3.803 \(\int \frac{x^2}{(a+b x)^{5/2} \sqrt{c+d x}} \, dx\)

Optimal. Leaf size=126 \[ -\frac{2 a^2 \sqrt{c+d x}}{3 b^2 (a+b x)^{3/2} (b c-a d)}+\frac{4 a \sqrt{c+d x} (3 b c-2 a d)}{3 b^2 \sqrt{a+b x} (b c-a d)^2}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{5/2} \sqrt{d}} \]

[Out]

(-2*a^2*Sqrt[c + d*x])/(3*b^2*(b*c - a*d)*(a + b*x)^(3/2)) + (4*a*(3*b*c - 2*a*d)*Sqrt[c + d*x])/(3*b^2*(b*c -
a*d)^2*Sqrt[a + b*x]) + (2*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(b^(5/2)*Sqrt[d])

________________________________________________________________________________________

Rubi [A]  time = 0.0802058, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {89, 78, 63, 217, 206} \[ -\frac{2 a^2 \sqrt{c+d x}}{3 b^2 (a+b x)^{3/2} (b c-a d)}+\frac{4 a \sqrt{c+d x} (3 b c-2 a d)}{3 b^2 \sqrt{a+b x} (b c-a d)^2}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{5/2} \sqrt{d}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/((a + b*x)^(5/2)*Sqrt[c + d*x]),x]

[Out]

(-2*a^2*Sqrt[c + d*x])/(3*b^2*(b*c - a*d)*(a + b*x)^(3/2)) + (4*a*(3*b*c - 2*a*d)*Sqrt[c + d*x])/(3*b^2*(b*c -
a*d)^2*Sqrt[a + b*x]) + (2*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(b^(5/2)*Sqrt[d])

Rule 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 63

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 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{x^2}{(a+b x)^{5/2} \sqrt{c+d x}} \, dx &=-\frac{2 a^2 \sqrt{c+d x}}{3 b^2 (b c-a d) (a+b x)^{3/2}}+\frac{2 \int \frac{-\frac{1}{2} a (3 b c-a d)+\frac{3}{2} b (b c-a d) x}{(a+b x)^{3/2} \sqrt{c+d x}} \, dx}{3 b^2 (b c-a d)}\\ &=-\frac{2 a^2 \sqrt{c+d x}}{3 b^2 (b c-a d) (a+b x)^{3/2}}+\frac{4 a (3 b c-2 a d) \sqrt{c+d x}}{3 b^2 (b c-a d)^2 \sqrt{a+b x}}+\frac{\int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{b^2}\\ &=-\frac{2 a^2 \sqrt{c+d x}}{3 b^2 (b c-a d) (a+b x)^{3/2}}+\frac{4 a (3 b c-2 a d) \sqrt{c+d x}}{3 b^2 (b c-a d)^2 \sqrt{a+b x}}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{c-\frac{a d}{b}+\frac{d x^2}{b}}} \, dx,x,\sqrt{a+b x}\right )}{b^3}\\ &=-\frac{2 a^2 \sqrt{c+d x}}{3 b^2 (b c-a d) (a+b x)^{3/2}}+\frac{4 a (3 b c-2 a d) \sqrt{c+d x}}{3 b^2 (b c-a d)^2 \sqrt{a+b x}}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{1-\frac{d x^2}{b}} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{b^3}\\ &=-\frac{2 a^2 \sqrt{c+d x}}{3 b^2 (b c-a d) (a+b x)^{3/2}}+\frac{4 a (3 b c-2 a d) \sqrt{c+d x}}{3 b^2 (b c-a d)^2 \sqrt{a+b x}}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{5/2} \sqrt{d}}\\ \end{align*}

