Integrand size = 42, antiderivative size = 140 \[ \int \frac {(d+e x)^2 (f+g x)}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=-\frac {2 (e f-d g) (d+e x)^2}{e \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 \left (a e^2 g+c d (2 e f-3 d g)\right ) (d+e x)^3}{3 e \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \] Output:
-2*(-d*g+e*f)*(e*x+d)^2/e/(-a*e^2+c*d^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2) ^(3/2)+2/3*(a*e^2*g+c*d*(-3*d*g+2*e*f))*(e*x+d)^3/e/(-a*e^2+c*d^2)^2/(a*d* e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)
Time = 0.19 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.56 \[ \int \frac {(d+e x)^2 (f+g x)}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=-\frac {2 (d+e x)^2 (-a e (3 e f-2 d g+e g x)+c d (-2 e f x+d (f+3 g x)))}{3 \left (c d^2-a e^2\right )^2 ((a e+c d x) (d+e x))^{3/2}} \] Input:
Integrate[((d + e*x)^2*(f + g*x))/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^ (5/2),x]
Output:
(-2*(d + e*x)^2*(-(a*e*(3*e*f - 2*d*g + e*g*x)) + c*d*(-2*e*f*x + d*(f + 3 *g*x))))/(3*(c*d^2 - a*e^2)^2*((a*e + c*d*x)*(d + e*x))^(3/2))
Time = 0.33 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.06, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1218, 1124, 24}
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.
\(\displaystyle \int \frac {(d+e x)^2 (f+g x)}{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 1218 |
\(\displaystyle -\frac {\left (a e^2 g+c d (2 e f-3 d g)\right ) \int \frac {d+e x}{\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{3 c d \left (c d^2-a e^2\right )}-\frac {2 (d+e x)^2 (c d f-a e g)}{3 c d \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 1124 |
\(\displaystyle -\frac {\left (a e^2 g+c d (2 e f-3 d g)\right ) \left (e^2 \int 0dx-\frac {2 (d+e x)}{\left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{3 c d \left (c d^2-a e^2\right )}-\frac {2 (d+e x)^2 (c d f-a e g)}{3 c d \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {2 (d+e x) \left (a e^2 g+c d (2 e f-3 d g)\right )}{3 c d \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {2 (d+e x)^2 (c d f-a e g)}{3 c d \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
Input:
Int[((d + e*x)^2*(f + g*x))/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2), x]
Output:
(-2*(c*d*f - a*e*g)*(d + e*x)^2)/(3*c*d*(c*d^2 - a*e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (2*(a*e^2*g + c*d*(2*e*f - 3*d*g))*(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[((d_.) + (e_.)*(x_))^(m_.)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x _Symbol] :> Simp[-2*e*(2*c*d - b*e)^(m - 2)*((d + e*x)/(c^(m - 1)*Sqrt[a + b*x + c*x^2])), x] + Simp[e^2/c^(m - 1) Int[(1/Sqrt[a + b*x + c*x^2])*Exp andToSum[((2*c*d - b*e)^(m - 1) - c^(m - 1)*(d + e*x)^(m - 1))/(c*d - b*e - c*e*x), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d^2 - b*d*e + a*e ^2, 0] && IGtQ[m, 0]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + ( c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(g*(c*d - b*e) + c*e*f)*(d + e*x)^m*(( a + b*x + c*x^2)^(p + 1)/(c*(p + 1)*(2*c*d - b*e))), x] - Simp[e*((m*(g*(c* d - b*e) + c*e*f) + e*(p + 1)*(2*c*f - b*g))/(c*(p + 1)*(2*c*d - b*e))) I nt[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d , e, f, g}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0]
Time = 2.10 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.78
method | result | size |
trager | \(-\frac {2 \left (-a \,e^{2} g x +3 c \,d^{2} g x -2 c d e f x +2 a d e g -3 a \,e^{2} f +c \,d^{2} f \right ) \sqrt {c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e}}{3 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \left (c d x +a e \right )^{2}}\) | \(109\) |
gosper | \(-\frac {2 \left (e x +d \right )^{3} \left (c d x +a e \right ) \left (-a \,e^{2} g x +3 c \,d^{2} g x -2 c d e f x +2 a d e g -3 a \,e^{2} f +c \,d^{2} f \right )}{3 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \left (c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}\) | \(114\) |
orering | \(-\frac {2 \left (-a \,e^{2} g x +3 c \,d^{2} g x -2 c d e f x +2 a d e g -3 a \,e^{2} f +c \,d^{2} f \right ) \left (e x +d \right )^{3} \left (c d x +a e \right )}{3 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {5}{2}}}\) | \(115\) |
default | \(\text {Expression too large to display}\) | \(1546\) |
Input:
int((e*x+d)^2*(g*x+f)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/2),x,method=_RE TURNVERBOSE)
Output:
-2/3*(-a*e^2*g*x+3*c*d^2*g*x-2*c*d*e*f*x+2*a*d*e*g-3*a*e^2*f+c*d^2*f)/(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 = 2.54 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.31 \[ \int \frac {(d+e x)^2 (f+g x)}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=-\frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d e g + {\left (c d^{2} - 3 \, a e^{2}\right )} f - {\left (2 \, c d e f - {\left (3 \, c d^{2} - a e^{2}\right )} g\right )} x\right )}}{3 \, {\left (a^{2} c^{2} d^{4} e^{2} - 2 \, a^{3} c d^{2} e^{4} + a^{4} e^{6} + {\left (c^{4} d^{6} - 2 \, a c^{3} d^{4} e^{2} + a^{2} c^{2} d^{2} e^{4}\right )} x^{2} + 2 \, {\left (a c^{3} d^{5} e - 2 \, a^{2} c^{2} d^{3} e^{3} + a^{3} c d e^{5}\right )} x\right )}} \] Input:
integrate((e*x+d)^2*(g*x+f)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, alg orithm="fricas")
Output:
-2/3*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*a*d*e*g + (c*d^2 - 3*a *e^2)*f - (2*c*d*e*f - (3*c*d^2 - a*e^2)*g)*x)/(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)
\[ \int \frac {(d+e x)^2 (f+g x)}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\int \frac {\left (d + e x\right )^{2} \left (f + g x\right )}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((e*x+d)**2*(g*x+f)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x )
Output:
Integral((d + e*x)**2*(f + g*x)/((d + e*x)*(a*e + c*d*x))**(5/2), x)
Exception generated. \[ \int \frac {(d+e x)^2 (f+g x)}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((e*x+d)^2*(g*x+f)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, alg orithm="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. \[ \int \frac {(d+e x)^2 (f+g x)}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((e*x+d)^2*(g*x+f)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, alg orithm="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 = 10.93 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.69 \[ \int \frac {(d+e x)^2 (f+g x)}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {2\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (3\,a\,e^2\,f-c\,d^2\,f+a\,e^2\,g\,x-3\,c\,d^2\,g\,x-2\,a\,d\,e\,g+2\,c\,d\,e\,f\,x\right )}{3\,{\left (a\,e+c\,d\,x\right )}^2\,{\left (a\,e^2-c\,d^2\right )}^2} \] Input:
int(((f + g*x)*(d + e*x)^2)/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2), x)
Output:
(2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*(3*a*e^2*f - c*d^2*f + a* e^2*g*x - 3*c*d^2*g*x - 2*a*d*e*g + 2*c*d*e*f*x))/(3*(a*e + c*d*x)^2*(a*e^ 2 - c*d^2)^2)
Time = 0.55 (sec) , antiderivative size = 329, normalized size of antiderivative = 2.35 \[ \int \frac {(d+e x)^2 (f+g x)}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {\frac {2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, a^{2} e^{3} g}{3}+\frac {2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, a c \,d^{2} e g}{3}-\frac {4 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, a c d \,e^{2} f}{3}+\frac {2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, a c d \,e^{2} g x}{3}+\frac {2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, c^{2} d^{3} g x}{3}-\frac {4 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, c^{2} d^{2} e f x}{3}-\frac {4 \sqrt {e x +d}\, a \,c^{2} d^{3} e g}{3}+2 \sqrt {e x +d}\, a \,c^{2} d^{2} e^{2} f +\frac {2 \sqrt {e x +d}\, a \,c^{2} d^{2} e^{2} g x}{3}-\frac {2 \sqrt {e x +d}\, c^{3} d^{4} f}{3}-2 \sqrt {e x +d}\, c^{3} d^{4} g x +\frac {4 \sqrt {e x +d}\, c^{3} d^{3} e f x}{3}}{\sqrt {c d x +a e}\, c^{2} d^{2} \left (a^{2} c d \,e^{4} x -2 a \,c^{2} d^{3} e^{2} x +c^{3} d^{5} x +a^{3} e^{5}-2 a^{2} c \,d^{2} e^{3}+a \,c^{2} d^{4} e \right )} \] Input:
int((e*x+d)^2*(g*x+f)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x)
Output:
(2*(sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a**2*e**3*g + sqrt(e)*sqrt(d )*sqrt(c)*sqrt(a*e + c*d*x)*a*c*d**2*e*g - 2*sqrt(e)*sqrt(d)*sqrt(c)*sqrt( a*e + c*d*x)*a*c*d*e**2*f + sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a*c* d*e**2*g*x + sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*c**2*d**3*g*x - 2*s qrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*c**2*d**2*e*f*x - 2*sqrt(d + e*x) *a*c**2*d**3*e*g + 3*sqrt(d + e*x)*a*c**2*d**2*e**2*f + sqrt(d + e*x)*a*c* *2*d**2*e**2*g*x - sqrt(d + e*x)*c**3*d**4*f - 3*sqrt(d + e*x)*c**3*d**4*g *x + 2*sqrt(d + e*x)*c**3*d**3*e*f*x))/(3*sqrt(a*e + c*d*x)*c**2*d**2*(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))