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

Optimal. Leaf size=336 \[ \frac{1048576 b^{10} \sqrt{a x+b x^{2/3}}}{29393 a^{12} \sqrt [3]{x}}-\frac{524288 b^9 \sqrt{a x+b x^{2/3}}}{29393 a^{11}}+\frac{393216 b^8 \sqrt [3]{x} \sqrt{a x+b x^{2/3}}}{29393 a^{10}}-\frac{327680 b^7 x^{2/3} \sqrt{a x+b x^{2/3}}}{29393 a^9}+\frac{40960 b^6 x \sqrt{a x+b x^{2/3}}}{4199 a^8}-\frac{36864 b^5 x^{4/3} \sqrt{a x+b x^{2/3}}}{4199 a^7}+\frac{33792 b^4 x^{5/3} \sqrt{a x+b x^{2/3}}}{4199 a^6}-\frac{16896 b^3 x^2 \sqrt{a x+b x^{2/3}}}{2261 a^5}+\frac{15840 b^2 x^{7/3} \sqrt{a x+b x^{2/3}}}{2261 a^4}-\frac{880 b x^{8/3} \sqrt{a x+b x^{2/3}}}{133 a^3}+\frac{44 x^3 \sqrt{a x+b x^{2/3}}}{7 a^2}-\frac{6 x^4}{a \sqrt{a x+b x^{2/3}}} \]

[Out]

(-6*x^4)/(a*Sqrt[b*x^(2/3) + a*x]) - (524288*b^9*Sqrt[b*x^(2/3) + a*x])/(29393*a^11) + (1048576*b^10*Sqrt[b*x^
(2/3) + a*x])/(29393*a^12*x^(1/3)) + (393216*b^8*x^(1/3)*Sqrt[b*x^(2/3) + a*x])/(29393*a^10) - (327680*b^7*x^(
2/3)*Sqrt[b*x^(2/3) + a*x])/(29393*a^9) + (40960*b^6*x*Sqrt[b*x^(2/3) + a*x])/(4199*a^8) - (36864*b^5*x^(4/3)*
Sqrt[b*x^(2/3) + a*x])/(4199*a^7) + (33792*b^4*x^(5/3)*Sqrt[b*x^(2/3) + a*x])/(4199*a^6) - (16896*b^3*x^2*Sqrt
[b*x^(2/3) + a*x])/(2261*a^5) + (15840*b^2*x^(7/3)*Sqrt[b*x^(2/3) + a*x])/(2261*a^4) - (880*b*x^(8/3)*Sqrt[b*x
^(2/3) + a*x])/(133*a^3) + (44*x^3*Sqrt[b*x^(2/3) + a*x])/(7*a^2)

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Rubi [A]  time = 0.599225, antiderivative size = 336, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2015, 2016, 2002, 2014} \[ \frac{1048576 b^{10} \sqrt{a x+b x^{2/3}}}{29393 a^{12} \sqrt [3]{x}}-\frac{524288 b^9 \sqrt{a x+b x^{2/3}}}{29393 a^{11}}+\frac{393216 b^8 \sqrt [3]{x} \sqrt{a x+b x^{2/3}}}{29393 a^{10}}-\frac{327680 b^7 x^{2/3} \sqrt{a x+b x^{2/3}}}{29393 a^9}+\frac{40960 b^6 x \sqrt{a x+b x^{2/3}}}{4199 a^8}-\frac{36864 b^5 x^{4/3} \sqrt{a x+b x^{2/3}}}{4199 a^7}+\frac{33792 b^4 x^{5/3} \sqrt{a x+b x^{2/3}}}{4199 a^6}-\frac{16896 b^3 x^2 \sqrt{a x+b x^{2/3}}}{2261 a^5}+\frac{15840 b^2 x^{7/3} \sqrt{a x+b x^{2/3}}}{2261 a^4}-\frac{880 b x^{8/3} \sqrt{a x+b x^{2/3}}}{133 a^3}+\frac{44 x^3 \sqrt{a x+b x^{2/3}}}{7 a^2}-\frac{6 x^4}{a \sqrt{a x+b x^{2/3}}} \]

Antiderivative was successfully verified.

