Integrand size = 46, antiderivative size = 223 \[ \int \frac {f+g x}{\sqrt {d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {2 (c e f+c d g-b e g) \sqrt {d+e x}}{e^2 (2 c d-b e)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {(e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (2 c d-b e)^2 (d+e x)^{3/2}}-\frac {(3 c e f+c d g-2 b e g) \text {arctanh}\left (\frac {\sqrt {2 c d-b e} \sqrt {d+e x}}{\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{e^2 (2 c d-b e)^{5/2}} \] Output:
2*(-b*e*g+c*d*g+c*e*f)*(e*x+d)^(1/2)/e^2/(-b*e+2*c*d)^2/(d*(-b*e+c*d)-b*e^ 2*x-c*e^2*x^2)^(1/2)-(-d*g+e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/e^2 /(-b*e+2*c*d)^2/(e*x+d)^(3/2)-(-2*b*e*g+c*d*g+3*c*e*f)*arctanh((-b*e+2*c*d )^(1/2)*(e*x+d)^(1/2)/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2))/e^2/(-b*e+2* c*d)^(5/2)
Time = 0.69 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.77 \[ \int \frac {f+g x}{\sqrt {d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {\sqrt {-2 c d+b e} \left (b e (-3 d g+e (f-2 g x))+c \left (3 d^2 g+3 e^2 f x+d e (f+g x)\right )\right )+(3 c e f+c d g-2 b e g) (d+e x) \sqrt {c d-b e-c e x} \arctan \left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {-2 c d+b e}}\right )}{e^2 (-2 c d+b e)^{5/2} \sqrt {d+e x} \sqrt {(d+e x) (-b e+c (d-e x))}} \] Input:
Integrate[(f + g*x)/(Sqrt[d + e*x]*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^( 3/2)),x]
Output:
(Sqrt[-2*c*d + b*e]*(b*e*(-3*d*g + e*(f - 2*g*x)) + c*(3*d^2*g + 3*e^2*f*x + d*e*(f + g*x))) + (3*c*e*f + c*d*g - 2*b*e*g)*(d + e*x)*Sqrt[c*d - b*e - c*e*x]*ArcTan[Sqrt[c*d - b*e - c*e*x]/Sqrt[-2*c*d + b*e]])/(e^2*(-2*c*d + b*e)^(5/2)*Sqrt[d + e*x]*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))])
Time = 0.77 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {1220, 1132, 1136, 221}
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.
| \(\displaystyle \int \frac {f+g x}{\sqrt {d+e x} \left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1220 |
| \(\displaystyle \frac {(-2 b e g+c d g+3 c e f) \int \frac {\sqrt {d+e x}}{\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}dx}{2 e (2 c d-b e)}-\frac {e f-d g}{e^2 \sqrt {d+e x} (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\) |
| \(\Big \downarrow \) 1132 |
| \(\displaystyle \frac {(-2 b e g+c d g+3 c e f) \left (\frac {\int \frac {1}{\sqrt {d+e x} \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{2 c d-b e}+\frac {2 \sqrt {d+e x}}{e (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{2 e (2 c d-b e)}-\frac {e f-d g}{e^2 \sqrt {d+e x} (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\) |
| \(\Big \downarrow \) 1136 |
| \(\displaystyle \frac {(-2 b e g+c d g+3 c e f) \left (\frac {2 e \int \frac {1}{\frac {e^2 \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )}{d+e x}-e^2 (2 c d-b e)}d\frac {\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}{\sqrt {d+e x}}}{2 c d-b e}+\frac {2 \sqrt {d+e x}}{e (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{2 e (2 c d-b e)}-\frac {e f-d g}{e^2 \sqrt {d+e x} (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\left (\frac {2 \sqrt {d+e x}}{e (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{e (2 c d-b e)^{3/2}}\right ) (-2 b e g+c d g+3 c e f)}{2 e (2 c d-b e)}-\frac {e f-d g}{e^2 \sqrt {d+e x} (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\) |
Input:
Int[(f + g*x)/(Sqrt[d + e*x]*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2)), x]
Output:
-((e*f - d*g)/(e^2*(2*c*d - b*e)*Sqrt[d + e*x]*Sqrt[d*(c*d - b*e) - b*e^2* x - c*e^2*x^2])) + ((3*c*e*f + c*d*g - 2*b*e*g)*((2*Sqrt[d + e*x])/(e*(2*c *d - b*e)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) - (2*ArcTanh[Sqrt[d*( c*d - b*e) - b*e^2*x - c*e^2*x^2]/(Sqrt[2*c*d - b*e]*Sqrt[d + e*x])])/(e*( 2*c*d - b*e)^(3/2))))/(2*e*(2*c*d - b*e))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(2*c*d - b*e)*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(e*(p + 1)*(b^2 - 4*a*c))), x] - Simp[(2*c*d - b*e)*((m + 2*p + 2)/((p + 1)*(b^2 - 4*a*c))) Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1), x], x] /; Free Q[{a, b, c, d, e}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && LtQ [0, m, 1] && IntegerQ[2*p]
Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x _Symbol] :> Simp[2*e Subst[Int[1/(2*c*d - b*e + e^2*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d*g - e*f)*(d + e*x)^m*((a + b*x + c*x ^2)^(p + 1)/((2*c*d - b*e)*(m + p + 1))), x] + Simp[(m*(g*(c*d - b*e) + c*e *f) + e*(p + 1)*(2*c*f - b*g))/(e*(2*c*d - b*e)*(m + p + 1)) Int[(d + e*x )^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ((LtQ[m, -1] && !IGtQ[m + p + 1, 0 ]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0 ]
Leaf count of result is larger than twice the leaf count of optimal. \(470\) vs. \(2(207)=414\).
