3.646 \(\int \frac{x^3}{\sqrt{a+b \sqrt{c+d x}}} \, dx\)

Optimal. Leaf size=324 \[ \frac{4 \left (-30 a^2 b^2 c+35 a^4+3 b^4 c^2\right ) \left (a+b \sqrt{c+d x}\right )^{7/2}}{7 b^8 d^4}+\frac{12 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{11/2}}{11 b^8 d^4}-\frac{20 a \left (7 a^2-3 b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{9/2}}{9 b^8 d^4}-\frac{12 a \left (7 a^2-3 b^2 c\right ) \left (a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{5/2}}{5 b^8 d^4}+\frac{4 \left (a^2-b^2 c\right )^2 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{3/2}}{3 b^8 d^4}-\frac{4 a \left (a^2-b^2 c\right )^3 \sqrt{a+b \sqrt{c+d x}}}{b^8 d^4}+\frac{4 \left (a+b \sqrt{c+d x}\right )^{15/2}}{15 b^8 d^4}-\frac{28 a \left (a+b \sqrt{c+d x}\right )^{13/2}}{13 b^8 d^4} \]

[Out]

(-4*a*(a^2 - b^2*c)^3*Sqrt[a + b*Sqrt[c + d*x]])/(b^8*d^4) + (4*(a^2 - b^2*c)^2*(7*a^2 - b^2*c)*(a + b*Sqrt[c
+ d*x])^(3/2))/(3*b^8*d^4) - (12*a*(7*a^2 - 3*b^2*c)*(a^2 - b^2*c)*(a + b*Sqrt[c + d*x])^(5/2))/(5*b^8*d^4) +
(4*(35*a^4 - 30*a^2*b^2*c + 3*b^4*c^2)*(a + b*Sqrt[c + d*x])^(7/2))/(7*b^8*d^4) - (20*a*(7*a^2 - 3*b^2*c)*(a +
b*Sqrt[c + d*x])^(9/2))/(9*b^8*d^4) + (12*(7*a^2 - b^2*c)*(a + b*Sqrt[c + d*x])^(11/2))/(11*b^8*d^4) - (28*a*
(a + b*Sqrt[c + d*x])^(13/2))/(13*b^8*d^4) + (4*(a + b*Sqrt[c + d*x])^(15/2))/(15*b^8*d^4)

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Rubi [A]  time = 0.229957, antiderivative size = 324, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {371, 1398, 772} \[ \frac{4 \left (-30 a^2 b^2 c+35 a^4+3 b^4 c^2\right ) \left (a+b \sqrt{c+d x}\right )^{7/2}}{7 b^8 d^4}+\frac{12 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{11/2}}{11 b^8 d^4}-\frac{20 a \left (7 a^2-3 b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{9/2}}{9 b^8 d^4}-\frac{12 a \left (7 a^2-3 b^2 c\right ) \left (a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{5/2}}{5 b^8 d^4}+\frac{4 \left (a^2-b^2 c\right )^2 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{3/2}}{3 b^8 d^4}-\frac{4 a \left (a^2-b^2 c\right )^3 \sqrt{a+b \sqrt{c+d x}}}{b^8 d^4}+\frac{4 \left (a+b \sqrt{c+d x}\right )^{15/2}}{15 b^8 d^4}-\frac{28 a \left (a+b \sqrt{c+d x}\right )^{13/2}}{13 b^8 d^4} \]

Antiderivative was successfully verified.

[In]

Int[x^3/Sqrt[a + b*Sqrt[c + d*x]],x]

[Out]

(-4*a*(a^2 - b^2*c)^3*Sqrt[a + b*Sqrt[c + d*x]])/(b^8*d^4) + (4*(a^2 - b^2*c)^2*(7*a^2 - b^2*c)*(a + b*Sqrt[c
+ d*x])^(3/2))/(3*b^8*d^4) - (12*a*(7*a^2 - 3*b^2*c)*(a^2 - b^2*c)*(a + b*Sqrt[c + d*x])^(5/2))/(5*b^8*d^4) +
(4*(35*a^4 - 30*a^2*b^2*c + 3*b^4*c^2)*(a + b*Sqrt[c + d*x])^(7/2))/(7*b^8*d^4) - (20*a*(7*a^2 - 3*b^2*c)*(a +
b*Sqrt[c + d*x])^(9/2))/(9*b^8*d^4) + (12*(7*a^2 - b^2*c)*(a + b*Sqrt[c + d*x])^(11/2))/(11*b^8*d^4) - (28*a*
(a + b*Sqrt[c + d*x])^(13/2))/(13*b^8*d^4) + (4*(a + b*Sqrt[c + d*x])^(15/2))/(15*b^8*d^4)

