Integrand size = 44, antiderivative size = 210 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^9} \, dx=-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 e^2 (2 c d-b e) (d+e x)^9}-\frac {2 (4 c e f+18 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{99 e^2 (2 c d-b e)^2 (d+e x)^8}-\frac {4 c (4 c e f+18 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{693 e^2 (2 c d-b e)^3 (d+e x)^7} \]
-2/11*(-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)^9-2/99*(-11*b*e*g+18*c*d*g+4*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)^8-4/693*c*(-11*b*e*g+18*c*d*g+4*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)^7
Time = 0.24 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.80 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^9} \, dx=-\frac {2 (-c d+b e+c e x)^3 \sqrt {(d+e x) (-b e+c (d-e x))} \left (7 b^2 e^2 (9 e f+2 d g+11 e g x)-2 b c e \left (25 d^2 g+e^2 x (14 f+11 g x)+2 d e (70 f+81 g x)\right )+4 c^2 \left (9 d^3 g+2 e^3 f x^2+9 d e^2 x (2 f+g x)+d^2 e (79 f+81 g x)\right )\right )}{693 e^2 (-2 c d+b e)^3 (d+e x)^6} \]
(-2*(-(c*d) + b*e + c*e*x)^3*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(7*b^2 *e^2*(9*e*f + 2*d*g + 11*e*g*x) - 2*b*c*e*(25*d^2*g + e^2*x*(14*f + 11*g*x ) + 2*d*e*(70*f + 81*g*x)) + 4*c^2*(9*d^3*g + 2*e^3*f*x^2 + 9*d*e^2*x*(2*f + g*x) + d^2*e*(79*f + 81*g*x))))/(693*e^2*(-2*c*d + b*e)^3*(d + e*x)^6)
Time = 0.43 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used = {1220, 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.
| \(\displaystyle \int \frac {(f+g x) \left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{5/2}}{(d+e x)^9} \, dx\) |
| \(\Big \downarrow \) 1220 |
| \(\displaystyle \frac {(-11 b e g+18 c d g+4 c e f) \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}}{(d+e x)^8}dx}{11 e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 e^2 (d+e x)^9 (2 c d-b e)}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {(-11 b e g+18 c d g+4 c e f) \left (\frac {2 c \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}}{(d+e x)^7}dx}{9 (2 c d-b e)}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 e (d+e x)^8 (2 c d-b e)}\right )}{11 e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 e^2 (d+e x)^9 (2 c d-b e)}\) |
| \(\Big \downarrow \) 1123 |
| \(\displaystyle \frac {\left (-\frac {4 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{63 e (d+e x)^7 (2 c d-b e)^2}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 e (d+e x)^8 (2 c d-b e)}\right ) (-11 b e g+18 c d g+4 c e f)}{11 e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 e^2 (d+e x)^9 (2 c d-b e)}\) |
(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(11*e^2*(2*c* d - b*e)*(d + e*x)^9) + ((4*c*e*f + 18*c*d*g - 11*b*e*g)*((-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*e*(2*c*d - b*e))
