3.683 \(\int \frac{x (a+b x)^{5/2}}{(c+d x)^{3/2}} \, dx\)

Optimal. Leaf size=218 \[ \frac{(a+b x)^{5/2} \sqrt{c+d x} (7 b c-a d)}{3 d^2 (b c-a d)}-\frac{5 (a+b x)^{3/2} \sqrt{c+d x} (7 b c-a d)}{12 d^3}+\frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d) (7 b c-a d)}{8 d^4}-\frac{5 (b c-a d)^2 (7 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 \sqrt{b} d^{9/2}}-\frac{2 c (a+b x)^{7/2}}{d \sqrt{c+d x} (b c-a d)} \]

[Out]

(-2*c*(a + b*x)^(7/2))/(d*(b*c - a*d)*Sqrt[c + d*x]) + (5*(b*c - a*d)*(7*b*c - a*d)*Sqrt[a + b*x]*Sqrt[c + d*x
])/(8*d^4) - (5*(7*b*c - a*d)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(12*d^3) + ((7*b*c - a*d)*(a + b*x)^(5/2)*Sqrt[c
+ d*x])/(3*d^2*(b*c - a*d)) - (5*(b*c - a*d)^2*(7*b*c - a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c +
d*x])])/(8*Sqrt[b]*d^(9/2))

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Rubi [A]  time = 0.128319, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {78, 50, 63, 217, 206} \[ \frac{(a+b x)^{5/2} \sqrt{c+d x} (7 b c-a d)}{3 d^2 (b c-a d)}-\frac{5 (a+b x)^{3/2} \sqrt{c+d x} (7 b c-a d)}{12 d^3}+\frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d) (7 b c-a d)}{8 d^4}-\frac{5 (b c-a d)^2 (7 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 \sqrt{b} d^{9/2}}-\frac{2 c (a+b x)^{7/2}}{d \sqrt{c+d x} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[(x*(a + b*x)^(5/2))/(c + d*x)^(3/2),x]

[Out]

(-2*c*(a + b*x)^(7/2))/(d*(b*c - a*d)*Sqrt[c + d*x]) + (5*(b*c - a*d)*(7*b*c - a*d)*Sqrt[a + b*x]*Sqrt[c + d*x
])/(8*d^4) - (5*(7*b*c - a*d)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(12*d^3) + ((7*b*c - a*d)*(a + b*x)^(5/2)*Sqrt[c
+ d*x])/(3*d^2*(b*c - a*d)) - (5*(b*c - a*d)^2*(7*b*c - a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c +
d*x])])/(8*Sqrt[b]*d^(9/2))

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 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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 (a+b x)^{5/2}}{(c+d x)^{3/2}} \, dx &=-\frac{2 c (a+b x)^{7/2}}{d (b c-a d) \sqrt{c+d x}}+\frac{(7 b c-a d) \int \frac{(a+b x)^{5/2}}{\sqrt{c+d x}} \, dx}{d (b c-a d)}\\ &=-\frac{2 c (a+b x)^{7/2}}{d (b c-a d) \sqrt{c+d x}}+\frac{(7 b c-a d) (a+b x)^{5/2} \sqrt{c+d x}}{3 d^2 (b c-a d)}-\frac{(5 (7 b c-a d)) \int \frac{(a+b x)^{3/2}}{\sqrt{c+d x}} \, dx}{6 d^2}\\ &=-\frac{2 c (a+b x)^{7/2}}{d (b c-a d) \sqrt{c+d x}}-\frac{5 (7 b c-a d) (a+b x)^{3/2} \sqrt{c+d x}}{12 d^3}+\frac{(7 b c-a d) (a+b x)^{5/2} \sqrt{c+d x}}{3 d^2 (b c-a d)}+\frac{(5 (b c-a d) (7 b c-a d)) \int \frac{\sqrt{a+b x}}{\sqrt{c+d x}} \, dx}{8 d^3}\\ &=-\frac{2 c (a+b x)^{7/2}}{d (b c-a d) \sqrt{c+d x}}+\frac{5 (b c-a d) (7 b c-a d) \sqrt{a+b x} \sqrt{c+d x}}{8 d^4}-\frac{5 (7 b c-a d) (a+b x)^{3/2} \sqrt{c+d x}}{12 d^3}+\frac{(7 b c-a d) (a+b x)^{5/2} \sqrt{c+d x}}{3 d^2 (b c-a d)}-\frac{\left (5 (b c-a d)^2 (7 b c-a d)\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{16 d^4}\\ &=-\frac{2 c (a+b x)^{7/2}}{d (b c-a d) \sqrt{c+d x}}+\frac{5 (b c-a d) (7 b c-a d) \sqrt{a+b x} \sqrt{c+d x}}{8 d^4}-\frac{5 (7 b c-a d) (a+b x)^{3/2} \sqrt{c+d x}}{12 d^3}+\frac{(7 b c-a d) (a+b x)^{5/2} \sqrt{c+d x}}{3 d^2 (b c-a d)}-\frac{\left (5 (b c-a d)^2 (7 b c-a d)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c-\frac{a d}{b}+\frac{d x^2}{b}}} \, dx,x,\sqrt{a+b x}\right )}{8 b d^4}\\ &=-\frac{2 c (a+b x)^{7/2}}{d (b c-a d) \sqrt{c+d x}}+\frac{5 (b c-a d) (7 b c-a d) \sqrt{a+b x} \sqrt{c+d x}}{8 d^4}-\frac{5 (7 b c-a d) (a+b x)^{3/2} \sqrt{c+d x}}{12 d^3}+\frac{(7 b c-a d) (a+b x)^{5/2} \sqrt{c+d x}}{3 d^2 (b c-a d)}-\frac{\left (5 (b c-a d)^2 (7 b c-a d)\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{d x^2}{b}} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{8 b d^4}\\ &=-\frac{2 c (a+b x)^{7/2}}{d (b c-a d) \sqrt{c+d x}}+\frac{5 (b c-a d) (7 b c-a d) \sqrt{a+b x} \sqrt{c+d x}}{8 d^4}-\frac{5 (7 b c-a d) (a+b x)^{3/2} \sqrt{c+d x}}{12 d^3}+\frac{(7 b c-a d) (a+b x)^{5/2} \sqrt{c+d x}}{3 d^2 (b c-a d)}-\frac{5 (b c-a d)^2 (7 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 \sqrt{b} d^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.561016, size = 221, normalized size = 1.01 \[ \frac{\frac{\sqrt{d} \left (a^2 b d \left (-190 c^2+13 c d x+59 d^2 x^2\right )+3 a^3 d^2 (27 c+11 d x)+a b^2 \left (-155 c^2 d x+105 c^3-82 c d^2 x^2+34 d^3 x^3\right )+b^3 x \left (35 c^2 d x+105 c^3-14 c d^2 x^2+8 d^3 x^3\right )\right )}{\sqrt{a+b x}}-\frac{15 (b c-a d)^{5/2} (7 b c-a d) \sqrt{\frac{b (c+d x)}{b c-a d}} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )}{b}}{24 d^{9/2} \sqrt{c+d x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x*(a + b*x)^(5/2))/(c + d*x)^(3/2),x]

