Integrand size = 37, antiderivative size = 231 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^8} \, dx=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{11 \left (c d^2-a e^2\right ) (d+e x)^8}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{33 \left (c d^2-a e^2\right )^2 (d+e x)^7}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{231 \left (c d^2-a e^2\right )^3 (d+e x)^6}+\frac {32 c^3 d^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{1155 \left (c d^2-a e^2\right )^4 (d+e x)^5} \]
2/11*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(-a*e^2+c*d^2)/(e*x+d)^8+4/33 *c*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(-a*e^2+c*d^2)^2/(e*x+d)^7+16 /231*c^2*d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(-a*e^2+c*d^2)^3/(e*x +d)^6+32/1155*c^3*d^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(-a*e^2+c*d^ 2)^4/(e*x+d)^5
Time = 1.28 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.60 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^8} \, dx=\frac {2 ((a e+c d x) (d+e x))^{5/2} \left (-105 a^3 e^6+35 a^2 c d e^4 (11 d+2 e x)-5 a c^2 d^2 e^2 \left (99 d^2+44 d e x+8 e^2 x^2\right )+c^3 d^3 \left (231 d^3+198 d^2 e x+88 d e^2 x^2+16 e^3 x^3\right )\right )}{1155 \left (c d^2-a e^2\right )^4 (d+e x)^8} \]
(2*((a*e + c*d*x)*(d + e*x))^(5/2)*(-105*a^3*e^6 + 35*a^2*c*d*e^4*(11*d + 2*e*x) - 5*a*c^2*d^2*e^2*(99*d^2 + 44*d*e*x + 8*e^2*x^2) + c^3*d^3*(231*d^ 3 + 198*d^2*e*x + 88*d*e^2*x^2 + 16*e^3*x^3)))/(1155*(c*d^2 - a*e^2)^4*(d + e*x)^8)
Time = 0.41 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.13, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {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.
| \(\displaystyle \int \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{(d+e x)^8} \, dx\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {6 c d \int \frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}{(d+e x)^7}dx}{11 \left (c d^2-a e^2\right )}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{11 (d+e x)^8 \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {6 c d \left (\frac {4 c d \int \frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}{(d+e x)^6}dx}{9 \left (c d^2-a e^2\right )}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{9 (d+e x)^7 \left (c d^2-a e^2\right )}\right )}{11 \left (c d^2-a e^2\right )}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{11 (d+e x)^8 \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {6 c d \left (\frac {4 c d \left (\frac {2 c d \int \frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}{(d+e x)^5}dx}{7 \left (c d^2-a e^2\right )}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 (d+e x)^6 \left (c d^2-a e^2\right )}\right )}{9 \left (c d^2-a e^2\right )}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{9 (d+e x)^7 \left (c d^2-a e^2\right )}\right )}{11 \left (c d^2-a e^2\right )}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{11 (d+e x)^8 \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 1123 |
