3.1968 \(\int (a+b x) (d+e x)^7 (a^2+2 a b x+b^2 x^2)^{3/2} \, dx\)

Optimal. Leaf size=254 \[ \frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{12}}{12 e^5 (a+b x)}-\frac{4 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11} (b d-a e)}{11 e^5 (a+b x)}+\frac{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{10} (b d-a e)^2}{5 e^5 (a+b x)}-\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^9 (b d-a e)^3}{9 e^5 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^8 (b d-a e)^4}{8 e^5 (a+b x)} \]

[Out]

((b*d - a*e)^4*(d + e*x)^8*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*e^5*(a + b*x)) - (4*b*(b*d - a*e)^3*(d + e*x)^9*S
qrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^5*(a + b*x)) + (3*b^2*(b*d - a*e)^2*(d + e*x)^10*Sqrt[a^2 + 2*a*b*x + b^2*x
^2])/(5*e^5*(a + b*x)) - (4*b^3*(b*d - a*e)*(d + e*x)^11*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^5*(a + b*x)) + (
b^4*(d + e*x)^12*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(12*e^5*(a + b*x))

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Rubi [A]  time = 0.3519, antiderivative size = 254, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {770, 21, 43} \[ \frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{12}}{12 e^5 (a+b x)}-\frac{4 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11} (b d-a e)}{11 e^5 (a+b x)}+\frac{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{10} (b d-a e)^2}{5 e^5 (a+b x)}-\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^9 (b d-a e)^3}{9 e^5 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^8 (b d-a e)^4}{8 e^5 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(d + e*x)^7*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

((b*d - a*e)^4*(d + e*x)^8*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*e^5*(a + b*x)) - (4*b*(b*d - a*e)^3*(d + e*x)^9*S
qrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^5*(a + b*x)) + (3*b^2*(b*d - a*e)^2*(d + e*x)^10*Sqrt[a^2 + 2*a*b*x + b^2*x
^2])/(5*e^5*(a + b*x)) - (4*b^3*(b*d - a*e)*(d + e*x)^11*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^5*(a + b*x)) + (
b^4*(d + e*x)^12*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(12*e^5*(a + b*x))

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int (a+b x) (d+e x)^7 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int (a+b x) \left (a b+b^2 x\right )^3 (d+e x)^7 \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac{\left (b \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int (a+b x)^4 (d+e x)^7 \, dx}{a b+b^2 x}\\ &=\frac{\left (b \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac{(-b d+a e)^4 (d+e x)^7}{e^4}-\frac{4 b (b d-a e)^3 (d+e x)^8}{e^4}+\frac{6 b^2 (b d-a e)^2 (d+e x)^9}{e^4}-\frac{4 b^3 (b d-a e) (d+e x)^{10}}{e^4}+\frac{b^4 (d+e x)^{11}}{e^4}\right ) \, dx}{a b+b^2 x}\\ &=\frac{(b d-a e)^4 (d+e x)^8 \sqrt{a^2+2 a b x+b^2 x^2}}{8 e^5 (a+b x)}-\frac{4 b (b d-a e)^3 (d+e x)^9 \sqrt{a^2+2 a b x+b^2 x^2}}{9 e^5 (a+b x)}+\frac{3 b^2 (b d-a e)^2 (d+e x)^{10} \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^5 (a+b x)}-\frac{4 b^3 (b d-a e) (d+e x)^{11} \sqrt{a^2+2 a b x+b^2 x^2}}{11 e^5 (a+b x)}+\frac{b^4 (d+e x)^{12} \sqrt{a^2+2 a b x+b^2 x^2}}{12 e^5 (a+b x)}\\ \end{align*}