[Out]

((Sqrt[d]*(3*a^3*d^2*(27*c + 11*d*x) + a^2*b*d*(-190*c^2 + 13*c*d*x + 59*d^2*x^2) + b^3*x*(105*c^3 + 35*c^2*d*
x - 14*c*d^2*x^2 + 8*d^3*x^3) + a*b^2*(105*c^3 - 155*c^2*d*x - 82*c*d^2*x^2 + 34*d^3*x^3)))/Sqrt[a + b*x] - (1
5*(b*c - a*d)^(5/2)*(7*b*c - a*d)*Sqrt[(b*(c + d*x))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c - a
*d]])/b)/(24*d^(9/2)*Sqrt[c + d*x])

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Maple [B]  time = 0.023, size = 689, normalized size = 3.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(b*x+a)^(5/2)/(d*x+c)^(3/2),x)

[Out]

1/48*(b*x+a)^(1/2)*(16*x^3*b^2*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+15*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*d^4-135*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*c*d^3+225*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^2*c^2*d^2-105*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*b^3*c^3*
d+52*x^2*a*b*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-28*x^2*b^2*c*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+15*l
n(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*c*d^3-135*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*c^2*d^2+225*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*b^2*c^3*d-105*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))*b^3*c^4+66*x*a^2*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-136*x*a*b*c*d^2*((b*x+a)*(d*x+c)
)^(1/2)*(b*d)^(1/2)+70*x*b^2*c^2*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+162*a^2*c*d^2*((b*x+a)*(d*x+c))^(1/2)*(
b*d)^(1/2)-380*a*b*c^2*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+210*b^2*c^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2))/
((b*x+a)*(d*x+c))^(1/2)/(b*d)^(1/2)/(d*x+c)^(1/2)/d^4

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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*(b*x+a)^(5/2)/(d*x+c)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 5.35238, size = 1318, normalized size = 6.05 \begin{align*} \left [-\frac{15 \,{\left (7 \, b^{3} c^{4} - 15 \, a b^{2} c^{3} d + 9 \, a^{2} b c^{2} d^{2} - a^{3} c d^{3} +{\left (7 \, b^{3} c^{3} d - 15 \, a b^{2} c^{2} d^{2} + 9 \, a^{2} b c d^{3} - a^{3} d^{4}\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 (8 \, b^{3} d^{4} x^{3} + 105 \, b^{3} c^{3} d - 190 \, a b^{2} c^{2} d^{2} + 81 \, a^{2} b c d^{3} - 2 \,{\left (7 \, b^{3} c d^{3} - 13 \, a b^{2} d^{4}\right )} x^{2} +{\left (35 \, b^{3} c^{2} d^{2} - 68 \, a b^{2} c d^{3} + 33 \, a^{2} b d^{4}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{96 \,{\left (b d^{6} x + b c d^{5}\right )}}, \frac{15 \,{\left (7 \, b^{3} c^{4} - 15 \, a b^{2} c^{3} d + 9 \, a^{2} b c^{2} d^{2} - a^{3} c d^{3} +{\left (7 \, b^{3} c^{3} d - 15 \, a b^{2} c^{2} d^{2} + 9 \, a^{2} b c d^{3} - a^{3} d^{4}\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 (8 \, b^{3} d^{4} x^{3} + 105 \, b^{3} c^{3} d - 190 \, a b^{2} c^{2} d^{2} + 81 \, a^{2} b c d^{3} - 2 \,{\left (7 \, b^{3} c d^{3} - 13 \, a b^{2} d^{4}\right )} x^{2} +{\left (35 \, b^{3} c^{2} d^{2} - 68 \, a b^{2} c d^{3} + 33 \, a^{2} b d^{4}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{48 \,{\left (b d^{6} x + b c d^{5}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)^(5/2)/(d*x+c)^(3/2),x, algorithm="fricas")

