3.1511 \(\int \frac{1}{(a+b x)^{11/2} (c+d x)^{3/2}} \, dx\)

Optimal. Leaf size=206 \[ -\frac{512 d^5 \sqrt{a+b x}}{63 \sqrt{c+d x} (b c-a d)^6}-\frac{256 d^4}{63 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^5}+\frac{64 d^3}{63 (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)^4}-\frac{32 d^2}{63 (a+b x)^{5/2} \sqrt{c+d x} (b c-a d)^3}+\frac{20 d}{63 (a+b x)^{7/2} \sqrt{c+d x} (b c-a d)^2}-\frac{2}{9 (a+b x)^{9/2} \sqrt{c+d x} (b c-a d)} \]

[Out]

-2/(9*(b*c - a*d)*(a + b*x)^(9/2)*Sqrt[c + d*x]) + (20*d)/(63*(b*c - a*d)^2*(a + b*x)^(7/2)*Sqrt[c + d*x]) - (
32*d^2)/(63*(b*c - a*d)^3*(a + b*x)^(5/2)*Sqrt[c + d*x]) + (64*d^3)/(63*(b*c - a*d)^4*(a + b*x)^(3/2)*Sqrt[c +
d*x]) - (256*d^4)/(63*(b*c - a*d)^5*Sqrt[a + b*x]*Sqrt[c + d*x]) - (512*d^5*Sqrt[a + b*x])/(63*(b*c - a*d)^6*
Sqrt[c + d*x])

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Rubi [A]  time = 0.0565066, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {45, 37} \[ -\frac{512 d^5 \sqrt{a+b x}}{63 \sqrt{c+d x} (b c-a d)^6}-\frac{256 d^4}{63 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^5}+\frac{64 d^3}{63 (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)^4}-\frac{32 d^2}{63 (a+b x)^{5/2} \sqrt{c+d x} (b c-a d)^3}+\frac{20 d}{63 (a+b x)^{7/2} \sqrt{c+d x} (b c-a d)^2}-\frac{2}{9 (a+b x)^{9/2} \sqrt{c+d x} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x)^(11/2)*(c + d*x)^(3/2)),x]

[Out]

-2/(9*(b*c - a*d)*(a + b*x)^(9/2)*Sqrt[c + d*x]) + (20*d)/(63*(b*c - a*d)^2*(a + b*x)^(7/2)*Sqrt[c + d*x]) - (
32*d^2)/(63*(b*c - a*d)^3*(a + b*x)^(5/2)*Sqrt[c + d*x]) + (64*d^3)/(63*(b*c - a*d)^4*(a + b*x)^(3/2)*Sqrt[c +
d*x]) - (256*d^4)/(63*(b*c - a*d)^5*Sqrt[a + b*x]*Sqrt[c + d*x]) - (512*d^5*Sqrt[a + b*x])/(63*(b*c - a*d)^6*
Sqrt[c + d*x])

Rule 45

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

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{1}{(a+b x)^{11/2} (c+d x)^{3/2}} \, dx &=-\frac{2}{9 (b c-a d) (a+b x)^{9/2} \sqrt{c+d x}}-\frac{(10 d) \int \frac{1}{(a+b x)^{9/2} (c+d x)^{3/2}} \, dx}{9 (b c-a d)}\\ &=-\frac{2}{9 (b c-a d) (a+b x)^{9/2} \sqrt{c+d x}}+\frac{20 d}{63 (b c-a d)^2 (a+b x)^{7/2} \sqrt{c+d x}}+\frac{\left (80 d^2\right ) \int \frac{1}{(a+b x)^{7/2} (c+d x)^{3/2}} \, dx}{63 (b c-a d)^2}\\ &=-\frac{2}{9 (b c-a d) (a+b x)^{9/2} \sqrt{c+d x}}+\frac{20 d}{63 (b c-a d)^2 (a+b x)^{7/2} \sqrt{c+d x}}-\frac{32 d^2}{63 (b c-a d)^3 (a+b x)^{5/2} \sqrt{c+d x}}-\frac{\left (32 d^3\right ) \int \frac{1}{(a+b x)^{5/2} (c+d x)^{3/2}} \, dx}{21 (b c-a d)^3}\\ &=-\frac{2}{9 (b c-a d) (a+b x)^{9/2} \sqrt{c+d x}}+\frac{20 d}{63 (b c-a d)^2 (a+b x)^{7/2} \sqrt{c+d x}}-\frac{32 d^2}{63 (b c-a d)^3 (a+b x)^{5/2} \sqrt{c+d x}}+\frac{64 d^3}{63 (b c-a d)^4 (a+b x)^{3/2} \sqrt{c+d x}}+\frac{\left (128 d^4\right ) \int \frac{1}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx}{63 (b c-a d)^4}\\ &=-\frac{2}{9 (b c-a d) (a+b x)^{9/2} \sqrt{c+d x}}+\frac{20 d}{63 (b c-a d)^2 (a+b x)^{7/2} \sqrt{c+d x}}-\frac{32 d^2}{63 (b c-a d)^3 (a+b x)^{5/2} \sqrt{c+d x}}+\frac{64 d^3}{63 (b c-a d)^4 (a+b x)^{3/2} \sqrt{c+d x}}-\frac{256 d^4}{63 (b c-a d)^5 \sqrt{a+b x} \sqrt{c+d x}}-\frac{\left (256 d^5\right ) \int \frac{1}{\sqrt{a+b x} (c+d x)^{3/2}} \, dx}{63 (b c-a d)^5}\\ &=-\frac{2}{9 (b c-a d) (a+b x)^{9/2} \sqrt{c+d x}}+\frac{20 d}{63 (b c-a d)^2 (a+b x)^{7/2} \sqrt{c+d x}}-\frac{32 d^2}{63 (b c-a d)^3 (a+b x)^{5/2} \sqrt{c+d x}}+\frac{64 d^3}{63 (b c-a d)^4 (a+b x)^{3/2} \sqrt{c+d x}}-\frac{256 d^4}{63 (b c-a d)^5 \sqrt{a+b x} \sqrt{c+d x}}-\frac{512 d^5 \sqrt{a+b x}}{63 (b c-a d)^6 \sqrt{c+d x}}\\ \end{align*}

