3.73 \(\int \frac{x^7 (A+B x^3)}{(a+b x^3)^2} \, dx\)

Optimal. Leaf size=215 \[ -\frac{a^{2/3} (5 A b-8 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{11/3}}+\frac{a^{2/3} (5 A b-8 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{11/3}}+\frac{a^{2/3} (5 A b-8 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} b^{11/3}}-\frac{x^5 (5 A b-8 a B)}{15 a b^2}+\frac{x^2 (5 A b-8 a B)}{6 b^3}+\frac{x^8 (A b-a B)}{3 a b \left (a+b x^3\right )} \]

[Out]

((5*A*b - 8*a*B)*x^2)/(6*b^3) - ((5*A*b - 8*a*B)*x^5)/(15*a*b^2) + ((A*b - a*B)*x^8)/(3*a*b*(a + b*x^3)) + (a^
(2/3)*(5*A*b - 8*a*B)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*b^(11/3)) + (a^(2/3)*(5*A*
b - 8*a*B)*Log[a^(1/3) + b^(1/3)*x])/(9*b^(11/3)) - (a^(2/3)*(5*A*b - 8*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x +
b^(2/3)*x^2])/(18*b^(11/3))

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Rubi [A]  time = 0.139865, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {457, 302, 292, 31, 634, 617, 204, 628} \[ -\frac{a^{2/3} (5 A b-8 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{11/3}}+\frac{a^{2/3} (5 A b-8 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{11/3}}+\frac{a^{2/3} (5 A b-8 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} b^{11/3}}-\frac{x^5 (5 A b-8 a B)}{15 a b^2}+\frac{x^2 (5 A b-8 a B)}{6 b^3}+\frac{x^8 (A b-a B)}{3 a b \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((5*A*b - 8*a*B)*x^2)/(6*b^3) - ((5*A*b - 8*a*B)*x^5)/(15*a*b^2) + ((A*b - a*B)*x^8)/(3*a*b*(a + b*x^3)) + (a^
(2/3)*(5*A*b - 8*a*B)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*b^(11/3)) + (a^(2/3)*(5*A*
b - 8*a*B)*Log[a^(1/3) + b^(1/3)*x])/(9*b^(11/3)) - (a^(2/3)*(5*A*b - 8*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x +
b^(2/3)*x^2])/(18*b^(11/3))

Rule 457

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d
)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*b*e*n*(p + 1)), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*b
*n*(p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) ||  !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0] &&
LeQ[-1, m, -(n*(p + 1))]))

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 292

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> -Dist[(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3
]^2*x^2), x], x] /; FreeQ[{a, b}, x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{x^7 \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx &=\frac{(A b-a B) x^8}{3 a b \left (a+b x^3\right )}+\frac{(-5 A b+8 a B) \int \frac{x^7}{a+b x^3} \, dx}{3 a b}\\ &=\frac{(A b-a B) x^8}{3 a b \left (a+b x^3\right )}+\frac{(-5 A b+8 a B) \int \left (-\frac{a x}{b^2}+\frac{x^4}{b}+\frac{a^2 x}{b^2 \left (a+b x^3\right )}\right ) \, dx}{3 a b}\\ &=\frac{(5 A b-8 a B) x^2}{6 b^3}-\frac{(5 A b-8 a B) x^5}{15 a b^2}+\frac{(A b-a B) x^8}{3 a b \left (a+b x^3\right )}-\frac{(a (5 A b-8 a B)) \int \frac{x}{a+b x^3} \, dx}{3 b^3}\\ &=\frac{(5 A b-8 a B) x^2}{6 b^3}-\frac{(5 A b-8 a B) x^5}{15 a b^2}+\frac{(A b-a B) x^8}{3 a b \left (a+b x^3\right )}+\frac{\left (a^{2/3} (5 A b-8 a B)\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 b^{10/3}}-\frac{\left (a^{2/3} (5 A b-8 a B)\right ) \int \frac{\sqrt [3]{a}+\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{9 b^{10/3}}\\ &=\frac{(5 A b-8 a B) x^2}{6 b^3}-\frac{(5 A b-8 a B) x^5}{15 a b^2}+\frac{(A b-a B) x^8}{3 a b \left (a+b x^3\right )}+\frac{a^{2/3} (5 A b-8 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{11/3}}-\frac{\left (a^{2/3} (5 A b-8 a B)\right ) \int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 b^{11/3}}-\frac{(a (5 A b-8 a B)) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 b^{10/3}}\\ &=\frac{(5 A b-8 a B) x^2}{6 b^3}-\frac{(5 A b-8 a B) x^5}{15 a b^2}+\frac{(A b-a B) x^8}{3 a b \left (a+b x^3\right )}+\frac{a^{2/3} (5 A b-8 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{11/3}}-\frac{a^{2/3} (5 A b-8 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{11/3}}-\frac{\left (a^{2/3} (5 A b-8 a B)\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 b^{11/3}}\\ &=\frac{(5 A b-8 a B) x^2}{6 b^3}-\frac{(5 A b-8 a B) x^5}{15 a b^2}+\frac{(A b-a B) x^8}{3 a b \left (a+b x^3\right )}+\frac{a^{2/3} (5 A b-8 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} b^{11/3}}+\frac{a^{2/3} (5 A b-8 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{11/3}}-\frac{a^{2/3} (5 A b-8 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{11/3}}\\ \end{align*}

