3.193 \(\int \frac{1}{x^4 \sqrt{b x^{2/3}+a x}} \, dx\)

Optimal. Leaf size=329 \[ \frac{138567 a^9 \sqrt{a x+b x^{2/3}}}{131072 b^{10} x^{2/3}}-\frac{46189 a^8 \sqrt{a x+b x^{2/3}}}{65536 b^9 x}+\frac{46189 a^7 \sqrt{a x+b x^{2/3}}}{81920 b^8 x^{4/3}}-\frac{138567 a^6 \sqrt{a x+b x^{2/3}}}{286720 b^7 x^{5/3}}+\frac{46189 a^5 \sqrt{a x+b x^{2/3}}}{107520 b^6 x^2}-\frac{4199 a^4 \sqrt{a x+b x^{2/3}}}{10752 b^5 x^{7/3}}+\frac{323 a^3 \sqrt{a x+b x^{2/3}}}{896 b^4 x^{8/3}}-\frac{323 a^2 \sqrt{a x+b x^{2/3}}}{960 b^3 x^3}-\frac{138567 a^{10} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{131072 b^{21/2}}+\frac{19 a \sqrt{a x+b x^{2/3}}}{60 b^2 x^{10/3}}-\frac{3 \sqrt{a x+b x^{2/3}}}{10 b x^{11/3}} \]

[Out]

(-3*Sqrt[b*x^(2/3) + a*x])/(10*b*x^(11/3)) + (19*a*Sqrt[b*x^(2/3) + a*x])/(60*b^2*x^(10/3)) - (323*a^2*Sqrt[b*
x^(2/3) + a*x])/(960*b^3*x^3) + (323*a^3*Sqrt[b*x^(2/3) + a*x])/(896*b^4*x^(8/3)) - (4199*a^4*Sqrt[b*x^(2/3) +
a*x])/(10752*b^5*x^(7/3)) + (46189*a^5*Sqrt[b*x^(2/3) + a*x])/(107520*b^6*x^2) - (138567*a^6*Sqrt[b*x^(2/3) +
a*x])/(286720*b^7*x^(5/3)) + (46189*a^7*Sqrt[b*x^(2/3) + a*x])/(81920*b^8*x^(4/3)) - (46189*a^8*Sqrt[b*x^(2/3
) + a*x])/(65536*b^9*x) + (138567*a^9*Sqrt[b*x^(2/3) + a*x])/(131072*b^10*x^(2/3)) - (138567*a^10*ArcTanh[(Sqr
t[b]*x^(1/3))/Sqrt[b*x^(2/3) + a*x]])/(131072*b^(21/2))

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Rubi [A]  time = 0.577139, antiderivative size = 329, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2025, 2029, 206} \[ \frac{138567 a^9 \sqrt{a x+b x^{2/3}}}{131072 b^{10} x^{2/3}}-\frac{46189 a^8 \sqrt{a x+b x^{2/3}}}{65536 b^9 x}+\frac{46189 a^7 \sqrt{a x+b x^{2/3}}}{81920 b^8 x^{4/3}}-\frac{138567 a^6 \sqrt{a x+b x^{2/3}}}{286720 b^7 x^{5/3}}+\frac{46189 a^5 \sqrt{a x+b x^{2/3}}}{107520 b^6 x^2}-\frac{4199 a^4 \sqrt{a x+b x^{2/3}}}{10752 b^5 x^{7/3}}+\frac{323 a^3 \sqrt{a x+b x^{2/3}}}{896 b^4 x^{8/3}}-\frac{323 a^2 \sqrt{a x+b x^{2/3}}}{960 b^3 x^3}-\frac{138567 a^{10} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{131072 b^{21/2}}+\frac{19 a \sqrt{a x+b x^{2/3}}}{60 b^2 x^{10/3}}-\frac{3 \sqrt{a x+b x^{2/3}}}{10 b x^{11/3}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*Sqrt[b*x^(2/3) + a*x]),x]

[Out]

