3.1745 \(\int \frac{(A+B x) (a^2+2 a b x+b^2 x^2)^{5/2}}{d+e x} \, dx\)

Optimal. Leaf size=340 \[ -\frac{b x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 (B d-A e)}{e^6 (a+b x)}+\frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{2 e^5}-\frac{(a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (B d-A e)}{3 e^4}+\frac{(a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e)}{4 e^3}-\frac{(a+b x)^4 \sqrt{a^2+2 a b x+b^2 x^2} (B d-A e)}{5 e^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e) \log (d+e x)}{e^7 (a+b x)}+\frac{B (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 b e} \]

[Out]

-((b*(b*d - a*e)^4*(B*d - A*e)*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x))) + ((b*d - a*e)^3*(B*d - A*e)*
(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^5) - ((b*d - a*e)^2*(B*d - A*e)*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x +
b^2*x^2])/(3*e^4) + ((b*d - a*e)*(B*d - A*e)*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^3) - ((B*d - A*e
)*(a + b*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^2) + (B*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*b*e) +
((b*d - a*e)^5*(B*d - A*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Log[d + e*x])/(e^7*(a + b*x))

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Rubi [A]  time = 0.253587, antiderivative size = 340, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061, Rules used = {770, 77} \[ -\frac{b x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 (B d-A e)}{e^6 (a+b x)}+\frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{2 e^5}-\frac{(a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (B d-A e)}{3 e^4}+\frac{(a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e)}{4 e^3}-\frac{(a+b x)^4 \sqrt{a^2+2 a b x+b^2 x^2} (B d-A e)}{5 e^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e) \log (d+e x)}{e^7 (a+b x)}+\frac{B (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 b e} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x),x]

[Out]

-((b*(b*d - a*e)^4*(B*d - A*e)*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x))) + ((b*d - a*e)^3*(B*d - A*e)*
(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^5) - ((b*d - a*e)^2*(B*d - A*e)*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x +
b^2*x^2])/(3*e^4) + ((b*d - a*e)*(B*d - A*e)*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^3) - ((B*d - A*e
)*(a + b*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^2) + (B*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*b*e) +
((b*d - a*e)^5*(B*d - A*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Log[d + e*x])/(e^7*(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 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

Mathematica [A]  time = 0.255295, size = 386, normalized size = 1.14 \[ \frac{\sqrt{(a+b x)^2} \left (e x \left (50 a^2 b^3 e^2 \left (2 A e \left (6 d^2-3 d e x+2 e^2 x^2\right )+B \left (6 d^2 e x-12 d^3-4 d e^2 x^2+3 e^3 x^3\right )\right )+100 a^3 b^2 e^3 \left (3 A e (e x-2 d)+B \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )+150 a^4 b e^4 (2 A e-2 B d+B e x)+60 a^5 B e^5+5 a b^4 e \left (5 A e \left (6 d^2 e x-12 d^3-4 d e^2 x^2+3 e^3 x^3\right )+B \left (20 d^2 e^2 x^2-30 d^3 e x+60 d^4-15 d e^3 x^3+12 e^4 x^4\right )\right )+b^5 \left (A e \left (20 d^2 e^2 x^2-30 d^3 e x+60 d^4-15 d e^3 x^3+12 e^4 x^4\right )+B \left (-20 d^3 e^2 x^2+15 d^2 e^3 x^3+30 d^4 e x-60 d^5-12 d e^4 x^4+10 e^5 x^5\right )\right )\right )+60 (b d-a e)^5 (B d-A e) \log (d+e x)\right )}{60 e^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x),x]

[Out]

(Sqrt[(a + b*x)^2]*(e*x*(60*a^5*B*e^5 + 150*a^4*b*e^4*(-2*B*d + 2*A*e + B*e*x) + 100*a^3*b^2*e^3*(3*A*e*(-2*d
+ e*x) + B*(6*d^2 - 3*d*e*x + 2*e^2*x^2)) + 50*a^2*b^3*e^2*(2*A*e*(6*d^2 - 3*d*e*x + 2*e^2*x^2) + B*(-12*d^3 +
6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3)) + 5*a*b^4*e*(5*A*e*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + B*
(60*d^4 - 30*d^3*e*x + 20*d^2*e^2*x^2 - 15*d*e^3*x^3 + 12*e^4*x^4)) + b^5*(A*e*(60*d^4 - 30*d^3*e*x + 20*d^2*e
^2*x^2 - 15*d*e^3*x^3 + 12*e^4*x^4) + B*(-60*d^5 + 30*d^4*e*x - 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 - 12*d*e^4*x^4
+ 10*e^5*x^5))) + 60*(b*d - a*e)^5*(B*d - A*e)*Log[d + e*x]))/(60*e^7*(a + b*x))

