Integrand size = 42, antiderivative size = 124 \[ \int \frac {(d+e x) (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {2 (c e f+c d g-b e g) (d+e x)}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {2 g \arctan \left (\frac {\sqrt {c} (d+e x)}{\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{c^{3/2} e^2} \] Output:
2*(-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)^(1/2)-2*g*arctan(c^(1/2)*(e*x+d)/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2) ^(1/2))/c^(3/2)/e^2
Time = 0.31 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.12 \[ \int \frac {(d+e x) (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=-\frac {2 \left (\sqrt {c} (c e f+c d g-b e g) (d+e x)+(2 c d-b e) g \sqrt {d+e x} \sqrt {c d-b e-c e x} \arctan \left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {c} \sqrt {d+e x}}\right )\right )}{c^{3/2} e^2 (-2 c d+b e) \sqrt {(d+e x) (-b e+c (d-e x))}} \] Input:
Integrate[((d + e*x)*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2 ),x]
Output:
(-2*(Sqrt[c]*(c*e*f + c*d*g - b*e*g)*(d + e*x) + (2*c*d - b*e)*g*Sqrt[d + e*x]*Sqrt[c*d - b*e - c*e*x]*ArcTan[Sqrt[c*d - b*e - c*e*x]/(Sqrt[c]*Sqrt[ d + e*x])]))/(c^(3/2)*e^2*(-2*c*d + b*e)*Sqrt[(d + e*x)*(-(b*e) + c*(d - e *x))])
Time = 0.47 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {1211, 27, 1092, 217}
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.
| \(\displaystyle \int \frac {(d+e x) (f+g x)}{\left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1211 |
| \(\displaystyle \frac {2 (d+e x) (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {\int \frac {g}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{c e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 (d+e x) (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {g \int \frac {1}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{c e}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {2 (d+e x) (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {2 g \int \frac {1}{-\frac {(b+2 c x)^2 e^4}{-c x^2 e^2-b x e^2+d (c d-b e)}-4 c e^2}d\left (-\frac {e^2 (b+2 c x)}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}\right )}{c e}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2 (d+e x) (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {g \arctan \left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{c^{3/2} e^2}\) |
Input:
Int[((d + e*x)*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2),x]
Output:
(2*(c*e*f + c*d*g - b*e*g)*(d + e*x))/(c*e^2*(2*c*d - b*e)*Sqrt[d*(c*d - b *e) - b*e^2*x - c*e^2*x^2]) - (g*ArcTan[(e*(b + 2*c*x))/(2*Sqrt[c]*Sqrt[d* (c*d - b*e) - b*e^2*x - c*e^2*x^2])])/(c^(3/2)*e^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* (x_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[-2*(2*c*d - b*e)^(m - 2)*(c*( e*f + d*g) - b*e*g)^n*((d + e*x)/(c^(m + n - 1)*e^(n - 1)*Sqrt[a + b*x + c* x^2])), x] + Simp[1/(c^(m + n - 1)*e^(n - 2)) Int[ExpandToSum[((2*c*d - b *e)^(m - 1)*(c*(e*f + d*g) - b*e*g)^n - c^(m + n - 1)*e^n*(d + e*x)^(m - 1) *(f + g*x)^n)/(c*d - b*e - c*e*x), x]/Sqrt[a + b*x + c*x^2], x], x] /; Free Q[{a, b, c, d, e, f, g}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[m, 0] && IGtQ[n, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(420\) vs. \(2(116)=232\).
