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

Optimal. Leaf size=222 \[ -\frac{2 (c+d x)^{5/2} (3 b c-7 a d)}{3 b^2 \sqrt{a+b x} (b c-a d)}+\frac{5 d \sqrt{a+b x} (c+d x)^{3/2} (3 b c-7 a d)}{6 b^3 (b c-a d)}+\frac{5 d \sqrt{a+b x} \sqrt{c+d x} (3 b c-7 a d)}{4 b^4}+\frac{5 \sqrt{d} (3 b c-7 a d) (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 b^{9/2}}+\frac{2 a (c+d x)^{7/2}}{3 b (a+b x)^{3/2} (b c-a d)} \]

[Out]

(5*d*(3*b*c - 7*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(4*b^4) + (5*d*(3*b*c - 7*a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2)
)/(6*b^3*(b*c - a*d)) - (2*(3*b*c - 7*a*d)*(c + d*x)^(5/2))/(3*b^2*(b*c - a*d)*Sqrt[a + b*x]) + (2*a*(c + d*x)
^(7/2))/(3*b*(b*c - a*d)*(a + b*x)^(3/2)) + (5*Sqrt[d]*(3*b*c - 7*a*d)*(b*c - a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b
*x])/(Sqrt[b]*Sqrt[c + d*x])])/(4*b^(9/2))

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Rubi [A]  time = 0.119782, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {78, 47, 50, 63, 217, 206} \[ -\frac{2 (c+d x)^{5/2} (3 b c-7 a d)}{3 b^2 \sqrt{a+b x} (b c-a d)}+\frac{5 d \sqrt{a+b x} (c+d x)^{3/2} (3 b c-7 a d)}{6 b^3 (b c-a d)}+\frac{5 d \sqrt{a+b x} \sqrt{c+d x} (3 b c-7 a d)}{4 b^4}+\frac{5 \sqrt{d} (3 b c-7 a d) (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 b^{9/2}}+\frac{2 a (c+d x)^{7/2}}{3 b (a+b x)^{3/2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*d*(3*b*c - 7*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(4*b^4) + (5*d*(3*b*c - 7*a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2)
)/(6*b^3*(b*c - a*d)) - (2*(3*b*c - 7*a*d)*(c + d*x)^(5/2))/(3*b^2*(b*c - a*d)*Sqrt[a + b*x]) + (2*a*(c + d*x)
^(7/2))/(3*b*(b*c - a*d)*(a + b*x)^(3/2)) + (5*Sqrt[d]*(3*b*c - 7*a*d)*(b*c - a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b
*x])/(Sqrt[b]*Sqrt[c + d*x])])/(4*b^(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 47

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

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

Mathematica [C]  time = 0.113562, size = 122, normalized size = 0.55 \[ \frac{2 \sqrt{c+d x} \left (a b^3 (c+d x)^3-\frac{(a+b x) (3 b c-7 a d) (b c-a d)^2 \, _2F_1\left (-\frac{5}{2},-\frac{1}{2};\frac{1}{2};\frac{d (a+b x)}{a d-b c}\right )}{\sqrt{\frac{b (c+d x)}{b c-a d}}}\right )}{3 b^4 (a+b x)^{3/2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[c + d*x]*(a*b^3*(c + d*x)^3 - ((3*b*c - 7*a*d)*(b*c - a*d)^2*(a + b*x)*Hypergeometric2F1[-5/2, -1/2, 1
/2, (d*(a + b*x))/(-(b*c) + a*d)])/Sqrt[(b*(c + d*x))/(b*c - a*d)]))/(3*b^4*(b*c - a*d)*(a + b*x)^(3/2))