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Maple [B]  time = 0.013, size = 754, normalized size = 2.2 \begin{align*}{\frac{-300\,B\ln \left ( ex+d \right ) a{b}^{4}{d}^{5}e-150\,B{x}^{2}a{b}^{4}{d}^{3}{e}^{3}-600\,Ax{a}^{3}{b}^{2}d{e}^{5}+600\,Ax{a}^{2}{b}^{3}{d}^{2}{e}^{4}-300\,Axa{b}^{4}{d}^{3}{e}^{3}-300\,Bx{a}^{4}bd{e}^{5}+600\,Bx{a}^{3}{b}^{2}{d}^{2}{e}^{4}+300\,B{x}^{2}{a}^{2}{b}^{3}{d}^{2}{e}^{4}-300\,B{x}^{2}{a}^{3}{b}^{2}d{e}^{5}+600\,A\ln \left ( ex+d \right ){a}^{3}{b}^{2}{d}^{2}{e}^{4}-600\,A\ln \left ( ex+d \right ){a}^{2}{b}^{3}{d}^{3}{e}^{3}-300\,A\ln \left ( ex+d \right ){a}^{4}bd{e}^{5}-600\,Bx{a}^{2}{b}^{3}{d}^{3}{e}^{3}+300\,Bxa{b}^{4}{d}^{4}{e}^{2}-75\,B{x}^{4}a{b}^{4}d{e}^{5}-100\,A{x}^{3}a{b}^{4}d{e}^{5}-200\,B{x}^{3}{a}^{2}{b}^{3}d{e}^{5}+100\,B{x}^{3}a{b}^{4}{d}^{2}{e}^{4}-300\,A{x}^{2}{a}^{2}{b}^{3}d{e}^{5}+150\,A{x}^{2}a{b}^{4}{d}^{2}{e}^{4}+600\,B\ln \left ( ex+d \right ){a}^{2}{b}^{3}{d}^{4}{e}^{2}+300\,A\ln \left ( ex+d \right ) a{b}^{4}{d}^{4}{e}^{2}+300\,B\ln \left ( ex+d \right ){a}^{4}b{d}^{2}{e}^{4}-600\,B\ln \left ( ex+d \right ){a}^{3}{b}^{2}{d}^{3}{e}^{3}+60\,A\ln \left ( ex+d \right ){a}^{5}{e}^{6}+60\,B\ln \left ( ex+d \right ){b}^{5}{d}^{6}+10\,B{x}^{6}{b}^{5}{e}^{6}+12\,A{x}^{5}{b}^{5}{e}^{6}+60\,Bx{a}^{5}{e}^{6}+15\,B{x}^{4}{b}^{5}{d}^{2}{e}^{4}-20\,B{x}^{3}{b}^{5}{d}^{3}{e}^{3}+300\,A{x}^{2}{a}^{3}{b}^{2}{e}^{6}+60\,B{x}^{5}a{b}^{4}{e}^{6}-12\,B{x}^{5}{b}^{5}d{e}^{5}+75\,A{x}^{4}a{b}^{4}{e}^{6}-15\,A{x}^{4}{b}^{5}d{e}^{5}+150\,B{x}^{4}{a}^{2}{b}^{3}{e}^{6}-60\,Bx{b}^{5}{d}^{5}e+200\,A{x}^{3}{a}^{2}{b}^{3}{e}^{6}+20\,A{x}^{3}{b}^{5}{d}^{2}{e}^{4}+200\,B{x}^{3}{a}^{3}{b}^{2}{e}^{6}+300\,Ax{a}^{4}b{e}^{6}+60\,Ax{b}^{5}{d}^{4}{e}^{2}-30\,A{x}^{2}{b}^{5}{d}^{3}{e}^{3}+150\,B{x}^{2}{a}^{4}b{e}^{6}+30\,B{x}^{2}{b}^{5}{d}^{4}{e}^{2}-60\,A\ln \left ( ex+d \right ){b}^{5}{d}^{5}e-60\,B\ln \left ( ex+d \right ){a}^{5}d{e}^{5}}{60\, \left ( bx+a \right ) ^{5}{e}^{7}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d),x)

[Out]

