Integrand size = 37, antiderivative size = 171 \[ \int \frac {1}{(d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 \left (c d^2-a e^2\right ) (d+e x)^3}+\frac {8 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{15 \left (c d^2-a e^2\right )^2 (d+e x)^2}+\frac {16 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{15 \left (c d^2-a e^2\right )^3 (d+e x)} \] Output:
2/5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e^2+c*d^2)/(e*x+d)^3+8/15* c*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e^2+c*d^2)^2/(e*x+d)^2+16/ 15*c^2*d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e^2+c*d^2)^3/(e*x+d )
Time = 0.14 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.55 \[ \int \frac {1}{(d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \sqrt {(a e+c d x) (d+e x)} \left (3 a^2 e^4-2 a c d e^2 (5 d+2 e x)+c^2 d^2 \left (15 d^2+20 d e x+8 e^2 x^2\right )\right )}{15 \left (c d^2-a e^2\right )^3 (d+e x)^3} \] Input:
Integrate[1/((d + e*x)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]
Output:
(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(3*a^2*e^4 - 2*a*c*d*e^2*(5*d + 2*e*x) + c^2*d^2*(15*d^2 + 20*d*e*x + 8*e^2*x^2)))/(15*(c*d^2 - a*e^2)^3*(d + e*x)^ 3)
Time = 0.53 (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.
\(\displaystyle \int \frac {1}{(d+e x)^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \, dx\) |
\(\Big \downarrow \) 1129 |
\(\displaystyle \frac {4 c d \int \frac {1}{(d+e x)^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{5 \left (c d^2-a e^2\right )}+\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 (d+e x)^3 \left (c d^2-a e^2\right )}\) |
\(\Big \downarrow \) 1129 |
\(\displaystyle \frac {4 c d \left (\frac {2 c d \int \frac {1}{(d+e x) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{3 \left (c d^2-a e^2\right )}+\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 (d+e x)^2 \left (c d^2-a e^2\right )}\right )}{5 \left (c d^2-a e^2\right )}+\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 (d+e x)^3 \left (c d^2-a e^2\right )}\) |
\(\Big \downarrow \) 1123 |
\(\displaystyle \frac {4 c d \left (\frac {4 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 (d+e x) \left (c d^2-a e^2\right )^2}+\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 (d+e x)^2 \left (c d^2-a e^2\right )}\right )}{5 \left (c d^2-a e^2\right )}+\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 (d+e x)^3 \left (c d^2-a e^2\right )}\) |
Input:
Int[1/((d + e*x)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]
Output:
(2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(5*(c*d^2 - a*e^2)*(d + e* x)^3) + (4*c*d*((2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(3*(c*d^2 - a*e^2)*(d + e*x)^2) + (4*c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2] )/(3*(c*d^2 - a*e^2)^2*(d + e*x))))/(5*(c*d^2 - a*e^2))
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 = 1.33 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.81
method | result | size |
trager | \(-\frac {2 \left (8 x^{2} c^{2} d^{2} e^{2}-4 x a c d \,e^{3}+20 x \,c^{2} d^{3} e +3 a^{2} e^{4}-10 a c \,d^{2} e^{2}+15 c^{2} d^{4}\right ) \sqrt {c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e}}{15 \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} a \,c^{2}-d^{6} c^{3}\right ) \left (e x +d \right )^{3}}\) | \(138\) |
