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

Optimal. Leaf size=326 \[ \frac{4 \left (-30 a^2 b^2 c+35 a^4+3 b^4 c^2\right ) \left (a+b \sqrt{c+d x}\right )^{9/2}}{9 b^8 d^4}+\frac{12 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{13/2}}{13 b^8 d^4}-\frac{20 a \left (7 a^2-3 b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{11/2}}{11 b^8 d^4}-\frac{12 a \left (7 a^2-3 b^2 c\right ) \left (a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{7/2}}{7 b^8 d^4}+\frac{4 \left (a^2-b^2 c\right )^2 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{5/2}}{5 b^8 d^4}-\frac{4 a \left (a^2-b^2 c\right )^3 \left (a+b \sqrt{c+d x}\right )^{3/2}}{3 b^8 d^4}+\frac{4 \left (a+b \sqrt{c+d x}\right )^{17/2}}{17 b^8 d^4}-\frac{28 a \left (a+b \sqrt{c+d x}\right )^{15/2}}{15 b^8 d^4} \]

[Out]

(-4*a*(a^2 - b^2*c)^3*(a + b*Sqrt[c + d*x])^(3/2))/(3*b^8*d^4) + (4*(a^2 - b^2*c)^2*(7*a^2 - b^2*c)*(a + b*Sqr
t[c + d*x])^(5/2))/(5*b^8*d^4) - (12*a*(7*a^2 - 3*b^2*c)*(a^2 - b^2*c)*(a + b*Sqrt[c + d*x])^(7/2))/(7*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])^(9/2))/(9*b^8*d^4) - (20*a*(7*a^2 - 3*b^2*c)*
(a + b*Sqrt[c + d*x])^(11/2))/(11*b^8*d^4) + (12*(7*a^2 - b^2*c)*(a + b*Sqrt[c + d*x])^(13/2))/(13*b^8*d^4) -
(28*a*(a + b*Sqrt[c + d*x])^(15/2))/(15*b^8*d^4) + (4*(a + b*Sqrt[c + d*x])^(17/2))/(17*b^8*d^4)

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Rubi [A]  time = 0.242677, antiderivative size = 326, 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} \[ \frac{4 \left (-30 a^2 b^2 c+35 a^4+3 b^4 c^2\right ) \left (a+b \sqrt{c+d x}\right )^{9/2}}{9 b^8 d^4}+\frac{12 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{13/2}}{13 b^8 d^4}-\frac{20 a \left (7 a^2-3 b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{11/2}}{11 b^8 d^4}-\frac{12 a \left (7 a^2-3 b^2 c\right ) \left (a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{7/2}}{7 b^8 d^4}+\frac{4 \left (a^2-b^2 c\right )^2 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{5/2}}{5 b^8 d^4}-\frac{4 a \left (a^2-b^2 c\right )^3 \left (a+b \sqrt{c+d x}\right )^{3/2}}{3 b^8 d^4}+\frac{4 \left (a+b \sqrt{c+d x}\right )^{17/2}}{17 b^8 d^4}-\frac{28 a \left (a+b \sqrt{c+d x}\right )^{15/2}}{15 b^8 d^4} \]