Mathematica [A]  time = 0.0781708, size = 226, normalized size = 1.1 \[ \frac{512 d^5 \sqrt{a+b x}}{63 \sqrt{c+d x} (b c-a d)^5 (a d-b c)}+\frac{256 d^4}{63 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^4 (a d-b c)}+\frac{64 d^3}{63 (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)^4}-\frac{32 d^2}{63 (a+b x)^{5/2} \sqrt{c+d x} (b c-a d)^3}+\frac{20 d}{63 (a+b x)^{7/2} \sqrt{c+d x} (b c-a d)^2}-\frac{2}{9 (a+b x)^{9/2} \sqrt{c+d x} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x)^(11/2)*(c + d*x)^(3/2)),x]

[Out]

-2/(9*(b*c - a*d)*(a + b*x)^(9/2)*Sqrt[c + d*x]) + (20*d)/(63*(b*c - a*d)^2*(a + b*x)^(7/2)*Sqrt[c + d*x]) - (
32*d^2)/(63*(b*c - a*d)^3*(a + b*x)^(5/2)*Sqrt[c + d*x]) + (64*d^3)/(63*(b*c - a*d)^4*(a + b*x)^(3/2)*Sqrt[c +
d*x]) + (256*d^4)/(63*(b*c - a*d)^4*(-(b*c) + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x]) + (512*d^5*Sqrt[a + b*x])/(63
*(b*c - a*d)^5*(-(b*c) + a*d)*Sqrt[c + d*x])

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Maple [B]  time = 0.011, size = 356, normalized size = 1.7 \begin{align*} -{\frac{512\,{b}^{5}{d}^{5}{x}^{5}+2304\,a{b}^{4}{d}^{5}{x}^{4}+256\,{b}^{5}c{d}^{4}{x}^{4}+4032\,{a}^{2}{b}^{3}{d}^{5}{x}^{3}+1152\,a{b}^{4}c{d}^{4}{x}^{3}-64\,{b}^{5}{c}^{2}{d}^{3}{x}^{3}+3360\,{a}^{3}{b}^{2}{d}^{5}{x}^{2}+2016\,{a}^{2}{b}^{3}c{d}^{4}{x}^{2}-288\,a{b}^{4}{c}^{2}{d}^{3}{x}^{2}+32\,{b}^{5}{c}^{3}{d}^{2}{x}^{2}+1260\,{a}^{4}b{d}^{5}x+1680\,{a}^{3}{b}^{2}c{d}^{4}x-504\,{a}^{2}{b}^{3}{c}^{2}{d}^{3}x+144\,a{b}^{4}{c}^{3}{d}^{2}x-20\,{b}^{5}{c}^{4}dx+126\,{a}^{5}{d}^{5}+630\,{a}^{4}bc{d}^{4}-420\,{a}^{3}{b}^{2}{c}^{2}{d}^{3}+252\,{a}^{2}{b}^{3}{c}^{3}{d}^{2}-90\,a{b}^{4}{c}^{4}d+14\,{b}^{5}{c}^{5}}{63\,{d}^{6}{a}^{6}-378\,b{d}^{5}c{a}^{5}+945\,{b}^{2}{d}^{4}{c}^{2}{a}^{4}-1260\,{b}^{3}{d}^{3}{c}^{3}{a}^{3}+945\,{b}^{4}{d}^{2}{c}^{4}{a}^{2}-378\,{b}^{5}d{c}^{5}a+63\,{b}^{6}{c}^{6}} \left ( bx+a \right ) ^{-{\frac{9}{2}}}{\frac{1}{\sqrt{dx+c}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)^(11/2)/(d*x+c)^(3/2),x)