gosper | \(-\frac {2 \left (c d x +a e \right ) \left (8 x^{2} c^{2} d^{2} e^{2}-4 x a c d \,e^{3}+20 x \,c^{2} d^{3} e +3 a^{2} e^{4}-10 a c \,d^{2} e^{2}+15 c^{2} d^{4}\right )}{15 \left (e x +d \right )^{2} \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} a \,c^{2}-d^{6} c^{3}\right ) \sqrt {c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e}}\) | \(146\) |
orering | \(-\frac {2 \left (8 x^{2} c^{2} d^{2} e^{2}-4 x a c d \,e^{3}+20 x \,c^{2} d^{3} e +3 a^{2} e^{4}-10 a c \,d^{2} e^{2}+15 c^{2} d^{4}\right ) \left (c d x +a e \right )}{15 \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} a \,c^{2}-d^{6} c^{3}\right ) \left (e x +d \right )^{2} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}\) | \(147\) |
default | \(\frac {-\frac {2 \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}{5 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{3}}-\frac {4 d e c \left (-\frac {2 \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}{3 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2}}+\frac {4 d e c \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}{3 \left (a \,e^{2}-c \,d^{2}\right )^{2} \left (x +\frac {d}{e}\right )}\right )}{5 \left (a \,e^{2}-c \,d^{2}\right )}}{e^{3}}\) | \(212\) |
Input:
int(1/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2),x,method=_RETURNVE RBOSE)
Output:
-2/15*(8*c^2*d^2*e^2*x^2-4*a*c*d*e^3*x+20*c^2*d^3*e*x+3*a^2*e^4-10*a*c*d^2 *e^2+15*c^2*d^4)/(a^3*e^6-3*a^2*c*d^2*e^4+3*a*c^2*d^4*e^2-c^3*d^6)/(e*x+d) ^3*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)
Time = 1.29 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.63 \[ \int \frac {1}{(d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \, {\left (8 \, c^{2} d^{2} e^{2} x^{2} + 15 \, c^{2} d^{4} - 10 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4} + 4 \, {\left (5 \, c^{2} d^{3} e - a c d e^{3}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{15 \, {\left (c^{3} d^{9} - 3 \, a c^{2} d^{7} e^{2} + 3 \, a^{2} c d^{5} e^{4} - a^{3} d^{3} e^{6} + {\left (c^{3} d^{6} e^{3} - 3 \, a c^{2} d^{4} e^{5} + 3 \, a^{2} c d^{2} e^{7} - a^{3} e^{9}\right )} x^{3} + 3 \, {\left (c^{3} d^{7} e^{2} - 3 \, a c^{2} d^{5} e^{4} + 3 \, a^{2} c d^{3} e^{6} - a^{3} d e^{8}\right )} x^{2} + 3 \, {\left (c^{3} d^{8} e - 3 \, a c^{2} d^{6} e^{3} + 3 \, a^{2} c d^{4} e^{5} - a^{3} d^{2} e^{7}\right )} x\right )}} \] Input:
integrate(1/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm ="fricas")
Output:
2/15*(8*c^2*d^2*e^2*x^2 + 15*c^2*d^4 - 10*a*c*d^2*e^2 + 3*a^2*e^4 + 4*(5*c ^2*d^3*e - a*c*d*e^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(c^3* d^9 - 3*a*c^2*d^7*e^2 + 3*a^2*c*d^5*e^4 - a^3*d^3*e^6 + (c^3*d^6*e^3 - 3*a *c^2*d^4*e^5 + 3*a^2*c*d^2*e^7 - a^3*e^9)*x^3 + 3*(c^3*d^7*e^2 - 3*a*c^2*d ^5*e^4 + 3*a^2*c*d^3*e^6 - a^3*d*e^8)*x^2 + 3*(c^3*d^8*e - 3*a*c^2*d^6*e^3 + 3*a^2*c*d^4*e^5 - a^3*d^2*e^7)*x)
\[ \int \frac {1}{(d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {1}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )} \left (d + e x\right )^{3}}\, dx \] Input:
integrate(1/(e*x+d)**3/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)
Output:
Integral(1/(sqrt((d + e*x)*(a*e + c*d*x))*(d + e*x)**3), x)