3.23.6.3.1 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 = 8.74 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.12
| method | result | size |
| gosper | \(-\frac {2 \left (x c e +b e -c d \right ) \left (-22 b c \,e^{3} g \,x^{2}+36 c^{2} d \,e^{2} g \,x^{2}+8 c^{2} e^{3} f \,x^{2}+77 b^{2} e^{3} g x -324 b c d \,e^{2} g x -28 b c \,e^{3} f x +324 c^{2} d^{2} e g x +72 c^{2} d \,e^{2} f x +14 b^{2} d \,e^{2} g +63 b^{2} e^{3} f -50 b c \,d^{2} e g -280 b c d \,e^{2} f +36 c^{2} d^{3} g +316 c^{2} d^{2} e f \right ) \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {5}{2}}}{693 \left (e x +d \right )^{8} \left (b^{3} e^{3}-6 b^{2} c d \,e^{2}+12 b \,c^{2} d^{2} e -8 c^{3} d^{3}\right ) e^{2}}\) | \(236\) |
| default | \(\frac {g \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{9 \left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )^{8}}-\frac {4 c \,e^{2} \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{63 \left (-b \,e^{2}+2 c d e \right )^{2} \left (x +\frac {d}{e}\right )^{7}}\right )}{e^{9}}+\frac {\left (-d g +e f \right ) \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{11 \left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )^{9}}+\frac {4 c \,e^{2} \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{9 \left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )^{8}}-\frac {4 c \,e^{2} \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{63 \left (-b \,e^{2}+2 c d e \right )^{2} \left (x +\frac {d}{e}\right )^{7}}\right )}{11 \left (-b \,e^{2}+2 c d e \right )}\right )}{e^{10}}\) | \(365\) |
| trager | \(-\frac {2 \left (-22 b \,c^{4} e^{6} g \,x^{5}+36 c^{5} d \,e^{5} g \,x^{5}+8 c^{5} e^{6} f \,x^{5}+11 b^{2} c^{3} e^{6} g \,x^{4}-150 b \,c^{4} d \,e^{5} g \,x^{4}-4 b \,c^{4} e^{6} f \,x^{4}+216 c^{5} d^{2} e^{4} g \,x^{4}+48 c^{5} d \,e^{5} f \,x^{4}+165 b^{3} c^{2} e^{6} g \,x^{3}-949 b^{2} c^{3} d \,e^{5} g \,x^{3}+3 b^{2} c^{3} e^{6} f \,x^{3}+1612 b \,c^{4} d^{2} e^{4} g \,x^{3}-28 b \,c^{4} d \,e^{5} f \,x^{3}-828 c^{5} d^{3} e^{3} g \,x^{3}+124 c^{5} d^{2} e^{4} f \,x^{3}+209 b^{4} c \,e^{6} g \,x^{2}-1290 b^{3} c^{2} d \,e^{5} g \,x^{2}+113 b^{3} c^{2} e^{6} f \,x^{2}+2781 b^{2} c^{3} d^{2} e^{4} g \,x^{2}-669 b^{2} c^{3} d \,e^{5} f \,x^{2}-2528 b \,c^{4} d^{3} e^{3} g \,x^{2}+1296 b \,c^{4} d^{2} e^{4} f \,x^{2}+828 c^{5} d^{4} e^{2} g \,x^{2}-740 c^{5} d^{3} e^{3} f \,x^{2}+77 b^{5} e^{6} g x -513 b^{4} c d \,e^{5} g x +161 b^{4} c \,e^{6} f x +1293 b^{3} c^{2} d^{2} e^{4} g x -1062 b^{3} c^{2} d \,e^{5} f x -1571 b^{2} c^{3} d^{3} e^{3} g x +2517 b^{2} c^{3} d^{2} e^{4} f x +930 b \,c^{4} d^{4} e^{2} g x -2492 b \,c^{4} d^{3} e^{3} f x -216 c^{5} d^{5} e g x +876 c^{5} d^{4} e^{2} f x +14 b^{5} d \,e^{5} g +63 b^{5} e^{6} f -92 b^{4} c \,d^{2} e^{4} g -469 b^{4} c d \,e^{5} f +228 b^{3} c^{2} d^{3} e^{3} g +1345 b^{3} c^{2} d^{2} e^{4} f -272 b^{2} c^{3} d^{4} e^{2} g -1851 b^{2} c^{3} d^{3} e^{3} f +158 b \,c^{4} d^{5} e g +1228 b \,c^{4} d^{4} e^{2} f -36 c^{5} d^{6} g -316 c^{5} d^{5} e f \right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}{693 \left (b^{3} e^{3}-6 b^{2} c d \,e^{2}+12 b \,c^{2} d^{2} e -8 c^{3} d^{3}\right ) e^{2} \left (e x +d \right )^{6}}\) | \(728\) |