[In]

Int[x^4/(b*x^(2/3) + a*x)^(3/2),x]

[Out]

(-6*x^4)/(a*Sqrt[b*x^(2/3) + a*x]) - (524288*b^9*Sqrt[b*x^(2/3) + a*x])/(29393*a^11) + (1048576*b^10*Sqrt[b*x^
(2/3) + a*x])/(29393*a^12*x^(1/3)) + (393216*b^8*x^(1/3)*Sqrt[b*x^(2/3) + a*x])/(29393*a^10) - (327680*b^7*x^(
2/3)*Sqrt[b*x^(2/3) + a*x])/(29393*a^9) + (40960*b^6*x*Sqrt[b*x^(2/3) + a*x])/(4199*a^8) - (36864*b^5*x^(4/3)*
Sqrt[b*x^(2/3) + a*x])/(4199*a^7) + (33792*b^4*x^(5/3)*Sqrt[b*x^(2/3) + a*x])/(4199*a^6) - (16896*b^3*x^2*Sqrt
[b*x^(2/3) + a*x])/(2261*a^5) + (15840*b^2*x^(7/3)*Sqrt[b*x^(2/3) + a*x])/(2261*a^4) - (880*b*x^(8/3)*Sqrt[b*x
^(2/3) + a*x])/(133*a^3) + (44*x^3*Sqrt[b*x^(2/3) + a*x])/(7*a^2)

Rule 2015

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*(n - j)*(p + 1)), x] + Dist[(c^j*(m + n*p + n - j + 1))/(a*(n - j)*(p + 1)),
Int[(c*x)^(m - j)*(a*x^j + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, j, m, n}, x] &&  !IntegerQ[p] && NeQ[n, j
] && ILtQ[Simplify[(m + n*p + n - j + 1)/(n - j)], 0] && LtQ[p, -1] && (IntegerQ[j] || GtQ[c, 0])

Rule 2016

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, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[
n, j] && ILtQ[Simplify[(m + n*p + n - j + 1)/(n - j)], 0] && NeQ[m + j*p + 1, 0] && (IntegersQ[j, n] || GtQ[c,
0])

Rule 2002

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(a*x^j + b*x^n)^(p + 1)/(a*(j*p + 1)*x^(j -
1)), x] - Dist[(b*(n*p + n - j + 1))/(a*(j*p + 1)), Int[x^(n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, j,
n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && ILtQ[Simplify[(n*p + n - j + 1)/(n - j)], 0] && NeQ[j*p + 1, 0]

Rule 2014

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*(n - j)*(p + 1)), x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && N
eQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