Mathematica [A]  time = 0.139763, size = 432, normalized size = 1.7 \[ \frac{x \sqrt{(a+b x)^2} \left (66 a^2 b^2 x^2 \left (1512 d^5 e^2 x^2+2100 d^4 e^3 x^3+1800 d^3 e^4 x^4+945 d^2 e^5 x^5+630 d^6 e x+120 d^7+280 d e^6 x^6+36 e^7 x^7\right )+220 a^3 b x \left (378 d^5 e^2 x^2+504 d^4 e^3 x^3+420 d^3 e^4 x^4+216 d^2 e^5 x^5+168 d^6 e x+36 d^7+63 d e^6 x^6+8 e^7 x^7\right )+495 a^4 \left (56 d^5 e^2 x^2+70 d^4 e^3 x^3+56 d^3 e^4 x^4+28 d^2 e^5 x^5+28 d^6 e x+8 d^7+8 d e^6 x^6+e^7 x^7\right )+12 a b^3 x^3 \left (4620 d^5 e^2 x^2+6600 d^4 e^3 x^3+5775 d^3 e^4 x^4+3080 d^2 e^5 x^5+1848 d^6 e x+330 d^7+924 d e^6 x^6+120 e^7 x^7\right )+b^4 x^4 \left (11880 d^5 e^2 x^2+17325 d^4 e^3 x^3+15400 d^3 e^4 x^4+8316 d^2 e^5 x^5+4620 d^6 e x+792 d^7+2520 d e^6 x^6+330 e^7 x^7\right )\right )}{3960 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*(d + e*x)^7*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(x*Sqrt[(a + b*x)^2]*(495*a^4*(8*d^7 + 28*d^6*e*x + 56*d^5*e^2*x^2 + 70*d^4*e^3*x^3 + 56*d^3*e^4*x^4 + 28*d^2*
e^5*x^5 + 8*d*e^6*x^6 + e^7*x^7) + 220*a^3*b*x*(36*d^7 + 168*d^6*e*x + 378*d^5*e^2*x^2 + 504*d^4*e^3*x^3 + 420
*d^3*e^4*x^4 + 216*d^2*e^5*x^5 + 63*d*e^6*x^6 + 8*e^7*x^7) + 66*a^2*b^2*x^2*(120*d^7 + 630*d^6*e*x + 1512*d^5*
e^2*x^2 + 2100*d^4*e^3*x^3 + 1800*d^3*e^4*x^4 + 945*d^2*e^5*x^5 + 280*d*e^6*x^6 + 36*e^7*x^7) + 12*a*b^3*x^3*(
330*d^7 + 1848*d^6*e*x + 4620*d^5*e^2*x^2 + 6600*d^4*e^3*x^3 + 5775*d^3*e^4*x^4 + 3080*d^2*e^5*x^5 + 924*d*e^6
*x^6 + 120*e^7*x^7) + b^4*x^4*(792*d^7 + 4620*d^6*e*x + 11880*d^5*e^2*x^2 + 17325*d^4*e^3*x^3 + 15400*d^3*e^4*
x^4 + 8316*d^2*e^5*x^5 + 2520*d*e^6*x^6 + 330*e^7*x^7)))/(3960*(a + b*x))