-2/693*(c*e*x+b*e-c*d)*(-22*b*c*e^3*g*x^2+36*c^2*d*e^2*g*x^2+8*c^2*e^3*f*x ^2+77*b^2*e^3*g*x-324*b*c*d*e^2*g*x-28*b*c*e^3*f*x+324*c^2*d^2*e*g*x+72*c^ 2*d*e^2*f*x+14*b^2*d*e^2*g+63*b^2*e^3*f-50*b*c*d^2*e*g-280*b*c*d*e^2*f+36* c^2*d^3*g+316*c^2*d^2*e*f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^ 8/(b^3*e^3-6*b^2*c*d*e^2+12*b*c^2*d^2*e-8*c^3*d^3)/e^2
Timed out. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^9} \, dx=\text {Timed out} \]
\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^9} \, dx=\int \frac {\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {5}{2}} \left (f + g x\right )}{\left (d + e x\right )^{9}}\, dx \]
Exception generated. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^9} \, dx=\text {Exception raised: ValueError} \]
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. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^9} \, dx=\text {Timed out} \]
Time = 63.30 (sec) , antiderivative size = 25236, normalized size of antiderivative = 120.17 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^9} \, dx=\text {Too large to display} \]
(((d*((1632*b^2*c^6*e^3*f - 16704*c^8*d^3*g + 2784*b^3*c^5*e^3*g + 4672*c^ 8*d^2*e*f - 5504*b*c^7*d*e^2*f + 27392*b*c^7*d^2*e*g - 15072*b^2*c^6*d*e^2 *g)/(10395*e*(b*e - 2*c*d)^6) - (d*((32*c^6*(51*b^2*e^2*g + 146*c^2*d^2*g + 16*b*c*e^2*f - 26*c^2*d*e*f - 172*b*c*d*e*g))/(10395*(b*e - 2*c*d)^6) - (d*((64*c^7*e*(8*b*e*g - 13*c*d*g + c*e*f))/(10395*(b*e - 2*c*d)^6) - (64* c^8*d*e*g)/(10395*(b*e - 2*c*d)^6)))/e))/e))/e - (696*b^3*c^5*e^3*f + 1044 *b^4*c^4*e^3*g - 8352*b*c^7*d^3*g + 2336*b*c^7*d^2*e*f - 2544*b^2*c^6*d*e^ 2*f + 12528*b^2*c^6*d^2*e*g - 6264*b^3*c^5*d*e^2*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*((1952* b^2*c^6*e^3*f - 20928*c^8*d^3*g + 3456*b^3*c^5*e^3*g + 5696*c^8*d^2*e*f - 6656*b*c^7*d*e^2*f + 34240*b*c^7*d^2*e*g - 18784*b^2*c^6*d*e^2*g)/(10395*e *(b*e - 2*c*d)^6) - (d*((32*c^6*(61*b^2*e^2*g + 178*c^2*d^2*g + 18*b*c*e^2 *f - 30*c^2*d*e*f - 208*b*c*d*e*g))/(10395*(b*e - 2*c*d)^6) - (d*((64*c^7* e*(9*b*e*g - 15*c*d*g + c*e*f))/(10395*(b*e - 2*c*d)^6) - (64*c^8*d*e*g)/( 10395*(b*e - 2*c*d)^6)))/e))/e))/e - (840*b^3*c^5*e^3*f + 1308*b^4*c^4*e^3 *g - 10464*b*c^7*d^3*g + 2848*b*c^7*d^2*e*f - 3088*b^2*c^6*d*e^2*f + 15696 *b^2*c^6*d^2*e*g - 7848*b^3*c^5*d*e^2*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*((2272*b^2*c^6*e^3 *f - 25152*c^8*d^3*g + 4128*b^3*c^5*e^3*g + 6720*c^8*d^2*e*f - 7808*b*c^7* d*e^2*f + 41088*b*c^7*d^2*e*g - 22496*b^2*c^6*d*e^2*g)/(10395*e*(b*e - ...