1/60*((b*x+a)^2)^(5/2)*(-300*B*ln(e*x+d)*a*b^4*d^5*e-150*B*x^2*a*b^4*d^3*e^3-600*A*x*a^3*b^2*d*e^5+600*A*x*a^2
*b^3*d^2*e^4-300*A*x*a*b^4*d^3*e^3-300*B*x*a^4*b*d*e^5+600*B*x*a^3*b^2*d^2*e^4+300*B*x^2*a^2*b^3*d^2*e^4-300*B
*x^2*a^3*b^2*d*e^5+600*A*ln(e*x+d)*a^3*b^2*d^2*e^4-600*A*ln(e*x+d)*a^2*b^3*d^3*e^3-300*A*ln(e*x+d)*a^4*b*d*e^5
-600*B*x*a^2*b^3*d^3*e^3+300*B*x*a*b^4*d^4*e^2-75*B*x^4*a*b^4*d*e^5-100*A*x^3*a*b^4*d*e^5-200*B*x^3*a^2*b^3*d*
e^5+100*B*x^3*a*b^4*d^2*e^4-300*A*x^2*a^2*b^3*d*e^5+150*A*x^2*a*b^4*d^2*e^4+600*B*ln(e*x+d)*a^2*b^3*d^4*e^2+30
0*A*ln(e*x+d)*a*b^4*d^4*e^2+300*B*ln(e*x+d)*a^4*b*d^2*e^4-600*B*ln(e*x+d)*a^3*b^2*d^3*e^3+60*A*ln(e*x+d)*a^5*e
^6+60*B*ln(e*x+d)*b^5*d^6+10*B*x^6*b^5*e^6+12*A*x^5*b^5*e^6+60*B*x*a^5*e^6+15*B*x^4*b^5*d^2*e^4-20*B*x^3*b^5*d
^3*e^3+300*A*x^2*a^3*b^2*e^6+60*B*x^5*a*b^4*e^6-12*B*x^5*b^5*d*e^5+75*A*x^4*a*b^4*e^6-15*A*x^4*b^5*d*e^5+150*B
*x^4*a^2*b^3*e^6-60*B*x*b^5*d^5*e+200*A*x^3*a^2*b^3*e^6+20*A*x^3*b^5*d^2*e^4+200*B*x^3*a^3*b^2*e^6+300*A*x*a^4
*b*e^6+60*A*x*b^5*d^4*e^2-30*A*x^2*b^5*d^3*e^3+150*B*x^2*a^4*b*e^6+30*B*x^2*b^5*d^4*e^2-60*A*ln(e*x+d)*b^5*d^5
*e-60*B*ln(e*x+d)*a^5*d*e^5)/(b*x+a)^5/e^7

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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)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.56451, size = 1126, normalized size = 3.31 \begin{align*} \frac{10 \, B b^{5} e^{6} x^{6} - 12 \,{\left (B b^{5} d e^{5} -{\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} + 15 \,{\left (B b^{5} d^{2} e^{4} -{\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + 5 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} - 20 \,{\left (B b^{5} d^{3} e^{3} -{\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 5 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} - 10 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} + 30 \,{\left (B b^{5} d^{4} e^{2} -{\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 5 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} - 10 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} + 5 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} - 60 \,{\left (B b^{5} d^{5} e -{\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 5 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} - 10 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} + 5 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} -{\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x + 60 \,{\left (B b^{5} d^{6} + A a^{5} e^{6} -{\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e + 5 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} - 10 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} + 5 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} -{\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5}\right )} \log \left (e x + d\right )}{60 \, e^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d),x, algorithm="fricas")

[Out]

1/60*(10*B*b^5*e^6*x^6 - 12*(B*b^5*d*e^5 - (5*B*a*b^4 + A*b^5)*e^6)*x^5 + 15*(B*b^5*d^2*e^4 - (5*B*a*b^4 + A*b
^5)*d*e^5 + 5*(2*B*a^2*b^3 + A*a*b^4)*e^6)*x^4 - 20*(B*b^5*d^3*e^3 - (5*B*a*b^4 + A*b^5)*d^2*e^4 + 5*(2*B*a^2*
b^3 + A*a*b^4)*d*e^5 - 10*(B*a^3*b^2 + A*a^2*b^3)*e^6)*x^3 + 30*(B*b^5*d^4*e^2 - (5*B*a*b^4 + A*b^5)*d^3*e^3 +
5*(2*B*a^2*b^3 + A*a*b^4)*d^2*e^4 - 10*(B*a^3*b^2 + A*a^2*b^3)*d*e^5 + 5*(B*a^4*b + 2*A*a^3*b^2)*e^6)*x^2 - 6
0*(B*b^5*d^5*e - (5*B*a*b^4 + A*b^5)*d^4*e^2 + 5*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^3 - 10*(B*a^3*b^2 + A*a^2*b^3)*
d^2*e^4 + 5*(B*a^4*b + 2*A*a^3*b^2)*d*e^5 - (B*a^5 + 5*A*a^4*b)*e^6)*x + 60*(B*b^5*d^6 + A*a^5*e^6 - (5*B*a*b^
4 + A*b^5)*d^5*e + 5*(2*B*a^2*b^3 + A*a*b^4)*d^4*e^2 - 10*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^3 + 5*(B*a^4*b + 2*A*a
^3*b^2)*d^2*e^4 - (B*a^5 + 5*A*a^4*b)*d*e^5)*log(e*x + d))/e^7