Rule 371

Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coefficient[v, x, 0], d = Coefficient[v,
x, 1]}, Dist[1/d^(m + 1), Subst[Int[SimplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]
] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]

Rule 1398

Int[((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{g = Denominator[n]}, D
ist[g, Subst[Int[x^(g - 1)*(d + e*x^(g*n))^q*(a + c*x^(2*g*n))^p, x], x, x^(1/g)], x]] /; FreeQ[{a, c, d, e, p
, q}, x] && EqQ[n2, 2*n] && FractionQ[n]

Rule 772

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{x^3}{\sqrt{a+b \sqrt{c+d x}}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(-c+x)^3}{\sqrt{a+b \sqrt{x}}} \, dx,x,c+d x\right )}{d^4}\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{x \left (-c+x^2\right )^3}{\sqrt{a+b x}} \, dx,x,\sqrt{c+d x}\right )}{d^4}\\ &=\frac{2 \operatorname{Subst}\left (\int \left (-\frac{a \left (a^2-b^2 c\right )^3}{b^7 \sqrt{a+b x}}-\frac{\left (-7 a^2+b^2 c\right ) \left (-a^2+b^2 c\right )^2 \sqrt{a+b x}}{b^7}-\frac{3 \left (7 a^5-10 a^3 b^2 c+3 a b^4 c^2\right ) (a+b x)^{3/2}}{b^7}+\frac{\left (35 a^4-30 a^2 b^2 c+3 b^4 c^2\right ) (a+b x)^{5/2}}{b^7}-\frac{5 a \left (7 a^2-3 b^2 c\right ) (a+b x)^{7/2}}{b^7}-\frac{3 \left (-7 a^2+b^2 c\right ) (a+b x)^{9/2}}{b^7}-\frac{7 a (a+b x)^{11/2}}{b^7}+\frac{(a+b x)^{13/2}}{b^7}\right ) \, dx,x,\sqrt{c+d x}\right )}{d^4}\\ &=-\frac{4 a \left (a^2-b^2 c\right )^3 \sqrt{a+b \sqrt{c+d x}}}{b^8 d^4}+\frac{4 \left (a^2-b^2 c\right )^2 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{3/2}}{3 b^8 d^4}-\frac{12 a \left (7 a^2-3 b^2 c\right ) \left (a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{5/2}}{5 b^8 d^4}+\frac{4 \left (35 a^4-30 a^2 b^2 c+3 b^4 c^2\right ) \left (a+b \sqrt{c+d x}\right )^{7/2}}{7 b^8 d^4}-\frac{20 a \left (7 a^2-3 b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{9/2}}{9 b^8 d^4}+\frac{12 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{11/2}}{11 b^8 d^4}-\frac{28 a \left (a+b \sqrt{c+d x}\right )^{13/2}}{13 b^8 d^4}+\frac{4 \left (a+b \sqrt{c+d x}\right )^{15/2}}{15 b^8 d^4}\\ \end{align*}

Mathematica [A]  time = 0.355469, size = 232, normalized size = 0.72 \[ \frac{4 \sqrt{a+b \sqrt{c+d x}} \left (-16 a^3 b^4 \left (2936 c^2-680 c d x+245 d^2 x^2\right )+24 a^2 b^5 \sqrt{c+d x} \left (784 c^2-356 c d x+147 d^2 x^2\right )+768 a^5 b^2 (58 c-7 d x)-640 a^4 b^3 (32 c-7 d x) \sqrt{c+d x}+7168 a^6 b \sqrt{c+d x}-14336 a^7+6 a b^6 \left (-928 c^2 d x+2880 c^3+658 c d^2 x^2-539 d^3 x^3\right )-39 b^7 \sqrt{c+d x} \left (-96 c^2 d x+128 c^3+84 c d^2 x^2-77 d^3 x^3\right )\right )}{45045 b^8 d^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/Sqrt[a + b*Sqrt[c + d*x]],x]

