3.655 \(\int x (a+b \sqrt{c+d x})^p \, dx\)
Optimal. Leaf size=145 \[ -\frac{2 a \left (a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{p+1}}{b^4 d^2 (p+1)}+\frac{2 \left (3 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{p+2}}{b^4 d^2 (p+2)}-\frac{6 a \left (a+b \sqrt{c+d x}\right )^{p+3}}{b^4 d^2 (p+3)}+\frac{2 \left (a+b \sqrt{c+d x}\right )^{p+4}}{b^4 d^2 (p+4)} \]
[Out]
(-2*a*(a^2 - b^2*c)*(a + b*Sqrt[c + d*x])^(1 + p))/(b^4*d^2*(1 + p)) + (2*(3*a^2 - b^2*c)*(a + b*Sqrt[c + d*x]
)^(2 + p))/(b^4*d^2*(2 + p)) - (6*a*(a + b*Sqrt[c + d*x])^(3 + p))/(b^4*d^2*(3 + p)) + (2*(a + b*Sqrt[c + d*x]
)^(4 + p))/(b^4*d^2*(4 + p))
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Rubi [A] time = 0.108389, antiderivative size = 145, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used =
{371, 1398, 772} \[ -\frac{2 a \left (a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{p+1}}{b^4 d^2 (p+1)}+\frac{2 \left (3 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{p+2}}{b^4 d^2 (p+2)}-\frac{6 a \left (a+b \sqrt{c+d x}\right )^{p+3}}{b^4 d^2 (p+3)}+\frac{2 \left (a+b \sqrt{c+d x}\right )^{p+4}}{b^4 d^2 (p+4)} \]
Antiderivative was successfully verified.
[In]
Int[x*(a + b*Sqrt[c + d*x])^p,x]
[Out]
(-2*a*(a^2 - b^2*c)*(a + b*Sqrt[c + d*x])^(1 + p))/(b^4*d^2*(1 + p)) + (2*(3*a^2 - b^2*c)*(a + b*Sqrt[c + d*x]
)^(2 + p))/(b^4*d^2*(2 + p)) - (6*a*(a + b*Sqrt[c + d*x])^(3 + p))/(b^4*d^2*(3 + p)) + (2*(a + b*Sqrt[c + d*x]
)^(4 + p))/(b^4*d^2*(4 + p))
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 \left (a+b \sqrt{c+d x}\right )^p \, dx &=\frac{\operatorname{Subst}\left (\int \left (a+b \sqrt{x}\right )^p (-c+x) \, dx,x,c+d x\right )}{d^2}\\ &=\frac{2 \operatorname{Subst}\left (\int x (a+b x)^p \left (-c+x^2\right ) \, dx,x,\sqrt{c+d x}\right )}{d^2}\\ &=\frac{2 \operatorname{Subst}\left (\int \left (\frac{\left (-a^3+a b^2 c\right ) (a+b x)^p}{b^3}+\frac{\left (3 a^2-b^2 c\right ) (a+b x)^{1+p}}{b^3}-\frac{3 a (a+b x)^{2+p}}{b^3}+\frac{(a+b x)^{3+p}}{b^3}\right ) \, dx,x,\sqrt{c+d x}\right )}{d^2}\\ &=-\frac{2 a \left (a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{1+p}}{b^4 d^2 (1+p)}+\frac{2 \left (3 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{2+p}}{b^4 d^2 (2+p)}-\frac{6 a \left (a+b \sqrt{c+d x}\right )^{3+p}}{b^4 d^2 (3+p)}+\frac{2 \left (a+b \sqrt{c+d x}\right )^{4+p}}{b^4 d^2 (4+p)}\\ \end{align*}
Mathematica [A] time = 0.168909, size = 128, normalized size = 0.88 \[ -\frac{2 \left (a+b \sqrt{c+d x}\right )^{p+1} \left (-6 a^2 b (p+1) \sqrt{c+d x}+6 a^3+a b^2 \left (2 c \left (p^2+p-3\right )+3 d \left (p^2+3 p+2\right ) x\right )-b^3 \left (p^2+4 p+3\right ) \sqrt{c+d x} (d (p+2) x-2 c)\right )}{b^4 d^2 (p+1) (p+2) (p+3) (p+4)} \]
Antiderivative was successfully verified.