Time = 2.33 (sec) , antiderivative size = 421, normalized size of antiderivative = 3.40
| method | result | size |
| default | \(\frac {2 d f \left (-2 c \,e^{2} x -b \,e^{2}\right )}{\left (-4 c \,e^{2} \left (-b d e +c \,d^{2}\right )-b^{2} e^{4}\right ) \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}+\left (d g +e f \right ) \left (\frac {1}{c \,e^{2} \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}-\frac {b \left (-2 c \,e^{2} x -b \,e^{2}\right )}{c \left (-4 c \,e^{2} \left (-b d e +c \,d^{2}\right )-b^{2} e^{4}\right ) \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}\right )+e g \left (\frac {x}{c \,e^{2} \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}-\frac {b \left (\frac {1}{c \,e^{2} \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}-\frac {b \left (-2 c \,e^{2} x -b \,e^{2}\right )}{c \left (-4 c \,e^{2} \left (-b d e +c \,d^{2}\right )-b^{2} e^{4}\right ) \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}\right )}{2 c}-\frac {\arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}\right )}{c \,e^{2} \sqrt {c \,e^{2}}}\right )\) | \(421\) |
Input:
int((e*x+d)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x,method=_RETUR NVERBOSE)
Output:
2*d*f*(-2*c*e^2*x-b*e^2)/(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/(-c*e^2*x^2-b*e ^2*x-b*d*e+c*d^2)^(1/2)+(d*g+e*f)*(1/c/e^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2 )^(1/2)-b/c*(-2*c*e^2*x-b*e^2)/(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/(-c*e^2*x ^2-b*e^2*x-b*d*e+c*d^2)^(1/2))+e*g*(x/c/e^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^ 2)^(1/2)-1/2*b/c*(1/c/e^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-b/c*(-2*c *e^2*x-b*e^2)/(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/(-c*e^2*x^2-b*e^2*x-b*d*e+ c*d^2)^(1/2))-1/c/e^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-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. 235 vs. \(2 (116) = 232\).
Time = 0.35 (sec) , antiderivative size = 483, normalized size of antiderivative = 3.90 \[ \int \frac {(d+e x) (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\left [\frac {{\left ({\left (2 \, c^{2} d e - b c e^{2}\right )} g x - {\left (2 \, c^{2} d^{2} - 3 \, b c d e + b^{2} e^{2}\right )} g\right )} \sqrt {-c} \log \left (8 \, c^{2} e^{2} x^{2} + 8 \, b c e^{2} x - 4 \, c^{2} d^{2} + 4 \, b c d e + b^{2} e^{2} + 4 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, c e x + b e\right )} \sqrt {-c}\right ) + 4 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (c^{2} e f + {\left (c^{2} d - b c e\right )} g\right )}}{2 \, {\left (2 \, c^{4} d^{2} e^{2} - 3 \, b c^{3} d e^{3} + b^{2} c^{2} e^{4} - {\left (2 \, c^{4} d e^{3} - b c^{3} e^{4}\right )} x\right )}}, -\frac {{\left ({\left (2 \, c^{2} d e - b c e^{2}\right )} g x - {\left (2 \, c^{2} d^{2} - 3 \, b c d e + b^{2} e^{2}\right )} g\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, c e x + b e\right )} \sqrt {c}}{2 \, {\left (c^{2} e^{2} x^{2} + b c e^{2} x - c^{2} d^{2} + b c d e\right )}}\right ) - 2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (c^{2} e f + {\left (c^{2} d - b c e\right )} g\right )}}{2 \, c^{4} d^{2} e^{2} - 3 \, b c^{3} d e^{3} + b^{2} c^{2} e^{4} - {\left (2 \, c^{4} d e^{3} - b c^{3} e^{4}\right )} x}\right ] \] Input:
integrate((e*x+d)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algori thm="fricas")
Output:
[1/2*(((2*c^2*d*e - b*c*e^2)*g*x - (2*c^2*d^2 - 3*b*c*d*e + b^2*e^2)*g)*sq rt(-c)*log(8*c^2*e^2*x^2 + 8*b*c*e^2*x - 4*c^2*d^2 + 4*b*c*d*e + b^2*e^2 + 4*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*e*x + b*e)*sqrt(-c)) + 4*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(c^2*e*f + (c^2*d - b*c*e)*g) )/(2*c^4*d^2*e^2 - 3*b*c^3*d*e^3 + b^2*c^2*e^4 - (2*c^4*d*e^3 - b*c^3*e^4) *x), -(((2*c^2*d*e - b*c*e^2)*g*x - (2*c^2*d^2 - 3*b*c*d*e + b^2*e^2)*g)*s qrt(c)*arctan(1/2*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*e*x + b* e)*sqrt(c)/(c^2*e^2*x^2 + b*c*e^2*x - c^2*d^2 + b*c*d*e)) - 2*sqrt(-c*e^2* x^2 - b*e^2*x + c*d^2 - b*d*e)*(c^2*e*f + (c^2*d - b*c*e)*g))/(2*c^4*d^2*e ^2 - 3*b*c^3*d*e^3 + b^2*c^2*e^4 - (2*c^4*d*e^3 - b*c^3*e^4)*x)]
\[ \int \frac {(d+e x) (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\int \frac {\left (d + e x\right ) \left (f + g x\right )}{\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((e*x+d)*(g*x+f)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(3/2),x)
Output:
Integral((d + e*x)*(f + g*x)/(-(d + e*x)*(b*e - c*d + c*e*x))**(3/2), x)
Exception generated. \[ \int \frac {(d+e x) (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((e*x+d)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),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
Leaf count of result is larger than twice the leaf count of optimal. 374 vs. \(2 (116) = 232\).
