Integrand size = 37, antiderivative size = 116 \[ \int \frac {(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=-\frac {2 (d+e x)}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 e \left (c d^2+a e^2+2 c d e x\right )}{3 c d \left (c d^2-a e^2\right )^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \] Output:
1/3*(-2*e*x-2*d)/c/d/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+2/3*e*(2*c*d* e*x+a*e^2+c*d^2)/c/d/(-a*e^2+c*d^2)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1 /2)
Time = 0.12 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.51 \[ \int \frac {(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=-\frac {2 (d+e x)^2 \left (-3 a e^2+c d (d-2 e x)\right )}{3 \left (c d^2-a e^2\right )^2 ((a e+c d x) (d+e x))^{3/2}} \] Input:
Integrate[(d + e*x)^2/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]
Output:
(-2*(d + e*x)^2*(-3*a*e^2 + c*d*(d - 2*e*x)))/(3*(c*d^2 - a*e^2)^2*((a*e + c*d*x)*(d + e*x))^(3/2))
Time = 0.24 (sec) , antiderivative size = 116, 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 = {1126, 1088}
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.
\(\displaystyle \int \frac {(d+e x)^2}{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 1126 |
\(\displaystyle -\frac {e \int \frac {1}{\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{3 c d}-\frac {2 (d+e x)}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 1088 |
\(\displaystyle \frac {2 e \left (a e^2+c d^2+2 c d e x\right )}{3 c d \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {2 (d+e x)}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
Input:
Int[(d + e*x)^2/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]
Output:
(-2*(d + e*x))/(3*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (2* e*(c*d^2 + a*e^2 + 2*c*d*e*x))/(3*c*d*(c*d^2 - a*e^2)^2*Sqrt[a*d*e + (c*d^ 2 + a*e^2)*x + c*d*e*x^2])
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[-2*((b + 2*c*x)/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Int[((d_.) + (e_.)*(x_))^2*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symb ol] :> Simp[e*(d + e*x)*((a + b*x + c*x^2)^(p + 1)/(c*(p + 1))), x] - Simp[ e^2*((p + 2)/(c*(p + 1))) Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[ {a, b, c, d, e, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1]
Time = 1.58 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.73
method | result | size |
trager | \(\frac {2 \left (2 c d x e +3 a \,e^{2}-c \,d^{2}\right ) \sqrt {c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e}}{3 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \left (c d x +a e \right )^{2}}\) | \(85\) |
gosper | \(\frac {2 \left (e x +d \right )^{3} \left (c d x +a e \right ) \left (2 c d x e +3 a \,e^{2}-c \,d^{2}\right )}{3 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \left (c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}\) | \(90\) |
orering | \(\frac {2 \left (2 c d x e +3 a \,e^{2}-c \,d^{2}\right ) \left (e x +d \right )^{3} \left (c d x +a e \right )}{3 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {5}{2}}}\) | \(91\) |
default | \(d^{2} \left (\frac {\frac {4}{3} c d x e +\frac {2}{3} a \,e^{2}+\frac {2}{3} c \,d^{2}}{\left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {3}{2}}}+\frac {16 d e c \left (2 c d x e +a \,e^{2}+c \,d^{2}\right )}{3 \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right )^{2} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}\right )+e^{2} \left (-\frac {x}{2 d e c {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {3}{2}}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (-\frac {1}{3 d e c {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {3}{2}}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (\frac {\frac {4}{3} c d x e +\frac {2}{3} a \,e^{2}+\frac {2}{3} c \,d^{2}}{\left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {3}{2}}}+\frac {16 d e c \left (2 c d x e +a \,e^{2}+c \,d^{2}\right )}{3 \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right )^{2} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}\right )}{2 d e c}\right )}{4 d e c}+\frac {a \left (\frac {\frac {4}{3} c d x e +\frac {2}{3} a \,e^{2}+\frac {2}{3} c \,d^{2}}{\left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {3}{2}}}+\frac {16 d e c \left (2 c d x e +a \,e^{2}+c \,d^{2}\right )}{3 \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right )^{2} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}\right )}{2 c}\right )+2 d e \left (-\frac {1}{3 d e c {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {3}{2}}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (\frac {\frac {4}{3} c d x e +\frac {2}{3} a \,e^{2}+\frac {2}{3} c \,d^{2}}{\left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {3}{2}}}+\frac {16 d e c \left (2 c d x e +a \,e^{2}+c \,d^{2}\right )}{3 \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right )^{2} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}\right )}{2 d e c}\right )\) | \(812\) |
Input:
int((e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/2),x,method=_RETURNVERB OSE)
Output:
2/3*(2*c*d*e*x+3*a*e^2-c*d^2)/(a^2*e^4-2*a*c*d^2*e^2+c^2*d^4)/(c*d*x+a*e)^ 2*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)
Time = 0.45 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.34 \[ \int \frac {(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x - c d^{2} + 3 \, a e^{2}\right )}}{3 \, {\left (a^{2} c^{2} d^{4} e^{2} - 2 \, a^{3} c d^{2} e^{4} + a^{4} e^{6} + {\left (c^{4} d^{6} - 2 \, a c^{3} d^{4} e^{2} + a^{2} c^{2} d^{2} e^{4}\right )} x^{2} + 2 \, {\left (a c^{3} d^{5} e - 2 \, a^{2} c^{2} d^{3} e^{3} + a^{3} c d e^{5}\right )} x\right )}} \] Input:
integrate((e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm=" fricas")
Output:
2/3*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*c*d*e*x - c*d^2 + 3*a*e ^2)/(a^2*c^2*d^4*e^2 - 2*a^3*c*d^2*e^4 + a^4*e^6 + (c^4*d^6 - 2*a*c^3*d^4* e^2 + a^2*c^2*d^2*e^4)*x^2 + 2*(a*c^3*d^5*e - 2*a^2*c^2*d^3*e^3 + a^3*c*d* e^5)*x)
\[ \int \frac {(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\int \frac {\left (d + e x\right )^{2}}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((e*x+d)**2/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)
Output:
Integral((d + e*x)**2/((d + e*x)*(a*e + c*d*x))**(5/2), x)
Exception generated. \[ \int \frac {(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),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(a*e^2-c*d^2>0)', see `assume?` f or more de
Exception generated. \[ \int \frac {(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),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,[2,2,0]%%%},[4,0]%%%}+%%%{%%{[%%%{-4,[1,1,1]%%%},0]: [1,0,%%%{
Time = 10.71 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.62 \[ \int \frac {(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {2\,\left (-c\,d^2+2\,c\,x\,d\,e+3\,a\,e^2\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{3\,{\left (a\,e+c\,d\,x\right )}^2\,{\left (a\,e^2-c\,d^2\right )}^2} \] Input:
int((d + e*x)^2/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2),x)
Output:
(2*(3*a*e^2 - c*d^2 + 2*c*d*e*x)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^( 1/2))/(3*(a*e + c*d*x)^2*(a*e^2 - c*d^2)^2)
Time = 0.51 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.45 \[ \int \frac {(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {-\frac {4 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, a \,e^{2}}{3}-\frac {4 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, c d e x}{3}+2 \sqrt {e x +d}\, a c d \,e^{2}-\frac {2 \sqrt {e x +d}\, c^{2} d^{3}}{3}+\frac {4 \sqrt {e x +d}\, c^{2} d^{2} e x}{3}}{\sqrt {c d x +a e}\, c d \left (a^{2} c d \,e^{4} x -2 a \,c^{2} d^{3} e^{2} x +c^{3} d^{5} x +a^{3} e^{5}-2 a^{2} c \,d^{2} e^{3}+a \,c^{2} d^{4} e \right )} \] Input:
int((e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x)
Output:
(2*( - 2*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a*e**2 - 2*sqrt(e)*sqrt (d)*sqrt(c)*sqrt(a*e + c*d*x)*c*d*e*x + 3*sqrt(d + e*x)*a*c*d*e**2 - sqrt( d + e*x)*c**2*d**3 + 2*sqrt(d + e*x)*c**2*d**2*e*x))/(3*sqrt(a*e + c*d*x)* c*d*(a**3*e**5 - 2*a**2*c*d**2*e**3 + a**2*c*d*e**4*x + a*c**2*d**4*e - 2* a*c**2*d**3*e**2*x + c**3*d**5*x))