[In]
Integrate[x*(a + b*Sqrt[c + d*x])^p,x]
[Out]
(-2*(a + b*Sqrt[c + d*x])^(1 + p)*(6*a^3 - 6*a^2*b*(1 + p)*Sqrt[c + d*x] - b^3*(3 + 4*p + p^2)*Sqrt[c + d*x]*(
-2*c + d*(2 + p)*x) + a*b^2*(2*c*(-3 + p + p^2) + 3*d*(2 + 3*p + p^2)*x)))/(b^4*d^2*(1 + p)*(2 + p)*(3 + p)*(4
+ p))
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Maple [F] time = 0.004, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b\sqrt{dx+c} \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x*(a+b*(d*x+c)^(1/2))^p,x)
[Out]
int(x*(a+b*(d*x+c)^(1/2))^p,x)
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Maxima [A] time = 1.14836, size = 252, normalized size = 1.74 \begin{align*} -\frac{2 \,{\left (\frac{{\left ({\left (d x + c\right )} b^{2}{\left (p + 1\right )} + \sqrt{d x + c} a b p - a^{2}\right )}{\left (\sqrt{d x + c} b + a\right )}^{p} c}{{\left (p^{2} + 3 \, p + 2\right )} b^{2}} - \frac{{\left ({\left (p^{3} + 6 \, p^{2} + 11 \, p + 6\right )}{\left (d x + c\right )}^{2} b^{4} +{\left (p^{3} + 3 \, p^{2} + 2 \, p\right )}{\left (d x + c\right )}^{\frac{3}{2}} a b^{3} - 3 \,{\left (p^{2} + p\right )}{\left (d x + c\right )} a^{2} b^{2} + 6 \, \sqrt{d x + c} a^{3} b p - 6 \, a^{4}\right )}{\left (\sqrt{d x + c} b + a\right )}^{p}}{{\left (p^{4} + 10 \, p^{3} + 35 \, p^{2} + 50 \, p + 24\right )} b^{4}}\right )}}{d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(a+b*(d*x+c)^(1/2))^p,x, algorithm="maxima")
[Out]
-2*(((d*x + c)*b^2*(p + 1) + sqrt(d*x + c)*a*b*p - a^2)*(sqrt(d*x + c)*b + a)^p*c/((p^2 + 3*p + 2)*b^2) - ((p^
3 + 6*p^2 + 11*p + 6)*(d*x + c)^2*b^4 + (p^3 + 3*p^2 + 2*p)*(d*x + c)^(3/2)*a*b^3 - 3*(p^2 + p)*(d*x + c)*a^2*
b^2 + 6*sqrt(d*x + c)*a^3*b*p - 6*a^4)*(sqrt(d*x + c)*b + a)^p/((p^4 + 10*p^3 + 35*p^2 + 50*p + 24)*b^4))/d^2
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Fricas [B] time = 2.09734, size = 599, normalized size = 4.13 \begin{align*} -\frac{2 \,{\left (6 \, b^{4} c^{2} - 12 \, a^{2} b^{2} c + 6 \, a^{4} + 2 \,{\left (b^{4} c^{2} + a^{2} b^{2} c\right )} p^{2} -{\left (b^{4} d^{2} p^{3} + 6 \, b^{4} d^{2} p^{2} + 11 \, b^{4} d^{2} p + 6 \, b^{4} d^{2}\right )} x^{2} + 4 \,{\left (2 \, b^{4} c^{2} - a^{2} b^{2} c\right )} p -{\left (b^{4} c d p^{3} +{\left (4 \, b^{4} c - 3 \, a^{2} b^{2}\right )} d p^{2} + 3 \,{\left (b^{4} c - a^{2} b^{2}\right )} d p\right )} x +{\left (4 \, a b^{3} c p^{2} + 2 \,{\left (5 \, a b^{3} c - 3 \, a^{3} b\right )} p -{\left (a b^{3} d p^{3} + 3 \, a b^{3} d p^{2} + 2 \, a b^{3} d p\right )} x\right )} \sqrt{d x + c}\right )}{\left (\sqrt{d x + c} b + a\right )}^{p}}{b^{4} d^{2} p^{4} + 10 \, b^{4} d^{2} p^{3} + 35 \, b^{4} d^{2} p^{2} + 50 \, b^{4} d^{2} p + 24 \, b^{4} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(a+b*(d*x+c)^(1/2))^p,x, algorithm="fricas")
[Out]
-2*(6*b^4*c^2 - 12*a^2*b^2*c + 6*a^4 + 2*(b^4*c^2 + a^2*b^2*c)*p^2 - (b^4*d^2*p^3 + 6*b^4*d^2*p^2 + 11*b^4*d^2
