Integrand size = 37, antiderivative size = 111 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^8} \, dx=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{9 \left (c d^2-a e^2\right ) (d+e x)^8}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{63 \left (c d^2-a e^2\right )^2 (d+e x)^7} \] Output:
2/9*(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)^8+4/63* 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)^7
Time = 0.30 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.55 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^8} \, dx=\frac {2 ((a e+c d x) (d+e x))^{7/2} \left (-7 a e^2+c d (9 d+2 e x)\right )}{63 \left (c d^2-a e^2\right )^2 (d+e x)^8} \] Input:
Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^8,x]
Output:
(2*((a*e + c*d*x)*(d + e*x))^(7/2)*(-7*a*e^2 + c*d*(9*d + 2*e*x)))/(63*(c* d^2 - a*e^2)^2*(d + e*x)^8)
Time = 0.39 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {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.
\(\displaystyle \int \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{(d+e x)^8} \, dx\) |
\(\Big \downarrow \) 1129 |
\(\displaystyle \frac {2 c d \int \frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{5/2}}{(d+e x)^7}dx}{9 \left (c d^2-a e^2\right )}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 (d+e x)^8 \left (c d^2-a e^2\right )}\) |
\(\Big \downarrow \) 1123 |
\(\displaystyle \frac {4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{63 (d+e x)^7 \left (c d^2-a e^2\right )^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 (d+e x)^8 \left (c d^2-a e^2\right )}\) |
Input:
Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^8,x]
Output:
(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)
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 = 7.37 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.81
method | result | size |
gosper | \(-\frac {2 \left (c d x +a e \right ) \left (-2 c d x e +7 a \,e^{2}-9 c \,d^{2}\right ) \left (c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}{63 \left (e x +d \right )^{7} \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}\) | \(90\) |
orering | \(-\frac {2 \left (-2 c d x e +7 a \,e^{2}-9 c \,d^{2}\right ) \left (c d x +a e \right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {5}{2}}}{63 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \left (e x +d \right )^{7}}\) | \(91\) |
default | \(\frac {-\frac {2 \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{9 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{8}}+\frac {4 d e c \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{63 \left (a \,e^{2}-c \,d^{2}\right )^{2} \left (x +\frac {d}{e}\right )^{7}}}{e^{8}}\) | \(131\) |
trager | \(-\frac {2 \left (-2 c^{4} d^{4} e \,x^{4}+a \,c^{3} d^{3} e^{2} x^{3}-9 c^{4} d^{5} x^{3}+15 a^{2} c^{2} d^{2} e^{3} x^{2}-27 a \,c^{3} d^{4} e \,x^{2}+19 a^{3} c d \,e^{4} x -27 a^{2} c^{2} d^{3} e^{2} x +7 a^{4} e^{5}-9 a^{3} c \,d^{2} e^{3}\right ) \sqrt {c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e}}{63 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \left (e x +d \right )^{5}}\) | \(177\) |
Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/2)/(e*x+d)^8,x,method=_RETURNVERB OSE)
Output:
-2/63*(c*d*x+a*e)*(-2*c*d*e*x+7*a*e^2-9*c*d^2)*(c*d*e*x^2+a*e^2*x+c*d^2*x+ a*d*e)^(5/2)/(e*x+d)^7/(a^2*e^4-2*a*c*d^2*e^2+c^2*d^4)
Leaf count of result is larger than twice the leaf count of optimal. 344 vs. \(2 (103) = 206\).
