Integrand size = 37, antiderivative size = 171 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^9} \, dx=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{11 \left (c d^2-a e^2\right ) (d+e x)^9}+\frac {8 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{99 \left (c d^2-a e^2\right )^2 (d+e x)^8}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{693 \left (c d^2-a e^2\right )^3 (d+e x)^7} \]
2/11*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(-a*e^2+c*d^2)/(e*x+d)^9+8/99 *c*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(-a*e^2+c*d^2)^2/(e*x+d)^8+16 /693*c^2*d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(-a*e^2+c*d^2)^3/(e*x +d)^7
Time = 0.65 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.61 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^9} \, dx=\frac {2 (a e+c d x)^3 \sqrt {(a e+c d x) (d+e x)} \left (63 a^2 e^4-14 a c d e^2 (11 d+2 e x)+c^2 d^2 \left (99 d^2+44 d e x+8 e^2 x^2\right )\right )}{693 \left (c d^2-a e^2\right )^3 (d+e x)^6} \]
(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(63*a^2*e^4 - 14*a*c*d*e^ 2*(11*d + 2*e*x) + c^2*d^2*(99*d^2 + 44*d*e*x + 8*e^2*x^2)))/(693*(c*d^2 - a*e^2)^3*(d + e*x)^6)
Time = 0.32 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.09, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {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.
\(\displaystyle \int \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{(d+e x)^9} \, dx\) |
\(\Big \downarrow \) 1129 |
\(\displaystyle \frac {4 c d \int \frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{5/2}}{(d+e x)^8}dx}{11 \left (c d^2-a e^2\right )}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{11 (d+e x)^9 \left (c d^2-a e^2\right )}\) |
\(\Big \downarrow \) 1129 |
\(\displaystyle \frac {4 c d \left (\frac {2 c d \int \frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{5/2}}{(d+e x)^7}dx}{9 \left (c d^2-a e^2\right )}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 (d+e x)^8 \left (c d^2-a e^2\right )}\right )}{11 \left (c d^2-a e^2\right )}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{11 (d+e x)^9 \left (c d^2-a e^2\right )}\) |
\(\Big \downarrow \) 1123 |
\(\displaystyle \frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{11 (d+e x)^9 \left (c d^2-a e^2\right )}+\frac {4 c d \left (\frac {4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{63 (d+e x)^7 \left (c d^2-a e^2\right )^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 (d+e x)^8 \left (c d^2-a e^2\right )}\right )}{11 \left (c d^2-a e^2\right )}\) |
(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(11*(c*d^2 - a*e^2)*(d + e*x)^9) + (4*c*d*((2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(9*(c *d^2 - a*e^2)*(d + e*x)^8) + (4*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2 )^(7/2))/(63*(c*d^2 - a*e^2)^2*(d + e*x)^7)))/(11*(c*d^2 - a*e^2))
3.20.44.3.1 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 = 8.39 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.85
method | result | size |
gosper | \(-\frac {2 \left (c d x +a e \right ) \left (8 x^{2} c^{2} d^{2} e^{2}-28 x a c d \,e^{3}+44 x \,c^{2} d^{3} e +63 a^{2} e^{4}-154 a c \,d^{2} e^{2}+99 c^{2} d^{4}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}{693 \left (e x +d \right )^{8} \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}\right )}\) | \(146\) |
default | \(\frac {-\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{11 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{9}}-\frac {4 c d e \left (-\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{9 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{8}}+\frac {4 c d e \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{63 \left (e^{2} a -c \,d^{2}\right )^{2} \left (x +\frac {d}{e}\right )^{7}}\right )}{11 \left (e^{2} a -c \,d^{2}\right )}}{e^{9}}\) | \(212\) |
trager | \(-\frac {2 \left (8 c^{5} d^{5} e^{2} x^{5}-4 a \,c^{4} d^{4} e^{3} x^{4}+44 c^{5} d^{6} e \,x^{4}+3 a^{2} c^{3} d^{3} e^{4} x^{3}-22 a \,c^{4} d^{5} e^{2} x^{3}+99 c^{5} d^{7} x^{3}+113 a^{3} c^{2} d^{2} e^{5} x^{2}-330 a^{2} c^{3} d^{4} e^{3} x^{2}+297 a \,c^{4} d^{6} e \,x^{2}+161 a^{4} c d \,e^{6} x -418 a^{3} c^{2} d^{3} e^{4} x +297 a^{2} c^{3} d^{5} e^{2} x +63 a^{5} e^{7}-154 a^{4} c \,d^{2} e^{5}+99 a^{3} c^{2} d^{4} e^{3}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{693 \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}\right ) \left (e x +d \right )^{6}}\) | \(285\) |
-2/693*(c*d*x+a*e)*(8*c^2*d^2*e^2*x^2-28*a*c*d*e^3*x+44*c^2*d^3*e*x+63*a^2 *e^4-154*a*c*d^2*e^2+99*c^2*d^4)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(5/2)/( e*x+d)^8/(a^3*e^6-3*a^2*c*d^2*e^4+3*a*c^2*d^4*e^2-c^3*d^6)
Leaf count of result is larger than twice the leaf count of optimal. 569 vs. \(2 (159) = 318\).