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Maple [B]  time = 0.007, size = 564, normalized size = 2.2 \begin{align*}{\frac{x \left ( 330\,{b}^{4}{e}^{7}{x}^{11}+1440\,{x}^{10}a{b}^{3}{e}^{7}+2520\,{x}^{10}{b}^{4}d{e}^{6}+2376\,{x}^{9}{a}^{2}{b}^{2}{e}^{7}+11088\,{x}^{9}a{b}^{3}d{e}^{6}+8316\,{x}^{9}{b}^{4}{d}^{2}{e}^{5}+1760\,{x}^{8}{a}^{3}b{e}^{7}+18480\,{x}^{8}{a}^{2}{b}^{2}d{e}^{6}+36960\,{x}^{8}a{b}^{3}{d}^{2}{e}^{5}+15400\,{x}^{8}{b}^{4}{d}^{3}{e}^{4}+495\,{x}^{7}{a}^{4}{e}^{7}+13860\,{x}^{7}{a}^{3}bd{e}^{6}+62370\,{x}^{7}{a}^{2}{b}^{2}{d}^{2}{e}^{5}+69300\,{x}^{7}a{b}^{3}{d}^{3}{e}^{4}+17325\,{x}^{7}{b}^{4}{d}^{4}{e}^{3}+3960\,{a}^{4}d{e}^{6}{x}^{6}+47520\,{a}^{3}b{d}^{2}{e}^{5}{x}^{6}+118800\,{a}^{2}{b}^{2}{d}^{3}{e}^{4}{x}^{6}+79200\,a{b}^{3}{d}^{4}{e}^{3}{x}^{6}+11880\,{b}^{4}{d}^{5}{e}^{2}{x}^{6}+13860\,{x}^{5}{a}^{4}{d}^{2}{e}^{5}+92400\,{x}^{5}{a}^{3}b{d}^{3}{e}^{4}+138600\,{x}^{5}{a}^{2}{b}^{2}{d}^{4}{e}^{3}+55440\,{x}^{5}a{b}^{3}{d}^{5}{e}^{2}+4620\,{x}^{5}{b}^{4}{d}^{6}e+27720\,{x}^{4}{a}^{4}{d}^{3}{e}^{4}+110880\,{x}^{4}{a}^{3}b{d}^{4}{e}^{3}+99792\,{x}^{4}{a}^{2}{b}^{2}{d}^{5}{e}^{2}+22176\,{x}^{4}a{b}^{3}{d}^{6}e+792\,{x}^{4}{b}^{4}{d}^{7}+34650\,{x}^{3}{a}^{4}{d}^{4}{e}^{3}+83160\,{x}^{3}{a}^{3}b{d}^{5}{e}^{2}+41580\,{x}^{3}{a}^{2}{b}^{2}{d}^{6}e+3960\,{x}^{3}a{b}^{3}{d}^{7}+27720\,{x}^{2}{a}^{4}{d}^{5}{e}^{2}+36960\,{x}^{2}{a}^{3}b{d}^{6}e+7920\,{x}^{2}{a}^{2}{b}^{2}{d}^{7}+13860\,x{a}^{4}{d}^{6}e+7920\,x{a}^{3}b{d}^{7}+3960\,{a}^{4}{d}^{7} \right ) }{3960\, \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(e*x+d)^7*(b^2*x^2+2*a*b*x+a^2)^(3/2),x)

[Out]

1/3960*x*(330*b^4*e^7*x^11+1440*a*b^3*e^7*x^10+2520*b^4*d*e^6*x^10+2376*a^2*b^2*e^7*x^9+11088*a*b^3*d*e^6*x^9+
8316*b^4*d^2*e^5*x^9+1760*a^3*b*e^7*x^8+18480*a^2*b^2*d*e^6*x^8+36960*a*b^3*d^2*e^5*x^8+15400*b^4*d^3*e^4*x^8+
495*a^4*e^7*x^7+13860*a^3*b*d*e^6*x^7+62370*a^2*b^2*d^2*e^5*x^7+69300*a*b^3*d^3*e^4*x^7+17325*b^4*d^4*e^3*x^7+
3960*a^4*d*e^6*x^6+47520*a^3*b*d^2*e^5*x^6+118800*a^2*b^2*d^3*e^4*x^6+79200*a*b^3*d^4*e^3*x^6+11880*b^4*d^5*e^
2*x^6+13860*a^4*d^2*e^5*x^5+92400*a^3*b*d^3*e^4*x^5+138600*a^2*b^2*d^4*e^3*x^5+55440*a*b^3*d^5*e^2*x^5+4620*b^
4*d^6*e*x^5+27720*a^4*d^3*e^4*x^4+110880*a^3*b*d^4*e^3*x^4+99792*a^2*b^2*d^5*e^2*x^4+22176*a*b^3*d^6*e*x^4+792
*b^4*d^7*x^4+34650*a^4*d^4*e^3*x^3+83160*a^3*b*d^5*e^2*x^3+41580*a^2*b^2*d^6*e*x^3+3960*a*b^3*d^7*x^3+27720*a^
4*d^5*e^2*x^2+36960*a^3*b*d^6*e*x^2+7920*a^2*b^2*d^7*x^2+13860*a^4*d^6*e*x+7920*a^3*b*d^7*x+3960*a^4*d^7)*((b*
x+a)^2)^(3/2)/(b*x+a)^3