Mathematica [A]  time = 0.124517, size = 185, normalized size = 0.86 \[ \frac{5 a^{2/3} (8 a B-5 A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-10 a^{2/3} (8 a B-5 A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-10 \sqrt{3} a^{2/3} (8 a B-5 A b) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )+45 b^{2/3} x^2 (A b-2 a B)+\frac{30 a b^{2/3} x^2 (A b-a B)}{a+b x^3}+18 b^{5/3} B x^5}{90 b^{11/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(45*b^(2/3)*(A*b - 2*a*B)*x^2 + 18*b^(5/3)*B*x^5 + (30*a*b^(2/3)*(A*b - a*B)*x^2)/(a + b*x^3) - 10*Sqrt[3]*a^(
2/3)*(-5*A*b + 8*a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] - 10*a^(2/3)*(-5*A*b + 8*a*B)*Log[a^(1/3) +
b^(1/3)*x] + 5*a^(2/3)*(-5*A*b + 8*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(90*b^(11/3))

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Maple [A]  time = 0.009, size = 266, normalized size = 1.2 \begin{align*}{\frac{B{x}^{5}}{5\,{b}^{2}}}+{\frac{A{x}^{2}}{2\,{b}^{2}}}-{\frac{B{x}^{2}a}{{b}^{3}}}+{\frac{aA{x}^{2}}{3\,{b}^{2} \left ( b{x}^{3}+a \right ) }}-{\frac{B{x}^{2}{a}^{2}}{3\,{b}^{3} \left ( b{x}^{3}+a \right ) }}+{\frac{5\,aA}{9\,{b}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{5\,aA}{18\,{b}^{3}}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{5\,aA\sqrt{3}}{9\,{b}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{8\,{a}^{2}B}{9\,{b}^{4}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{4\,{a}^{2}B}{9\,{b}^{4}}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{8\,{a}^{2}B\sqrt{3}}{9\,{b}^{4}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(B*x^3+A)/(b*x^3+a)^2,x)

[Out]

1/5/b^2*B*x^5+1/2/b^2*A*x^2-1/b^3*B*x^2*a+1/3*a/b^2*x^2/(b*x^3+a)*A-1/3*a^2/b^3*x^2/(b*x^3+a)*B+5/9*a/b^3*A/(a
/b)^(1/3)*ln(x+(a/b)^(1/3))-5/18*a/b^3*A/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))-5/9*a/b^3*A*3^(1/2)/(a/
b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-8/9*a^2/b^4*B/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+4/9*a^2/b^4*B/(a/
b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+8/9*a^2/b^4*B*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)
*x-1))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.65707, size = 599, normalized size = 2.79 \begin{align*} \frac{18 \, B b^{2} x^{8} - 9 \,{\left (8 \, B a b - 5 \, A b^{2}\right )} x^{5} - 15 \,{\left (8 \, B a^{2} - 5 \, A a b\right )} x^{2} + 10 \, \sqrt{3}{\left ({\left (8 \, B a b - 5 \, A b^{2}\right )} x^{3} + 8 \, B a^{2} - 5 \, A a b\right )} \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} b x \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} - \sqrt{3} a}{3 \, a}\right ) + 5 \,{\left ({\left (8 \, B a b - 5 \, A b^{2}\right )} x^{3} + 8 \, B a^{2} - 5 \, A a b\right )} \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a x^{2} - b x \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}} + a \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}}\right ) - 10 \,{\left ({\left (8 \, B a b - 5 \, A b^{2}\right )} x^{3} + 8 \, B a^{2} - 5 \, A a b\right )} \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a x + b \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}\right )}{90 \,{\left (b^{4} x^{3} + a b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/90*(18*B*b^2*x^8 - 9*(8*B*a*b - 5*A*b^2)*x^5 - 15*(8*B*a^2 - 5*A*a*b)*x^2 + 10*sqrt(3)*((8*B*a*b - 5*A*b^2)*
x^3 + 8*B*a^2 - 5*A*a*b)*(a^2/b^2)^(1/3)*arctan(1/3*(2*sqrt(3)*b*x*(a^2/b^2)^(1/3) - sqrt(3)*a)/a) + 5*((8*B*a
*b - 5*A*b^2)*x^3 + 8*B*a^2 - 5*A*a*b)*(a^2/b^2)^(1/3)*log(a*x^2 - b*x*(a^2/b^2)^(2/3) + a*(a^2/b^2)^(1/3)) -
10*((8*B*a*b - 5*A*b^2)*x^3 + 8*B*a^2 - 5*A*a*b)*(a^2/b^2)^(1/3)*log(a*x + b*(a^2/b^2)^(2/3)))/(b^4*x^3 + a*b^
3)