Time = 22.49 (sec) , antiderivative size = 569, normalized size of antiderivative = 3.33 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^9} \, dx=\frac {2 \, {\left (8 \, c^{5} d^{5} e^{2} x^{5} + 99 \, a^{3} c^{2} d^{4} e^{3} - 154 \, a^{4} c d^{2} e^{5} + 63 \, a^{5} e^{7} + 4 \, {\left (11 \, c^{5} d^{6} e - a c^{4} d^{4} e^{3}\right )} x^{4} + {\left (99 \, c^{5} d^{7} - 22 \, a c^{4} d^{5} e^{2} + 3 \, a^{2} c^{3} d^{3} e^{4}\right )} x^{3} + {\left (297 \, a c^{4} d^{6} e - 330 \, a^{2} c^{3} d^{4} e^{3} + 113 \, a^{3} c^{2} d^{2} e^{5}\right )} x^{2} + {\left (297 \, a^{2} c^{3} d^{5} e^{2} - 418 \, a^{3} c^{2} d^{3} e^{4} + 161 \, a^{4} c d e^{6}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{693 \, {\left (c^{3} d^{12} - 3 \, a c^{2} d^{10} e^{2} + 3 \, a^{2} c d^{8} e^{4} - a^{3} d^{6} e^{6} + {\left (c^{3} d^{6} e^{6} - 3 \, a c^{2} d^{4} e^{8} + 3 \, a^{2} c d^{2} e^{10} - a^{3} e^{12}\right )} x^{6} + 6 \, {\left (c^{3} d^{7} e^{5} - 3 \, a c^{2} d^{5} e^{7} + 3 \, a^{2} c d^{3} e^{9} - a^{3} d e^{11}\right )} x^{5} + 15 \, {\left (c^{3} d^{8} e^{4} - 3 \, a c^{2} d^{6} e^{6} + 3 \, a^{2} c d^{4} e^{8} - a^{3} d^{2} e^{10}\right )} x^{4} + 20 \, {\left (c^{3} d^{9} e^{3} - 3 \, a c^{2} d^{7} e^{5} + 3 \, a^{2} c d^{5} e^{7} - a^{3} d^{3} e^{9}\right )} x^{3} + 15 \, {\left (c^{3} d^{10} e^{2} - 3 \, a c^{2} d^{8} e^{4} + 3 \, a^{2} c d^{6} e^{6} - a^{3} d^{4} e^{8}\right )} x^{2} + 6 \, {\left (c^{3} d^{11} e - 3 \, a c^{2} d^{9} e^{3} + 3 \, a^{2} c d^{7} e^{5} - a^{3} d^{5} e^{7}\right )} x\right )}} \]
2/693*(8*c^5*d^5*e^2*x^5 + 99*a^3*c^2*d^4*e^3 - 154*a^4*c*d^2*e^5 + 63*a^5 *e^7 + 4*(11*c^5*d^6*e - a*c^4*d^4*e^3)*x^4 + (99*c^5*d^7 - 22*a*c^4*d^5*e ^2 + 3*a^2*c^3*d^3*e^4)*x^3 + (297*a*c^4*d^6*e - 330*a^2*c^3*d^4*e^3 + 113 *a^3*c^2*d^2*e^5)*x^2 + (297*a^2*c^3*d^5*e^2 - 418*a^3*c^2*d^3*e^4 + 161*a ^4*c*d*e^6)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(c^3*d^12 - 3*a *c^2*d^10*e^2 + 3*a^2*c*d^8*e^4 - a^3*d^6*e^6 + (c^3*d^6*e^6 - 3*a*c^2*d^4 *e^8 + 3*a^2*c*d^2*e^10 - a^3*e^12)*x^6 + 6*(c^3*d^7*e^5 - 3*a*c^2*d^5*e^7 + 3*a^2*c*d^3*e^9 - a^3*d*e^11)*x^5 + 15*(c^3*d^8*e^4 - 3*a*c^2*d^6*e^6 + 3*a^2*c*d^4*e^8 - a^3*d^2*e^10)*x^4 + 20*(c^3*d^9*e^3 - 3*a*c^2*d^7*e^5 + 3*a^2*c*d^5*e^7 - a^3*d^3*e^9)*x^3 + 15*(c^3*d^10*e^2 - 3*a*c^2*d^8*e^4 + 3*a^2*c*d^6*e^6 - a^3*d^4*e^8)*x^2 + 6*(c^3*d^11*e - 3*a*c^2*d^9*e^3 + 3* a^2*c*d^7*e^5 - a^3*d^5*e^7)*x)
Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^9} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^9} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e*(a*e^2-c*d^2)>0)', see `assume ?` for mor
Exception generated. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^9} \, dx=\text {Exception raised: TypeError} \]
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 = 16.92 (sec) , antiderivative size = 4096, normalized size of antiderivative = 23.95 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^9} \, dx=\text {Too large to display} \]
(((d*((8*c^5*d^6)/(99*e*(a*e^2 - c*d^2)^2*(5*a*e^3 - 5*c*d^2*e)) - (4*c^4* d^4*(19*a*e^2 - 15*c*d^2))/(99*e*(a*e^2 - c*d^2)^2*(5*a*e^3 - 5*c*d^2*e))) )/e + (4*c^3*d^3*(33*a^2*e^4 + 16*c^2*d^4 - 47*a*c*d^2*e^2))/(99*e^2*(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 - (((2*a^3*e^4)/(11*a*e^3 - 11*c*d^2*e) - (d*((d*((2 *c^3*d^4)/(11*a*e^3 - 11*c*d^2*e) - (6*a*c^2*d^2*e^2)/(11*a*e^3 - 11*c*d^2 *e)))/e + (6*a^2*c*d*e^3)/(11*a*e^3 - 11*c*d^2*e)))/e)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^6 + (((d*((16*c^6*d^7)/(693*e*(a*e^2 - c*d^2)^3*(3*a*e^3 - 3*c*d^2*e)) - (8*c^5*d^5*(29*a*e^2 - 25*c*d^2))/(69 3*e*(a*e^2 - c*d^2)^3*(3*a*e^3 - 3*c*d^2*e))))/e + (8*c^4*d^4*(466*a^2*e^4 + 331*c^2*d^4 - 787*a*c*d^2*e^2))/(3465*e^2*(a*e^2 - c*d^2)^3*(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 + (((6*c^4*d^5 + 22*a*c^3*d^3*e^2)/(77*e^2*(a*e^2 - c*d^2)*(5*a*e^3 - 5*c*d^ 2*e)) - (4*c^4*d^5)/(11*e^2*(a*e^2 - c*d^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 + (((188*c^5*d^6 - 14 8*a*c^4*d^4*e^2)/(495*e^2*(a*e^2 - c*d^2)^2*(3*a*e^3 - 3*c*d^2*e)) - (8*c^ 5*d^6)/(99*e^2*(a*e^2 - c*d^2)^2*(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 + (((d*((d*((64*c^8*d^9)/(10395* e*(a*e^2 - c*d^2)^6) - (64*c^7*d^7*(23*a*e^2 - 20*c*d^2))/(10395*e*(a*e^2 - c*d^2)^6)))/e + (64*c^6*d^6*(218*a^2*e^4 + 175*c^2*d^4 - 390*a*c*d^2*...