Mathematica [A]  time = 0.457032, size = 198, normalized size = 1.57 \[ \frac{2 \sqrt{c+d x} \left (\frac{(a+b x) \left (3 b^2 c^2-a^2 d^2\right )}{d (b c-a d)^2}+\frac{a^2}{a d-b c}-\frac{3 (a+b x) \left (\sqrt{b c-a d} \sqrt{\frac{b (c+d x)}{b c-a d}}-\sqrt{d} \sqrt{a+b x} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )\right )}{d \sqrt{b c-a d} \sqrt{\frac{b (c+d x)}{b c-a d}}}\right )}{3 b^2 (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/((a + b*x)^(5/2)*Sqrt[c + d*x]),x]

[Out]

(2*Sqrt[c + d*x]*(a^2/(-(b*c) + a*d) + ((3*b^2*c^2 - a^2*d^2)*(a + b*x))/(d*(b*c - a*d)^2) - (3*(a + b*x)*(Sqr
t[b*c - a*d]*Sqrt[(b*(c + d*x))/(b*c - a*d)] - Sqrt[d]*Sqrt[a + b*x]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c
- a*d]]))/(d*Sqrt[b*c - a*d]*Sqrt[(b*(c + d*x))/(b*c - a*d)])))/(3*b^2*(a + b*x)^(3/2))

________________________________________________________________________________________

Maple [B]  time = 0.021, size = 604, normalized size = 4.8 \begin{align*}{\frac{1}{3\, \left ( ad-bc \right ) ^{2}{b}^{2}}\sqrt{dx+c} \left ( 3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}{a}^{2}{b}^{2}{d}^{2}-6\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}a{b}^{3}cd+3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}{b}^{4}{c}^{2}+6\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) x{a}^{3}b{d}^{2}-12\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) x{a}^{2}{b}^{2}cd+6\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) xa{b}^{3}{c}^{2}+3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{4}{d}^{2}-6\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{3}bcd+3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}{b}^{2}{c}^{2}-8\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}x{a}^{2}bd+12\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}xa{b}^{2}c-6\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}{a}^{3}d+10\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}{a}^{2}bc \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(b*x+a)^(5/2)/(d*x+c)^(1/2),x)

[Out]

1/3*(d*x+c)^(1/2)*(3*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x^2*a^2*b^2*d
^2-6*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x^2*a*b^3*c*d+3*ln(1/2*(2*b*d
*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x^2*b^4*c^2+6*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c
))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x*a^3*b*d^2-12*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2
)+a*d+b*c)/(b*d)^(1/2))*x*a^2*b^2*c*d+6*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(
1/2))*x*a*b^3*c^2+3*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^4*d^2-6*ln(1
/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^3*b*c*d+3*ln(1/2*(2*b*d*x+2*((b*x+a)
*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^2*b^2*c^2-8*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*x*a^2*b*d+
12*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*x*a*b^2*c-6*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^3*d+10*((b*x+a)*(d*x+
c))^(1/2)*(b*d)^(1/2)*a^2*b*c)/(b*d)^(1/2)/(a*d-b*c)^2/((b*x+a)*(d*x+c))^(1/2)/b^2/(b*x+a)^(3/2)

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(b*x+a)^(5/2)/(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 4.12302, size = 1423, normalized size = 11.29 \begin{align*} \left [\frac{3 \,{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} +{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2} + 2 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x\right )} \sqrt{b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \,{\left (2 \, b d x + b c + a d\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \,{\left (5 \, a^{2} b^{2} c d - 3 \, a^{3} b d^{2} + 2 \,{\left (3 \, a b^{3} c d - 2 \, a^{2} b^{2} d^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{6 \,{\left (a^{2} b^{5} c^{2} d - 2 \, a^{3} b^{4} c d^{2} + a^{4} b^{3} d^{3} +{\left (b^{7} c^{2} d - 2 \, a b^{6} c d^{2} + a^{2} b^{5} d^{3}\right )} x^{2} + 2 \,{\left (a b^{6} c^{2} d - 2 \, a^{2} b^{5} c d^{2} + a^{3} b^{4} d^{3}\right )} x\right )}}, -\frac{3 \,{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} +{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2} + 2 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x\right )} \sqrt{-b d} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c}}{2 \,{\left (b^{2} d^{2} x^{2} + a b c d +{\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 2 \,{\left (5 \, a^{2} b^{2} c d - 3 \, a^{3} b d^{2} + 2 \,{\left (3 \, a b^{3} c d - 2 \, a^{2} b^{2} d^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{3 \,{\left (a^{2} b^{5} c^{2} d - 2 \, a^{3} b^{4} c d^{2} + a^{4} b^{3} d^{3} +{\left (b^{7} c^{2} d - 2 \, a b^{6} c d^{2} + a^{2} b^{5} d^{3}\right )} x^{2} + 2 \,{\left (a b^{6} c^{2} d - 2 \, a^{2} b^{5} c d^{2} + a^{3} b^{4} d^{3}\right )} x\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(b*x+a)^(5/2)/(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

