Integrand size = 44, antiderivative size = 134 \[ \int \frac {(d+e x)^2 (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {2 (e f-d g) (d+e x)^2}{e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {2 (2 c e f-4 c d g+b e g) (d+e x)^3}{3 e^2 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \] Output:
2*(-d*g+e*f)*(e*x+d)^2/e^2/(-b*e+2*c*d)/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^( 3/2)-2/3*(b*e*g-4*c*d*g+2*c*e*f)*(e*x+d)^3/e^2/(-b*e+2*c*d)^2/(d*(-b*e+c*d )-b*e^2*x-c*e^2*x^2)^(3/2)
Time = 0.23 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.75 \[ \int \frac {(d+e x)^2 (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {2 (d+e x) \left (b e (3 e f-2 d g+e g x)+2 c \left (d^2 g+e^2 f x-2 d e (f+g x)\right )\right )}{3 e^2 (-2 c d+b e)^2 (-c d+b e+c e x) \sqrt {(d+e x) (-b e+c (d-e x))}} \] Input:
Integrate[((d + e*x)^2*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5 /2),x]
Output:
(2*(d + e*x)*(b*e*(3*e*f - 2*d*g + e*g*x) + 2*c*(d^2*g + e^2*f*x - 2*d*e*( f + g*x))))/(3*e^2*(-2*c*d + b*e)^2*(-(c*d) + b*e + c*e*x)*Sqrt[(d + e*x)* (-(b*e) + c*(d - e*x))])
Time = 0.57 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.09, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, 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.
\(\displaystyle \int \frac {(d+e x)^2 (f+g x)}{\left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 1218 |
\(\displaystyle \frac {(b e g-4 c d g+2 c e f) \int \frac {d+e x}{\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}dx}{3 c e (2 c d-b e)}+\frac {2 (d+e x)^2 (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 1124 |
\(\displaystyle \frac {(b e g-4 c d g+2 c e f) \left (e^2 \int 0dx+\frac {2 (d+e x)}{e (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{3 c e (2 c d-b e)}+\frac {2 (d+e x)^2 (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {2 (d+e x)^2 (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {2 (d+e x) (b e g-4 c d g+2 c e f)}{3 c e^2 (2 c d-b e)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\) |
Input:
Int[((d + e*x)^2*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2),x]
Output:
(2*(c*e*f + c*d*g - b*e*g)*(d + e*x)^2)/(3*c*e^2*(2*c*d - b*e)*(d*(c*d - b *e) - b*e^2*x - c*e^2*x^2)^(3/2)) + (2*(2*c*e*f - 4*c*d*g + b*e*g)*(d + e* x))/(3*c*e^2*(2*c*d - b*e)^2*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*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 = 2.74 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.92
method | result | size |
trager | \(\frac {2 \left (-b \,e^{2} g x +4 c d e g x -2 c \,e^{2} f x +2 b d e g -3 b \,e^{2} f -2 c \,d^{2} g +4 c d e f \right ) \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}{3 \left (b^{2} e^{2}-4 b c d e +4 c^{2} d^{2}\right ) \left (c e x +b e -c d \right )^{2} e^{2}}\) | \(123\) |
gosper | \(-\frac {2 \left (e x +d \right )^{3} \left (c e x +b e -c d \right ) \left (-b \,e^{2} g x +4 c d e g x -2 c \,e^{2} f x +2 b d e g -3 b \,e^{2} f -2 c \,d^{2} g +4 c d e f \right )}{3 e^{2} \left (b^{2} e^{2}-4 b c d e +4 c^{2} d^{2}\right ) \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {5}{2}}}\) | \(128\) |