[Out]

-2/63*(256*b^5*d^5*x^5+1152*a*b^4*d^5*x^4+128*b^5*c*d^4*x^4+2016*a^2*b^3*d^5*x^3+576*a*b^4*c*d^4*x^3-32*b^5*c^
2*d^3*x^3+1680*a^3*b^2*d^5*x^2+1008*a^2*b^3*c*d^4*x^2-144*a*b^4*c^2*d^3*x^2+16*b^5*c^3*d^2*x^2+630*a^4*b*d^5*x
+840*a^3*b^2*c*d^4*x-252*a^2*b^3*c^2*d^3*x+72*a*b^4*c^3*d^2*x-10*b^5*c^4*d*x+63*a^5*d^5+315*a^4*b*c*d^4-210*a^
3*b^2*c^2*d^3+126*a^2*b^3*c^3*d^2-45*a*b^4*c^4*d+7*b^5*c^5)/(b*x+a)^(9/2)/(d*x+c)^(1/2)/(a^6*d^6-6*a^5*b*c*d^5
+15*a^4*b^2*c^2*d^4-20*a^3*b^3*c^3*d^3+15*a^2*b^4*c^4*d^2-6*a*b^5*c^5*d+b^6*c^6)

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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(1/(b*x+a)^(11/2)/(d*x+c)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 88.2424, size = 1960, normalized size = 9.51 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(11/2)/(d*x+c)^(3/2),x, algorithm="fricas")

[Out]

-2/63*(256*b^5*d^5*x^5 + 7*b^5*c^5 - 45*a*b^4*c^4*d + 126*a^2*b^3*c^3*d^2 - 210*a^3*b^2*c^2*d^3 + 315*a^4*b*c*
d^4 + 63*a^5*d^5 + 128*(b^5*c*d^4 + 9*a*b^4*d^5)*x^4 - 32*(b^5*c^2*d^3 - 18*a*b^4*c*d^4 - 63*a^2*b^3*d^5)*x^3
+ 16*(b^5*c^3*d^2 - 9*a*b^4*c^2*d^3 + 63*a^2*b^3*c*d^4 + 105*a^3*b^2*d^5)*x^2 - 2*(5*b^5*c^4*d - 36*a*b^4*c^3*
d^2 + 126*a^2*b^3*c^2*d^3 - 420*a^3*b^2*c*d^4 - 315*a^4*b*d^5)*x)*sqrt(b*x + a)*sqrt(d*x + c)/(a^5*b^6*c^7 - 6
*a^6*b^5*c^6*d + 15*a^7*b^4*c^5*d^2 - 20*a^8*b^3*c^4*d^3 + 15*a^9*b^2*c^3*d^4 - 6*a^10*b*c^2*d^5 + a^11*c*d^6
+ (b^11*c^6*d - 6*a*b^10*c^5*d^2 + 15*a^2*b^9*c^4*d^3 - 20*a^3*b^8*c^3*d^4 + 15*a^4*b^7*c^2*d^5 - 6*a^5*b^6*c*
d^6 + a^6*b^5*d^7)*x^6 + (b^11*c^7 - a*b^10*c^6*d - 15*a^2*b^9*c^5*d^2 + 55*a^3*b^8*c^4*d^3 - 85*a^4*b^7*c^3*d
^4 + 69*a^5*b^6*c^2*d^5 - 29*a^6*b^5*c*d^6 + 5*a^7*b^4*d^7)*x^5 + 5*(a*b^10*c^7 - 4*a^2*b^9*c^6*d + 3*a^3*b^8*
c^5*d^2 + 10*a^4*b^7*c^4*d^3 - 25*a^5*b^6*c^3*d^4 + 24*a^6*b^5*c^2*d^5 - 11*a^7*b^4*c*d^6 + 2*a^8*b^3*d^7)*x^4
+ 10*(a^2*b^9*c^7 - 5*a^3*b^8*c^6*d + 9*a^4*b^7*c^5*d^2 - 5*a^5*b^6*c^4*d^3 - 5*a^6*b^5*c^3*d^4 + 9*a^7*b^4*c
^2*d^5 - 5*a^8*b^3*c*d^6 + a^9*b^2*d^7)*x^3 + 5*(2*a^3*b^8*c^7 - 11*a^4*b^7*c^6*d + 24*a^5*b^6*c^5*d^2 - 25*a^
6*b^5*c^4*d^3 + 10*a^7*b^4*c^3*d^4 + 3*a^8*b^3*c^2*d^5 - 4*a^9*b^2*c*d^6 + a^10*b*d^7)*x^2 + (5*a^4*b^7*c^7 -
29*a^5*b^6*c^6*d + 69*a^6*b^5*c^5*d^2 - 85*a^7*b^4*c^4*d^3 + 55*a^8*b^3*c^3*d^4 - 15*a^9*b^2*c^2*d^5 - a^10*b*
c*d^6 + a^11*d^7)*x)