Rubi steps

\begin{align*} \int \frac{x^4}{\left (b x^{2/3}+a x\right )^{3/2}} \, dx &=-\frac{6 x^4}{a \sqrt{b x^{2/3}+a x}}+\frac{22 \int \frac{x^3}{\sqrt{b x^{2/3}+a x}} \, dx}{a}\\ &=-\frac{6 x^4}{a \sqrt{b x^{2/3}+a x}}+\frac{44 x^3 \sqrt{b x^{2/3}+a x}}{7 a^2}-\frac{(440 b) \int \frac{x^{8/3}}{\sqrt{b x^{2/3}+a x}} \, dx}{21 a^2}\\ &=-\frac{6 x^4}{a \sqrt{b x^{2/3}+a x}}-\frac{880 b x^{8/3} \sqrt{b x^{2/3}+a x}}{133 a^3}+\frac{44 x^3 \sqrt{b x^{2/3}+a x}}{7 a^2}+\frac{\left (2640 b^2\right ) \int \frac{x^{7/3}}{\sqrt{b x^{2/3}+a x}} \, dx}{133 a^3}\\ &=-\frac{6 x^4}{a \sqrt{b x^{2/3}+a x}}+\frac{15840 b^2 x^{7/3} \sqrt{b x^{2/3}+a x}}{2261 a^4}-\frac{880 b x^{8/3} \sqrt{b x^{2/3}+a x}}{133 a^3}+\frac{44 x^3 \sqrt{b x^{2/3}+a x}}{7 a^2}-\frac{\left (42240 b^3\right ) \int \frac{x^2}{\sqrt{b x^{2/3}+a x}} \, dx}{2261 a^4}\\ &=-\frac{6 x^4}{a \sqrt{b x^{2/3}+a x}}-\frac{16896 b^3 x^2 \sqrt{b x^{2/3}+a x}}{2261 a^5}+\frac{15840 b^2 x^{7/3} \sqrt{b x^{2/3}+a x}}{2261 a^4}-\frac{880 b x^{8/3} \sqrt{b x^{2/3}+a x}}{133 a^3}+\frac{44 x^3 \sqrt{b x^{2/3}+a x}}{7 a^2}+\frac{\left (5632 b^4\right ) \int \frac{x^{5/3}}{\sqrt{b x^{2/3}+a x}} \, dx}{323 a^5}\\ &=-\frac{6 x^4}{a \sqrt{b x^{2/3}+a x}}+\frac{33792 b^4 x^{5/3} \sqrt{b x^{2/3}+a x}}{4199 a^6}-\frac{16896 b^3 x^2 \sqrt{b x^{2/3}+a x}}{2261 a^5}+\frac{15840 b^2 x^{7/3} \sqrt{b x^{2/3}+a x}}{2261 a^4}-\frac{880 b x^{8/3} \sqrt{b x^{2/3}+a x}}{133 a^3}+\frac{44 x^3 \sqrt{b x^{2/3}+a x}}{7 a^2}-\frac{\left (67584 b^5\right ) \int \frac{x^{4/3}}{\sqrt{b x^{2/3}+a x}} \, dx}{4199 a^6}\\ &=-\frac{6 x^4}{a \sqrt{b x^{2/3}+a x}}-\frac{36864 b^5 x^{4/3} \sqrt{b x^{2/3}+a x}}{4199 a^7}+\frac{33792 b^4 x^{5/3} \sqrt{b x^{2/3}+a x}}{4199 a^6}-\frac{16896 b^3 x^2 \sqrt{b x^{2/3}+a x}}{2261 a^5}+\frac{15840 b^2 x^{7/3} \sqrt{b x^{2/3}+a x}}{2261 a^4}-\frac{880 b x^{8/3} \sqrt{b x^{2/3}+a x}}{133 a^3}+\frac{44 x^3 \sqrt{b x^{2/3}+a x}}{7 a^2}+\frac{\left (61440 b^6\right ) \int \frac{x}{\sqrt{b x^{2/3}+a x}} \, dx}{4199 a^7}\\ &=-\frac{6 x^4}{a \sqrt{b x^{2/3}+a x}}+\frac{40960 b^6 x \sqrt{b x^{2/3}+a x}}{4199 a^8}-\frac{36864 b^5 x^{4/3} \sqrt{b x^{2/3}+a x}}{4199 a^7}+\frac{33792 b^4 x^{5/3} \sqrt{b x^{2/3}+a x}}{4199 a^6}-\frac{16896 b^3 x^2 \sqrt{b x^{2/3}+a x}}{2261 a^5}+\frac{15840 b^2 x^{7/3} \sqrt{b x^{2/3}+a x}}{2261 a^4}-\frac{880 b x^{8/3} \sqrt{b x^{2/3}+a x}}{133 a^3}+\frac{44 x^3 \sqrt{b x^{2/3}+a x}}{7 a^2}-\frac{\left (163840 b^7\right ) \int \frac{x^{2/3}}{\sqrt{b x^{2/3}+a x}} \, dx}{12597 a^8}\\ &=-\frac{6 x^4}{a \sqrt{b x^{2/3}+a x}}-\frac{327680 b^7 x^{2/3} \sqrt{b x^{2/3}+a x}}{29393 a^9}+\frac{40960 b^6 x \sqrt{b x^{2/3}+a x}}{4199 a^8}-\frac{36864 b^5 x^{4/3} \sqrt{b x^{2/3}+a x}}{4199 a^7}+\frac{33792 b^4 x^{5/3} \sqrt{b x^{2/3}+a x}}{4199 a^6}-\frac{16896 b^3 x^2 \sqrt{b x^{2/3}+a x}}{2261 a^5}+\frac{15840 b^2 x^{7/3} \sqrt{b x^{2/3}+a x}}{2261 a^4}-\frac{880 b x^{8/3} \sqrt{b x^{2/3}+a x}}{133 a^3}+\frac{44 x^3 \sqrt{b x^{2/3}+a x}}{7 a^2}+\frac{\left (327680 b^8\right ) \int \frac{\sqrt [3]{x}}{\sqrt{b x^{2/3}+a x}} \, dx}{29393 a^9}\\ &=-\frac{6 x^4}{a \sqrt{b x^{2/3}+a x}}+\frac{393216 b^8 \sqrt [3]{x} \sqrt{b x^{2/3}+a x}}{29393 a^{10}}-\frac{327680 b^7 x^{2/3} \sqrt{b x^{2/3}+a x}}{29393 a^9}+\frac{40960 b^6 x \sqrt{b x^{2/3}+a x}}{4199 a^8}-\frac{36864 b^5 x^{4/3} \sqrt{b x^{2/3}+a x}}{4199 a^7}+\frac{33792 b^4 x^{5/3} \sqrt{b x^{2/3}+a x}}{4199 a^6}-\frac{16896 b^3 x^2 \sqrt{b x^{2/3}+a x}}{2261 a^5}+\frac{15840 b^2 x^{7/3} \sqrt{b x^{2/3}+a x}}{2261 a^4}-\frac{880 b x^{8/3} \sqrt{b x^{2/3}+a x}}{133 a^3}+\frac{44 x^3 \sqrt{b x^{2/3}+a x}}{7 a^2}-\frac{\left (262144 b^9\right ) \int \frac{1}{\sqrt{b x^{2/3}+a x}} \, dx}{29393 a^{10}}\\ &=-\frac{6 x^4}{a \sqrt{b x^{2/3}+a x}}-\frac{524288 b^9 \sqrt{b x^{2/3}+a x}}{29393 a^{11}}+\frac{393216 b^8 \sqrt [3]{x} \sqrt{b x^{2/3}+a x}}{29393 a^{10}}-\frac{327680 b^7 x^{2/3} \sqrt{b x^{2/3}+a x}}{29393 a^9}+\frac{40960 b^6 x \sqrt{b x^{2/3}+a x}}{4199 a^8}-\frac{36864 b^5 x^{4/3} \sqrt{b x^{2/3}+a x}}{4199 a^7}+\frac{33792 b^4 x^{5/3} \sqrt{b x^{2/3}+a x}}{4199 a^6}-\frac{16896 b^3 x^2 \sqrt{b x^{2/3}+a x}}{2261 a^5}+\frac{15840 b^2 x^{7/3} \sqrt{b x^{2/3}+a x}}{2261 a^4}-\frac{880 b x^{8/3} \sqrt{b x^{2/3}+a x}}{133 a^3}+\frac{44 x^3 \sqrt{b x^{2/3}+a x}}{7 a^2}+\frac{\left (524288 b^{10}\right ) \int \frac{1}{\sqrt [3]{x} \sqrt{b x^{2/3}+a x}} \, dx}{88179 a^{11}}\\ &=-\frac{6 x^4}{a \sqrt{b x^{2/3}+a x}}-\frac{524288 b^9 \sqrt{b x^{2/3}+a x}}{29393 a^{11}}+\frac{1048576 b^{10} \sqrt{b x^{2/3}+a x}}{29393 a^{12} \sqrt [3]{x}}+\frac{393216 b^8 \sqrt [3]{x} \sqrt{b x^{2/3}+a x}}{29393 a^{10}}-\frac{327680 b^7 x^{2/3} \sqrt{b x^{2/3}+a x}}{29393 a^9}+\frac{40960 b^6 x \sqrt{b x^{2/3}+a x}}{4199 a^8}-\frac{36864 b^5 x^{4/3} \sqrt{b x^{2/3}+a x}}{4199 a^7}+\frac{33792 b^4 x^{5/3} \sqrt{b x^{2/3}+a x}}{4199 a^6}-\frac{16896 b^3 x^2 \sqrt{b x^{2/3}+a x}}{2261 a^5}+\frac{15840 b^2 x^{7/3} \sqrt{b x^{2/3}+a x}}{2261 a^4}-\frac{880 b x^{8/3} \sqrt{b x^{2/3}+a x}}{133 a^3}+\frac{44 x^3 \sqrt{b x^{2/3}+a x}}{7 a^2}\\ \end{align*}