*p + 6*b^4*d^2)*x^2 + 4*(2*b^4*c^2 - a^2*b^2*c)*p - (b^4*c*d*p^3 + (4*b^4*c - 3*a^2*b^2)*d*p^2 + 3*(b^4*c - a^
2*b^2)*d*p)*x + (4*a*b^3*c*p^2 + 2*(5*a*b^3*c - 3*a^3*b)*p - (a*b^3*d*p^3 + 3*a*b^3*d*p^2 + 2*a*b^3*d*p)*x)*sq
rt(d*x + c))*(sqrt(d*x + c)*b + a)^p/(b^4*d^2*p^4 + 10*b^4*d^2*p^3 + 35*b^4*d^2*p^2 + 50*b^4*d^2*p + 24*b^4*d^
2)
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \left (a + b \sqrt{c + d x}\right )^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(a+b*(d*x+c)**(1/2))**p,x)
[Out]
Integral(x*(a + b*sqrt(c + d*x))**p, x)
________________________________________________________________________________________
Giac [B] time = 2.57629, size = 5399, normalized size = 37.23 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(a+b*(d*x+c)^(1/2))^p,x, algorithm="giac")
[Out]
-2*((sqrt(d*x + c)*b + a)*((sqrt(d*x + c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b - a*b) - a*sgn((sqrt(d*x + c)*b +
a)*b - a*b) + a)^p*a*b^2*c*p^3*sgn((sqrt(d*x + c)*b + a)*b - a*b) - ((sqrt(d*x + c)*b + a)*sgn((sqrt(d*x + c)
*b + a)*b - a*b) - a*sgn((sqrt(d*x + c)*b + a)*b - a*b) + a)^p*a^2*b^2*c*p^3*sgn((sqrt(d*x + c)*b + a)*b - a*b
) + (sqrt(d*x + c)*b + a)^2*((sqrt(d*x + c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b - a*b) - a*sgn((sqrt(d*x + c)*b
+ a)*b - a*b) + a)^p*b^2*c*p^3 - 2*(sqrt(d*x + c)*b + a)*((sqrt(d*x + c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b -
a*b) - a*sgn((sqrt(d*x + c)*b + a)*b - a*b) + a)^p*a*b^2*c*p^3 + ((sqrt(d*x + c)*b + a)*sgn((sqrt(d*x + c)*b
+ a)*b - a*b) - a*sgn((sqrt(d*x + c)*b + a)*b - a*b) + a)^p*a^2*b^2*c*p^3 + 7*(sqrt(d*x + c)*b + a)*((sqrt(d*x
+ c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b - a*b) - a*sgn((sqrt(d*x + c)*b + a)*b - a*b) + a)^p*a*b^2*c*p^2*sgn(
(sqrt(d*x + c)*b + a)*b - a*b) - 7*((sqrt(d*x + c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b - a*b) - a*sgn((sqrt(d*x
+ c)*b + a)*b - a*b) + a)^p*a^2*b^2*c*p^2*sgn((sqrt(d*x + c)*b + a)*b - a*b) - (sqrt(d*x + c)*b + a)^3*((sqrt
(d*x + c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b - a*b) - a*sgn((sqrt(d*x + c)*b + a)*b - a*b) + a)^p*a*p^3*sgn((s
qrt(d*x + c)*b + a)*b - a*b) + 3*(sqrt(d*x + c)*b + a)^2*((sqrt(d*x + c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b -
a*b) - a*sgn((sqrt(d*x + c)*b + a)*b - a*b) + a)^p*a^2*p^3*sgn((sqrt(d*x + c)*b + a)*b - a*b) - 3*(sqrt(d*x +
c)*b + a)*((sqrt(d*x + c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b - a*b) - a*sgn((sqrt(d*x + c)*b + a)*b - a*b) + a
)^p*a^3*p^3*sgn((sqrt(d*x + c)*b + a)*b - a*b) + ((sqrt(d*x + c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b - a*b) - a
*sgn((sqrt(d*x + c)*b + a)*b - a*b) + a)^p*a^4*p^3*sgn((sqrt(d*x + c)*b + a)*b - a*b) + 8*(sqrt(d*x + c)*b + a
)^2*((sqrt(d*x + c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b - a*b) - a*sgn((sqrt(d*x + c)*b + a)*b - a*b) + a)^p*b^
2*c*p^2 - 16*(sqrt(d*x + c)*b + a)*((sqrt(d*x + c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b - a*b) - a*sgn((sqrt(d*x
