\(\int \frac {\sqrt {a+b x} (e+f x)}{x (c+d x)^2} \, dx\) [20]

Optimal result

Integrand size = 25, antiderivative size = 128 \[ \int \frac {\sqrt {a+b x} (e+f x)}{x (c+d x)^2} \, dx=\frac {(d e-c f) \sqrt {a+b x}}{c d (c+d x)}-\frac {\left (2 a d^2 e-b c (d e+c f)\right ) \arctan \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{c^2 d^{3/2} \sqrt {b c-a d}}-\frac {2 \sqrt {a} e \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{c^2} \]

[Out]

-2*e*arctanh((b*x+a)^(1/2)/a^(1/2))*a^(1/2)/c^2-(2*a*d^2*e-b*c*(c*f+d*e))*arctan(d^(1/2)*(b*x+a)^(1/2)/(-a*d+b
*c)^(1/2))/c^2/d^(3/2)/(-a*d+b*c)^(1/2)+(-c*f+d*e)*(b*x+a)^(1/2)/c/d/(d*x+c)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {154, 162, 65, 214, 211} \[ \int \frac {\sqrt {a+b x} (e+f x)}{x (c+d x)^2} \, dx=-\frac {\arctan \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right ) \left (2 a d^2 e-b c (c f+d e)\right )}{c^2 d^{3/2} \sqrt {b c-a d}}-\frac {2 \sqrt {a} e \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{c^2}+\frac {\sqrt {a+b x} (d e-c f)}{c d (c+d x)} \]

[In]

Int[(Sqrt[a + b*x]*(e + f*x))/(x*(c + d*x)^2),x]

[Out]

((d*e - c*f)*Sqrt[a + b*x])/(c*d*(c + d*x)) - ((2*a*d^2*e - b*c*(d*e + c*f))*ArcTan[(Sqrt[d]*Sqrt[a + b*x])/Sq
rt[b*c - a*d]])/(c^2*d^(3/2)*Sqrt[b*c - a*d]) - (2*Sqrt[a]*e*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/c^2

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 154

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]

Rule 162

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 214

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]

Rubi steps \begin{align*} \text {integral}& = \frac {(d e-c f) \sqrt {a+b x}}{c d (c+d x)}-\frac {\int \frac {-a d e-\frac {1}{2} b (d e+c f) x}{x \sqrt {a+b x} (c+d x)} \, dx}{c d} \\ & = \frac {(d e-c f) \sqrt {a+b x}}{c d (c+d x)}+\frac {(a e) \int \frac {1}{x \sqrt {a+b x}} \, dx}{c^2}-\frac {\left (2 a d^2 e-b c (d e+c f)\right ) \int \frac {1}{\sqrt {a+b x} (c+d x)} \, dx}{2 c^2 d} \\ & = \frac {(d e-c f) \sqrt {a+b x}}{c d (c+d x)}+\frac {(2 a e) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{b c^2}-\frac {\left (2 a d^2 e-b c (d e+c f)\right ) \text {Subst}\left (\int \frac {1}{c-\frac {a d}{b}+\frac {d x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{b c^2 d} \\ & = \frac {(d e-c f) \sqrt {a+b x}}{c d (c+d x)}-\frac {\left (2 a d^2 e-b c (d e+c f)\right ) \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{c^2 d^{3/2} \sqrt {b c-a d}}-\frac {2 \sqrt {a} e \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{c^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.46 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {a+b x} (e+f x)}{x (c+d x)^2} \, dx=\frac {\frac {c (d e-c f) \sqrt {a+b x}}{d (c+d x)}+\frac {\left (-2 a d^2 e+b c (d e+c f)\right ) \arctan \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{d^{3/2} \sqrt {b c-a d}}-2 \sqrt {a} e \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{c^2} \]

[In]

Integrate[(Sqrt[a + b*x]*(e + f*x))/(x*(c + d*x)^2),x]

[Out]

((c*(d*e - c*f)*Sqrt[a + b*x])/(d*(c + d*x)) + ((-2*a*d^2*e + b*c*(d*e + c*f))*ArcTan[(Sqrt[d]*Sqrt[a + b*x])/
Sqrt[b*c - a*d]])/(d^(3/2)*Sqrt[b*c - a*d]) - 2*Sqrt[a]*e*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/c^2

Maple [A] (verified)