Mathematica [A]  time = 0.132949, size = 161, normalized size = 0.48 \[ \frac{2 \sqrt [3]{x} \left (5720 a^9 b^2 x^3-6864 a^8 b^3 x^{8/3}+8448 a^7 b^4 x^{7/3}-10752 a^6 b^5 x^2+14336 a^5 b^6 x^{5/3}-20480 a^4 b^7 x^{4/3}-65536 a^2 b^9 x^{2/3}+32768 a^3 b^8 x-4862 a^{10} b x^{10/3}+4199 a^{11} x^{11/3}+262144 a b^{10} \sqrt [3]{x}+524288 b^{11}\right )}{29393 a^{12} \sqrt{a x+b x^{2/3}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(1/3)*(524288*b^11 + 262144*a*b^10*x^(1/3) - 65536*a^2*b^9*x^(2/3) + 32768*a^3*b^8*x - 20480*a^4*b^7*x^(4
/3) + 14336*a^5*b^6*x^(5/3) - 10752*a^6*b^5*x^2 + 8448*a^7*b^4*x^(7/3) - 6864*a^8*b^3*x^(8/3) + 5720*a^9*b^2*x
^3 - 4862*a^10*b*x^(10/3) + 4199*a^11*x^(11/3)))/(29393*a^12*Sqrt[b*x^(2/3) + a*x])