(-3*Sqrt[b*x^(2/3) + a*x])/(10*b*x^(11/3)) + (19*a*Sqrt[b*x^(2/3) + a*x])/(60*b^2*x^(10/3)) - (323*a^2*Sqrt[b*
x^(2/3) + a*x])/(960*b^3*x^3) + (323*a^3*Sqrt[b*x^(2/3) + a*x])/(896*b^4*x^(8/3)) - (4199*a^4*Sqrt[b*x^(2/3) +
a*x])/(10752*b^5*x^(7/3)) + (46189*a^5*Sqrt[b*x^(2/3) + a*x])/(107520*b^6*x^2) - (138567*a^6*Sqrt[b*x^(2/3) +
a*x])/(286720*b^7*x^(5/3)) + (46189*a^7*Sqrt[b*x^(2/3) + a*x])/(81920*b^8*x^(4/3)) - (46189*a^8*Sqrt[b*x^(2/3
) + a*x])/(65536*b^9*x) + (138567*a^9*Sqrt[b*x^(2/3) + a*x])/(131072*b^10*x^(2/3)) - (138567*a^10*ArcTanh[(Sqr
t[b]*x^(1/3))/Sqrt[b*x^(2/3) + a*x]])/(131072*b^(21/2))

Rule 2025

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(c^(j - 1)*(c*x)^(m - j +
1)*(a*x^j + b*x^n)^(p + 1))/(a*(m + j*p + 1)), x] - Dist[(b*(m + n*p + n - j + 1))/(a*c^(n - j)*(m + j*p + 1)
), Int[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IntegerQ[p] && LtQ[0, j,
n] && (IntegersQ[j, n] || GtQ[c, 0]) && LtQ[m + j*p + 1, 0]

Rule 2029

Int[(x_)^(m_.)/Sqrt[(a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Dist[-2/(n - j), Subst[Int[1/(1 - a*x^2
), x], x, x^(j/2)/Sqrt[a*x^j + b*x^n]], x] /; FreeQ[{a, b, j, n}, x] && EqQ[m, j/2 - 1] && NeQ[n, j]

