Integrand size = 37, antiderivative size = 111 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^4} \, dx=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 \left (c d^2-a e^2\right ) (d+e x)^4}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{15 \left (c d^2-a e^2\right )^2 (d+e x)^3} \] Output:
2/5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(-a*e^2+c*d^2)/(e*x+d)^4+4/15* c*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(-a*e^2+c*d^2)^2/(e*x+d)^3
Time = 0.24 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.73 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^4} \, dx=\frac {2 \sqrt {(a e+c d x) (d+e x)} \left (-3 a^2 e^3+a c d e (5 d-e x)+c^2 d^2 x (5 d+2 e x)\right )}{15 \left (c d^2-a e^2\right )^2 (d+e x)^3} \] Input:
Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(d + e*x)^4,x]
Output:
(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-3*a^2*e^3 + a*c*d*e*(5*d - e*x) + c^2*d ^2*x*(5*d + 2*e*x)))/(15*(c*d^2 - a*e^2)^2*(d + e*x)^3)
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.
\(\displaystyle \int \frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{(d+e x)^4} \, dx\) |
\(\Big \downarrow \) 1129 |
\(\displaystyle \frac {2 c d \int \frac {\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{(d+e x)^3}dx}{5 \left (c d^2-a e^2\right )}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 (d+e x)^4 \left (c d^2-a e^2\right )}\) |
\(\Big \downarrow \) 1123 |
\(\displaystyle \frac {4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{15 (d+e x)^3 \left (c d^2-a e^2\right )^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 (d+e x)^4 \left (c d^2-a e^2\right )}\) |
Input:
Int[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(d + e*x)^4,x]
Output:
(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(5*(c*d^2 - a*e^2)*(d + e*x)^4) + (4*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(15*(c*d^2 - a*e^2)^2*(d + e*x)^3)
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 = 2.44 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.81
method | result | size |
gosper | \(-\frac {2 \left (c d x +a e \right ) \left (-2 c d x e +3 a \,e^{2}-5 c \,d^{2}\right ) \sqrt {c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e}}{15 \left (e x +d \right )^{3} \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}\) | \(90\) |
orering | \(-\frac {2 \left (-2 c d x e +3 a \,e^{2}-5 c \,d^{2}\right ) \left (c d x +a e \right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{15 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \left (e x +d \right )^{3}}\) | \(91\) |
trager | \(-\frac {2 \left (-2 d^{2} e \,c^{2} x^{2}+a c d \,e^{2} x -5 c^{2} d^{3} x +3 a^{2} e^{3}-5 d^{2} e a c \right ) \sqrt {c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e}}{15 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \left (e x +d \right )^{3}}\) | \(109\) |
default | \(\frac {-\frac {2 \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{5 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{4}}+\frac {4 d e c \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{15 \left (a \,e^{2}-c \,d^{2}\right )^{2} \left (x +\frac {d}{e}\right )^{3}}}{e^{4}}\) | \(131\) |
Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)/(e*x+d)^4,x,method=_RETURNVERB OSE)
Output:
-2/15*(c*d*x+a*e)*(-2*c*d*e*x+3*a*e^2-5*c*d^2)*(c*d*e*x^2+a*e^2*x+c*d^2*x+ a*d*e)^(1/2)/(e*x+d)^3/(a^2*e^4-2*a*c*d^2*e^2+c^2*d^4)
Time = 0.43 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.85 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^4} \, dx=\frac {2 \, {\left (2 \, c^{2} d^{2} e x^{2} + 5 \, a c d^{2} e - 3 \, a^{2} e^{3} + {\left (5 \, c^{2} d^{3} - a c d e^{2}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{15 \, {\left (c^{2} d^{7} - 2 \, a c d^{5} e^{2} + a^{2} d^{3} e^{4} + {\left (c^{2} d^{4} e^{3} - 2 \, a c d^{2} e^{5} + a^{2} e^{7}\right )} x^{3} + 3 \, {\left (c^{2} d^{5} e^{2} - 2 \, a c d^{3} e^{4} + a^{2} d e^{6}\right )} x^{2} + 3 \, {\left (c^{2} d^{6} e - 2 \, a c d^{4} e^{3} + a^{2} d^{2} e^{5}\right )} x\right )}} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^4,x, algorithm=" fricas")
Output:
2/15*(2*c^2*d^2*e*x^2 + 5*a*c*d^2*e - 3*a^2*e^3 + (5*c^2*d^3 - a*c*d*e^2)* x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(c^2*d^7 - 2*a*c*d^5*e^2 + a^2*d^3*e^4 + (c^2*d^4*e^3 - 2*a*c*d^2*e^5 + a^2*e^7)*x^3 + 3*(c^2*d^5*e^2 - 2*a*c*d^3*e^4 + a^2*d*e^6)*x^2 + 3*(c^2*d^6*e - 2*a*c*d^4*e^3 + a^2*d^2 *e^5)*x)
\[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^4} \, dx=\int \frac {\sqrt {\left (d + e x\right ) \left (a e + c d x\right )}}{\left (d + e x\right )^{4}}\, dx \] Input:
integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(e*x+d)**4,x)