Time = 1.62 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.86

method	result	size
pseudoelliptic	\(\frac {-2 e \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) \sqrt {a}+\frac {-\frac {c \left (c f -d e \right ) \sqrt {b x +a}}{d x +c}+\frac {\left (2 a e \,d^{2}-c^{2} b f -b c d e \right ) \operatorname {arctanh}\left (\frac {d \sqrt {b x +a}}{\sqrt {\left (a d -b c \right ) d}}\right )}{\sqrt {\left (a d -b c \right ) d}}}{d}}{c^{2}}\)	\(110\)
derivativedivides	\(2 b \left (-\frac {e \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{b \,c^{2}}+\frac {\frac {b c \left (c f -d e \right ) \sqrt {b x +a}}{2 d \left (-d \left (b x +a \right )+a d -b c \right )}+\frac {\left (2 a e \,d^{2}-c^{2} b f -b c d e \right ) \operatorname {arctanh}\left (\frac {d \sqrt {b x +a}}{\sqrt {\left (a d -b c \right ) d}}\right )}{2 d \sqrt {\left (a d -b c \right ) d}}}{c^{2} b}\right )\)	\(137\)
default	\(2 b \left (-\frac {e \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{b \,c^{2}}+\frac {\frac {b c \left (c f -d e \right ) \sqrt {b x +a}}{2 d \left (-d \left (b x +a \right )+a d -b c \right )}+\frac {\left (2 a e \,d^{2}-c^{2} b f -b c d e \right ) \operatorname {arctanh}\left (\frac {d \sqrt {b x +a}}{\sqrt {\left (a d -b c \right ) d}}\right )}{2 d \sqrt {\left (a d -b c \right ) d}}}{c^{2} b}\right )\)	\(137\)

[In]

int((f*x+e)*(b*x+a)^(1/2)/x/(d*x+c)^2,x,method=_RETURNVERBOSE)

[Out]

1/c^2*(-2*e*arctanh((b*x+a)^(1/2)/a^(1/2))*a^(1/2)+1/d*(-c*(c*f-d*e)*(b*x+a)^(1/2)/(d*x+c)+(2*a*d^2*e-b*c^2*f-
b*c*d*e)/((a*d-b*c)*d)^(1/2)*arctanh(d*(b*x+a)^(1/2)/((a*d-b*c)*d)^(1/2))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (110) = 220\).

Time = 0.32 (sec) , antiderivative size = 1008, normalized size of antiderivative = 7.88 \[ \int \frac {\sqrt {a+b x} (e+f x)}{x (c+d x)^2} \, dx=\left [-\frac {{\left (b c^{3} f + {\left (b c^{2} d - 2 \, a c d^{2}\right )} e + {\left (b c^{2} d f + {\left (b c d^{2} - 2 \, a d^{3}\right )} e\right )} x\right )} \sqrt {-b c d + a d^{2}} \log \left (\frac {b d x - b c + 2 \, a d - 2 \, \sqrt {-b c d + a d^{2}} \sqrt {b x + a}}{d x + c}\right ) - 2 \, {\left ({\left (b c d^{3} - a d^{4}\right )} e x + {\left (b c^{2} d^{2} - a c d^{3}\right )} e\right )} \sqrt {a} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) - 2 \, {\left ({\left (b c^{2} d^{2} - a c d^{3}\right )} e - {\left (b c^{3} d - a c^{2} d^{2}\right )} f\right )} \sqrt {b x + a}}{2 \, {\left (b c^{4} d^{2} - a c^{3} d^{3} + {\left (b c^{3} d^{3} - a c^{2} d^{4}\right )} x\right )}}, \frac {4 \, {\left ({\left (b c d^{3} - a d^{4}\right )} e x + {\left (b c^{2} d^{2} - a c d^{3}\right )} e\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - {\left (b c^{3} f + {\left (b c^{2} d - 2 \, a c d^{2}\right )} e + {\left (b c^{2} d f + {\left (b c d^{2} - 2 \, a d^{3}\right )} e\right )} x\right )} \sqrt {-b c d + a d^{2}} \log \left (\frac {b d x - b c + 2 \, a d - 2 \, \sqrt {-b c d + a d^{2}} \sqrt {b x + a}}{d x + c}\right ) + 2 \, {\left ({\left (b c^{2} d^{2} - a c d^{3}\right )} e - {\left (b c^{3} d - a c^{2} d^{2}\right )} f\right )} \sqrt {b x + a}}{2 \, {\left (b c^{4} d^{2} - a c^{3} d^{3} + {\left (b c^{3} d^{3} - a c^{2} d^{4}\right )} x\right )}}, -\frac {{\left (b c^{3} f + {\left (b c^{2} d - 2 \, a c d^{2}\right )} e + {\left (b c^{2} d f + {\left (b c d^{2} - 2 \, a d^{3}\right )} e\right )} x\right )} \sqrt {b c d - a d^{2}} \arctan \left (\frac {\sqrt {b c d - a d^{2}} \sqrt {b x + a}}{b d x + a d}\right ) - {\left ({\left (b c d^{3} - a d^{4}\right )} e x + {\left (b c^{2} d^{2} - a c d^{3}\right )} e\right )} \sqrt {a} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) - {\left ({\left (b c^{2} d^{2} - a c d^{3}\right )} e - {\left (b c^{3} d - a c^{2} d^{2}\right )} f\right )} \sqrt {b x + a}}{b c^{4} d^{2} - a c^{3} d^{3} + {\left (b c^{3} d^{3} - a c^{2} d^{4}\right )} x}, -\frac {{\left (b c^{3} f + {\left (b c^{2} d - 2 \, a c d^{2}\right )} e + {\left (b c^{2} d f + {\left (b c d^{2} - 2 \, a d^{3}\right )} e\right )} x\right )} \sqrt {b c d - a d^{2}} \arctan \left (\frac {\sqrt {b c d - a d^{2}} \sqrt {b x + a}}{b d x + a d}\right ) - 2 \, {\left ({\left (b c d^{3} - a d^{4}\right )} e x + {\left (b c^{2} d^{2} - a c d^{3}\right )} e\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - {\left ({\left (b c^{2} d^{2} - a c d^{3}\right )} e - {\left (b c^{3} d - a c^{2} d^{2}\right )} f\right )} \sqrt {b x + a}}{b c^{4} d^{2} - a c^{3} d^{3} + {\left (b c^{3} d^{3} - a c^{2} d^{4}\right )} x}\right ] \]