Rule 206

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

Rubi steps

\begin{align*} \int \frac{1}{x^4 \sqrt{b x^{2/3}+a x}} \, dx &=-\frac{3 \sqrt{b x^{2/3}+a x}}{10 b x^{11/3}}-\frac{(19 a) \int \frac{1}{x^{11/3} \sqrt{b x^{2/3}+a x}} \, dx}{20 b}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{10 b x^{11/3}}+\frac{19 a \sqrt{b x^{2/3}+a x}}{60 b^2 x^{10/3}}+\frac{\left (323 a^2\right ) \int \frac{1}{x^{10/3} \sqrt{b x^{2/3}+a x}} \, dx}{360 b^2}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{10 b x^{11/3}}+\frac{19 a \sqrt{b x^{2/3}+a x}}{60 b^2 x^{10/3}}-\frac{323 a^2 \sqrt{b x^{2/3}+a x}}{960 b^3 x^3}-\frac{\left (323 a^3\right ) \int \frac{1}{x^3 \sqrt{b x^{2/3}+a x}} \, dx}{384 b^3}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{10 b x^{11/3}}+\frac{19 a \sqrt{b x^{2/3}+a x}}{60 b^2 x^{10/3}}-\frac{323 a^2 \sqrt{b x^{2/3}+a x}}{960 b^3 x^3}+\frac{323 a^3 \sqrt{b x^{2/3}+a x}}{896 b^4 x^{8/3}}+\frac{\left (4199 a^4\right ) \int \frac{1}{x^{8/3} \sqrt{b x^{2/3}+a x}} \, dx}{5376 b^4}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{10 b x^{11/3}}+\frac{19 a \sqrt{b x^{2/3}+a x}}{60 b^2 x^{10/3}}-\frac{323 a^2 \sqrt{b x^{2/3}+a x}}{960 b^3 x^3}+\frac{323 a^3 \sqrt{b x^{2/3}+a x}}{896 b^4 x^{8/3}}-\frac{4199 a^4 \sqrt{b x^{2/3}+a x}}{10752 b^5 x^{7/3}}-\frac{\left (46189 a^5\right ) \int \frac{1}{x^{7/3} \sqrt{b x^{2/3}+a x}} \, dx}{64512 b^5}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{10 b x^{11/3}}+\frac{19 a \sqrt{b x^{2/3}+a x}}{60 b^2 x^{10/3}}-\frac{323 a^2 \sqrt{b x^{2/3}+a x}}{960 b^3 x^3}+\frac{323 a^3 \sqrt{b x^{2/3}+a x}}{896 b^4 x^{8/3}}-\frac{4199 a^4 \sqrt{b x^{2/3}+a x}}{10752 b^5 x^{7/3}}+\frac{46189 a^5 \sqrt{b x^{2/3}+a x}}{107520 b^6 x^2}+\frac{\left (46189 a^6\right ) \int \frac{1}{x^2 \sqrt{b x^{2/3}+a x}} \, dx}{71680 b^6}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{10 b x^{11/3}}+\frac{19 a \sqrt{b x^{2/3}+a x}}{60 b^2 x^{10/3}}-\frac{323 a^2 \sqrt{b x^{2/3}+a x}}{960 b^3 x^3}+\frac{323 a^3 \sqrt{b x^{2/3}+a x}}{896 b^4 x^{8/3}}-\frac{4199 a^4 \sqrt{b x^{2/3}+a x}}{10752 b^5 x^{7/3}}+\frac{46189 a^5 \sqrt{b x^{2/3}+a x}}{107520 b^6 x^2}-\frac{138567 a^6 \sqrt{b x^{2/3}+a x}}{286720 b^7 x^{5/3}}-\frac{\left (46189 a^7\right ) \int \frac{1}{x^{5/3} \sqrt{b x^{2/3}+a x}} \, dx}{81920 b^7}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{10 b x^{11/3}}+\frac{19 a \sqrt{b x^{2/3}+a x}}{60 b^2 x^{10/3}}-\frac{323 a^2 \sqrt{b x^{2/3}+a x}}{960 b^3 x^3}+\frac{323 a^3 \sqrt{b x^{2/3}+a x}}{896 b^4 x^{8/3}}-\frac{4199 a^4 \sqrt{b x^{2/3}+a x}}{10752 b^5 x^{7/3}}+\frac{46189 a^5 \sqrt{b x^{2/3}+a x}}{107520 b^6 x^2}-\frac{138567 a^6 \sqrt{b x^{2/3}+a x}}{286720 b^7 x^{5/3}}+\frac{46189 a^7 \sqrt{b x^{2/3}+a x}}{81920 b^8 x^{4/3}}+\frac{\left (46189 a^8\right ) \int \frac{1}{x^{4/3} \sqrt{b x^{2/3}+a x}} \, dx}{98304 b^8}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{10 b x^{11/3}}+\frac{19 a \sqrt{b x^{2/3}+a x}}{60 b^2 x^{10/3}}-\frac{323 a^2 \sqrt{b x^{2/3}+a x}}{960 b^3 x^3}+\frac{323 a^3 \sqrt{b x^{2/3}+a x}}{896 b^4 x^{8/3}}-\frac{4199 a^4 \sqrt{b x^{2/3}+a x}}{10752 b^5 x^{7/3}}+\frac{46189 a^5 \sqrt{b x^{2/3}+a x}}{107520 b^6 x^2}-\frac{138567 a^6 \sqrt{b x^{2/3}+a x}}{286720 b^7 x^{5/3}}+\frac{46189 a^7 \sqrt{b x^{2/3}+a x}}{81920 b^8 x^{4/3}}-\frac{46189 a^8 \sqrt{b x^{2/3}+a x}}{65536 b^9 x}-\frac{\left (46189 a^9\right ) \int \frac{1}{x \sqrt{b x^{2/3}+a x}} \, dx}{131072 b^9}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{10 b x^{11/3}}+\frac{19 a \sqrt{b x^{2/3}+a x}}{60 b^2 x^{10/3}}-\frac{323 a^2 \sqrt{b x^{2/3}+a x}}{960 b^3 x^3}+\frac{323 a^3 \sqrt{b x^{2/3}+a x}}{896 b^4 x^{8/3}}-\frac{4199 a^4 \sqrt{b x^{2/3}+a x}}{10752 b^5 x^{7/3}}+\frac{46189 a^5 \sqrt{b x^{2/3}+a x}}{107520 b^6 x^2}-\frac{138567 a^6 \sqrt{b x^{2/3}+a x}}{286720 b^7 x^{5/3}}+\frac{46189 a^7 \sqrt{b x^{2/3}+a x}}{81920 b^8 x^{4/3}}-\frac{46189 a^8 \sqrt{b x^{2/3}+a x}}{65536 b^9 x}+\frac{138567 a^9 \sqrt{b x^{2/3}+a x}}{131072 b^{10} x^{2/3}}+\frac{\left (46189 a^{10}\right ) \int \frac{1}{x^{2/3} \sqrt{b x^{2/3}+a x}} \, dx}{262144 b^{10}}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{10 b x^{11/3}}+\frac{19 a \sqrt{b x^{2/3}+a x}}{60 b^2 x^{10/3}}-\frac{323 a^2 \sqrt{b x^{2/3}+a x}}{960 b^3 x^3}+\frac{323 a^3 \sqrt{b x^{2/3}+a x}}{896 b^4 x^{8/3}}-\frac{4199 a^4 \sqrt{b x^{2/3}+a x}}{10752 b^5 x^{7/3}}+\frac{46189 a^5 \sqrt{b x^{2/3}+a x}}{107520 b^6 x^2}-\frac{138567 a^6 \sqrt{b x^{2/3}+a x}}{286720 b^7 x^{5/3}}+\frac{46189 a^7 \sqrt{b x^{2/3}+a x}}{81920 b^8 x^{4/3}}-\frac{46189 a^8 \sqrt{b x^{2/3}+a x}}{65536 b^9 x}+\frac{138567 a^9 \sqrt{b x^{2/3}+a x}}{131072 b^{10} x^{2/3}}-\frac{\left (138567 a^{10}\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\sqrt [3]{x}}{\sqrt{b x^{2/3}+a x}}\right )}{131072 b^{10}}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{10 b x^{11/3}}+\frac{19 a \sqrt{b x^{2/3}+a x}}{60 b^2 x^{10/3}}-\frac{323 a^2 \sqrt{b x^{2/3}+a x}}{960 b^3 x^3}+\frac{323 a^3 \sqrt{b x^{2/3}+a x}}{896 b^4 x^{8/3}}-\frac{4199 a^4 \sqrt{b x^{2/3}+a x}}{10752 b^5 x^{7/3}}+\frac{46189 a^5 \sqrt{b x^{2/3}+a x}}{107520 b^6 x^2}-\frac{138567 a^6 \sqrt{b x^{2/3}+a x}}{286720 b^7 x^{5/3}}+\frac{46189 a^7 \sqrt{b x^{2/3}+a x}}{81920 b^8 x^{4/3}}-\frac{46189 a^8 \sqrt{b x^{2/3}+a x}}{65536 b^9 x}+\frac{138567 a^9 \sqrt{b x^{2/3}+a x}}{131072 b^{10} x^{2/3}}-\frac{138567 a^{10} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{b x^{2/3}+a x}}\right )}{131072 b^{21/2}}\\ \end{align*}