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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(1/(b*x+a)**(11/2)/(d*x+c)**(3/2),x)

[Out]

Timed out

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Giac [B]  time = 4.49973, size = 3291, normalized size = 15.98 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(11/2)/(d*x+c)^(3/2),x, algorithm="giac")

[Out]

-2*sqrt(b*x + a)*b^2*d^5/((b^6*c^6*abs(b) - 6*a*b^5*c^5*d*abs(b) + 15*a^2*b^4*c^4*d^2*abs(b) - 20*a^3*b^3*c^3*
d^3*abs(b) + 15*a^4*b^2*c^2*d^4*abs(b) - 6*a^5*b*c*d^5*abs(b) + a^6*d^6*abs(b))*sqrt(b^2*c + (b*x + a)*b*d - a
*b*d)) - 4/63*(193*sqrt(b*d)*b^18*c^8*d^4 - 1544*sqrt(b*d)*a*b^17*c^7*d^5 + 5404*sqrt(b*d)*a^2*b^16*c^6*d^6 -
10808*sqrt(b*d)*a^3*b^15*c^5*d^7 + 13510*sqrt(b*d)*a^4*b^14*c^4*d^8 - 10808*sqrt(b*d)*a^5*b^13*c^3*d^9 + 5404*
sqrt(b*d)*a^6*b^12*c^2*d^10 - 1544*sqrt(b*d)*a^7*b^11*c*d^11 + 193*sqrt(b*d)*a^8*b^10*d^12 - 1674*sqrt(b*d)*(s
qrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^16*c^7*d^4 + 11718*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^15*c^6*d^5 - 35154*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^14*c^5*d^6 + 58590*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^13*c^4*d^7 - 58590*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^12*c^3*d^8 + 35154*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d
- a*b*d))^2*a^5*b^11*c^2*d^9 - 11718*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d)
)^2*a^6*b^10*c*d^10 + 1674*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^7*b^9
*d^11 + 6318*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^14*c^6*d^4 - 37908*
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^13*c^5*d^5 + 94770*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^12*c^4*d^6 - 126360*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*b^11*c^3*d^7 + 94770*sqrt(b*d)*(sqrt(b*d)*sqrt(b*
x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^4*b^10*c^2*d^8 - 37908*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) -
sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^5*b^9*c*d^9 + 6318*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c +
(b*x + a)*b*d - a*b*d))^4*a^6*b^8*d^10 - 13314*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d
- a*b*d))^6*b^12*c^5*d^4 + 66570*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*
a*b^11*c^4*d^5 - 133140*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^2*b^10*c
^3*d^6 + 133140*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^3*b^9*c^2*d^7 -
66570*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^4*b^8*c*d^8 + 13314*sqrt(b
*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^5*b^7*d^9 + 16128*sqrt(b*d)*(sqrt(b*d)
*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*b^10*c^4*d^4 - 64512*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a
) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a*b^9*c^3*d^5 + 96768*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2
*c + (b*x + a)*b*d - a*b*d))^8*a^2*b^8*c^2*d^6 - 64512*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x
+ a)*b*d - a*b*d))^8*a^3*b^7*c*d^7 + 16128*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a
*b*d))^8*a^4*b^6*d^8 - 8190*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*b^8*c
^3*d^4 + 24570*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a*b^7*c^2*d^5 - 24
570*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^2*b^6*c*d^6 + 8190*sqrt(b*d
)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^3*b^5*d^7 + 2898*sqrt(b*d)*(sqrt(b*d)*s
qrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*b^6*c^2*d^4 - 5796*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) -
sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*a*b^5*c*d^5 + 2898*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c +
(b*x + a)*b*d - a*b*d))^12*a^2*b^4*d^6 - 630*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d -
a*b*d))^14*b^4*c*d^4 + 630*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^14*a*b^3
*d^5 + 63*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^16*b^2*d^4)/((b^5*c^5*abs(
b) - 5*a*b^4*c^4*d*abs(b) + 10*a^2*b^3*c^3*d^2*abs(b) - 10*a^3*b^2*c^2*d^3*abs(b) + 5*a^4*b*c*d^4*abs(b) - a^5
*d^5*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)^9)