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Maple [A]  time = 0.005, size = 143, normalized size = 0.4 \begin{align*}{\frac{2\,x}{29393\,{a}^{12}} \left ( b+a\sqrt [3]{x} \right ) \left ( 4199\,{x}^{11/3}{a}^{11}-4862\,{x}^{10/3}{a}^{10}b+5720\,{x}^{3}{a}^{9}{b}^{2}-6864\,{x}^{8/3}{a}^{8}{b}^{3}+8448\,{x}^{7/3}{a}^{7}{b}^{4}-10752\,{x}^{2}{a}^{6}{b}^{5}+14336\,{x}^{5/3}{a}^{5}{b}^{6}-20480\,{x}^{4/3}{a}^{4}{b}^{7}+32768\,x{a}^{3}{b}^{8}-65536\,{x}^{2/3}{a}^{2}{b}^{9}+262144\,\sqrt [3]{x}a{b}^{10}+524288\,{b}^{11} \right ) \left ( b{x}^{{\frac{2}{3}}}+ax \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/29393*x*(b+a*x^(1/3))*(4199*x^(11/3)*a^11-4862*x^(10/3)*a^10*b+5720*x^3*a^9*b^2-6864*x^(8/3)*a^8*b^3+8448*x^
(7/3)*a^7*b^4-10752*x^2*a^6*b^5+14336*x^(5/3)*a^5*b^6-20480*x^(4/3)*a^4*b^7+32768*x*a^3*b^8-65536*x^(2/3)*a^2*
b^9+262144*x^(1/3)*a*b^10+524288*b^11)/(b*x^(2/3)+a*x)^(3/2)/a^12

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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Giac [A]  time = 1.23366, size = 289, normalized size = 0.86 \begin{align*} -\frac{1048576 \, b^{\frac{21}{2}}}{29393 \, a^{12}} + \frac{6 \, b^{11}}{\sqrt{a x^{\frac{1}{3}} + b} a^{12}} + \frac{2 \,{\left (4199 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{21}{2}} a^{240} - 51051 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{19}{2}} a^{240} b + 285285 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{17}{2}} a^{240} b^{2} - 969969 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{15}{2}} a^{240} b^{3} + 2238390 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{13}{2}} a^{240} b^{4} - 3703518 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{11}{2}} a^{240} b^{5} + 4526522 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{9}{2}} a^{240} b^{6} - 4157010 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{7}{2}} a^{240} b^{7} + 2909907 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} a^{240} b^{8} - 1616615 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} a^{240} b^{9} + 969969 \, \sqrt{a x^{\frac{1}{3}} + b} a^{240} b^{10}\right )}}{29393 \, a^{252}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1048576/29393*b^(21/2)/a^12 + 6*b^11/(sqrt(a*x^(1/3) + b)*a^12) + 2/29393*(4199*(a*x^(1/3) + b)^(21/2)*a^240
- 51051*(a*x^(1/3) + b)^(19/2)*a^240*b + 285285*(a*x^(1/3) + b)^(17/2)*a^240*b^2 - 969969*(a*x^(1/3) + b)^(15/
2)*a^240*b^3 + 2238390*(a*x^(1/3) + b)^(13/2)*a^240*b^4 - 3703518*(a*x^(1/3) + b)^(11/2)*a^240*b^5 + 4526522*(
a*x^(1/3) + b)^(9/2)*a^240*b^6 - 4157010*(a*x^(1/3) + b)^(7/2)*a^240*b^7 + 2909907*(a*x^(1/3) + b)^(5/2)*a^240
*b^8 - 1616615*(a*x^(1/3) + b)^(3/2)*a^240*b^9 + 969969*sqrt(a*x^(1/3) + b)*a^240*b^10)/a^252