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*(a + b*Sqrt[c + d*x])^(3/2))/(3*b^8*d^4) + (4*(a^2 - b^2*c)^2*(7*a^2 - b^2*c)*(a + b*Sqr
t[c + d*x])^(5/2))/(5*b^8*d^4) - (12*a*(7*a^2 - 3*b^2*c)*(a^2 - b^2*c)*(a + b*Sqrt[c + d*x])^(7/2))/(7*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])^(9/2))/(9*b^8*d^4) - (20*a*(7*a^2 - 3*b^2*c)*
(a + b*Sqrt[c + d*x])^(11/2))/(11*b^8*d^4) + (12*(7*a^2 - b^2*c)*(a + b*Sqrt[c + d*x])^(13/2))/(13*b^8*d^4) -
(28*a*(a + b*Sqrt[c + d*x])^(15/2))/(15*b^8*d^4) + (4*(a + b*Sqrt[c + d*x])^(17/2))/(17*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 x^3 \sqrt{a+b \sqrt{c+d x}} \, dx &=\frac{\operatorname{Subst}\left (\int \sqrt{a+b \sqrt{x}} (-c+x)^3 \, dx,x,c+d x\right )}{d^4}\\ &=\frac{2 \operatorname{Subst}\left (\int x \sqrt{a+b x} \left (-c+x^2\right )^3 \, 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 \sqrt{a+b x}}{b^7}-\frac{\left (-7 a^2+b^2 c\right ) \left (-a^2+b^2 c\right )^2 (a+b x)^{3/2}}{b^7}-\frac{3 \left (7 a^5-10 a^3 b^2 c+3 a b^4 c^2\right ) (a+b x)^{5/2}}{b^7}+\frac{\left (35 a^4-30 a^2 b^2 c+3 b^4 c^2\right ) (a+b x)^{7/2}}{b^7}-\frac{5 a \left (7 a^2-3 b^2 c\right ) (a+b x)^{9/2}}{b^7}-\frac{3 \left (-7 a^2+b^2 c\right ) (a+b x)^{11/2}}{b^7}-\frac{7 a (a+b x)^{13/2}}{b^7}+\frac{(a+b x)^{15/2}}{b^7}\right ) \, dx,x,\sqrt{c+d x}\right )}{d^4}\\ &=-\frac{4 a \left (a^2-b^2 c\right )^3 \left (a+b \sqrt{c+d x}\right )^{3/2}}{3 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 )^{5/2}}{5 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 )^{7/2}}{7 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 )^{9/2}}{9 b^8 d^4}-\frac{20 a \left (7 a^2-3 b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{11/2}}{11 b^8 d^4}+\frac{12 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{13/2}}{13 b^8 d^4}-\frac{28 a \left (a+b \sqrt{c+d x}\right )^{15/2}}{15 b^8 d^4}+\frac{4 \left (a+b \sqrt{c+d x}\right )^{17/2}}{17 b^8 d^4}\\ \end{align*}

Mathematica [A]  time = 0.420491, size = 232, normalized size = 0.71 \[ \frac{4 \left (a+b \sqrt{c+d x}\right )^{3/2} \left (-48 a^3 b^4 \left (616 c^2-1080 c d x+735 d^2 x^2\right )+24 a^2 b^5 \sqrt{c+d x} \left (2960 c^2-2716 c d x+1617 d^2 x^2\right )+3840 a^5 b^2 (10 c-7 d x)-640 a^4 b^3 (104 c-49 d x) \sqrt{c+d x}+21504 a^6 b \sqrt{c+d x}-14336 a^7+6 a b^6 \left (-3936 c^2 d x+320 c^3+5754 c d^2 x^2-7007 d^3 x^3\right )-231 b^7 \sqrt{c+d x} \left (-160 c^2 d x+128 c^3+180 c d^2 x^2-195 d^3 x^3\right )\right )}{765765 b^8 d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(a + b*Sqrt[c + d*x])^(3/2)*(-14336*a^7 + 3840*a^5*b^2*(10*c - 7*d*x) + 21504*a^6*b*Sqrt[c + d*x] - 640*a^4
*b^3*(104*c - 49*d*x)*Sqrt[c + d*x] - 48*a^3*b^4*(616*c^2 - 1080*c*d*x + 735*d^2*x^2) + 24*a^2*b^5*Sqrt[c + d*
x]*(2960*c^2 - 2716*c*d*x + 1617*d^2*x^2) + 6*a*b^6*(320*c^3 - 3936*c^2*d*x + 5754*c*d^2*x^2 - 7007*d^3*x^3) -
231*b^7*Sqrt[c + d*x]*(128*c^3 - 160*c^2*d*x + 180*c*d^2*x^2 - 195*d^3*x^3)))/(765765*b^8*d^4)

