Integrand size = 42, antiderivative size = 146 \[ \int \frac {(d+e x) (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {2 (c e f+c d g-b e g) (d+e x)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {2 (4 c e f-2 c d g-b e g) (b+2 c x)}{3 c e (2 c d-b e)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}} \] Output:
2/3*(-b*e*g+c*d*g+c*e*f)*(e*x+d)/c/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-2*c*d*g+4*c*e*f)*(2*c*x+b)/c/e/(-b*e+2*c*d)^3 /(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)
Time = 0.31 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.03 \[ \int \frac {(d+e x) (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {6 b^2 e^2 (-e f+2 d g+e g x)-4 b c e \left (5 d^2 g-2 d e g x+e^2 x (6 f-g x)\right )+8 c^2 \left (d^3 g-2 e^3 f x^2+d^2 e (f-g x)+d e^2 x (2 f+g x)\right )}{3 e^2 (-2 c d+b e)^3 (-c d+b e+c e x) \sqrt {(d+e x) (-b e+c (d-e x))}} \] Input:
Integrate[((d + e*x)*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2 ),x]
Output:
(6*b^2*e^2*(-(e*f) + 2*d*g + e*g*x) - 4*b*c*e*(5*d^2*g - 2*d*e*g*x + e^2*x *(6*f - g*x)) + 8*c^2*(d^3*g - 2*e^3*f*x^2 + d^2*e*(f - g*x) + d*e^2*x*(2* f + g*x)))/(3*e^2*(-2*c*d + b*e)^3*(-(c*d) + b*e + c*e*x)*Sqrt[(d + e*x)*( -(b*e) + c*(d - e*x))])
Time = 0.51 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {1218, 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.
| \(\displaystyle \int \frac {(d+e x) (f+g x)}{\left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1218 |
| \(\displaystyle \frac {(-b e g-2 c d g+4 c e f) \int \frac {1}{\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}dx}{3 c e (2 c d-b e)}+\frac {2 (d+e x) (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1088 |
| \(\displaystyle \frac {2 (b+2 c x) (-b e g-2 c d g+4 c e f)}{3 c e (2 c d-b e)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (d+e x) (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\) |
Input:
Int[((d + e*x)*(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))/(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*(4*c*e*f - 2*c*d*g - b*e*g)*(b + 2*c* x))/(3*c*e*(2*c*d - b*e)^3*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*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_))^(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 = 2.37 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.54
| method | result | size |
| trager | \(-\frac {2 \left (2 b c \,e^{3} g \,x^{2}+4 c^{2} d \,e^{2} g \,x^{2}-8 f \,c^{2} e^{3} x^{2}+3 b^{2} e^{3} g x +4 b c d \,e^{2} g x -12 b c \,e^{3} f x -4 c^{2} d^{2} e g x +8 c^{2} d \,e^{2} f x +6 b^{2} d \,e^{2} g -3 b^{2} e^{3} f -10 b c \,d^{2} e g +4 c^{2} d^{3} g +4 c^{2} d^{2} e f \right ) \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}{3 \left (e x +d \right ) \left (b^{2} e^{2}-4 b c d e +4 c^{2} d^{2}\right ) \left (b e -2 c d \right ) \left (c e x +b e -c d \right )^{2} e^{2}}\) | \(225\) |
| gosper | \(\frac {2 \left (e x +d \right )^{2} \left (c e x +b e -c d \right ) \left (2 b c \,e^{3} g \,x^{2}+4 c^{2} d \,e^{2} g \,x^{2}-8 f \,c^{2} e^{3} x^{2}+3 b^{2} e^{3} g x +4 b c d \,e^{2} g x -12 b c \,e^{3} f x -4 c^{2} d^{2} e g x +8 c^{2} d \,e^{2} f x +6 b^{2} d \,e^{2} g -3 b^{2} e^{3} f -10 b c \,d^{2} e g +4 c^{2} d^{3} g +4 c^{2} d^{2} e f \right )}{3 \left (b^{3} e^{3}-6 d \,e^{2} b^{2} c +12 d^{2} e b \,c^{2}-8 d^{3} c^{3}\right ) e^{2} \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {5}{2}}}\) | \(227\) |