[In]

integrate((f*x+e)*(b*x+a)^(1/2)/x/(d*x+c)^2,x, algorithm="fricas")

[Out]

[-1/2*((b*c^3*f + (b*c^2*d - 2*a*c*d^2)*e + (b*c^2*d*f + (b*c*d^2 - 2*a*d^3)*e)*x)*sqrt(-b*c*d + a*d^2)*log((b
*d*x - b*c + 2*a*d - 2*sqrt(-b*c*d + a*d^2)*sqrt(b*x + a))/(d*x + c)) - 2*((b*c*d^3 - a*d^4)*e*x + (b*c^2*d^2
- a*c*d^3)*e)*sqrt(a)*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) - 2*((b*c^2*d^2 - a*c*d^3)*e - (b*c^3*d - a
*c^2*d^2)*f)*sqrt(b*x + a))/(b*c^4*d^2 - a*c^3*d^3 + (b*c^3*d^3 - a*c^2*d^4)*x), 1/2*(4*((b*c*d^3 - a*d^4)*e*x
+ (b*c^2*d^2 - a*c*d^3)*e)*sqrt(-a)*arctan(sqrt(b*x + a)*sqrt(-a)/a) - (b*c^3*f + (b*c^2*d - 2*a*c*d^2)*e + (
b*c^2*d*f + (b*c*d^2 - 2*a*d^3)*e)*x)*sqrt(-b*c*d + a*d^2)*log((b*d*x - b*c + 2*a*d - 2*sqrt(-b*c*d + a*d^2)*s
qrt(b*x + a))/(d*x + c)) + 2*((b*c^2*d^2 - a*c*d^3)*e - (b*c^3*d - a*c^2*d^2)*f)*sqrt(b*x + a))/(b*c^4*d^2 - a
*c^3*d^3 + (b*c^3*d^3 - a*c^2*d^4)*x), -((b*c^3*f + (b*c^2*d - 2*a*c*d^2)*e + (b*c^2*d*f + (b*c*d^2 - 2*a*d^3)
*e)*x)*sqrt(b*c*d - a*d^2)*arctan(sqrt(b*c*d - a*d^2)*sqrt(b*x + a)/(b*d*x + a*d)) - ((b*c*d^3 - a*d^4)*e*x +
(b*c^2*d^2 - a*c*d^3)*e)*sqrt(a)*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) - ((b*c^2*d^2 - a*c*d^3)*e - (b*
c^3*d - a*c^2*d^2)*f)*sqrt(b*x + a))/(b*c^4*d^2 - a*c^3*d^3 + (b*c^3*d^3 - a*c^2*d^4)*x), -((b*c^3*f + (b*c^2*
d - 2*a*c*d^2)*e + (b*c^2*d*f + (b*c*d^2 - 2*a*d^3)*e)*x)*sqrt(b*c*d - a*d^2)*arctan(sqrt(b*c*d - a*d^2)*sqrt(
b*x + a)/(b*d*x + a*d)) - 2*((b*c*d^3 - a*d^4)*e*x + (b*c^2*d^2 - a*c*d^3)*e)*sqrt(-a)*arctan(sqrt(b*x + a)*sq
rt(-a)/a) - ((b*c^2*d^2 - a*c*d^3)*e - (b*c^3*d - a*c^2*d^2)*f)*sqrt(b*x + a))/(b*c^4*d^2 - a*c^3*d^3 + (b*c^3
*d^3 - a*c^2*d^4)*x)]