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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((b*x+a)*(e*x+d)^7*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.59523, size = 1041, normalized size = 4.1 \begin{align*} \frac{1}{12} \, b^{4} e^{7} x^{12} + a^{4} d^{7} x + \frac{1}{11} \,{\left (7 \, b^{4} d e^{6} + 4 \, a b^{3} e^{7}\right )} x^{11} + \frac{1}{10} \,{\left (21 \, b^{4} d^{2} e^{5} + 28 \, a b^{3} d e^{6} + 6 \, a^{2} b^{2} e^{7}\right )} x^{10} + \frac{1}{9} \,{\left (35 \, b^{4} d^{3} e^{4} + 84 \, a b^{3} d^{2} e^{5} + 42 \, a^{2} b^{2} d e^{6} + 4 \, a^{3} b e^{7}\right )} x^{9} + \frac{1}{8} \,{\left (35 \, b^{4} d^{4} e^{3} + 140 \, a b^{3} d^{3} e^{4} + 126 \, a^{2} b^{2} d^{2} e^{5} + 28 \, a^{3} b d e^{6} + a^{4} e^{7}\right )} x^{8} +{\left (3 \, b^{4} d^{5} e^{2} + 20 \, a b^{3} d^{4} e^{3} + 30 \, a^{2} b^{2} d^{3} e^{4} + 12 \, a^{3} b d^{2} e^{5} + a^{4} d e^{6}\right )} x^{7} + \frac{7}{6} \,{\left (b^{4} d^{6} e + 12 \, a b^{3} d^{5} e^{2} + 30 \, a^{2} b^{2} d^{4} e^{3} + 20 \, a^{3} b d^{3} e^{4} + 3 \, a^{4} d^{2} e^{5}\right )} x^{6} + \frac{1}{5} \,{\left (b^{4} d^{7} + 28 \, a b^{3} d^{6} e + 126 \, a^{2} b^{2} d^{5} e^{2} + 140 \, a^{3} b d^{4} e^{3} + 35 \, a^{4} d^{3} e^{4}\right )} x^{5} + \frac{1}{4} \,{\left (4 \, a b^{3} d^{7} + 42 \, a^{2} b^{2} d^{6} e + 84 \, a^{3} b d^{5} e^{2} + 35 \, a^{4} d^{4} e^{3}\right )} x^{4} + \frac{1}{3} \,{\left (6 \, a^{2} b^{2} d^{7} + 28 \, a^{3} b d^{6} e + 21 \, a^{4} d^{5} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (4 \, a^{3} b d^{7} + 7 \, a^{4} d^{6} e\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^7*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="fricas")

[Out]

1/12*b^4*e^7*x^12 + a^4*d^7*x + 1/11*(7*b^4*d*e^6 + 4*a*b^3*e^7)*x^11 + 1/10*(21*b^4*d^2*e^5 + 28*a*b^3*d*e^6
+ 6*a^2*b^2*e^7)*x^10 + 1/9*(35*b^4*d^3*e^4 + 84*a*b^3*d^2*e^5 + 42*a^2*b^2*d*e^6 + 4*a^3*b*e^7)*x^9 + 1/8*(35
*b^4*d^4*e^3 + 140*a*b^3*d^3*e^4 + 126*a^2*b^2*d^2*e^5 + 28*a^3*b*d*e^6 + a^4*e^7)*x^8 + (3*b^4*d^5*e^2 + 20*a
*b^3*d^4*e^3 + 30*a^2*b^2*d^3*e^4 + 12*a^3*b*d^2*e^5 + a^4*d*e^6)*x^7 + 7/6*(b^4*d^6*e + 12*a*b^3*d^5*e^2 + 30
*a^2*b^2*d^4*e^3 + 20*a^3*b*d^3*e^4 + 3*a^4*d^2*e^5)*x^6 + 1/5*(b^4*d^7 + 28*a*b^3*d^6*e + 126*a^2*b^2*d^5*e^2
+ 140*a^3*b*d^4*e^3 + 35*a^4*d^3*e^4)*x^5 + 1/4*(4*a*b^3*d^7 + 42*a^2*b^2*d^6*e + 84*a^3*b*d^5*e^2 + 35*a^4*d
^4*e^3)*x^4 + 1/3*(6*a^2*b^2*d^7 + 28*a^3*b*d^6*e + 21*a^4*d^5*e^2)*x^3 + 1/2*(4*a^3*b*d^7 + 7*a^4*d^6*e)*x^2