Mathematica [C]  time = 0.0547898, size = 48, normalized size = 0.15 \[ -\frac{6 a^{10} \sqrt{a x+b x^{2/3}} \, _2F_1\left (\frac{1}{2},11;\frac{3}{2};\frac{\sqrt [3]{x} a}{b}+1\right )}{b^{11} \sqrt [3]{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*Sqrt[b*x^(2/3) + a*x]),x]

[Out]

(-6*a^10*Sqrt[b*x^(2/3) + a*x]*Hypergeometric2F1[1/2, 11, 3/2, 1 + (a*x^(1/3))/b])/(b^11*x^(1/3))

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Maple [A]  time = 0.007, size = 248, normalized size = 0.8 \begin{align*} -{\frac{1}{13762560\,{x}^{6}}\sqrt{b+a\sqrt [3]{x}} \left ( -4358144\,{b}^{19/2}{x}^{10/3}\sqrt{b+a\sqrt [3]{x}}a+4630528\,{b}^{17/2}{x}^{11/3}\sqrt{b+a\sqrt [3]{x}}{a}^{2}+5374720\,{b}^{13/2}{x}^{13/3}\sqrt{b+a\sqrt [3]{x}}{a}^{4}-5912192\,{b}^{11/2}{x}^{14/3}\sqrt{b+a\sqrt [3]{x}}{a}^{5}-7759752\,{b}^{7/2}{x}^{16/3}\sqrt{b+a\sqrt [3]{x}}{a}^{7}+9699690\,{b}^{5/2}{x}^{{\frac{17}{3}}}\sqrt{b+a\sqrt [3]{x}}{a}^{8}[phone]\,{\it Artanh} \left ({\frac{\sqrt{b+a\sqrt [3]{x}}}{\sqrt{b}}} \right ){x}^{{\frac{19}{3}}}{a}^{10}b+4128768\,\sqrt{b+a\sqrt [3]{x}}{b}^{21/2}{x}^{3}-4961280\,{b}^{15/2}{x}^{4}\sqrt{b+a\sqrt [3]{x}}{a}^{3}+6651216\,{b}^{9/2}{x}^{5}\sqrt{b+a\sqrt [3]{x}}{a}^{6}-14549535\,{b}^{3/2}{x}^{6}\sqrt{b+a\sqrt [3]{x}}{a}^{9} \right ){\frac{1}{\sqrt{b{x}^{{\frac{2}{3}}}+ax}}}{b}^{-{\frac{23}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(b*x^(2/3)+a*x)^(1/2),x)