[Out]

(4*Sqrt[a + b*Sqrt[c + d*x]]*(-14336*a^7 + 768*a^5*b^2*(58*c - 7*d*x) + 7168*a^6*b*Sqrt[c + d*x] - 640*a^4*b^3
*(32*c - 7*d*x)*Sqrt[c + d*x] + 24*a^2*b^5*Sqrt[c + d*x]*(784*c^2 - 356*c*d*x + 147*d^2*x^2) - 16*a^3*b^4*(293
6*c^2 - 680*c*d*x + 245*d^2*x^2) + 6*a*b^6*(2880*c^3 - 928*c^2*d*x + 658*c*d^2*x^2 - 539*d^3*x^3) - 39*b^7*Sqr
t[c + d*x]*(128*c^3 - 96*c^2*d*x + 84*c*d^2*x^2 - 77*d^3*x^3)))/(45045*b^8*d^4)

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Maple [A]  time = 0.003, size = 383, normalized size = 1.2 \begin{align*} 4\,{\frac{1}{{d}^{4}{b}^{8}} \left ( 1/15\, \left ( a+b\sqrt{dx+c} \right ) ^{15/2}-{\frac{7\,a \left ( a+b\sqrt{dx+c} \right ) ^{13/2}}{13}}+1/11\, \left ( -3\,{b}^{2}c+21\,{a}^{2} \right ) \left ( a+b\sqrt{dx+c} \right ) ^{11/2}+1/9\, \left ( -8\, \left ( -{b}^{2}c+{a}^{2} \right ) a-2\,a \left ( -2\,{b}^{2}c+6\,{a}^{2} \right ) - \left ( -3\,{b}^{2}c+15\,{a}^{2} \right ) a \right ) \left ( a+b\sqrt{dx+c} \right ) ^{9/2}+1/7\, \left ( \left ( -{b}^{2}c+{a}^{2} \right ) \left ( -2\,{b}^{2}c+6\,{a}^{2} \right ) +8\,{a}^{2} \left ( -{b}^{2}c+{a}^{2} \right ) + \left ( -{b}^{2}c+{a}^{2} \right ) ^{2}- \left ( -8\, \left ( -{b}^{2}c+{a}^{2} \right ) a-2\,a \left ( -2\,{b}^{2}c+6\,{a}^{2} \right ) \right ) a \right ) \left ( a+b\sqrt{dx+c} \right ) ^{7/2}+1/5\, \left ( -6\, \left ( -{b}^{2}c+{a}^{2} \right ) ^{2}a- \left ( \left ( -{b}^{2}c+{a}^{2} \right ) \left ( -2\,{b}^{2}c+6\,{a}^{2} \right ) +8\,{a}^{2} \left ( -{b}^{2}c+{a}^{2} \right ) + \left ( -{b}^{2}c+{a}^{2} \right ) ^{2} \right ) a \right ) \left ( a+b\sqrt{dx+c} \right ) ^{5/2}+1/3\, \left ( \left ( -{b}^{2}c+{a}^{2} \right ) ^{3}+6\, \left ( -{b}^{2}c+{a}^{2} \right ) ^{2}{a}^{2} \right ) \left ( a+b\sqrt{dx+c} \right ) ^{3/2}- \left ( -{b}^{2}c+{a}^{2} \right ) ^{3}a\sqrt{a+b\sqrt{dx+c}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/d^4/b^8*(1/15*(a+b*(d*x+c)^(1/2))^(15/2)-7/13*a*(a+b*(d*x+c)^(1/2))^(13/2)+1/11*(-3*b^2*c+21*a^2)*(a+b*(d*x+
c)^(1/2))^(11/2)+1/9*(-8*(-b^2*c+a^2)*a-2*a*(-2*b^2*c+6*a^2)-(-3*b^2*c+15*a^2)*a)*(a+b*(d*x+c)^(1/2))^(9/2)+1/
7*((-b^2*c+a^2)*(-2*b^2*c+6*a^2)+8*a^2*(-b^2*c+a^2)+(-b^2*c+a^2)^2-(-8*(-b^2*c+a^2)*a-2*a*(-2*b^2*c+6*a^2))*a)
*(a+b*(d*x+c)^(1/2))^(7/2)+1/5*(-6*(-b^2*c+a^2)^2*a-((-b^2*c+a^2)*(-2*b^2*c+6*a^2)+8*a^2*(-b^2*c+a^2)+(-b^2*c+
a^2)^2)*a)*(a+b*(d*x+c)^(1/2))^(5/2)+1/3*((-b^2*c+a^2)^3+6*(-b^2*c+a^2)^2*a^2)*(a+b*(d*x+c)^(1/2))^(3/2)-(-b^2
*c+a^2)^3*a*(a+b*(d*x+c)^(1/2))^(1/2))