Time = 7.88 (sec) , antiderivative size = 344, normalized size of antiderivative = 3.10 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^8} \, dx=\frac {2 \, {\left (2 \, c^{4} d^{4} e x^{4} + 9 \, a^{3} c d^{2} e^{3} - 7 \, a^{4} e^{5} + {\left (9 \, c^{4} d^{5} - a c^{3} d^{3} e^{2}\right )} x^{3} + 3 \, {\left (9 \, a c^{3} d^{4} e - 5 \, a^{2} c^{2} d^{2} e^{3}\right )} x^{2} + {\left (27 \, a^{2} c^{2} d^{3} e^{2} - 19 \, a^{3} c d e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{63 \, {\left (c^{2} d^{9} - 2 \, a c d^{7} e^{2} + a^{2} d^{5} e^{4} + {\left (c^{2} d^{4} e^{5} - 2 \, a c d^{2} e^{7} + a^{2} e^{9}\right )} x^{5} + 5 \, {\left (c^{2} d^{5} e^{4} - 2 \, a c d^{3} e^{6} + a^{2} d e^{8}\right )} x^{4} + 10 \, {\left (c^{2} d^{6} e^{3} - 2 \, a c d^{4} e^{5} + a^{2} d^{2} e^{7}\right )} x^{3} + 10 \, {\left (c^{2} d^{7} e^{2} - 2 \, a c d^{5} e^{4} + a^{2} d^{3} e^{6}\right )} x^{2} + 5 \, {\left (c^{2} d^{8} e - 2 \, a c d^{6} e^{3} + a^{2} d^{4} e^{5}\right )} x\right )}} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^8,x, algorithm=" fricas")
Output:
2/63*(2*c^4*d^4*e*x^4 + 9*a^3*c*d^2*e^3 - 7*a^4*e^5 + (9*c^4*d^5 - a*c^3*d ^3*e^2)*x^3 + 3*(9*a*c^3*d^4*e - 5*a^2*c^2*d^2*e^3)*x^2 + (27*a^2*c^2*d^3* e^2 - 19*a^3*c*d*e^4)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(c^2* d^9 - 2*a*c*d^7*e^2 + a^2*d^5*e^4 + (c^2*d^4*e^5 - 2*a*c*d^2*e^7 + a^2*e^9 )*x^5 + 5*(c^2*d^5*e^4 - 2*a*c*d^3*e^6 + a^2*d*e^8)*x^4 + 10*(c^2*d^6*e^3 - 2*a*c*d^4*e^5 + a^2*d^2*e^7)*x^3 + 10*(c^2*d^7*e^2 - 2*a*c*d^5*e^4 + a^2 *d^3*e^6)*x^2 + 5*(c^2*d^8*e - 2*a*c*d^6*e^3 + a^2*d^4*e^5)*x)
\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^8} \, dx=\int \frac {\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {5}{2}}}{\left (d + e x\right )^{8}}\, dx \] Input:
integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/(e*x+d)**8,x)
Output:
Integral(((d + e*x)*(a*e + c*d*x))**(5/2)/(d + e*x)**8, x)
Exception generated. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^8} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^8,x, algorithm=" 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(e*(a*e^2-c*d^2)>0)', see `assume ?` for mor
Exception generated. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^8} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^8,x, algorithm=" giac")
Output:
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,[0,0,5]%%%},[10]%%%}+%%%{%%{[%%%{-10,[0,1,4]%%%},0]: [1,0,%%%{
Time = 8.06 (sec) , antiderivative size = 3125, normalized size of antiderivative = 28.15 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^8} \, dx=\text {Too large to display} \] Input:
int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(d + e*x)^8,x)
Output:
(((d*((8*c^5*d^6)/(63*e*(a*e^2 - c*d^2)^2*(3*a*e^3 - 3*c*d^2*e)) - (4*c^4* d^4*(17*a*e^2 - 13*c*d^2))/(63*e*(a*e^2 - c*d^2)^2*(3*a*e^3 - 3*c*d^2*e))) )/e + (4*c^3*d^3*(148*a^2*e^4 + 73*c^2*d^4 - 211*a*c*d^2*e^2))/(315*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 - (((2*a^3*e^4)/(9*a*e^3 - 9*c*d^2*e) - (d*((d*(( 2*c^3*d^4)/(9*a*e^3 - 9*c*d^2*e) - (6*a*c^2*d^2*e^2)/(9*a*e^3 - 9*c*d^2*e) ))/e + (6*a^2*c*d*e^3)/(9*a*e^3 - 9*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)^5 + (((2*c^4*d^5 + 26*a*c^3*d^3*e^2)/(63* e^2*(a*e^2 - c*d^2)*(3*a*e^3 - 3*c*d^2*e)) - (4*c^4*d^5)/(9*e^2*(a*e^2 - c *d^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*((32*c^7*d^8)/(945*e*(a*e^2 - c*d^2)^5) - (32*c^ 6*d^6*(17*a*e^2 - 14*c*d^2))/(945*e*(a*e^2 - c*d^2)^5)))/e + (32*c^5*d^5*( 116*a^2*e^4 + 85*c^2*d^4 - 198*a*c*d^2*e^2))/(945*e^2*(a*e^2 - c*d^2)^5))) /e - (32*a*c^4*d^4*(100*a^2*e^4 + 85*c^2*d^4 - 184*a*c*d^2*e^2))/(945*e*(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) + (((d*((d*((4*c^4*d^5)/(9*(a*e^2 - c*d^2)*(7*a*e^3 - 7*c*d^2*e)) - (4*c^ 3*d^3*(8*a*e^2 - 5*c*d^2))/(9*(a*e^2 - c*d^2)*(7*a*e^3 - 7*c*d^2*e))))/e + (20*c^4*d^6 - 80*a*c^3*d^4*e^2 + 72*a^2*c^2*d^2*e^4)/(9*e*(a*e^2 - c*d^2) *(7*a*e^3 - 7*c*d^2*e))))/e - (4*a*c*d*(11*a^2*e^4 + 5*c^2*d^4 - 15*a*c*d^ 2*e^2))/(9*(a*e^2 - c*d^2)*(7*a*e^3 - 7*c*d^2*e)))*(x*(a*e^2 + c*d^2) +...
