \,d^{2} e^{4} g -2730 b^{4} c d \,e^{5} f +1778 b^{3} c^{2} d^{3} e^{3} g +9255 b^{3} c^{2} d^{2} e^{4} f -2614 b^{2} c^{3} d^{4} e^{2} g -15228 b^{2} c^{3} d^{3} e^{3} f +1824 b \,c^{4} d^{5} e g +12036 b \,c^{4} d^{4} e^{2} f -488 c^{5} d^{6} g -3648 c^{5} d^{5} e f \right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}{3465 \left (b^{4} e^{4}-8 b^{3} c d \,e^{3}+24 b^{2} c^{2} d^{2} e^{2}-32 b \,c^{3} d^{3} e +16 c^{4} d^{4}\right ) e^{2} \left (e x +d \right )^{6}}\) | \(742\) |
-2/3465*(c*e*x+b*e-c*d)*(88*b*c^2*e^4*g*x^3-128*c^3*d*e^3*g*x^3-48*c^3*e^4 *f*x^3-220*b^2*c*e^4*g*x^2+1024*b*c^2*d*e^3*g*x^2+120*b*c^2*e^4*f*x^2-1024 *c^3*d^2*e^2*g*x^2-384*c^3*d*e^3*f*x^2+385*b^3*e^4*g*x-2540*b^2*c*d*e^3*g* x-210*b^2*c*e^4*f*x+5564*b*c^2*d^2*e^2*g*x+1080*b*c^2*d*e^3*f*x-3904*c^3*d ^3*e*g*x-1464*c^3*d^2*e^2*f*x+70*b^3*d*e^3*g+315*b^3*e^4*f-430*b^2*c*d^2*e ^2*g-2100*b^2*c*d*e^3*f+848*b*c^2*d^3*e*g+4740*b*c^2*d^2*e^2*f-488*c^3*d^4 *g-3648*c^3*d^3*e*f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^7/e^2/ (b^4*e^4-8*b^3*c*d*e^3+24*b^2*c^2*d^2*e^2-32*b*c^3*d^3*e+16*c^4*d^4)
Timed out. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^8} \, dx=\text {Timed out} \]
\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^8} \, dx=\int \frac {\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {3}{2}} \left (f + g x\right )}{\left (d + e x\right )^{8}}\, dx \]
Exception generated. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^8} \, dx=\text {Exception raised: ValueError} \]
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. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^8} \, dx=\text {Timed out} \]
Time = 50.88 (sec) , antiderivative size = 16485, normalized size of antiderivative = 57.84 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^8} \, dx=\text {Too large to display} \]
(((d*((d*((64*c^6*(7*b*e*g - 12*c*d*g + c*e*f))/(10395*(b*e - 2*c*d)^6) - (64*c^7*d*g)/(10395*(b*e - 2*c*d)^6)))/e - (3904*c^7*d^2*g + 1184*b^2*c^5* e^2*g - 768*c^7*d*e*f + 448*b*c^6*e^2*f - 4288*b*c^6*d*e*g)/(10395*e*(b*e - 2*c*d)^6)))/e + (8*b*c^4*(61*b^2*e^2*g + 244*c^2*d^2*g + 26*b*c*e^2*f - 48*c^2*d*e*f - 244*b*c*d*e*g))/(10395*e*(b*e - 2*c*d)^6))*(c*d^2 - c*e^2*x ^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((d*((d*((32*c^6*(9*b*e*g - 14*c *d*g + 2*c*e*f))/(10395*(b*e - 2*c*d)^6) - (64*c^7*d*g)/(10395*(b*e - 2*c* d)^6)))/e - (1664*c^7*d^2*g + 544*b^2*c^5*e^2*g - 448*c^7*d*e*f + 288*b*c^ 6*e^2*f - 1888*b*c^6*d*e*g)/(10395*e*(b*e - 2*c*d)^6)))/e + (16*b*c^4*(13* b^2*e^2*g + 52*c^2*d^2*g + 8*b*c*e^2*f - 14*c^2*d*e*f - 52*b*c*d*e*g))/(10 395*e*(b*e - 2*c*d)^6))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) + (((d*((d*((64*c^6*(8*b*e*g - 14*c*d*g + c*e*f))/(10395*(b*e - 2*c*d )^6) - (64*c^7*d*g)/(10395*(b*e - 2*c*d)^6)))/e - (4800*c^7*d^2*g + 1440*b ^2*c^5*e^2*g - 896*c^7*d*e*f + 512*b*c^6*e^2*f - 5248*b*c^6*d*e*g)/(10395* e*(b*e - 2*c*d)^6)))/e + (8*b*c^4*(75*b^2*e^2*g + 300*c^2*d^2*g + 30*b*c*e ^2*f - 56*c^2*d*e*f - 300*b*c*d*e*g))/(10395*e*(b*e - 2*c*d)^6))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) + (((d*((d*((64*c^6*(9*b*e*g - 16*c*d*g + c*e*f))/(10395*(b*e - 2*c*d)^6) - (64*c^7*d*g)/(10395*(b*e - 2*c*d)^6)))/e - (5696*c^7*d^2*g + 1696*b^2*c^5*e^2*g - 1024*c^7*d*e*f + 5 76*b*c^6*e^2*f - 6208*b*c^6*d*e*g)/(10395*e*(b*e - 2*c*d)^6)))/e + (8*b...