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x} (e+f x)}{x (c+d x)^2} \, dx=\text {Timed out} \]

[In]

integrate((f*x+e)*(b*x+a)**(1/2)/x/(d*x+c)**2,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {a+b x} (e+f x)}{x (c+d x)^2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((f*x+e)*(b*x+a)^(1/2)/x/(d*x+c)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for
more detail

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt {a+b x} (e+f x)}{x (c+d x)^2} \, dx=\frac {2 \, a e \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} c^{2}} + \frac {{\left (b c d e - 2 \, a d^{2} e + b c^{2} f\right )} \arctan \left (\frac {\sqrt {b x + a} d}{\sqrt {b c d - a d^{2}}}\right )}{\sqrt {b c d - a d^{2}} c^{2} d} + \frac {\sqrt {b x + a} b d e - \sqrt {b x + a} b c f}{{\left (b c + {\left (b x + a\right )} d - a d\right )} c d} \]

[In]

integrate((f*x+e)*(b*x+a)^(1/2)/x/(d*x+c)^2,x, algorithm="giac")

[Out]

2*a*e*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*c^2) + (b*c*d*e - 2*a*d^2*e + b*c^2*f)*arctan(sqrt(b*x + a)*d/s
qrt(b*c*d - a*d^2))/(sqrt(b*c*d - a*d^2)*c^2*d) + (sqrt(b*x + a)*b*d*e - sqrt(b*x + a)*b*c*f)/((b*c + (b*x + a
)*d - a*d)*c*d)

Mupad [B] (verification not implemented)

Time = 3.39 (sec) , antiderivative size = 1814, normalized size of antiderivative = 14.17 \[ \int \frac {\sqrt {a+b x} (e+f x)}{x (c+d x)^2} \, dx=\text {Too large to display} \]

[In]

int(((e + f*x)*(a + b*x)^(1/2))/(x*(c + d*x)^2),x)

[Out]