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Sympy [A]  time = 2.51983, size = 151, normalized size = 0.7 \begin{align*} \frac{B x^{5}}{5 b^{2}} - \frac{x^{2} \left (- A a b + B a^{2}\right )}{3 a b^{3} + 3 b^{4} x^{3}} + \operatorname{RootSum}{\left (729 t^{3} b^{11} - 125 A^{3} a^{2} b^{3} + 600 A^{2} B a^{3} b^{2} - 960 A B^{2} a^{4} b + 512 B^{3} a^{5}, \left ( t \mapsto t \log{\left (\frac{81 t^{2} b^{7}}{25 A^{2} a b^{2} - 80 A B a^{2} b + 64 B^{2} a^{3}} + x \right )} \right )\right )} - \frac{x^{2} \left (- A b + 2 B a\right )}{2 b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7*(B*x**3+A)/(b*x**3+a)**2,x)

[Out]

B*x**5/(5*b**2) - x**2*(-A*a*b + B*a**2)/(3*a*b**3 + 3*b**4*x**3) + RootSum(729*_t**3*b**11 - 125*A**3*a**2*b*
*3 + 600*A**2*B*a**3*b**2 - 960*A*B**2*a**4*b + 512*B**3*a**5, Lambda(_t, _t*log(81*_t**2*b**7/(25*A**2*a*b**2
- 80*A*B*a**2*b + 64*B**2*a**3) + x))) - x**2*(-A*b + 2*B*a)/(2*b**3)

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Giac [A]  time = 1.10413, size = 319, normalized size = 1.48 \begin{align*} -\frac{{\left (8 \, B a^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 5 \, A a b \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a b^{3}} - \frac{\sqrt{3}{\left (8 \, \left (-a b^{2}\right )^{\frac{2}{3}} B a - 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} A b\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, b^{5}} - \frac{B a^{2} x^{2} - A a b x^{2}}{3 \,{\left (b x^{3} + a\right )} b^{3}} + \frac{{\left (8 \, \left (-a b^{2}\right )^{\frac{2}{3}} B a - 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} A b\right )} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, b^{5}} + \frac{2 \, B b^{8} x^{5} - 10 \, B a b^{7} x^{2} + 5 \, A b^{8} x^{2}}{10 \, b^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*(8*B*a^2*(-a/b)^(1/3) - 5*A*a*b*(-a/b)^(1/3))*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^3) - 1/9*sqrt(
3)*(8*(-a*b^2)^(2/3)*B*a - 5*(-a*b^2)^(2/3)*A*b)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/b^5 - 1
/3*(B*a^2*x^2 - A*a*b*x^2)/((b*x^3 + a)*b^3) + 1/18*(8*(-a*b^2)^(2/3)*B*a - 5*(-a*b^2)^(2/3)*A*b)*log(x^2 + x*
(-a/b)^(1/3) + (-a/b)^(2/3))/b^5 + 1/10*(2*B*b^8*x^5 - 10*B*a*b^7*x^2 + 5*A*b^8*x^2)/b^10