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Maxima [A]  time = 1.02297, size = 362, normalized size = 1.12 \begin{align*} \frac{4 \,{\left (3003 \,{\left (\sqrt{d x + c} b + a\right )}^{\frac{15}{2}} - 24255 \,{\left (\sqrt{d x + c} b + a\right )}^{\frac{13}{2}} a - 12285 \,{\left (b^{2} c - 7 \, a^{2}\right )}{\left (\sqrt{d x + c} b + a\right )}^{\frac{11}{2}} + 25025 \,{\left (3 \, a b^{2} c - 7 \, a^{3}\right )}{\left (\sqrt{d x + c} b + a\right )}^{\frac{9}{2}} + 6435 \,{\left (3 \, b^{4} c^{2} - 30 \, a^{2} b^{2} c + 35 \, a^{4}\right )}{\left (\sqrt{d x + c} b + a\right )}^{\frac{7}{2}} - 27027 \,{\left (3 \, a b^{4} c^{2} - 10 \, a^{3} b^{2} c + 7 \, a^{5}\right )}{\left (\sqrt{d x + c} b + a\right )}^{\frac{5}{2}} - 15015 \,{\left (b^{6} c^{3} - 9 \, a^{2} b^{4} c^{2} + 15 \, a^{4} b^{2} c - 7 \, a^{6}\right )}{\left (\sqrt{d x + c} b + a\right )}^{\frac{3}{2}} + 45045 \,{\left (a b^{6} c^{3} - 3 \, a^{3} b^{4} c^{2} + 3 \, a^{5} b^{2} c - a^{7}\right )} \sqrt{\sqrt{d x + c} b + a}\right )}}{45045 \, b^{8} d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/45045*(3003*(sqrt(d*x + c)*b + a)^(15/2) - 24255*(sqrt(d*x + c)*b + a)^(13/2)*a - 12285*(b^2*c - 7*a^2)*(sqr
t(d*x + c)*b + a)^(11/2) + 25025*(3*a*b^2*c - 7*a^3)*(sqrt(d*x + c)*b + a)^(9/2) + 6435*(3*b^4*c^2 - 30*a^2*b^
2*c + 35*a^4)*(sqrt(d*x + c)*b + a)^(7/2) - 27027*(3*a*b^4*c^2 - 10*a^3*b^2*c + 7*a^5)*(sqrt(d*x + c)*b + a)^(
5/2) - 15015*(b^6*c^3 - 9*a^2*b^4*c^2 + 15*a^4*b^2*c - 7*a^6)*(sqrt(d*x + c)*b + a)^(3/2) + 45045*(a*b^6*c^3 -
3*a^3*b^4*c^2 + 3*a^5*b^2*c - a^7)*sqrt(sqrt(d*x + c)*b + a))/(b^8*d^4)