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Maple [A]  time = 0.004, size = 383, normalized size = 1.2 \begin{align*} 4\,{\frac{1}{{d}^{4}{b}^{8}} \left ( 1/17\, \left ( a+b\sqrt{dx+c} \right ) ^{17/2}-{\frac{7\,a \left ( a+b\sqrt{dx+c} \right ) ^{15/2}}{15}}+1/13\, \left ( -3\,{b}^{2}c+21\,{a}^{2} \right ) \left ( a+b\sqrt{dx+c} \right ) ^{13/2}+1/11\, \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 ) ^{11/2}+1/9\, \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 ) ^{9/2}+1/7\, \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 ) ^{7/2}+1/5\, \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 ) ^{5/2}-1/3\, \left ( -{b}^{2}c+{a}^{2} \right ) ^{3}a \left ( a+b\sqrt{dx+c} \right ) ^{3/2} \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*(1/17*(a+b*(d*x+c)^(1/2))^(17/2)-7/15*a*(a+b*(d*x+c)^(1/2))^(15/2)+1/13*(-3*b^2*c+21*a^2)*(a+b*(d*x+
c)^(1/2))^(13/2)+1/11*(-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))^(11/2)+
1/9*((-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))^(9/2)+1/7*(-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))^(7/2)+1/5*((-b^2*c+a^2)^3+6*(-b^2*c+a^2)^2*a^2)*(a+b*(d*x+c)^(1/2))^(5/2)-1/3
*(-b^2*c+a^2)^3*a*(a+b*(d*x+c)^(1/2))^(3/2))

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Maxima [A]  time = 1.17555, size = 362, normalized size = 1.11 \begin{align*} \frac{4 \,{\left (45045 \,{\left (\sqrt{d x + c} b + a\right )}^{\frac{17}{2}} - 357357 \,{\left (\sqrt{d x + c} b + a\right )}^{\frac{15}{2}} a - 176715 \,{\left (b^{2} c - 7 \, a^{2}\right )}{\left (\sqrt{d x + c} b + a\right )}^{\frac{13}{2}} + 348075 \,{\left (3 \, a b^{2} c - 7 \, a^{3}\right )}{\left (\sqrt{d x + c} b + a\right )}^{\frac{11}{2}} + 85085 \,{\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{9}{2}} - 328185 \,{\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{7}{2}} - 153153 \,{\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{5}{2}} + 255255 \,{\left (a b^{6} c^{3} - 3 \, a^{3} b^{4} c^{2} + 3 \, a^{5} b^{2} c - a^{7}\right )}{\left (\sqrt{d x + c} b + a\right )}^{\frac{3}{2}}\right )}}{765765 \, 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]

4/765765*(45045*(sqrt(d*x + c)*b + a)^(17/2) - 357357*(sqrt(d*x + c)*b + a)^(15/2)*a - 176715*(b^2*c - 7*a^2)*
(sqrt(d*x + c)*b + a)^(13/2) + 348075*(3*a*b^2*c - 7*a^3)*(sqrt(d*x + c)*b + a)^(11/2) + 85085*(3*b^4*c^2 - 30
*a^2*b^2*c + 35*a^4)*(sqrt(d*x + c)*b + a)^(9/2) - 328185*(3*a*b^4*c^2 - 10*a^3*b^2*c + 7*a^5)*(sqrt(d*x + c)*
b + a)^(7/2) - 153153*(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)^(5/2) + 255255*(a
*b^6*c^3 - 3*a^3*b^4*c^2 + 3*a^5*b^2*c - a^7)*(sqrt(d*x + c)*b + a)^(3/2))/(b^8*d^4)

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Fricas [A]  time = 2.37572, size = 697, normalized size = 2.14 \begin{align*} \frac{4 \,{\left (45045 \, b^{8} d^{4} x^{4} - 29568 \, b^{8} c^{4} + 72960 \, a^{2} b^{6} c^{3} - 96128 \, a^{4} b^{4} c^{2} + 59904 \, a^{6} b^{2} c - 14336 \, a^{8} + 231 \,{\left (15 \, b^{8} c - 14 \, a^{2} b^{6}\right )} d^{3} x^{3} - 28 \,{\left (165 \, b^{8} c^{2} - 291 \, a^{2} b^{6} c + 140 \, a^{4} b^{4}\right )} d^{2} x^{2} + 32 \,{\left (231 \, b^{8} c^{3} - 555 \, a^{2} b^{6} c^{2} + 520 \, a^{4} b^{4} c - 168 \, a^{6} b^{2}\right )} d x +{\left (3003 \, a b^{7} d^{3} x^{3} - 27648 \, a b^{7} c^{3} + 41472 \, a^{3} b^{5} c^{2} - 28160 \, a^{5} b^{3} c + 7168 \, a^{7} b - 3528 \,{\left (2 \, a b^{7} c - a^{3} b^{5}\right )} d^{2} x^{2} + 32 \,{\left (417 \, a b^{7} c^{2} - 417 \, a^{3} b^{5} c + 140 \, a^{5} b^{3}\right )} d x\right )} \sqrt{d x + c}\right )} \sqrt{\sqrt{d x + c} b + a}}{765765 \, 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]