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b x\right ) \left (d + e x\right )^{7} \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)**7*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Integral((a + b*x)*(d + e*x)**7*((a + b*x)**2)**(3/2), x)

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Giac [B]  time = 1.145, size = 1027, normalized size = 4.04 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^7*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="giac")

[Out]

1/12*b^4*x^12*e^7*sgn(b*x + a) + 7/11*b^4*d*x^11*e^6*sgn(b*x + a) + 21/10*b^4*d^2*x^10*e^5*sgn(b*x + a) + 35/9
*b^4*d^3*x^9*e^4*sgn(b*x + a) + 35/8*b^4*d^4*x^8*e^3*sgn(b*x + a) + 3*b^4*d^5*x^7*e^2*sgn(b*x + a) + 7/6*b^4*d
^6*x^6*e*sgn(b*x + a) + 1/5*b^4*d^7*x^5*sgn(b*x + a) + 4/11*a*b^3*x^11*e^7*sgn(b*x + a) + 14/5*a*b^3*d*x^10*e^
6*sgn(b*x + a) + 28/3*a*b^3*d^2*x^9*e^5*sgn(b*x + a) + 35/2*a*b^3*d^3*x^8*e^4*sgn(b*x + a) + 20*a*b^3*d^4*x^7*
e^3*sgn(b*x + a) + 14*a*b^3*d^5*x^6*e^2*sgn(b*x + a) + 28/5*a*b^3*d^6*x^5*e*sgn(b*x + a) + a*b^3*d^7*x^4*sgn(b
*x + a) + 3/5*a^2*b^2*x^10*e^7*sgn(b*x + a) + 14/3*a^2*b^2*d*x^9*e^6*sgn(b*x + a) + 63/4*a^2*b^2*d^2*x^8*e^5*s
gn(b*x + a) + 30*a^2*b^2*d^3*x^7*e^4*sgn(b*x + a) + 35*a^2*b^2*d^4*x^6*e^3*sgn(b*x + a) + 126/5*a^2*b^2*d^5*x^
5*e^2*sgn(b*x + a) + 21/2*a^2*b^2*d^6*x^4*e*sgn(b*x + a) + 2*a^2*b^2*d^7*x^3*sgn(b*x + a) + 4/9*a^3*b*x^9*e^7*
sgn(b*x + a) + 7/2*a^3*b*d*x^8*e^6*sgn(b*x + a) + 12*a^3*b*d^2*x^7*e^5*sgn(b*x + a) + 70/3*a^3*b*d^3*x^6*e^4*s
gn(b*x + a) + 28*a^3*b*d^4*x^5*e^3*sgn(b*x + a) + 21*a^3*b*d^5*x^4*e^2*sgn(b*x + a) + 28/3*a^3*b*d^6*x^3*e*sgn
(b*x + a) + 2*a^3*b*d^7*x^2*sgn(b*x + a) + 1/8*a^4*x^8*e^7*sgn(b*x + a) + a^4*d*x^7*e^6*sgn(b*x + a) + 7/2*a^4
*d^2*x^6*e^5*sgn(b*x + a) + 7*a^4*d^3*x^5*e^4*sgn(b*x + a) + 35/4*a^4*d^4*x^4*e^3*sgn(b*x + a) + 7*a^4*d^5*x^3
*e^2*sgn(b*x + a) + 7/2*a^4*d^6*x^2*e*sgn(b*x + a) + a^4*d^7*x*sgn(b*x + a)