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Maple [B]  time = 0.02, size = 750, normalized size = 3.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(d*x+c)^(1/2)*(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^2*a^2*b^
2*d^3-150*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^2+45*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*d+12*x^3*b^3*d^2*((b*x+a)*(
d*x+c))^(1/2)*(b*d)^(1/2)+210*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^3-300*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^2+90*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*d-42*x^2*a*b^2*d^2*((b*x+
a)*(d*x+c))^(1/2)*(b*d)^(1/2)+54*x^2*b^3*c*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+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))*a^4*d^3-150*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^2+45*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*d-280*x*a^2*b*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+316*x*a*b^2*c*d*((b*x+a)*(d*x+c))^
(1/2)*(b*d)^(1/2)-48*x*b^3*c^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-210*a^3*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(
1/2)+230*a^2*b*c*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-32*a*b^2*c^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2))/((b*x
+a)*(d*x+c))^(1/2)/(b*d)^(1/2)/(b*x+a)^(3/2)/b^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*(d*x+c)^(5/2)/(b*x+a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 6.92776, size = 1364, normalized size = 6.14 \begin{align*} \left [\frac{15 \,{\left (3 \, a^{2} b^{2} c^{2} - 10 \, a^{3} b c d + 7 \, a^{4} d^{2} +{\left (3 \, b^{4} c^{2} - 10 \, a b^{3} c d + 7 \, a^{2} b^{2} d^{2}\right )} x^{2} + 2 \,{\left (3 \, a b^{3} c^{2} - 10 \, a^{2} b^{2} c d + 7 \, a^{3} b d^{2}\right )} x\right )} \sqrt{\frac{d}{b}} \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^{2} d x + b^{2} c + a b d\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{d}{b}} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \,{\left (6 \, b^{3} d^{2} x^{3} - 16 \, a b^{2} c^{2} + 115 \, a^{2} b c d - 105 \, a^{3} d^{2} + 3 \,{\left (9 \, b^{3} c d - 7 \, a b^{2} d^{2}\right )} x^{2} - 2 \,{\left (12 \, b^{3} c^{2} - 79 \, a b^{2} c d + 70 \, a^{2} b d^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{48 \,{\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}, -\frac{15 \,{\left (3 \, a^{2} b^{2} c^{2} - 10 \, a^{3} b c d + 7 \, a^{4} d^{2} +{\left (3 \, b^{4} c^{2} - 10 \, a b^{3} c d + 7 \, a^{2} b^{2} d^{2}\right )} x^{2} + 2 \,{\left (3 \, a b^{3} c^{2} - 10 \, a^{2} b^{2} c d + 7 \, a^{3} b d^{2}\right )} x\right )} \sqrt{-\frac{d}{b}} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{-\frac{d}{b}}}{2 \,{\left (b d^{2} x^{2} + a c d +{\left (b c d + a d^{2}\right )} x\right )}}\right ) - 2 \,{\left (6 \, b^{3} d^{2} x^{3} - 16 \, a b^{2} c^{2} + 115 \, a^{2} b c d - 105 \, a^{3} d^{2} + 3 \,{\left (9 \, b^{3} c d - 7 \, a b^{2} d^{2}\right )} x^{2} - 2 \,{\left (12 \, b^{3} c^{2} - 79 \, a b^{2} c d + 70 \, a^{2} b d^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{24 \,{\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/48*(15*(3*a^2*b^2*c^2 - 10*a^3*b*c*d + 7*a^4*d^2 + (3*b^4*c^2 - 10*a*b^3*c*d + 7*a^2*b^2*d^2)*x^2 + 2*(3*a*
b^3*c^2 - 10*a^2*b^2*c*d + 7*a^3*b*d^2)*x)*sqrt(d/b)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*
b^2*d*x + b^2*c + a*b*d)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(d/b) + 8*(b^2*c*d + a*b*d^2)*x) + 4*(6*b^3*d^2*x^3 -
16*a*b^2*c^2 + 115*a^2*b*c*d - 105*a^3*d^2 + 3*(9*b^3*c*d - 7*a*b^2*d^2)*x^2 - 2*(12*b^3*c^2 - 79*a*b^2*c*d +
70*a^2*b*d^2)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^6*x^2 + 2*a*b^5*x + a^2*b^4), -1/24*(15*(3*a^2*b^2*c^2 - 10*
a^3*b*c*d + 7*a^4*d^2 + (3*b^4*c^2 - 10*a*b^3*c*d + 7*a^2*b^2*d^2)*x^2 + 2*(3*a*b^3*c^2 - 10*a^2*b^2*c*d + 7*a
^3*b*d^2)*x)*sqrt(-d/b)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(-d/b)/(b*d^2*x^2 + a
*c*d + (b*c*d + a*d^2)*x)) - 2*(6*b^3*d^2*x^3 - 16*a*b^2*c^2 + 115*a^2*b*c*d - 105*a^3*d^2 + 3*(9*b^3*c*d - 7*
a*b^2*d^2)*x^2 - 2*(12*b^3*c^2 - 79*a*b^2*c*d + 70*a^2*b*d^2)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^6*x^2 + 2*a*b
^5*x + a^2*b^4)]

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

[Out]

Timed out

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Giac [B]  time = 1.87697, size = 1134, normalized size = 5.11 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)*(2*(b*x + a)*d^2*abs(b)/b^6 + (9*b^12*c*d^3*abs(b) - 13*
a*b^11*d^4*abs(b))/(b^17*d^2)) - 5/8*(3*sqrt(b*d)*b^2*c^2*abs(b) - 10*sqrt(b*d)*a*b*c*d*abs(b) + 7*sqrt(b*d)*a
^2*d^2*abs(b))*log((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/b^6 - 4/3*(3*sqrt(b*d)*b
^7*c^5*abs(b) - 22*sqrt(b*d)*a*b^6*c^4*d*abs(b) + 58*sqrt(b*d)*a^2*b^5*c^3*d^2*abs(b) - 72*sqrt(b*d)*a^3*b^4*c
^2*d^3*abs(b) + 43*sqrt(b*d)*a^4*b^3*c*d^4*abs(b) - 10*sqrt(b*d)*a^5*b^2*d^5*abs(b) - 6*sqrt(b*d)*(sqrt(b*d)*s
qrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^5*c^4*abs(b) + 36*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^4*c^3*d*abs(b) - 72*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^3*c^2*d^2*abs(b) + 60*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b
*x + a)*b*d - a*b*d))^2*a^3*b^2*c*d^3*abs(b) - 18*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*
b*d - a*b*d))^2*a^4*b*d^4*abs(b) + 3*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))
^4*b^3*c^3*abs(b) - 18*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^2*d
*abs(b) + 27*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^2*b*c*d^2*abs(b) -
12*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^3*d^3*abs(b))/((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^5)