+ c)*b + a)*b - a*b) + a)^p*a*b^2*c*p^2 + 7*((sqrt(d*x + c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b - a*b) - a*sgn
((sqrt(d*x + c)*b + a)*b - a*b) + a)^p*a^2*b^2*c*p^2 - (sqrt(d*x + c)*b + a)^4*((sqrt(d*x + c)*b + a)*sgn((sqr
t(d*x + c)*b + a)*b - a*b) - a*sgn((sqrt(d*x + c)*b + a)*b - a*b) + a)^p*p^3 + 4*(sqrt(d*x + c)*b + a)^3*((sqr
t(d*x + c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b - a*b) - a*sgn((sqrt(d*x + c)*b + a)*b - a*b) + a)^p*a*p^3 - 6*(
sqrt(d*x + c)*b + a)^2*((sqrt(d*x + c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b - a*b) - a*sgn((sqrt(d*x + c)*b + a)
*b - a*b) + a)^p*a^2*p^3 + 4*(sqrt(d*x + c)*b + a)*((sqrt(d*x + c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b - a*b) -
a*sgn((sqrt(d*x + c)*b + a)*b - a*b) + a)^p*a^3*p^3 - ((sqrt(d*x + c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b - a*
b) - a*sgn((sqrt(d*x + c)*b + a)*b - a*b) + a)^p*a^4*p^3 + 12*(sqrt(d*x + c)*b + a)*((sqrt(d*x + c)*b + a)*sgn
((sqrt(d*x + c)*b + a)*b - a*b) - a*sgn((sqrt(d*x + c)*b + a)*b - a*b) + a)^p*a*b^2*c*p*sgn((sqrt(d*x + c)*b +
a)*b - a*b) - 12*((sqrt(d*x + c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b - a*b) - a*sgn((sqrt(d*x + c)*b + a)*b -
a*b) + a)^p*a^2*b^2*c*p*sgn((sqrt(d*x + c)*b + a)*b - a*b) - 3*(sqrt(d*x + c)*b + a)^3*((sqrt(d*x + c)*b + a)*
sgn((sqrt(d*x + c)*b + a)*b - a*b) - a*sgn((sqrt(d*x + c)*b + a)*b - a*b) + a)^p*a*p^2*sgn((sqrt(d*x + c)*b +
a)*b - a*b) + 9*(sqrt(d*x + c)*b + a)^2*((sqrt(d*x + c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b - a*b) - a*sgn((sqr
t(d*x + c)*b + a)*b - a*b) + a)^p*a^2*p^2*sgn((sqrt(d*x + c)*b + a)*b - a*b) - 9*(sqrt(d*x + c)*b + a)*((sqrt(
d*x + c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b - a*b) - a*sgn((sqrt(d*x + c)*b + a)*b - a*b) + a)^p*a^3*p^2*sgn((
sqrt(d*x + c)*b + a)*b - a*b) + 3*((sqrt(d*x + c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b - a*b) - a*sgn((sqrt(d*x
+ c)*b + a)*b - a*b) + a)^p*a^4*p^2*sgn((sqrt(d*x + c)*b + a)*b - a*b) + 19*(sqrt(d*x + c)*b + a)^2*((sqrt(d*x
+ c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b - a*b) - a*sgn((sqrt(d*x + c)*b + a)*b - a*b) + a)^p*b^2*c*p - 38*(sq
rt(d*x + c)*b + a)*((sqrt(d*x + c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b - a*b) - a*sgn((sqrt(d*x + c)*b + a)*b -
a*b) + a)^p*a*b^2*c*p + 12*((sqrt(d*x + c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b - a*b) - a*sgn((sqrt(d*x + c)*b
+ a)*b - a*b) + a)^p*a^2*b^2*c*p - 6*(sqrt(d*x + c)*b + a)^4*((sqrt(d*x + c)*b + a)*sgn((sqrt(d*x + c)*b + a)
*b - a*b) - a*sgn((sqrt(d*x + c)*b + a)*b - a*b) + a)^p*p^2 + 24*(sqrt(d*x + c)*b + a)^3*((sqrt(d*x + c)*b + a
)*sgn((sqrt(d*x + c)*b + a)*b - a*b) - a*sgn((sqrt(d*x + c)*b + a)*b - a*b) + a)^p*a*p^2 - 33*(sqrt(d*x + c)*b
+ a)^2*((sqrt(d*x + c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b - a*b) - a*sgn((sqrt(d*x + c)*b + a)*b - a*b) + a)^