Time = 1.74 (sec) , antiderivative size = 471, normalized size of antiderivative = 2.11
| method | result | size |
| default | \(\frac {\sqrt {-\left (e x +d \right ) \left (c e x +b e -c d \right )}\, \left (2 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) \sqrt {-c e x -b e +c d}\, b \,e^{2} g x -\arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) \sqrt {-c e x -b e +c d}\, c d e g x -3 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) \sqrt {-c e x -b e +c d}\, c \,e^{2} f x +2 \sqrt {b e -2 c d}\, b \,e^{2} g x -\sqrt {b e -2 c d}\, c d e g x -3 \sqrt {b e -2 c d}\, c \,e^{2} f x +2 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) \sqrt {-c e x -b e +c d}\, b d e g -\arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) \sqrt {-c e x -b e +c d}\, c \,d^{2} g -3 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) \sqrt {-c e x -b e +c d}\, c d e f +3 \sqrt {b e -2 c d}\, b d e g -\sqrt {b e -2 c d}\, b \,e^{2} f -3 \sqrt {b e -2 c d}\, c \,d^{2} g -\sqrt {b e -2 c d}\, c d e f \right )}{\left (e x +d \right )^{\frac {3}{2}} \left (c e x +b e -c d \right ) e^{2} \left (b e -2 c d \right )^{\frac {5}{2}}}\) | \(471\) |
Input:
int((g*x+f)/(e*x+d)^(1/2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x,method= _RETURNVERBOSE)
Output:
1/(e*x+d)^(3/2)*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)*(2*arctan((-c*e*x-b*e+c*d )^(1/2)/(b*e-2*c*d)^(1/2))*(-c*e*x-b*e+c*d)^(1/2)*b*e^2*g*x-arctan((-c*e*x -b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*(-c*e*x-b*e+c*d)^(1/2)*c*d*e*g*x-3*arct an((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*(-c*e*x-b*e+c*d)^(1/2)*c*e^2* f*x+2*(b*e-2*c*d)^(1/2)*b*e^2*g*x-(b*e-2*c*d)^(1/2)*c*d*e*g*x-3*(b*e-2*c*d )^(1/2)*c*e^2*f*x+2*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*(-c*e *x-b*e+c*d)^(1/2)*b*d*e*g-arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2)) *(-c*e*x-b*e+c*d)^(1/2)*c*d^2*g-3*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d )^(1/2))*(-c*e*x-b*e+c*d)^(1/2)*c*d*e*f+3*(b*e-2*c*d)^(1/2)*b*d*e*g-(b*e-2 *c*d)^(1/2)*b*e^2*f-3*(b*e-2*c*d)^(1/2)*c*d^2*g-(b*e-2*c*d)^(1/2)*c*d*e*f) /(c*e*x+b*e-c*d)/e^2/(b*e-2*c*d)^(5/2)
Leaf count of result is larger than twice the leaf count of optimal. 661 vs. \(2 (207) = 414\).