Exception generated. \[ \int \frac {1}{(d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),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. \[ \int \frac {1}{(d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),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,[0,0,3]%%%},[6]%%%}+%%%{%%{[%%%{-6,[0,1,2]%%%},0]:[1 ,0,%%%{-1
Time = 6.17 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.64 \[ \int \frac {1}{(d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=-\frac {2\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (3\,a^2\,e^4-10\,a\,c\,d^2\,e^2-4\,a\,c\,d\,e^3\,x+15\,c^2\,d^4+20\,c^2\,d^3\,e\,x+8\,c^2\,d^2\,e^2\,x^2\right )}{15\,{\left (a\,e^2-c\,d^2\right )}^3\,{\left (d+e\,x\right )}^3} \] Input:
int(1/((d + e*x)^3*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)),x)
Output:
-(2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*(3*a^2*e^4 + 15*c^2*d^4 + 8*c^2*d^2*e^2*x^2 - 10*a*c*d^2*e^2 + 20*c^2*d^3*e*x - 4*a*c*d*e^3*x))/(1 5*(a*e^2 - c*d^2)^3*(d + e*x)^3)
Time = 0.26 (sec) , antiderivative size = 431, normalized size of antiderivative = 2.52 \[ \int \frac {1}{(d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {-\frac {2 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a^{2} e^{5}}{5}+\frac {4 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a c \,d^{2} e^{3}}{3}+\frac {8 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a c d \,e^{4} x}{15}-2 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{2} d^{4} e -\frac {8 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{2} d^{3} e^{2} x}{3}-\frac {16 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{2} d^{2} e^{3} x^{2}}{15}+\frac {16 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, c^{2} d^{5}}{15}+\frac {16 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, c^{2} d^{4} e x}{5}+\frac {16 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, c^{2} d^{3} e^{2} x^{2}}{5}+\frac {16 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, c^{2} d^{2} e^{3} x^{3}}{15}}{e \left (a^{3} e^{9} x^{3}-3 a^{2} c \,d^{2} e^{7} x^{3}+3 a \,c^{2} d^{4} e^{5} x^{3}-c^{3} d^{6} e^{3} x^{3}+3 a^{3} d \,e^{8} x^{2}-9 a^{2} c \,d^{3} e^{6} x^{2}+9 a \,c^{2} d^{5} e^{4} x^{2}-3 c^{3} d^{7} e^{2} x^{2}+3 a^{3} d^{2} e^{7} x -9 a^{2} c \,d^{4} e^{5} x +9 a \,c^{2} d^{6} e^{3} x -3 c^{3} d^{8} e x +a^{3} d^{3} e^{6}-3 a^{2} c \,d^{5} e^{4}+3 a \,c^{2} d^{7} e^{2}-c^{3} d^{9}\right )} \] Input:
int(1/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x)
Output:
(2*( - 3*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*e**5 + 10*sqrt(d + e*x)*sqrt (a*e + c*d*x)*a*c*d**2*e**3 + 4*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c*d*e**4 *x - 15*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**2*d**4*e - 20*sqrt(d + e*x)*sqr t(a*e + c*d*x)*c**2*d**3*e**2*x - 8*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**2*d **2*e**3*x**2 + 8*sqrt(e)*sqrt(d)*sqrt(c)*c**2*d**5 + 24*sqrt(e)*sqrt(d)*s qrt(c)*c**2*d**4*e*x + 24*sqrt(e)*sqrt(d)*sqrt(c)*c**2*d**3*e**2*x**2 + 8* sqrt(e)*sqrt(d)*sqrt(c)*c**2*d**2*e**3*x**3))/(15*e*(a**3*d**3*e**6 + 3*a* *3*d**2*e**7*x + 3*a**3*d*e**8*x**2 + a**3*e**9*x**3 - 3*a**2*c*d**5*e**4 - 9*a**2*c*d**4*e**5*x - 9*a**2*c*d**3*e**6*x**2 - 3*a**2*c*d**2*e**7*x**3 + 3*a*c**2*d**7*e**2 + 9*a*c**2*d**6*e**3*x + 9*a*c**2*d**5*e**4*x**2 + 3 *a*c**2*d**4*e**5*x**3 - c**3*d**9 - 3*c**3*d**8*e*x - 3*c**3*d**7*e**2*x* *2 - c**3*d**6*e**3*x**3))