[Out]

[-1/96*(15*(7*b^3*c^4 - 15*a*b^2*c^3*d + 9*a^2*b*c^2*d^2 - a^3*c*d^3 + (7*b^3*c^3*d - 15*a*b^2*c^2*d^2 + 9*a^2
*b*c*d^3 - a^3*d^4)*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)*s
qrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) - 4*(8*b^3*d^4*x^3 + 105*b^3*c^3*d - 190*a*b^2
*c^2*d^2 + 81*a^2*b*c*d^3 - 2*(7*b^3*c*d^3 - 13*a*b^2*d^4)*x^2 + (35*b^3*c^2*d^2 - 68*a*b^2*c*d^3 + 33*a^2*b*d
^4)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b*d^6*x + b*c*d^5), 1/48*(15*(7*b^3*c^4 - 15*a*b^2*c^3*d + 9*a^2*b*c^2*d^
2 - a^3*c*d^3 + (7*b^3*c^3*d - 15*a*b^2*c^2*d^2 + 9*a^2*b*c*d^3 - a^3*d^4)*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*(8*b^3
*d^4*x^3 + 105*b^3*c^3*d - 190*a*b^2*c^2*d^2 + 81*a^2*b*c*d^3 - 2*(7*b^3*c*d^3 - 13*a*b^2*d^4)*x^2 + (35*b^3*c
^2*d^2 - 68*a*b^2*c*d^3 + 33*a^2*b*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b*d^6*x + b*c*d^5)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)**(5/2)/(d*x+c)**(3/2),x)

[Out]

Timed out

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Giac [A]  time = 1.3966, size = 452, normalized size = 2.07 \begin{align*} \frac{{\left ({\left (2 \,{\left (\frac{4 \,{\left (b x + a\right )} b d^{6}{\left | b \right |}}{b^{10} c d^{8} - a b^{9} d^{9}} - \frac{7 \, b^{2} c d^{5}{\left | b \right |} - a b d^{6}{\left | b \right |}}{b^{10} c d^{8} - a b^{9} d^{9}}\right )}{\left (b x + a\right )} + \frac{5 \,{\left (7 \, b^{3} c^{2} d^{4}{\left | b \right |} - 8 \, a b^{2} c d^{5}{\left | b \right |} + a^{2} b d^{6}{\left | b \right |}\right )}}{b^{10} c d^{8} - a b^{9} d^{9}}\right )}{\left (b x + a\right )} + \frac{15 \,{\left (7 \, b^{4} c^{3} d^{3}{\left | b \right |} - 15 \, a b^{3} c^{2} d^{4}{\left | b \right |} + 9 \, a^{2} b^{2} c d^{5}{\left | b \right |} - a^{3} b d^{6}{\left | b \right |}\right )}}{b^{10} c d^{8} - a b^{9} d^{9}}\right )} \sqrt{b x + a}}{184320 \, \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}} + \frac{{\left (7 \, b^{2} c^{2}{\left | b \right |} - 8 \, a b c d{\left | b \right |} + a^{2} d^{2}{\left | b \right |}\right )} \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 |}\right )}{12288 \, \sqrt{b d} b^{8} d^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)^(5/2)/(d*x+c)^(3/2),x, algorithm="giac")

[Out]

1/184320*((2*(4*(b*x + a)*b*d^6*abs(b)/(b^10*c*d^8 - a*b^9*d^9) - (7*b^2*c*d^5*abs(b) - a*b*d^6*abs(b))/(b^10*
c*d^8 - a*b^9*d^9))*(b*x + a) + 5*(7*b^3*c^2*d^4*abs(b) - 8*a*b^2*c*d^5*abs(b) + a^2*b*d^6*abs(b))/(b^10*c*d^8
- a*b^9*d^9))*(b*x + a) + 15*(7*b^4*c^3*d^3*abs(b) - 15*a*b^3*c^2*d^4*abs(b) + 9*a^2*b^2*c*d^5*abs(b) - a^3*b
*d^6*abs(b))/(b^10*c*d^8 - a*b^9*d^9))*sqrt(b*x + a)/sqrt(b^2*c + (b*x + a)*b*d - a*b*d) + 1/12288*(7*b^2*c^2*
abs(b) - 8*a*b*c*d*abs(b) + a^2*d^2*abs(b))*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*
b*d)))/(sqrt(b*d)*b^8*d^5)