| orering | \(\frac {2 \left (e x +d \right )^{2} \left (c e x +b e -c d \right ) \left (2 b c \,e^{3} g \,x^{2}+4 c^{2} d \,e^{2} g \,x^{2}-8 f \,c^{2} e^{3} x^{2}+3 b^{2} e^{3} g x +4 b c d \,e^{2} g x -12 b c \,e^{3} f x -4 c^{2} d^{2} e g x +8 c^{2} d \,e^{2} f x +6 b^{2} d \,e^{2} g -3 b^{2} e^{3} f -10 b c \,d^{2} e g +4 c^{2} d^{3} g +4 c^{2} d^{2} e f \right )}{3 \left (b^{3} e^{3}-6 d \,e^{2} b^{2} c +12 d^{2} e b \,c^{2}-8 d^{3} c^{3}\right ) e^{2} \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {5}{2}}}\) | \(227\) |
| default | \(d f \left (\frac {-\frac {4}{3} c \,e^{2} x -\frac {2}{3} b \,e^{2}}{\left (-4 c \,e^{2} \left (-b d e +c \,d^{2}\right )-b^{2} e^{4}\right ) \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {3}{2}}}-\frac {16 c \,e^{2} \left (-2 c \,e^{2} x -b \,e^{2}\right )}{3 \left (-4 c \,e^{2} \left (-b d e +c \,d^{2}\right )-b^{2} e^{4}\right )^{2} \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}\right )+\left (d g +e f \right ) \left (\frac {1}{3 c \,e^{2} \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {3}{2}}}-\frac {b \left (\frac {-\frac {4}{3} c \,e^{2} x -\frac {2}{3} b \,e^{2}}{\left (-4 c \,e^{2} \left (-b d e +c \,d^{2}\right )-b^{2} e^{4}\right ) \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {3}{2}}}-\frac {16 c \,e^{2} \left (-2 c \,e^{2} x -b \,e^{2}\right )}{3 \left (-4 c \,e^{2} \left (-b d e +c \,d^{2}\right )-b^{2} e^{4}\right )^{2} \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}\right )}{2 c}\right )+e g \left (\frac {x}{2 c \,e^{2} \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {3}{2}}}-\frac {b \left (\frac {1}{3 c \,e^{2} \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {3}{2}}}-\frac {b \left (\frac {-\frac {4}{3} c \,e^{2} x -\frac {2}{3} b \,e^{2}}{\left (-4 c \,e^{2} \left (-b d e +c \,d^{2}\right )-b^{2} e^{4}\right ) \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {3}{2}}}-\frac {16 c \,e^{2} \left (-2 c \,e^{2} x -b \,e^{2}\right )}{3 \left (-4 c \,e^{2} \left (-b d e +c \,d^{2}\right )-b^{2} e^{4}\right )^{2} \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}\right )}{2 c}\right )}{4 c}-\frac {\left (-b d e +c \,d^{2}\right ) \left (\frac {-\frac {4}{3} c \,e^{2} x -\frac {2}{3} b \,e^{2}}{\left (-4 c \,e^{2} \left (-b d e +c \,d^{2}\right )-b^{2} e^{4}\right ) \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {3}{2}}}-\frac {16 c \,e^{2} \left (-2 c \,e^{2} x -b \,e^{2}\right )}{3 \left (-4 c \,e^{2} \left (-b d e +c \,d^{2}\right )-b^{2} e^{4}\right )^{2} \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}\right )}{2 c \,e^{2}}\right )\) | \(772\) |
Input:
int((e*x+d)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x,method=_RETUR NVERBOSE)
Output:
-2/3*(2*b*c*e^3*g*x^2+4*c^2*d*e^2*g*x^2-8*c^2*e^3*f*x^2+3*b^2*e^3*g*x+4*b* c*d*e^2*g*x-12*b*c*e^3*f*x-4*c^2*d^2*e*g*x+8*c^2*d*e^2*f*x+6*b^2*d*e^2*g-3 *b^2*e^3*f-10*b*c*d^2*e*g+4*c^2*d^3*g+4*c^2*d^2*e*f)/(e*x+d)/(b^2*e^2-4*b* c*d*e+4*c^2*d^2)/(b*e-2*c*d)/(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)
Leaf count of result is larger than twice the leaf count of optimal. 431 vs. \(2 (138) = 276\).