| \(\displaystyle \frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{11 (d+e x)^8 \left (c d^2-a e^2\right )}+\frac {6 c d \left (\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{9 (d+e x)^7 \left (c d^2-a e^2\right )}+\frac {4 c d \left (\frac {4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{35 (d+e x)^5 \left (c d^2-a e^2\right )^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 (d+e x)^6 \left (c d^2-a e^2\right )}\right )}{9 \left (c d^2-a e^2\right )}\right )}{11 \left (c d^2-a e^2\right )}\) |
(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(11*(c*d^2 - a*e^2)*(d + e*x)^8) + (6*c*d*((2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(9*(c *d^2 - a*e^2)*(d + e*x)^7) + (4*c*d*((2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e *x^2)^(5/2))/(7*(c*d^2 - a*e^2)*(d + e*x)^6) + (4*c*d*(a*d*e + (c*d^2 + a* e^2)*x + c*d*e*x^2)^(5/2))/(35*(c*d^2 - a*e^2)^2*(d + e*x)^5)))/(9*(c*d^2 - a*e^2))))/(11*(c*d^2 - a*e^2))
3.20.30.3.1 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]
Time = 4.93 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.94
| method | result | size |
| gosper | \(-\frac {2 \left (c d x +a e \right ) \left (-16 x^{3} c^{3} d^{3} e^{3}+40 x^{2} a \,c^{2} d^{2} e^{4}-88 x^{2} c^{3} d^{4} e^{2}-70 x \,a^{2} c d \,e^{5}+220 x a \,c^{2} d^{3} e^{3}-198 x \,c^{3} d^{5} e +105 e^{6} a^{3}-385 d^{2} e^{4} a^{2} c +495 d^{4} e^{2} c^{2} a -231 c^{3} d^{6}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}{1155 \left (e x +d \right )^{7} \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right )}\) | \(217\) |
| default | \(\frac {-\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{11 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{8}}-\frac {6 c d e \left (-\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{9 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{7}}-\frac {4 c d e \left (-\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{7 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{6}}+\frac {4 c d e \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{35 \left (e^{2} a -c \,d^{2}\right )^{2} \left (x +\frac {d}{e}\right )^{5}}\right )}{9 \left (e^{2} a -c \,d^{2}\right )}\right )}{11 \left (e^{2} a -c \,d^{2}\right )}}{e^{8}}\) | \(293\) |
| trager | \(-\frac {2 \left (-16 c^{5} d^{5} e^{3} x^{5}+8 a \,c^{4} d^{4} e^{4} x^{4}-88 c^{5} d^{6} e^{2} x^{4}-6 a^{2} c^{3} d^{3} e^{5} x^{3}+44 a \,c^{4} d^{5} e^{3} x^{3}-198 c^{5} d^{7} e \,x^{3}+5 a^{3} c^{2} d^{2} e^{6} x^{2}-33 a^{2} c^{3} d^{4} e^{4} x^{2}+99 a \,c^{4} d^{6} e^{2} x^{2}-231 c^{5} d^{8} x^{2}+140 d \,e^{7} c \,a^{4} x -550 a^{3} c^{2} d^{3} e^{5} x +792 a^{2} c^{3} d^{5} e^{3} x -462 a \,c^{4} d^{7} e x +105 a^{5} e^{8}-385 a^{4} c \,d^{2} e^{6}+495 a^{3} c^{2} d^{4} e^{4}-231 a^{2} c^{3} d^{6} e^{2}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{1155 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \left (e x +d \right )^{6}}\) | \(339\) |
-2/1155*(c*d*x+a*e)*(-16*c^3*d^3*e^3*x^3+40*a*c^2*d^2*e^4*x^2-88*c^3*d^4*e ^2*x^2-70*a^2*c*d*e^5*x+220*a*c^2*d^3*e^3*x-198*c^3*d^5*e*x+105*a^3*e^6-38 5*a^2*c*d^2*e^4+495*a*c^2*d^4*e^2-231*c^3*d^6)*(c*d*e*x^2+a*e^2*x+c*d^2*x+ a*d*e)^(3/2)/(e*x+d)^7/(a^4*e^8-4*a^3*c*d^2*e^6+6*a^2*c^2*d^4*e^4-4*a*c^3* d^6*e^2+c^4*d^8)
Leaf count of result is larger than twice the leaf count of optimal. 699 vs. \(2 (215) = 430\).