4/765765*(45045*b^8*d^4*x^4 - 29568*b^8*c^4 + 72960*a^2*b^6*c^3 - 96128*a^4*b^4*c^2 + 59904*a^6*b^2*c - 14336*
a^8 + 231*(15*b^8*c - 14*a^2*b^6)*d^3*x^3 - 28*(165*b^8*c^2 - 291*a^2*b^6*c + 140*a^4*b^4)*d^2*x^2 + 32*(231*b
^8*c^3 - 555*a^2*b^6*c^2 + 520*a^4*b^4*c - 168*a^6*b^2)*d*x + (3003*a*b^7*d^3*x^3 - 27648*a*b^7*c^3 + 41472*a^
3*b^5*c^2 - 28160*a^5*b^3*c + 7168*a^7*b - 3528*(2*a*b^7*c - a^3*b^5)*d^2*x^2 + 32*(417*a*b^7*c^2 - 417*a^3*b^
5*c + 140*a^5*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 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 = 1.34757, size = 1548, normalized size = 4.75 \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]

-4/765765*(153153*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^2*b^8*c^3*sgn((sqrt(d*x + c)*b + a)*b
- a*b) - 255255*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)*a*b^8*c^3*sgn((sqrt(d*x + c)*b + a)*b -
a*b) - 255255*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^4*b^6*c^2*sgn((sqrt(d*x + c)*b + a)*b - a*
b) + 984555*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^3*a*b^6*c^2*sgn((sqrt(d*x + c)*b + a)*b - a*
b) - 1378377*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^2*a^2*b^6*c^2*sgn((sqrt(d*x + c)*b + a)*b -
a*b) + 765765*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)*a^3*b^6*c^2*sgn((sqrt(d*x + c)*b + a)*b -
a*b) + 176715*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^6*b^4*c*sgn((sqrt(d*x + c)*b + a)*b - a*b
) - 1044225*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^5*a*b^4*c*sgn((sqrt(d*x + c)*b + a)*b - a*b)
+ 2552550*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^4*a^2*b^4*c*sgn((sqrt(d*x + c)*b + a)*b - a*b
) - 3281850*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^3*a^3*b^4*c*sgn((sqrt(d*x + c)*b + a)*b - a*
b) + 2297295*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^2*a^4*b^4*c*sgn((sqrt(d*x + c)*b + a)*b - a
*b) - 765765*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)*a^5*b^4*c*sgn((sqrt(d*x + c)*b + a)*b - a*b
) - 45045*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^8*b^2*sgn((sqrt(d*x + c)*b + a)*b - a*b) + 357
357*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^7*a*b^2*sgn((sqrt(d*x + c)*b + a)*b - a*b) - 1237005
*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^6*a^2*b^2*sgn((sqrt(d*x + c)*b + a)*b - a*b) + 2436525*
sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^5*a^3*b^2*sgn((sqrt(d*x + c)*b + a)*b - a*b) - 2977975*s
qrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^4*a^4*b^2*sgn((sqrt(d*x + c)*b + a)*b - a*b) + 2297295*sq
rt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^3*a^5*b^2*sgn((sqrt(d*x + c)*b + a)*b - a*b) - 1072071*sqr
t((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^2*a^6*b^2*sgn((sqrt(d*x + c)*b + a)*b - a*b) + 255255*sqrt(
(sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)*a^7*b^2*sgn((sqrt(d*x + c)*b + a)*b - a*b))*abs(b)/(b^12*d^4)