Time = 5.19 (sec) , antiderivative size = 431, normalized size of antiderivative = 2.95 \[ \int \frac {(d+e x) (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=-\frac {2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, {\left (4 \, c^{2} e^{3} f - {\left (2 \, c^{2} d e^{2} + b c e^{3}\right )} g\right )} x^{2} - {\left (4 \, c^{2} d^{2} e - 3 \, b^{2} e^{3}\right )} f - 2 \, {\left (2 \, c^{2} d^{3} - 5 \, b c d^{2} e + 3 \, b^{2} d e^{2}\right )} g - {\left (4 \, {\left (2 \, c^{2} d e^{2} - 3 \, b c e^{3}\right )} f - {\left (4 \, c^{2} d^{2} e - 4 \, b c d e^{2} - 3 \, b^{2} e^{3}\right )} g\right )} x\right )}}{3 \, {\left (8 \, c^{5} d^{6} e^{2} - 28 \, b c^{4} d^{5} e^{3} + 38 \, b^{2} c^{3} d^{4} e^{4} - 25 \, b^{3} c^{2} d^{3} e^{5} + 8 \, b^{4} c d^{2} e^{6} - b^{5} d e^{7} + {\left (8 \, c^{5} d^{3} e^{5} - 12 \, b c^{4} d^{2} e^{6} + 6 \, b^{2} c^{3} d e^{7} - b^{3} c^{2} e^{8}\right )} x^{3} - {\left (8 \, c^{5} d^{4} e^{4} - 28 \, b c^{4} d^{3} e^{5} + 30 \, b^{2} c^{3} d^{2} e^{6} - 13 \, b^{3} c^{2} d e^{7} + 2 \, b^{4} c e^{8}\right )} x^{2} - {\left (8 \, c^{5} d^{5} e^{3} - 12 \, b c^{4} d^{4} e^{4} - 2 \, b^{2} c^{3} d^{3} e^{5} + 11 \, b^{3} c^{2} d^{2} e^{6} - 6 \, b^{4} c d e^{7} + b^{5} e^{8}\right )} x\right )}} \] Input:
integrate((e*x+d)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algori thm="fricas")
Output:
-2/3*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*(4*c^2*e^3*f - (2*c^2*d *e^2 + b*c*e^3)*g)*x^2 - (4*c^2*d^2*e - 3*b^2*e^3)*f - 2*(2*c^2*d^3 - 5*b* c*d^2*e + 3*b^2*d*e^2)*g - (4*(2*c^2*d*e^2 - 3*b*c*e^3)*f - (4*c^2*d^2*e - 4*b*c*d*e^2 - 3*b^2*e^3)*g)*x)/(8*c^5*d^6*e^2 - 28*b*c^4*d^5*e^3 + 38*b^2 *c^3*d^4*e^4 - 25*b^3*c^2*d^3*e^5 + 8*b^4*c*d^2*e^6 - b^5*d*e^7 + (8*c^5*d ^3*e^5 - 12*b*c^4*d^2*e^6 + 6*b^2*c^3*d*e^7 - b^3*c^2*e^8)*x^3 - (8*c^5*d^ 4*e^4 - 28*b*c^4*d^3*e^5 + 30*b^2*c^3*d^2*e^6 - 13*b^3*c^2*d*e^7 + 2*b^4*c *e^8)*x^2 - (8*c^5*d^5*e^3 - 12*b*c^4*d^4*e^4 - 2*b^2*c^3*d^3*e^5 + 11*b^3 *c^2*d^2*e^6 - 6*b^4*c*d*e^7 + b^5*e^8)*x)
\[ \int \frac {(d+e x) (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\int \frac {\left (d + e x\right ) \left (f + g x\right )}{\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((e*x+d)*(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)*(f + g*x)/(-(d + e*x)*(b*e - c*d + c*e*x))**(5/2), x)
Exception generated. \[ \int \frac {(d+e x) (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((e*x+d)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algori thm="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
\[ \int \frac {(d+e x) (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\int { \frac {{\left (e x + d\right )} {\left (g x + f\right )}}{{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((e*x+d)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algori thm="giac")
Output:
integrate((e*x + d)*(g*x + f)/(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2) , x)
Time = 11.88 (sec) , antiderivative size = 795, normalized size of antiderivative = 5.45 \[ \int \frac {(d+e x) (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=-\frac {8\,c^2\,d^3\,g\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}-6\,b^2\,e^3\,f\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}-16\,c^2\,e^3\,f\,x^2\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}+12\,b^2\,d\,e^2\,g\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}+8\,c^2\,d^2\,e\,f\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}+6\,b^2\,e^3\,g\,x\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}+4\,b\,c\,e^3\,g\,x^2\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}+16\,c^2\,d\,e^2\,f\,x\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}-8\,c^2\,d^2\,e\,g\,x\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}+8\,c^2\,d\,e^2\,g\,x^2\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}-20\,b\,c\,d^2\,e\,g\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}-24\,b\,c\,e^3\,f\,x\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}+8\,b\,c\,d\,e^2\,g\,x\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{3\,b^5\,d\,e^7+3\,b^5\,e^8\,x-24\,b^4\,c\,d^2\,e^6-18\,b^4\,c\,d\,e^7\,x+6\,b^4\,c\,e^8\,x^2+75\,b^3\,c^2\,d^3\,e^5+33\,b^3\,c^2\,d^2\,e^6\,x-39\,b^3\,c^2\,d\,e^7\,x^2+3\,b^3\,c^2\,e^8\,x^3-114\,b^2\,c^3\,d^4\,e^4-6\,b^2\,c^3\,d^3\,e^5\,x+90\,b^2\,c^3\,d^2\,e^6\,x^2-18\,b^2\,c^3\,d\,e^7\,x^3+84\,b\,c^4\,d^5\,e^3-36\,b\,c^4\,d^4\,e^4\,x-84\,b\,c^4\,d^3\,e^5\,x^2+36\,b\,c^4\,d^2\,e^6\,x^3-24\,c^5\,d^6\,e^2+24\,c^5\,d^5\,e^3\,x+24\,c^5\,d^4\,e^4\,x^2-24\,c^5\,d^3\,e^5\,x^3} \] Input:
int(((f + g*x)*(d + e*x))/(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(5/2),x)
Output:
-(8*c^2*d^3*g*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2) - 6*b^2*e^3*f*(c *d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2) - 16*c^2*e^3*f*x^2*(c*d^2 - c*e^ 2*x^2 - b*d*e - b*e^2*x)^(1/2) + 12*b^2*d*e^2*g*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2) + 8*c^2*d^2*e*f*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1 /2) + 6*b^2*e^3*g*x*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2) + 4*b*c*e^ 3*g*x^2*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2) + 16*c^2*d*e^2*f*x*(c* d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2) - 8*c^2*d^2*e*g*x*(c*d^2 - c*e^2* x^2 - b*d*e - b*e^2*x)^(1/2) + 8*c^2*d*e^2*g*x^2*(c*d^2 - c*e^2*x^2 - b*d* e - b*e^2*x)^(1/2) - 20*b*c*d^2*e*g*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^ (1/2) - 24*b*c*e^3*f*x*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2) + 8*b*c *d*e^2*g*x*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(3*b^5*d*e^7 + 3*b ^5*e^8*x - 24*c^5*d^6*e^2 + 84*b*c^4*d^5*e^3 - 24*b^4*c*d^2*e^6 + 6*b^4*c* e^8*x^2 + 24*c^5*d^5*e^3*x - 114*b^2*c^3*d^4*e^4 + 75*b^3*c^2*d^3*e^5 + 3* b^3*c^2*e^8*x^3 + 24*c^5*d^4*e^4*x^2 - 24*c^5*d^3*e^5*x^3 - 18*b^4*c*d*e^7 *x + 90*b^2*c^3*d^2*e^6*x^2 - 36*b*c^4*d^4*e^4*x - 6*b^2*c^3*d^3*e^5*x + 3 3*b^3*c^2*d^2*e^6*x - 84*b*c^4*d^3*e^5*x^2 - 39*b^3*c^2*d*e^7*x^2 + 36*b*c ^4*d^2*e^6*x^3 - 18*b^2*c^3*d*e^7*x^3)
Time = 0.25 (sec) , antiderivative size = 1117, normalized size of antiderivative = 7.65 \[ \int \frac {(d+e x) (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:
int((e*x+d)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x)
Output:
(2*i*( - 2*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b**3*d*e**3*g - 2*sqrt(c)*sq rt( - b*e + c*d - c*e*x)*b**3*e**4*g*x + 2*sqrt(c)*sqrt( - b*e + c*d - c*e *x)*b**2*c*d**2*e**2*g + 8*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b**2*c*d*e** 3*f + 8*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b**2*c*e**4*f*x - 2*sqrt(c)*sqr t( - b*e + c*d - c*e*x)*b**2*c*e**4*g*x**2 + 8*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b*c**2*d**3*e*g - 24*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b*c**2*d** 2*e**2*f + 8*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b*c**2*d**2*e**2*g*x - 16* sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b*c**2*d*e**3*f*x + 8*sqrt(c)*sqrt( - b *e + c*d - c*e*x)*b*c**2*e**4*f*x**2 - 8*sqrt(c)*sqrt( - b*e + c*d - c*e*x )*c**3*d**4*g + 16*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*c**3*d**3*e*f + 8*sq rt(c)*sqrt( - b*e + c*d - c*e*x)*c**3*d**2*e**2*g*x**2 - 16*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*c**3*d*e**3*f*x**2 + 6*sqrt(d + e*x)*sqrt(b*e - 2*c*d )*sqrt( - b*e + 2*c*d)*b**2*c*d*e**2*g - 3*sqrt(d + e*x)*sqrt(b*e - 2*c*d) *sqrt( - b*e + 2*c*d)*b**2*c*e**3*f + 3*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sq rt( - b*e + 2*c*d)*b**2*c*e**3*g*x - 10*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sq rt( - b*e + 2*c*d)*b*c**2*d**2*e*g + 4*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqr t( - b*e + 2*c*d)*b*c**2*d*e**2*g*x - 12*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*s qrt( - b*e + 2*c*d)*b*c**2*e**3*f*x + 2*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sq rt( - b*e + 2*c*d)*b*c**2*e**3*g*x**2 + 4*sqrt(d + e*x)*sqrt(b*e - 2*c*d)* sqrt( - b*e + 2*c*d)*c**3*d**3*g + 4*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sq...