[1/6*(3*(a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2 + (b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*x^2 + 2*(a*b^3*c^2 - 2*a^
2*b^2*c*d + a^3*b*d^2)*x)*sqrt(b*d)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b*d*x + b*c + a*d
)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) + 4*(5*a^2*b^2*c*d - 3*a^3*b*d^2 + 2*(3*a*b
^3*c*d - 2*a^2*b^2*d^2)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^2*b^5*c^2*d - 2*a^3*b^4*c*d^2 + a^4*b^3*d^3 + (b^7*
c^2*d - 2*a*b^6*c*d^2 + a^2*b^5*d^3)*x^2 + 2*(a*b^6*c^2*d - 2*a^2*b^5*c*d^2 + a^3*b^4*d^3)*x), -1/3*(3*(a^2*b^
2*c^2 - 2*a^3*b*c*d + a^4*d^2 + (b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*x^2 + 2*(a*b^3*c^2 - 2*a^2*b^2*c*d + a^3
*b*d^2)*x)*sqrt(-b*d)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c)/(b^2*d^2*x^2 + a
*b*c*d + (b^2*c*d + a*b*d^2)*x)) - 2*(5*a^2*b^2*c*d - 3*a^3*b*d^2 + 2*(3*a*b^3*c*d - 2*a^2*b^2*d^2)*x)*sqrt(b*
x + a)*sqrt(d*x + c))/(a^2*b^5*c^2*d - 2*a^3*b^4*c*d^2 + a^4*b^3*d^3 + (b^7*c^2*d - 2*a*b^6*c*d^2 + a^2*b^5*d^
3)*x^2 + 2*(a*b^6*c^2*d - 2*a^2*b^5*c*d^2 + a^3*b^4*d^3)*x)]

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(b*x+a)**(5/2)/(d*x+c)**(1/2),x)

[Out]

Integral(x**2/((a + b*x)**(5/2)*sqrt(c + d*x)), x)

________________________________________________________________________________________

Giac [B]  time = 1.43973, size = 423, normalized size = 3.36 \begin{align*} -\frac{\log \left ({\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{\sqrt{b d} b{\left | b \right |}} + \frac{8 \,{\left (3 \, \sqrt{b d} a b^{4} c^{2} - 5 \, \sqrt{b d} a^{2} b^{3} c d + 2 \, \sqrt{b d} a^{3} b^{2} d^{2} - 6 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{2} c + 3 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b d + 3 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{4} a\right )}}{3 \,{\left (b^{2} c - a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{3} b{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(b*x+a)^(5/2)/(d*x+c)^(1/2),x, algorithm="giac")

[Out]

-log((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/(sqrt(b*d)*b*abs(b)) + 8/3*(3*sqrt(b*d
)*a*b^4*c^2 - 5*sqrt(b*d)*a^2*b^3*c*d + 2*sqrt(b*d)*a^3*b^2*d^2 - 6*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(
b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^2*c + 3*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d
- a*b*d))^2*a^2*b*d + 3*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a)/((b^2*c
- a*b*d - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)^3*b*abs(b))