Time = 0.32 (sec) , antiderivative size = 374, normalized size of antiderivative = 3.02 \[ \int \frac {(d+e x) (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {g \log \left ({\left | b c^{2} d^{2} e^{2} - 2 \, b^{2} c d e^{3} + b^{3} e^{4} - 2 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )} \sqrt {-c} c^{2} d^{2} {\left | e \right |} + 6 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )} b \sqrt {-c} c d e {\left | e \right |} - 4 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )} b^{2} \sqrt {-c} e^{2} {\left | e \right |} + 4 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{2} c^{2} d e - 5 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{2} b c e^{2} + 2 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{3} \sqrt {-c} c {\left | e \right |} \right |}\right )}{3 \, \sqrt {-c} c e {\left | e \right |}} \] Input:
integrate((e*x+d)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algori thm="giac")
Output:
1/3*g*log(abs(b*c^2*d^2*e^2 - 2*b^2*c*d*e^3 + b^3*e^4 - 2*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))*sqrt(-c)*c^2*d^2*abs(e) + 6* (sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))*b*sqrt(-c)*c *d*e*abs(e) - 4*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d* e))*b^2*sqrt(-c)*e^2*abs(e) + 4*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2* x + c*d^2 - b*d*e))^2*c^2*d*e - 5*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^ 2*x + c*d^2 - b*d*e))^2*b*c*e^2 + 2*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b* e^2*x + c*d^2 - b*d*e))^3*sqrt(-c)*c*abs(e)))/(sqrt(-c)*c*e*abs(e))
Time = 12.88 (sec) , antiderivative size = 344, normalized size of antiderivative = 2.77 \[ \int \frac {(d+e x) (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {4\,c\,d^3\,g+2\,b\,d\,e^2\,f-4\,b\,d^2\,e\,g-2\,b\,d\,e^2\,g\,x+4\,c\,d\,e^2\,f\,x}{\left (b^2\,e^4+4\,c\,e^2\,\left (c\,d^2-b\,d\,e\right )\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}+\frac {e\,g\,\ln \left (b\,e^2-2\,\sqrt {-c\,e^2}\,\sqrt {-\left (d+e\,x\right )\,\left (b\,e-c\,d+c\,e\,x\right )}+2\,c\,e^2\,x\right )}{{\left (-c\,e^2\right )}^{3/2}}-\frac {e\,f\,\left (-4\,c\,d^2+4\,b\,d\,e+2\,b\,x\,e^2\right )}{\left (b^2\,e^4+4\,c\,e^2\,\left (c\,d^2-b\,d\,e\right )\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}+\frac {g\,\left (x\,\left (\frac {b^2\,e^4}{2}+c\,e^2\,\left (c\,d^2-b\,d\,e\right )\right )-\frac {b\,e^2\,\left (c\,d^2-b\,d\,e\right )}{2}\right )}{c\,e\,\left (\frac {b^2\,e^4}{4}+c\,e^2\,\left (c\,d^2-b\,d\,e\right )\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}} \] Input:
int(((f + g*x)*(d + e*x))/(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2),x)
Output:
(4*c*d^3*g + 2*b*d*e^2*f - 4*b*d^2*e*g - 2*b*d*e^2*g*x + 4*c*d*e^2*f*x)/(( b^2*e^4 + 4*c*e^2*(c*d^2 - b*d*e))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^( 1/2)) + (e*g*log(b*e^2 - 2*(-c*e^2)^(1/2)*(-(d + e*x)*(b*e - c*d + c*e*x)) ^(1/2) + 2*c*e^2*x))/(-c*e^2)^(3/2) - (e*f*(4*b*d*e - 4*c*d^2 + 2*b*e^2*x) )/((b^2*e^4 + 4*c*e^2*(c*d^2 - b*d*e))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2* x)^(1/2)) + (g*(x*((b^2*e^4)/2 + c*e^2*(c*d^2 - b*d*e)) - (b*e^2*(c*d^2 - b*d*e))/2))/(c*e*((b^2*e^4)/4 + c*e^2*(c*d^2 - b*d*e))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))
Time = 0.26 (sec) , antiderivative size = 424, normalized size of antiderivative = 3.42 \[ \int \frac {(d+e x) (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {2 i \left (-\sqrt {c}\, \sqrt {-c e x -b e +c d}\, \mathit {asinh} \left (\frac {\sqrt {-c e x -b e +c d}\, i}{\sqrt {-b e +2 c d}}\right ) b^{2} e^{2} g +4 \sqrt {c}\, \sqrt {-c e x -b e +c d}\, \mathit {asinh} \left (\frac {\sqrt {-c e x -b e +c d}\, i}{\sqrt {-b e +2 c d}}\right ) b c d e g -4 \sqrt {c}\, \sqrt {-c e x -b e +c d}\, \mathit {asinh} \left (\frac {\sqrt {-c e x -b e +c d}\, i}{\sqrt {-b e +2 c d}}\right ) c^{2} d^{2} g +\sqrt {c}\, \sqrt {-c e x -b e +c d}\, b^{2} e^{2} g -3 \sqrt {c}\, \sqrt {-c e x -b e +c d}\, b c d e g -\sqrt {c}\, \sqrt {-c e x -b e +c d}\, b c \,e^{2} f +2 \sqrt {c}\, \sqrt {-c e x -b e +c d}\, c^{2} d^{2} g +2 \sqrt {c}\, \sqrt {-c e x -b e +c d}\, c^{2} d e f +\sqrt {e x +d}\, \sqrt {b e -2 c d}\, \sqrt {-b e +2 c d}\, b c e g -\sqrt {e x +d}\, \sqrt {b e -2 c d}\, \sqrt {-b e +2 c d}\, c^{2} d g -\sqrt {e x +d}\, \sqrt {b e -2 c d}\, \sqrt {-b e +2 c d}\, c^{2} e f \right )}{\sqrt {-c e x -b e +c d}\, c^{2} e^{2} \left (b^{2} e^{2}-4 b c d e +4 c^{2} d^{2}\right )} \] Input:
int((e*x+d)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x)
Output:
(2*i*( - sqrt(c)*sqrt( - b*e + c*d - c*e*x)*asinh((sqrt( - b*e + c*d - c*e *x)*i)/sqrt( - b*e + 2*c*d))*b**2*e**2*g + 4*sqrt(c)*sqrt( - b*e + c*d - c *e*x)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b*c*d*e*g - 4*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*asinh((sqrt( - b*e + c*d - c*e*x)* i)/sqrt( - b*e + 2*c*d))*c**2*d**2*g + sqrt(c)*sqrt( - b*e + c*d - c*e*x)* b**2*e**2*g - 3*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b*c*d*e*g - sqrt(c)*sqr t( - b*e + c*d - c*e*x)*b*c*e**2*f + 2*sqrt(c)*sqrt( - b*e + c*d - c*e*x)* c**2*d**2*g + 2*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*c**2*d*e*f + sqrt(d + e *x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*b*c*e*g - sqrt(d + e*x)*sqrt(b* e - 2*c*d)*sqrt( - b*e + 2*c*d)*c**2*d*g - sqrt(d + e*x)*sqrt(b*e - 2*c*d) *sqrt( - b*e + 2*c*d)*c**2*e*f))/(sqrt( - b*e + c*d - c*e*x)*c**2*e**2*(b* *2*e**2 - 4*b*c*d*e + 4*c**2*d**2))