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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((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d),x)

[Out]

Timed out

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Giac [B]  time = 1.19558, size = 1242, normalized size = 3.65 \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)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d),x, algorithm="giac")

[Out]

(B*b^5*d^6*sgn(b*x + a) - 5*B*a*b^4*d^5*e*sgn(b*x + a) - A*b^5*d^5*e*sgn(b*x + a) + 10*B*a^2*b^3*d^4*e^2*sgn(b
*x + a) + 5*A*a*b^4*d^4*e^2*sgn(b*x + a) - 10*B*a^3*b^2*d^3*e^3*sgn(b*x + a) - 10*A*a^2*b^3*d^3*e^3*sgn(b*x +
a) + 5*B*a^4*b*d^2*e^4*sgn(b*x + a) + 10*A*a^3*b^2*d^2*e^4*sgn(b*x + a) - B*a^5*d*e^5*sgn(b*x + a) - 5*A*a^4*b
*d*e^5*sgn(b*x + a) + A*a^5*e^6*sgn(b*x + a))*e^(-7)*log(abs(x*e + d)) + 1/60*(10*B*b^5*x^6*e^5*sgn(b*x + a) -
12*B*b^5*d*x^5*e^4*sgn(b*x + a) + 15*B*b^5*d^2*x^4*e^3*sgn(b*x + a) - 20*B*b^5*d^3*x^3*e^2*sgn(b*x + a) + 30*
B*b^5*d^4*x^2*e*sgn(b*x + a) - 60*B*b^5*d^5*x*sgn(b*x + a) + 60*B*a*b^4*x^5*e^5*sgn(b*x + a) + 12*A*b^5*x^5*e^
5*sgn(b*x + a) - 75*B*a*b^4*d*x^4*e^4*sgn(b*x + a) - 15*A*b^5*d*x^4*e^4*sgn(b*x + a) + 100*B*a*b^4*d^2*x^3*e^3
*sgn(b*x + a) + 20*A*b^5*d^2*x^3*e^3*sgn(b*x + a) - 150*B*a*b^4*d^3*x^2*e^2*sgn(b*x + a) - 30*A*b^5*d^3*x^2*e^
2*sgn(b*x + a) + 300*B*a*b^4*d^4*x*e*sgn(b*x + a) + 60*A*b^5*d^4*x*e*sgn(b*x + a) + 150*B*a^2*b^3*x^4*e^5*sgn(
b*x + a) + 75*A*a*b^4*x^4*e^5*sgn(b*x + a) - 200*B*a^2*b^3*d*x^3*e^4*sgn(b*x + a) - 100*A*a*b^4*d*x^3*e^4*sgn(
b*x + a) + 300*B*a^2*b^3*d^2*x^2*e^3*sgn(b*x + a) + 150*A*a*b^4*d^2*x^2*e^3*sgn(b*x + a) - 600*B*a^2*b^3*d^3*x
*e^2*sgn(b*x + a) - 300*A*a*b^4*d^3*x*e^2*sgn(b*x + a) + 200*B*a^3*b^2*x^3*e^5*sgn(b*x + a) + 200*A*a^2*b^3*x^
3*e^5*sgn(b*x + a) - 300*B*a^3*b^2*d*x^2*e^4*sgn(b*x + a) - 300*A*a^2*b^3*d*x^2*e^4*sgn(b*x + a) + 600*B*a^3*b
^2*d^2*x*e^3*sgn(b*x + a) + 600*A*a^2*b^3*d^2*x*e^3*sgn(b*x + a) + 150*B*a^4*b*x^2*e^5*sgn(b*x + a) + 300*A*a^
3*b^2*x^2*e^5*sgn(b*x + a) - 300*B*a^4*b*d*x*e^4*sgn(b*x + a) - 600*A*a^3*b^2*d*x*e^4*sgn(b*x + a) + 60*B*a^5*
x*e^5*sgn(b*x + a) + 300*A*a^4*b*x*e^5*sgn(b*x + a))*e^(-6)