Output:
Integral(sqrt((d + e*x)*(a*e + c*d*x))/(d + e*x)**4, x)
Exception generated. \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^4} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^4,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. \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^4} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^4,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 = 5.66 (sec) , antiderivative size = 562, normalized size of antiderivative = 5.06 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^4} \, dx=\frac {\left (\frac {4\,c^2\,d^3}{5\,\left (a\,e^2-c\,d^2\right )\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}-\frac {4\,a\,c\,d\,e^2}{5\,\left (a\,e^2-c\,d^2\right )\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^2}-\frac {\left (\frac {2\,a\,e^2}{5\,a\,e^3-5\,c\,d^2\,e}-\frac {2\,c\,d^2}{5\,a\,e^3-5\,c\,d^2\,e}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^3}+\frac {\left (\frac {4\,c^3\,d^4+4\,a\,c^2\,d^2\,e^2}{15\,e\,{\left (a\,e^2-c\,d^2\right )}^3}-\frac {8\,c^3\,d^4}{15\,e\,{\left (a\,e^2-c\,d^2\right )}^3}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{d+e\,x}+\frac {\left (\frac {2\,c^2\,d^3+2\,a\,c\,d\,e^2}{5\,\left (a\,e^2-c\,d^2\right )\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}-\frac {4\,c^2\,d^3}{5\,\left (a\,e^2-c\,d^2\right )\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^2}+\frac {\left (\frac {8\,c^3\,d^4}{15\,e\,{\left (a\,e^2-c\,d^2\right )}^3}-\frac {8\,a\,c^2\,d^2\,e}{15\,{\left (a\,e^2-c\,d^2\right )}^3}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{d+e\,x}+\frac {8\,c^2\,d^2\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{15\,e\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (d+e\,x\right )} \] Input:
int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)/(d + e*x)^4,x)
Output:
(((4*c^2*d^3)/(5*(a*e^2 - c*d^2)*(3*a*e^3 - 3*c*d^2*e)) - (4*a*c*d*e^2)/(5 *(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 - (((2*a*e^2)/(5*a*e^3 - 5*c*d^2*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 + (((4*c^3*d^4 + 4*a*c^2*d^2*e^2)/(15*e*(a*e^2 - c*d^2)^3) - (8*c^ 3*d^4)/(15*e*(a*e^2 - c*d^2)^3))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^( 1/2))/(d + e*x) + (((2*c^2*d^3 + 2*a*c*d*e^2)/(5*(a*e^2 - c*d^2)*(3*a*e^3 - 3*c*d^2*e)) - (4*c^2*d^3)/(5*(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 + (((8*c^3*d^4)/(1 5*e*(a*e^2 - c*d^2)^3) - (8*a*c^2*d^2*e)/(15*(a*e^2 - c*d^2)^3))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x) + (8*c^2*d^2*(x*(a*e^2 + c *d^2) + a*d*e + c*d*e*x^2)^(1/2))/(15*e*(a*e^2 - c*d^2)^2*(d + e*x))
Time = 0.25 (sec) , antiderivative size = 342, normalized size of antiderivative = 3.08 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^4} \, dx=\frac {-\frac {2 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a^{2} e^{5}}{5}+\frac {2 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a c \,d^{2} e^{3}}{3}-\frac {2 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a c d \,e^{4} x}{15}+\frac {2 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{2} d^{3} e^{2} x}{3}+\frac {4 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{2} d^{2} e^{3} x^{2}}{15}-\frac {4 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, c^{2} d^{5}}{15}-\frac {4 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, c^{2} d^{4} e x}{5}-\frac {4 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, c^{2} d^{3} e^{2} x^{2}}{5}-\frac {4 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, c^{2} d^{2} e^{3} x^{3}}{15}}{e^{2} \left (a^{2} e^{7} x^{3}-2 a c \,d^{2} e^{5} x^{3}+c^{2} d^{4} e^{3} x^{3}+3 a^{2} d \,e^{6} x^{2}-6 a c \,d^{3} e^{4} x^{2}+3 c^{2} d^{5} e^{2} x^{2}+3 a^{2} d^{2} e^{5} x -6 a c \,d^{4} e^{3} x +3 c^{2} d^{6} e x +a^{2} d^{3} e^{4}-2 a c \,d^{5} e^{2}+c^{2} d^{7}\right )} \] Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^4,x)
Output:
(2*( - 3*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*e**5 + 5*sqrt(d + e*x)*sqrt( a*e + c*d*x)*a*c*d**2*e**3 - sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c*d*e**4*x + 5*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**2*d**3*e**2*x + 2*sqrt(d + e*x)*sqr t(a*e + c*d*x)*c**2*d**2*e**3*x**2 - 2*sqrt(e)*sqrt(d)*sqrt(c)*c**2*d**5 - 6*sqrt(e)*sqrt(d)*sqrt(c)*c**2*d**4*e*x - 6*sqrt(e)*sqrt(d)*sqrt(c)*c**2* d**3*e**2*x**2 - 2*sqrt(e)*sqrt(d)*sqrt(c)*c**2*d**2*e**3*x**3))/(15*e**2* (a**2*d**3*e**4 + 3*a**2*d**2*e**5*x + 3*a**2*d*e**6*x**2 + a**2*e**7*x**3 - 2*a*c*d**5*e**2 - 6*a*c*d**4*e**3*x - 6*a*c*d**3*e**4*x**2 - 2*a*c*d**2 *e**5*x**3 + c**2*d**7 + 3*c**2*d**6*e*x + 3*c**2*d**5*e**2*x**2 + c**2*d* *4*e**3*x**3))