[Out]

-1/13762560*(b+a*x^(1/3))^(1/2)*(-4358144*b^(19/2)*x^(10/3)*(b+a*x^(1/3))^(1/2)*a+4630528*b^(17/2)*x^(11/3)*(b
+a*x^(1/3))^(1/2)*a^2+5374720*b^(13/2)*x^(13/3)*(b+a*x^(1/3))^(1/2)*a^4-5912192*b^(11/2)*x^(14/3)*(b+a*x^(1/3)
)^(1/2)*a^5-7759752*b^(7/2)*x^(16/3)*(b+a*x^(1/3))^(1/2)*a^7+9699690*b^(5/2)*x^(17/3)*(b+a*x^(1/3))^(1/2)*a^8+
14549535*arctanh((b+a*x^(1/3))^(1/2)/b^(1/2))*x^(19/3)*a^10*b+4128768*(b+a*x^(1/3))^(1/2)*b^(21/2)*x^3-4961280
*b^(15/2)*x^4*(b+a*x^(1/3))^(1/2)*a^3+6651216*b^(9/2)*x^5*(b+a*x^(1/3))^(1/2)*a^6-14549535*b^(3/2)*x^6*(b+a*x^
(1/3))^(1/2)*a^9)/x^6/(b*x^(2/3)+a*x)^(1/2)/b^(23/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a x + b x^{\frac{2}{3}}} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^(2/3)+a*x)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(a*x + b*x^(2/3))*x^4), x)

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Fricas [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(1/x^4/(b*x^(2/3)+a*x)^(1/2),x, algorithm="fricas")

[Out]

Timed out

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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(1/x**4/(b*x**(2/3)+a*x)**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 1.35374, size = 285, normalized size = 0.87 \begin{align*} \frac{\frac{14549535 \, a^{11} \arctan \left (\frac{\sqrt{a x^{\frac{1}{3}} + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{10}} + \frac{14549535 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{19}{2}} a^{11} - 140645505 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{17}{2}} a^{11} b + 609140532 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{15}{2}} a^{11} b^{2} - 1554721740 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{13}{2}} a^{11} b^{3} + 2585198330 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{11}{2}} a^{11} b^{4} - 2918514950 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{9}{2}} a^{11} b^{5} + 2255541300 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{7}{2}} a^{11} b^{6} - 1168982220 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} a^{11} b^{7} + 382331775 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} a^{11} b^{8} - 68025825 \, \sqrt{a x^{\frac{1}{3}} + b} a^{11} b^{9}}{a^{10} b^{10} x^{\frac{10}{3}}}}{13762560 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^(2/3)+a*x)^(1/2),x, algorithm="giac")

[Out]

1/13762560*(14549535*a^11*arctan(sqrt(a*x^(1/3) + b)/sqrt(-b))/(sqrt(-b)*b^10) + (14549535*(a*x^(1/3) + b)^(19
/2)*a^11 - 140645505*(a*x^(1/3) + b)^(17/2)*a^11*b + 609140532*(a*x^(1/3) + b)^(15/2)*a^11*b^2 - 1554721740*(a
*x^(1/3) + b)^(13/2)*a^11*b^3 + 2585198330*(a*x^(1/3) + b)^(11/2)*a^11*b^4 - 2918514950*(a*x^(1/3) + b)^(9/2)*
a^11*b^5 + 2255541300*(a*x^(1/3) + b)^(7/2)*a^11*b^6 - 1168982220*(a*x^(1/3) + b)^(5/2)*a^11*b^7 + 382331775*(
a*x^(1/3) + b)^(3/2)*a^11*b^8 - 68025825*sqrt(a*x^(1/3) + b)*a^11*b^9)/(a^10*b^10*x^(10/3)))/a