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Fricas [A]  time = 2.27983, size = 568, normalized size = 1.75 \begin{align*} -\frac{4 \,{\left (3234 \, a b^{6} d^{3} x^{3} - 17280 \, a b^{6} c^{3} + 46976 \, a^{3} b^{4} c^{2} - 44544 \, a^{5} b^{2} c + 14336 \, a^{7} - 28 \,{\left (141 \, a b^{6} c - 140 \, a^{3} b^{4}\right )} d^{2} x^{2} + 64 \,{\left (87 \, a b^{6} c^{2} - 170 \, a^{3} b^{4} c + 84 \, a^{5} b^{2}\right )} d x -{\left (3003 \, b^{7} d^{3} x^{3} - 4992 \, b^{7} c^{3} + 18816 \, a^{2} b^{5} c^{2} - 20480 \, a^{4} b^{3} c + 7168 \, a^{6} b - 252 \,{\left (13 \, b^{7} c - 14 \, a^{2} b^{5}\right )} d^{2} x^{2} + 32 \,{\left (117 \, b^{7} c^{2} - 267 \, a^{2} b^{5} c + 140 \, a^{4} b^{3}\right )} d x\right )} \sqrt{d x + c}\right )} \sqrt{\sqrt{d x + c} b + a}}{45045 \, b^{8} d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/45045*(3234*a*b^6*d^3*x^3 - 17280*a*b^6*c^3 + 46976*a^3*b^4*c^2 - 44544*a^5*b^2*c + 14336*a^7 - 28*(141*a*b
^6*c - 140*a^3*b^4)*d^2*x^2 + 64*(87*a*b^6*c^2 - 170*a^3*b^4*c + 84*a^5*b^2)*d*x - (3003*b^7*d^3*x^3 - 4992*b^
7*c^3 + 18816*a^2*b^5*c^2 - 20480*a^4*b^3*c + 7168*a^6*b - 252*(13*b^7*c - 14*a^2*b^5)*d^2*x^2 + 32*(117*b^7*c
^2 - 267*a^2*b^5*c + 140*a^4*b^3)*d*x)*sqrt(d*x + c))*sqrt(sqrt(d*x + c)*b + a)/(b^8*d^4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\sqrt{a + b \sqrt{c + d x}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3/sqrt(a + b*sqrt(c + d*x)), x)

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Giac [B]  time = 1.33864, size = 1449, normalized size = 4.47 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/45045*(15015*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)*b^6*c^3*sgn((sqrt(d*x + c)*b + a)*b - a*
b) - 45045*sqrt((sqrt(d*x + c)*b + a)*b^2)*a*b^6*c^3*sgn((sqrt(d*x + c)*b + a)*b - a*b) - 19305*sqrt((sqrt(d*x
+ c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^3*b^4*c^2*sgn((sqrt(d*x + c)*b + a)*b - a*b) + 81081*sqrt((sqrt(d*x +
c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^2*a*b^4*c^2*sgn((sqrt(d*x + c)*b + a)*b - a*b) - 135135*sqrt((sqrt(d*x +
c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)*a^2*b^4*c^2*sgn((sqrt(d*x + c)*b + a)*b - a*b) + 135135*sqrt((sqrt(d*x +
c)*b + a)*b^2)*a^3*b^4*c^2*sgn((sqrt(d*x + c)*b + a)*b - a*b) + 12285*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*
x + c)*b + a)^5*b^2*c*sgn((sqrt(d*x + c)*b + a)*b - a*b) - 75075*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c
)*b + a)^4*a*b^2*c*sgn((sqrt(d*x + c)*b + a)*b - a*b) + 193050*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*
b + a)^3*a^2*b^2*c*sgn((sqrt(d*x + c)*b + a)*b - a*b) - 270270*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*
b + a)^2*a^3*b^2*c*sgn((sqrt(d*x + c)*b + a)*b - a*b) + 225225*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*
b + a)*a^4*b^2*c*sgn((sqrt(d*x + c)*b + a)*b - a*b) - 135135*sqrt((sqrt(d*x + c)*b + a)*b^2)*a^5*b^2*c*sgn((sq
rt(d*x + c)*b + a)*b - a*b) - 3003*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^7*sgn((sqrt(d*x + c)*
b + a)*b - a*b) + 24255*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^6*a*sgn((sqrt(d*x + c)*b + a)*b
- a*b) - 85995*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^5*a^2*sgn((sqrt(d*x + c)*b + a)*b - a*b)
+ 175175*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^4*a^3*sgn((sqrt(d*x + c)*b + a)*b - a*b) - 2252
25*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^3*a^4*sgn((sqrt(d*x + c)*b + a)*b - a*b) + 189189*sqr
t((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^2*a^5*sgn((sqrt(d*x + c)*b + a)*b - a*b) - 105105*sqrt((sqr
t(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)*a^6*sgn((sqrt(d*x + c)*b + a)*b - a*b) + 45045*sqrt((sqrt(d*x + c
)*b + a)*b^2)*a^7*sgn((sqrt(d*x + c)*b + a)*b - a*b))/(b^8*d^4*abs(b))