Time = 16.98 (sec) , antiderivative size = 699, normalized size of antiderivative = 3.03 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^8} \, dx=\frac {2 \, {\left (16 \, c^{5} d^{5} e^{3} x^{5} + 231 \, a^{2} c^{3} d^{6} e^{2} - 495 \, a^{3} c^{2} d^{4} e^{4} + 385 \, a^{4} c d^{2} e^{6} - 105 \, a^{5} e^{8} + 8 \, {\left (11 \, c^{5} d^{6} e^{2} - a c^{4} d^{4} e^{4}\right )} x^{4} + 2 \, {\left (99 \, c^{5} d^{7} e - 22 \, a c^{4} d^{5} e^{3} + 3 \, a^{2} c^{3} d^{3} e^{5}\right )} x^{3} + {\left (231 \, c^{5} d^{8} - 99 \, a c^{4} d^{6} e^{2} + 33 \, a^{2} c^{3} d^{4} e^{4} - 5 \, a^{3} c^{2} d^{2} e^{6}\right )} x^{2} + 2 \, {\left (231 \, a c^{4} d^{7} e - 396 \, a^{2} c^{3} d^{5} e^{3} + 275 \, a^{3} c^{2} d^{3} e^{5} - 70 \, a^{4} c d e^{7}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{1155 \, {\left (c^{4} d^{14} - 4 \, a c^{3} d^{12} e^{2} + 6 \, a^{2} c^{2} d^{10} e^{4} - 4 \, a^{3} c d^{8} e^{6} + a^{4} d^{6} e^{8} + {\left (c^{4} d^{8} e^{6} - 4 \, a c^{3} d^{6} e^{8} + 6 \, a^{2} c^{2} d^{4} e^{10} - 4 \, a^{3} c d^{2} e^{12} + a^{4} e^{14}\right )} x^{6} + 6 \, {\left (c^{4} d^{9} e^{5} - 4 \, a c^{3} d^{7} e^{7} + 6 \, a^{2} c^{2} d^{5} e^{9} - 4 \, a^{3} c d^{3} e^{11} + a^{4} d e^{13}\right )} x^{5} + 15 \, {\left (c^{4} d^{10} e^{4} - 4 \, a c^{3} d^{8} e^{6} + 6 \, a^{2} c^{2} d^{6} e^{8} - 4 \, a^{3} c d^{4} e^{10} + a^{4} d^{2} e^{12}\right )} x^{4} + 20 \, {\left (c^{4} d^{11} e^{3} - 4 \, a c^{3} d^{9} e^{5} + 6 \, a^{2} c^{2} d^{7} e^{7} - 4 \, a^{3} c d^{5} e^{9} + a^{4} d^{3} e^{11}\right )} x^{3} + 15 \, {\left (c^{4} d^{12} e^{2} - 4 \, a c^{3} d^{10} e^{4} + 6 \, a^{2} c^{2} d^{8} e^{6} - 4 \, a^{3} c d^{6} e^{8} + a^{4} d^{4} e^{10}\right )} x^{2} + 6 \, {\left (c^{4} d^{13} e - 4 \, a c^{3} d^{11} e^{3} + 6 \, a^{2} c^{2} d^{9} e^{5} - 4 \, a^{3} c d^{7} e^{7} + a^{4} d^{5} e^{9}\right )} x\right )}} \]
2/1155*(16*c^5*d^5*e^3*x^5 + 231*a^2*c^3*d^6*e^2 - 495*a^3*c^2*d^4*e^4 + 3 85*a^4*c*d^2*e^6 - 105*a^5*e^8 + 8*(11*c^5*d^6*e^2 - a*c^4*d^4*e^4)*x^4 + 2*(99*c^5*d^7*e - 22*a*c^4*d^5*e^3 + 3*a^2*c^3*d^3*e^5)*x^3 + (231*c^5*d^8 - 99*a*c^4*d^6*e^2 + 33*a^2*c^3*d^4*e^4 - 5*a^3*c^2*d^2*e^6)*x^2 + 2*(231 *a*c^4*d^7*e - 396*a^2*c^3*d^5*e^3 + 275*a^3*c^2*d^3*e^5 - 70*a^4*c*d*e^7) *x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(c^4*d^14 - 4*a*c^3*d^12*e ^2 + 6*a^2*c^2*d^10*e^4 - 4*a^3*c*d^8*e^6 + a^4*d^6*e^8 + (c^4*d^8*e^6 - 4 *a*c^3*d^6*e^8 + 6*a^2*c^2*d^4*e^10 - 4*a^3*c*d^2*e^12 + a^4*e^14)*x^6 + 6 *(c^4*d^9*e^5 - 4*a*c^3*d^7*e^7 + 6*a^2*c^2*d^5*e^9 - 4*a^3*c*d^3*e^11 + a ^4*d*e^13)*x^5 + 15*(c^4*d^10*e^4 - 4*a*c^3*d^8*e^6 + 6*a^2*c^2*d^6*e^8 - 4*a^3*c*d^4*e^10 + a^4*d^2*e^12)*x^4 + 20*(c^4*d^11*e^3 - 4*a*c^3*d^9*e^5 + 6*a^2*c^2*d^7*e^7 - 4*a^3*c*d^5*e^9 + a^4*d^3*e^11)*x^3 + 15*(c^4*d^12*e ^2 - 4*a*c^3*d^10*e^4 + 6*a^2*c^2*d^8*e^6 - 4*a^3*c*d^6*e^8 + a^4*d^4*e^10 )*x^2 + 6*(c^4*d^13*e - 4*a*c^3*d^11*e^3 + 6*a^2*c^2*d^9*e^5 - 4*a^3*c*d^7 *e^7 + a^4*d^5*e^9)*x)
Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^8} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^8} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e*(a*e^2-c*d^2)>0)', see `assume ?` for mor
Exception generated. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^8} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{%%%{1,[0,0,6]%%%},[12]%%%}+%%%{%%{[%%%{-12,[0,1,5]%%%},0]: [1,0,%%%{
Time = 14.89 (sec) , antiderivative size = 2657, normalized size of antiderivative = 11.50 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^8} \, dx=\text {Too large to display} \]
(((d*((4*c^3*d^4)/(11*(a*e^2 - c*d^2)*(9*a*e^3 - 9*c*d^2*e)) - (2*c^2*d^2* (5*a*e^2 - c*d^2))/(11*(a*e^2 - c*d^2)*(9*a*e^3 - 9*c*d^2*e))))/e + (2*a*c ^2*d^3*e^2 - 2*c^3*d^5 + 4*a^2*c*d*e^4)/(11*e*(a*e^2 - c*d^2)*(9*a*e^3 - 9 *c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^5 - ( ((228*c^4*d^5 - 284*a*c^3*d^3*e^2)/(693*e*(a*e^2 - c*d^2)^2*(5*a*e^3 - 5*c *d^2*e)) + (8*c^4*d^5)/(99*e*(a*e^2 - c*d^2)^2*(5*a*e^3 - 5*c*d^2*e)))*(x* (a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^3 - (((d*((16*c^5*d^ 6)/(693*(a*e^2 - c*d^2)^3*(5*a*e^3 - 5*c*d^2*e)) - (16*c^4*d^4*(17*a*e^2 - 15*c*d^2))/(693*(a*e^2 - c*d^2)^3*(5*a*e^3 - 5*c*d^2*e))))/e + (16*a*c^3* d^3*e*(16*a*e^2 - 15*c*d^2))/(693*(a*e^2 - c*d^2)^3*(5*a*e^3 - 5*c*d^2*e)) )*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^3 - (((d*((32*c ^6*d^7)/(3465*(a*e^2 - c*d^2)^4*(3*a*e^3 - 3*c*d^2*e)) - (64*c^5*d^5*(10*a *e^2 - 9*c*d^2))/(3465*(a*e^2 - c*d^2)^4*(3*a*e^3 - 3*c*d^2*e))))/e + (32* a*c^4*d^4*e*(19*a*e^2 - 18*c*d^2))/(3465*(a*e^2 - c*d^2)^4*(3*a*e^3 - 3*c* d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^2 - (((2 2*c^3*d^4 - 58*a*c^2*d^2*e^2)/(99*e*(a*e^2 - c*d^2)*(7*a*e^3 - 7*c*d^2*e)) + (4*c^3*d^4)/(11*e*(a*e^2 - c*d^2)*(7*a*e^3 - 7*c*d^2*e)))*(x*(a*e^2 + c *d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^4 - (((32*c^6*d^7)/(3465*e^2*( a*e^2 - c*d^2)^5) - (16*c^5*d^5*(71*a*e^2 - 65*c*d^2))/(10395*e^2*(a*e^2 - c*d^2)^5))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x) - ...