(atan(((((((2*(2*a*b^3*c^4*d^3*e - 2*a*b^3*c^5*d^2*f))/(c^3*d) + ((4*b^3*c^5*d^3 - 8*a*b^2*c^4*d^4)*(d^3*(a*d
- b*c))^(1/2)*(a + b*x)^(1/2)*(b*c^2*f - 2*a*d^2*e + b*c*d*e))/(c^2*d*(a*c^2*d^4 - b*c^3*d^3)))*(d^3*(a*d - b*
c))^(1/2)*(b*c^2*f - 2*a*d^2*e + b*c*d*e))/(2*(a*c^2*d^4 - b*c^3*d^3)) + (2*(a + b*x)^(1/2)*(b^4*c^4*f^2 + 8*a
^2*b^2*d^4*e^2 + b^4*c^2*d^2*e^2 + 2*b^4*c^3*d*e*f - 4*a*b^3*c*d^3*e^2 - 4*a*b^3*c^2*d^2*e*f))/(c^2*d))*(d^3*(
a*d - b*c))^(1/2)*(b*c^2*f - 2*a*d^2*e + b*c*d*e)*1i)/(2*(a*c^2*d^4 - b*c^3*d^3)) - (((((2*(2*a*b^3*c^4*d^3*e
- 2*a*b^3*c^5*d^2*f))/(c^3*d) - ((4*b^3*c^5*d^3 - 8*a*b^2*c^4*d^4)*(d^3*(a*d - b*c))^(1/2)*(a + b*x)^(1/2)*(b*
c^2*f - 2*a*d^2*e + b*c*d*e))/(c^2*d*(a*c^2*d^4 - b*c^3*d^3)))*(d^3*(a*d - b*c))^(1/2)*(b*c^2*f - 2*a*d^2*e +
b*c*d*e))/(2*(a*c^2*d^4 - b*c^3*d^3)) - (2*(a + b*x)^(1/2)*(b^4*c^4*f^2 + 8*a^2*b^2*d^4*e^2 + b^4*c^2*d^2*e^2
+ 2*b^4*c^3*d*e*f - 4*a*b^3*c*d^3*e^2 - 4*a*b^3*c^2*d^2*e*f))/(c^2*d))*(d^3*(a*d - b*c))^(1/2)*(b*c^2*f - 2*a*
d^2*e + b*c*d*e)*1i)/(2*(a*c^2*d^4 - b*c^3*d^3)))/((4*(a*b^4*c*d^2*e^3 - 2*a^2*b^3*d^3*e^3 + a*b^4*c^3*e*f^2 -
2*a^2*b^3*c*d^2*e^2*f + 2*a*b^4*c^2*d*e^2*f))/(c^3*d) + (((((2*(2*a*b^3*c^4*d^3*e - 2*a*b^3*c^5*d^2*f))/(c^3*
d) + ((4*b^3*c^5*d^3 - 8*a*b^2*c^4*d^4)*(d^3*(a*d - b*c))^(1/2)*(a + b*x)^(1/2)*(b*c^2*f - 2*a*d^2*e + b*c*d*e
))/(c^2*d*(a*c^2*d^4 - b*c^3*d^3)))*(d^3*(a*d - b*c))^(1/2)*(b*c^2*f - 2*a*d^2*e + b*c*d*e))/(2*(a*c^2*d^4 - b
*c^3*d^3)) + (2*(a + b*x)^(1/2)*(b^4*c^4*f^2 + 8*a^2*b^2*d^4*e^2 + b^4*c^2*d^2*e^2 + 2*b^4*c^3*d*e*f - 4*a*b^3
*c*d^3*e^2 - 4*a*b^3*c^2*d^2*e*f))/(c^2*d))*(d^3*(a*d - b*c))^(1/2)*(b*c^2*f - 2*a*d^2*e + b*c*d*e))/(2*(a*c^2
*d^4 - b*c^3*d^3)) + (((((2*(2*a*b^3*c^4*d^3*e - 2*a*b^3*c^5*d^2*f))/(c^3*d) - ((4*b^3*c^5*d^3 - 8*a*b^2*c^4*d
^4)*(d^3*(a*d - b*c))^(1/2)*(a + b*x)^(1/2)*(b*c^2*f - 2*a*d^2*e + b*c*d*e))/(c^2*d*(a*c^2*d^4 - b*c^3*d^3)))*
(d^3*(a*d - b*c))^(1/2)*(b*c^2*f - 2*a*d^2*e + b*c*d*e))/(2*(a*c^2*d^4 - b*c^3*d^3)) - (2*(a + b*x)^(1/2)*(b^4
*c^4*f^2 + 8*a^2*b^2*d^4*e^2 + b^4*c^2*d^2*e^2 + 2*b^4*c^3*d*e*f - 4*a*b^3*c*d^3*e^2 - 4*a*b^3*c^2*d^2*e*f))/(
c^2*d))*(d^3*(a*d - b*c))^(1/2)*(b*c^2*f - 2*a*d^2*e + b*c*d*e))/(2*(a*c^2*d^4 - b*c^3*d^3))))*(d^3*(a*d - b*c
))^(1/2)*(b*c^2*f - 2*a*d^2*e + b*c*d*e)*1i)/(a*c^2*d^4 - b*c^3*d^3) - (2*a^(1/2)*e*atanh((4*a^(1/2)*b^4*e*f^2
*(a + b*x)^(1/2))/(4*a*b^4*e*f^2 + (4*a*b^4*d^2*e^3)/c^2 - (16*a^2*b^3*d^2*e^2*f)/c^2 + (8*a*b^4*d*e^2*f)/c) +
(8*a^(1/2)*b^4*e^2*f*(a + b*x)^(1/2))/(8*a*b^4*e^2*f + (4*a*b^4*d*e^3)/c - (16*a^2*b^3*d*e^2*f)/c + (4*a*b^4*
c*e*f^2)/d) + (4*a^(1/2)*b^4*d*e^3*(a + b*x)^(1/2))/(4*a*b^4*d*e^3 + 8*a*b^4*c*e^2*f - 16*a^2*b^3*d*e^2*f + (4
*a*b^4*c^2*e*f^2)/d) - (16*a^(3/2)*b^3*d*e^2*f*(a + b*x)^(1/2))/(4*a*b^4*d*e^3 + 8*a*b^4*c*e^2*f - 16*a^2*b^3*
d*e^2*f + (4*a*b^4*c^2*e*f^2)/d)))/c^2 - ((b*c*f - b*d*e)*(a + b*x)^(1/2))/(c*d*(b*c - a*d + d*(a + b*x)))