p*a^2*p^2 + 18*(sqrt(d*x + c)*b + a)*((sqrt(d*x + c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b - a*b) - a*sgn((sqrt(d
*x + c)*b + a)*b - a*b) + a)^p*a^3*p^2 - 3*((sqrt(d*x + c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b - a*b) - a*sgn((
sqrt(d*x + c)*b + a)*b - a*b) + a)^p*a^4*p^2 - 2*(sqrt(d*x + c)*b + a)^3*((sqrt(d*x + c)*b + a)*sgn((sqrt(d*x
+ c)*b + a)*b - a*b) - a*sgn((sqrt(d*x + c)*b + a)*b - a*b) + a)^p*a*p*sgn((sqrt(d*x + c)*b + a)*b - a*b) + 6*
(sqrt(d*x + c)*b + a)^2*((sqrt(d*x + c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b - a*b) - a*sgn((sqrt(d*x + c)*b + a
)*b - a*b) + a)^p*a^2*p*sgn((sqrt(d*x + c)*b + a)*b - a*b) - 12*(sqrt(d*x + c)*b + a)*((sqrt(d*x + c)*b + a)*s
gn((sqrt(d*x + c)*b + a)*b - a*b) - a*sgn((sqrt(d*x + c)*b + a)*b - a*b) + a)^p*a^3*p*sgn((sqrt(d*x + c)*b + a
)*b - a*b) + 8*((sqrt(d*x + c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b - a*b) - a*sgn((sqrt(d*x + c)*b + a)*b - a*b
) + a)^p*a^4*p*sgn((sqrt(d*x + c)*b + a)*b - a*b) + 12*(sqrt(d*x + c)*b + a)^2*((sqrt(d*x + c)*b + a)*sgn((sqr
t(d*x + c)*b + a)*b - a*b) - a*sgn((sqrt(d*x + c)*b + a)*b - a*b) + a)^p*b^2*c - 24*(sqrt(d*x + c)*b + a)*((sq
rt(d*x + c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b - a*b) - a*sgn((sqrt(d*x + c)*b + a)*b - a*b) + a)^p*a*b^2*c -
11*(sqrt(d*x + c)*b + a)^4*((sqrt(d*x + c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b - a*b) - a*sgn((sqrt(d*x + c)*b
+ a)*b - a*b) + a)^p*p + 44*(sqrt(d*x + c)*b + a)^3*((sqrt(d*x + c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b - a*b)
- a*sgn((sqrt(d*x + c)*b + a)*b - a*b) + a)^p*a*p - 63*(sqrt(d*x + c)*b + a)^2*((sqrt(d*x + c)*b + a)*sgn((sqr
t(d*x + c)*b + a)*b - a*b) - a*sgn((sqrt(d*x + c)*b + a)*b - a*b) + a)^p*a^2*p + 38*(sqrt(d*x + c)*b + a)*((sq
rt(d*x + c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b - a*b) - a*sgn((sqrt(d*x + c)*b + a)*b - a*b) + a)^p*a^3*p - 8*
((sqrt(d*x + c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b - a*b) - a*sgn((sqrt(d*x + c)*b + a)*b - a*b) + a)^p*a^4*p
- 6*(sqrt(d*x + c)*b + a)^4*((sqrt(d*x + c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b - a*b) - a*sgn((sqrt(d*x + c)*b
+ a)*b - a*b) + a)^p + 24*(sqrt(d*x + c)*b + a)^3*((sqrt(d*x + c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b - a*b) -
a*sgn((sqrt(d*x + c)*b + a)*b - a*b) + a)^p*a - 36*(sqrt(d*x + c)*b + a)^2*((sqrt(d*x + c)*b + a)*sgn((sqrt(d
*x + c)*b + a)*b - a*b) - a*sgn((sqrt(d*x + c)*b + a)*b - a*b) + a)^p*a^2 + 24*(sqrt(d*x + c)*b + a)*((sqrt(d*
x + c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b - a*b) - a*sgn((sqrt(d*x + c)*b + a)*b - a*b) + a)^p*a^3)/((b*p^4*sg
n((sqrt(d*x + c)*b + a)*b - a*b) + 10*b*p^3*sgn((sqrt(d*x + c)*b + a)*b - a*b) + 35*b*p^2*sgn((sqrt(d*x + c)*b
+ a)*b - a*b) + 50*b*p*sgn((sqrt(d*x + c)*b + a)*b - a*b) + 24*b*sgn((sqrt(d*x + c)*b + a)*b - a*b))*b^3*d^2)