Time = 0.39 (sec) , antiderivative size = 592, normalized size of antiderivative = 5.33 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^8} \, dx=\frac {-\frac {2 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a^{4} e^{9}}{9}+\frac {2 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a^{3} c \,d^{2} e^{7}}{7}-\frac {38 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a^{3} c d \,e^{8} x}{63}+\frac {6 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a^{2} c^{2} d^{3} e^{6} x}{7}-\frac {10 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a^{2} c^{2} d^{2} e^{7} x^{2}}{21}+\frac {6 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a \,c^{3} d^{4} e^{5} x^{2}}{7}-\frac {2 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a \,c^{3} d^{3} e^{6} x^{3}}{63}+\frac {2 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{4} d^{5} e^{4} x^{3}}{7}+\frac {4 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{4} d^{4} e^{5} x^{4}}{63}-\frac {4 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, c^{4} d^{9}}{63}-\frac {20 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, c^{4} d^{8} e x}{63}-\frac {40 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, c^{4} d^{7} e^{2} x^{2}}{63}-\frac {40 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, c^{4} d^{6} e^{3} x^{3}}{63}-\frac {20 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, c^{4} d^{5} e^{4} x^{4}}{63}-\frac {4 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, c^{4} d^{4} e^{5} x^{5}}{63}}{e^{4} \left (a^{2} e^{9} x^{5}-2 a c \,d^{2} e^{7} x^{5}+c^{2} d^{4} e^{5} x^{5}+5 a^{2} d \,e^{8} x^{4}-10 a c \,d^{3} e^{6} x^{4}+5 c^{2} d^{5} e^{4} x^{4}+10 a^{2} d^{2} e^{7} x^{3}-20 a c \,d^{4} e^{5} x^{3}+10 c^{2} d^{6} e^{3} x^{3}+10 a^{2} d^{3} e^{6} x^{2}-20 a c \,d^{5} e^{4} x^{2}+10 c^{2} d^{7} e^{2} x^{2}+5 a^{2} d^{4} e^{5} x -10 a c \,d^{6} e^{3} x +5 c^{2} d^{8} e x +a^{2} d^{5} e^{4}-2 a c \,d^{7} e^{2}+c^{2} d^{9}\right )} \] Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^8,x)
Output:
(2*( - 7*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**4*e**9 + 9*sqrt(d + e*x)*sqrt( a*e + c*d*x)*a**3*c*d**2*e**7 - 19*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c* d*e**8*x + 27*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**2*d**3*e**6*x - 15*s qrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**2*d**2*e**7*x**2 + 27*sqrt(d + e*x) *sqrt(a*e + c*d*x)*a*c**3*d**4*e**5*x**2 - sqrt(d + e*x)*sqrt(a*e + c*d*x) *a*c**3*d**3*e**6*x**3 + 9*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**4*d**5*e**4* x**3 + 2*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**4*d**4*e**5*x**4 - 2*sqrt(e)*s qrt(d)*sqrt(c)*c**4*d**9 - 10*sqrt(e)*sqrt(d)*sqrt(c)*c**4*d**8*e*x - 20*s qrt(e)*sqrt(d)*sqrt(c)*c**4*d**7*e**2*x**2 - 20*sqrt(e)*sqrt(d)*sqrt(c)*c* *4*d**6*e**3*x**3 - 10*sqrt(e)*sqrt(d)*sqrt(c)*c**4*d**5*e**4*x**4 - 2*sqr t(e)*sqrt(d)*sqrt(c)*c**4*d**4*e**5*x**5))/(63*e**4*(a**2*d**5*e**4 + 5*a* *2*d**4*e**5*x + 10*a**2*d**3*e**6*x**2 + 10*a**2*d**2*e**7*x**3 + 5*a**2* d*e**8*x**4 + a**2*e**9*x**5 - 2*a*c*d**7*e**2 - 10*a*c*d**6*e**3*x - 20*a *c*d**5*e**4*x**2 - 20*a*c*d**4*e**5*x**3 - 10*a*c*d**3*e**6*x**4 - 2*a*c* d**2*e**7*x**5 + c**2*d**9 + 5*c**2*d**8*e*x + 10*c**2*d**7*e**2*x**2 + 10 *c**2*d**6*e**3*x**3 + 5*c**2*d**5*e**4*x**4 + c**2*d**4*e**5*x**5))