Time = 0.17 (sec) , antiderivative size = 1353, normalized size of antiderivative = 6.07 \[ \int \frac {f+g x}{\sqrt {d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate((g*x+f)/(e*x+d)^(1/2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algorithm="fricas")
Output:
[1/2*(((3*c^2*e^4*f + (c^2*d*e^3 - 2*b*c*e^4)*g)*x^3 + (3*(c^2*d*e^3 + b*c *e^4)*f + (c^2*d^2*e^2 - b*c*d*e^3 - 2*b^2*e^4)*g)*x^2 - 3*(c^2*d^3*e - b* c*d^2*e^2)*f - (c^2*d^4 - 3*b*c*d^3*e + 2*b^2*d^2*e^2)*g - (3*(c^2*d^2*e^2 - 2*b*c*d*e^3)*f + (c^2*d^3*e - 4*b*c*d^2*e^2 + 4*b^2*d*e^3)*g)*x)*sqrt(2 *c*d - b*e)*log(-(c*e^2*x^2 - 3*c*d^2 + 2*b*d*e - 2*(c*d*e - b*e^2)*x - 2* sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(2*c*d - b*e)*sqrt(e*x + d) )/(e^2*x^2 + 2*d*e*x + d^2)) + 2*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e )*((2*c^2*d^2*e + b*c*d*e^2 - b^2*e^3)*f + 3*(2*c^2*d^3 - 3*b*c*d^2*e + b^ 2*d*e^2)*g + (3*(2*c^2*d*e^2 - b*c*e^3)*f + (2*c^2*d^2*e - 5*b*c*d*e^2 + 2 *b^2*e^3)*g)*x)*sqrt(e*x + d))/(8*c^4*d^6*e^2 - 20*b*c^3*d^5*e^3 + 18*b^2* c^2*d^4*e^4 - 7*b^3*c*d^3*e^5 + b^4*d^2*e^6 - (8*c^4*d^3*e^5 - 12*b*c^3*d^ 2*e^6 + 6*b^2*c^2*d*e^7 - b^3*c*e^8)*x^3 - (8*c^4*d^4*e^4 - 4*b*c^3*d^3*e^ 5 - 6*b^2*c^2*d^2*e^6 + 5*b^3*c*d*e^7 - b^4*e^8)*x^2 + (8*c^4*d^5*e^3 - 28 *b*c^3*d^4*e^4 + 30*b^2*c^2*d^3*e^5 - 13*b^3*c*d^2*e^6 + 2*b^4*d*e^7)*x), (((3*c^2*e^4*f + (c^2*d*e^3 - 2*b*c*e^4)*g)*x^3 + (3*(c^2*d*e^3 + b*c*e^4) *f + (c^2*d^2*e^2 - b*c*d*e^3 - 2*b^2*e^4)*g)*x^2 - 3*(c^2*d^3*e - b*c*d^2 *e^2)*f - (c^2*d^4 - 3*b*c*d^3*e + 2*b^2*d^2*e^2)*g - (3*(c^2*d^2*e^2 - 2* b*c*d*e^3)*f + (c^2*d^3*e - 4*b*c*d^2*e^2 + 4*b^2*d*e^3)*g)*x)*sqrt(-2*c*d + b*e)*arctan(-sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(-2*c*d + b *e)*sqrt(e*x + d)/(2*c*d^2 - b*d*e + (2*c*d*e - b*e^2)*x)) + sqrt(-c*e^...
\[ \int \frac {f+g x}{\sqrt {d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\int \frac {f + g x}{\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {3}{2}} \sqrt {d + e x}}\, dx \] Input:
integrate((g*x+f)/(e*x+d)**(1/2)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(3/ 2),x)
Output:
Integral((f + g*x)/((-(d + e*x)*(b*e - c*d + c*e*x))**(3/2)*sqrt(d + e*x)) , x)
\[ \int \frac {f+g x}{\sqrt {d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\int { \frac {g x + f}{{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {3}{2}} \sqrt {e x + d}} \,d x } \] Input:
integrate((g*x+f)/(e*x+d)^(1/2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algorithm="maxima")
Output:
integrate((g*x + f)/((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(3/2)*sqrt(e*x + d)), x)
Time = 0.37 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.29 \[ \int \frac {f+g x}{\sqrt {d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {\frac {{\left (3 \, c e f + c d g - 2 \, b e g\right )} \arctan \left (\frac {\sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right )}{{\left (4 \, c^{2} d^{2} e - 4 \, b c d e^{2} + b^{2} e^{3}\right )} \sqrt {-2 \, c d + b e}} + \frac {4 \, c^{2} d e f - 2 \, b c e^{2} f + 4 \, c^{2} d^{2} g - 6 \, b c d e g + 2 \, b^{2} e^{2} g + 3 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )} c e f + {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )} c d g - 2 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )} b e g}{{\left (4 \, c^{2} d^{2} e - 4 \, b c d e^{2} + b^{2} e^{3}\right )} {\left (2 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c d - \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b e - {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}}\right )}}}{e} \] Input:
integrate((g*x+f)/(e*x+d)^(1/2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algorithm="giac")
Output:
((3*c*e*f + c*d*g - 2*b*e*g)*arctan(sqrt(-(e*x + d)*c + 2*c*d - b*e)/sqrt( -2*c*d + b*e))/((4*c^2*d^2*e - 4*b*c*d*e^2 + b^2*e^3)*sqrt(-2*c*d + b*e)) + (4*c^2*d*e*f - 2*b*c*e^2*f + 4*c^2*d^2*g - 6*b*c*d*e*g + 2*b^2*e^2*g + 3 *((e*x + d)*c - 2*c*d + b*e)*c*e*f + ((e*x + d)*c - 2*c*d + b*e)*c*d*g - 2 *((e*x + d)*c - 2*c*d + b*e)*b*e*g)/((4*c^2*d^2*e - 4*b*c*d*e^2 + b^2*e^3) *(2*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c*d - sqrt(-(e*x + d)*c + 2*c*d - b*e )*b*e - (-(e*x + d)*c + 2*c*d - b*e)^(3/2))))/e
Timed out. \[ \int \frac {f+g x}{\sqrt {d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\int \frac {f+g\,x}{\sqrt {d+e\,x}\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{3/2}} \,d x \] Input:
int((f + g*x)/((d + e*x)^(1/2)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2) ),x)
Output:
int((f + g*x)/((d + e*x)^(1/2)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2) ), x)
Time = 0.22 (sec) , antiderivative size = 550, normalized size of antiderivative = 2.47 \[ \int \frac {f+g x}{\sqrt {d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {-2 \sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b d e g -2 \sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b \,e^{2} g x +\sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c \,d^{2} g +3 \sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c d e f +\sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c d e g x +3 \sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c \,e^{2} f x -3 b^{2} d \,e^{2} g +b^{2} e^{3} f -2 b^{2} e^{3} g x +9 b c \,d^{2} e g -b c d \,e^{2} f +5 b c d \,e^{2} g x +3 b c \,e^{3} f x -6 c^{2} d^{3} g -2 c^{2} d^{2} e f -2 c^{2} d^{2} e g x -6 c^{2} d \,e^{2} f x}{\sqrt {-c e x -b e +c d}\, e^{2} \left (b^{3} e^{4} x -6 b^{2} c d \,e^{3} x +12 b \,c^{2} d^{2} e^{2} x -8 c^{3} d^{3} e x +b^{3} d \,e^{3}-6 b^{2} c \,d^{2} e^{2}+12 b \,c^{2} d^{3} e -8 c^{3} d^{4}\right )} \] Input:
int((g*x+f)/(e*x+d)^(1/2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x)
Output:
( - 2*sqrt(b*e - 2*c*d)*sqrt( - b*e + c*d - c*e*x)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*b*d*e*g - 2*sqrt(b*e - 2*c*d)*sqrt( - b*e + c* d - c*e*x)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*b*e**2*g*x + sqrt(b*e - 2*c*d)*sqrt( - b*e + c*d - c*e*x)*atan(sqrt( - b*e + c*d - c*e *x)/sqrt(b*e - 2*c*d))*c*d**2*g + 3*sqrt(b*e - 2*c*d)*sqrt( - b*e + c*d - c*e*x)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*c*d*e*f + sqrt(b *e - 2*c*d)*sqrt( - b*e + c*d - c*e*x)*atan(sqrt( - b*e + c*d - c*e*x)/sqr t(b*e - 2*c*d))*c*d*e*g*x + 3*sqrt(b*e - 2*c*d)*sqrt( - b*e + c*d - c*e*x) *atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*c*e**2*f*x - 3*b**2*d* e**2*g + b**2*e**3*f - 2*b**2*e**3*g*x + 9*b*c*d**2*e*g - b*c*d*e**2*f + 5 *b*c*d*e**2*g*x + 3*b*c*e**3*f*x - 6*c**2*d**3*g - 2*c**2*d**2*e*f - 2*c** 2*d**2*e*g*x - 6*c**2*d*e**2*f*x)/(sqrt( - b*e + c*d - c*e*x)*e**2*(b**3*d *e**3 + b**3*e**4*x - 6*b**2*c*d**2*e**2 - 6*b**2*c*d*e**3*x + 12*b*c**2*d **3*e + 12*b*c**2*d**2*e**2*x - 8*c**3*d**4 - 8*c**3*d**3*e*x))