orering | \(-\frac {2 \left (e x +d \right )^{3} \left (c e x +b e -c d \right ) \left (-b \,e^{2} g x +4 c d e g x -2 c \,e^{2} f x +2 b d e g -3 b \,e^{2} f -2 c \,d^{2} g +4 c d e f \right )}{3 e^{2} \left (b^{2} e^{2}-4 b c d e +4 c^{2} d^{2}\right ) \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {5}{2}}}\) | \(128\) |
default | \(\text {Expression too large to display}\) | \(1458\) |
Input:
int((e*x+d)^2*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x,method=_RET URNVERBOSE)
Output:
2/3*(-b*e^2*g*x+4*c*d*e*g*x-2*c*e^2*f*x+2*b*d*e*g-3*b*e^2*f-2*c*d^2*g+4*c* d*e*f)/(b^2*e^2-4*b*c*d*e+4*c^2*d^2)/(c*e*x+b*e-c*d)^2/e^2*(-c*e^2*x^2-b*e ^2*x-b*d*e+c*d^2)^(1/2)
Time = 1.27 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.69 \[ \int \frac {(d+e x)^2 (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left ({\left (4 \, c d e - 3 \, b e^{2}\right )} f - 2 \, {\left (c d^{2} - b d e\right )} g - {\left (2 \, c e^{2} f - {\left (4 \, c d e - b e^{2}\right )} g\right )} x\right )}}{3 \, {\left (4 \, c^{4} d^{4} e^{2} - 12 \, b c^{3} d^{3} e^{3} + 13 \, b^{2} c^{2} d^{2} e^{4} - 6 \, b^{3} c d e^{5} + b^{4} e^{6} + {\left (4 \, c^{4} d^{2} e^{4} - 4 \, b c^{3} d e^{5} + b^{2} c^{2} e^{6}\right )} x^{2} - 2 \, {\left (4 \, c^{4} d^{3} e^{3} - 8 \, b c^{3} d^{2} e^{4} + 5 \, b^{2} c^{2} d e^{5} - b^{3} c e^{6}\right )} x\right )}} \] Input:
integrate((e*x+d)^2*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algo rithm="fricas")
Output:
2/3*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*((4*c*d*e - 3*b*e^2)*f - 2* (c*d^2 - b*d*e)*g - (2*c*e^2*f - (4*c*d*e - b*e^2)*g)*x)/(4*c^4*d^4*e^2 - 12*b*c^3*d^3*e^3 + 13*b^2*c^2*d^2*e^4 - 6*b^3*c*d*e^5 + b^4*e^6 + (4*c^4*d ^2*e^4 - 4*b*c^3*d*e^5 + b^2*c^2*e^6)*x^2 - 2*(4*c^4*d^3*e^3 - 8*b*c^3*d^2 *e^4 + 5*b^2*c^2*d*e^5 - b^3*c*e^6)*x)
\[ \int \frac {(d+e x)^2 (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\int \frac {\left (d + e x\right )^{2} \left (f + g x\right )}{\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((e*x+d)**2*(g*x+f)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2),x )
Output:
Integral((d + e*x)**2*(f + g*x)/(-(d + e*x)*(b*e - c*d + c*e*x))**(5/2), x )
Exception generated. \[ \int \frac {(d+e x)^2 (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((e*x+d)^2*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algo rithm="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*(b*e-2*c*d)>0)', see `assume?` for more
Exception generated. \[ \int \frac {(d+e x)^2 (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((e*x+d)^2*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algo rithm="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,[6,0,10,2]%%%}+%%%{-10,[5,1,9,3]%%%}+%%%{41,[4,2,8,4 ]%%%}+%%%
Time = 11.55 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.80 \[ \int \frac {(d+e x)^2 (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=-\frac {2\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}\,\left (3\,b\,e^2\,f+2\,c\,d^2\,g+b\,e^2\,g\,x+2\,c\,e^2\,f\,x-2\,b\,d\,e\,g-4\,c\,d\,e\,f-4\,c\,d\,e\,g\,x\right )}{3\,e^2\,{\left (b\,e-2\,c\,d\right )}^2\,{\left (b\,e-c\,d+c\,e\,x\right )}^2} \] Input:
int(((f + g*x)*(d + e*x)^2)/(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(5/2),x)
Output:
-(2*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2)*(3*b*e^2*f + 2*c*d^2*g + b *e^2*g*x + 2*c*e^2*f*x - 2*b*d*e*g - 4*c*d*e*f - 4*c*d*e*g*x))/(3*e^2*(b*e - 2*c*d)^2*(b*e - c*d + c*e*x)^2)
Time = 0.22 (sec) , antiderivative size = 628, normalized size of antiderivative = 4.69 \[ \int \frac {(d+e x)^2 (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {2 i \left (\sqrt {c}\, \sqrt {-c e x -b e +c d}\, b^{3} e^{3} g -3 \sqrt {c}\, \sqrt {-c e x -b e +c d}\, b^{2} c d \,e^{2} g -2 \sqrt {c}\, \sqrt {-c e x -b e +c d}\, b^{2} c \,e^{3} f +\sqrt {c}\, \sqrt {-c e x -b e +c d}\, b^{2} c \,e^{3} g x +2 \sqrt {c}\, \sqrt {-c e x -b e +c d}\, b \,c^{2} d^{2} e g +6 \sqrt {c}\, \sqrt {-c e x -b e +c d}\, b \,c^{2} d \,e^{2} f -2 \sqrt {c}\, \sqrt {-c e x -b e +c d}\, b \,c^{2} d \,e^{2} g x -2 \sqrt {c}\, \sqrt {-c e x -b e +c d}\, b \,c^{2} e^{3} f x -4 \sqrt {c}\, \sqrt {-c e x -b e +c d}\, c^{3} d^{2} e f +4 \sqrt {c}\, \sqrt {-c e x -b e +c d}\, c^{3} d \,e^{2} f x -2 \sqrt {e x +d}\, \sqrt {b e -2 c d}\, \sqrt {-b e +2 c d}\, b \,c^{2} d e g +3 \sqrt {e x +d}\, \sqrt {b e -2 c d}\, \sqrt {-b e +2 c d}\, b \,c^{2} e^{2} f +\sqrt {e x +d}\, \sqrt {b e -2 c d}\, \sqrt {-b e +2 c d}\, b \,c^{2} e^{2} g x +2 \sqrt {e x +d}\, \sqrt {b e -2 c d}\, \sqrt {-b e +2 c d}\, c^{3} d^{2} g -4 \sqrt {e x +d}\, \sqrt {b e -2 c d}\, \sqrt {-b e +2 c d}\, c^{3} d e f -4 \sqrt {e x +d}\, \sqrt {b e -2 c d}\, \sqrt {-b e +2 c d}\, c^{3} d e g x +2 \sqrt {e x +d}\, \sqrt {b e -2 c d}\, \sqrt {-b e +2 c d}\, c^{3} e^{2} f x \right )}{3 \sqrt {-c e x -b e +c d}\, c^{2} e^{2} \left (b^{3} c \,e^{4} x -6 b^{2} c^{2} d \,e^{3} x +12 b \,c^{3} d^{2} e^{2} x -8 c^{4} d^{3} e x +b^{4} e^{4}-7 b^{3} c d \,e^{3}+18 b^{2} c^{2} d^{2} e^{2}-20 b \,c^{3} d^{3} e +8 c^{4} d^{4}\right )} \] Input:
int((e*x+d)^2*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x)
Output:
(2*i*(sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b**3*e**3*g - 3*sqrt(c)*sqrt( - b *e + c*d - c*e*x)*b**2*c*d*e**2*g - 2*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b **2*c*e**3*f + sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b**2*c*e**3*g*x + 2*sqrt (c)*sqrt( - b*e + c*d - c*e*x)*b*c**2*d**2*e*g + 6*sqrt(c)*sqrt( - b*e + c *d - c*e*x)*b*c**2*d*e**2*f - 2*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b*c**2* d*e**2*g*x - 2*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b*c**2*e**3*f*x - 4*sqrt (c)*sqrt( - b*e + c*d - c*e*x)*c**3*d**2*e*f + 4*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*c**3*d*e**2*f*x - 2*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*b*c**2*d*e*g + 3*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2* c*d)*b*c**2*e**2*f + sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)* b*c**2*e**2*g*x + 2*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*c **3*d**2*g - 4*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*c**3*d *e*f - 4*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*c**3*d*e*g*x + 2*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*c**3*e**2*f*x))/ (3*sqrt( - b*e + c*d - c*e*x)*c**2*e**2*(b**4*e**4 - 7*b**3*c*d*e**3 + b** 3*c*e**4*x + 18*b**2*c**2*d**2*e**2 - 6*b**2*c**2*d*e**3*x - 20*b*c**3*d** 3*e + 12*b*c**3*d**2*e**2*x + 8*c**4*d**4 - 8*c**4*d**3*e*x))