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

Optimal. Leaf size=343 \[ -\frac{1048576 b^{11} \left (a x+b x^{2/3}\right )^{5/2}}{152108775 a^{12} x^{5/3}}+\frac{524288 b^{10} \left (a x+b x^{2/3}\right )^{5/2}}{30421755 a^{11} x^{4/3}}-\frac{131072 b^9 \left (a x+b x^{2/3}\right )^{5/2}}{4345965 a^{10} x}+\frac{65536 b^8 \left (a x+b x^{2/3}\right )^{5/2}}{1448655 a^9 x^{2/3}}-\frac{90112 b^7 \left (a x+b x^{2/3}\right )^{5/2}}{1448655 a^8 \sqrt [3]{x}}+\frac{45056 b^6 \left (a x+b x^{2/3}\right )^{5/2}}{557175 a^7}-\frac{11264 b^5 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{5/2}}{111435 a^6}+\frac{5632 b^4 x^{2/3} \left (a x+b x^{2/3}\right )^{5/2}}{45885 a^5}-\frac{352 b^3 x \left (a x+b x^{2/3}\right )^{5/2}}{2415 a^4}+\frac{176 b^2 x^{4/3} \left (a x+b x^{2/3}\right )^{5/2}}{1035 a^3}-\frac{44 b x^{5/3} \left (a x+b x^{2/3}\right )^{5/2}}{225 a^2}+\frac{2 x^2 \left (a x+b x^{2/3}\right )^{5/2}}{9 a} \]

[Out]

(45056*b^6*(b*x^(2/3) + a*x)^(5/2))/(557175*a^7) - (1048576*b^11*(b*x^(2/3) + a*x)^(5/2))/(152108775*a^12*x^(5
/3)) + (524288*b^10*(b*x^(2/3) + a*x)^(5/2))/(30421755*a^11*x^(4/3)) - (131072*b^9*(b*x^(2/3) + a*x)^(5/2))/(4
345965*a^10*x) + (65536*b^8*(b*x^(2/3) + a*x)^(5/2))/(1448655*a^9*x^(2/3)) - (90112*b^7*(b*x^(2/3) + a*x)^(5/2
))/(1448655*a^8*x^(1/3)) - (11264*b^5*x^(1/3)*(b*x^(2/3) + a*x)^(5/2))/(111435*a^6) + (5632*b^4*x^(2/3)*(b*x^(
2/3) + a*x)^(5/2))/(45885*a^5) - (352*b^3*x*(b*x^(2/3) + a*x)^(5/2))/(2415*a^4) + (176*b^2*x^(4/3)*(b*x^(2/3)
+ a*x)^(5/2))/(1035*a^3) - (44*b*x^(5/3)*(b*x^(2/3) + a*x)^(5/2))/(225*a^2) + (2*x^2*(b*x^(2/3) + a*x)^(5/2))/
(9*a)

________________________________________________________________________________________

Rubi [A]  time = 0.616286, antiderivative size = 343, 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 = {2016, 2002, 2014} \[ -\frac{1048576 b^{11} \left (a x+b x^{2/3}\right )^{5/2}}{152108775 a^{12} x^{5/3}}+\frac{524288 b^{10} \left (a x+b x^{2/3}\right )^{5/2}}{30421755 a^{11} x^{4/3}}-\frac{131072 b^9 \left (a x+b x^{2/3}\right )^{5/2}}{4345965 a^{10} x}+\frac{65536 b^8 \left (a x+b x^{2/3}\right )^{5/2}}{1448655 a^9 x^{2/3}}-\frac{90112 b^7 \left (a x+b x^{2/3}\right )^{5/2}}{1448655 a^8 \sqrt [3]{x}}+\frac{45056 b^6 \left (a x+b x^{2/3}\right )^{5/2}}{557175 a^7}-\frac{11264 b^5 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{5/2}}{111435 a^6}+\frac{5632 b^4 x^{2/3} \left (a x+b x^{2/3}\right )^{5/2}}{45885 a^5}-\frac{352 b^3 x \left (a x+b x^{2/3}\right )^{5/2}}{2415 a^4}+\frac{176 b^2 x^{4/3} \left (a x+b x^{2/3}\right )^{5/2}}{1035 a^3}-\frac{44 b x^{5/3} \left (a x+b x^{2/3}\right )^{5/2}}{225 a^2}+\frac{2 x^2 \left (a x+b x^{2/3}\right )^{5/2}}{9 a} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(b*x^(2/3) + a*x)^(3/2),x]

[Out]

(45056*b^6*(b*x^(2/3) + a*x)^(5/2))/(557175*a^7) - (1048576*b^11*(b*x^(2/3) + a*x)^(5/2))/(152108775*a^12*x^(5
/3)) + (524288*b^10*(b*x^(2/3) + a*x)^(5/2))/(30421755*a^11*x^(4/3)) - (131072*b^9*(b*x^(2/3) + a*x)^(5/2))/(4
345965*a^10*x) + (65536*b^8*(b*x^(2/3) + a*x)^(5/2))/(1448655*a^9*x^(2/3)) - (90112*b^7*(b*x^(2/3) + a*x)^(5/2
))/(1448655*a^8*x^(1/3)) - (11264*b^5*x^(1/3)*(b*x^(2/3) + a*x)^(5/2))/(111435*a^6) + (5632*b^4*x^(2/3)*(b*x^(
2/3) + a*x)^(5/2))/(45885*a^5) - (352*b^3*x*(b*x^(2/3) + a*x)^(5/2))/(2415*a^4) + (176*b^2*x^(4/3)*(b*x^(2/3)
+ a*x)^(5/2))/(1035*a^3) - (44*b*x^(5/3)*(b*x^(2/3) + a*x)^(5/2))/(225*a^2) + (2*x^2*(b*x^(2/3) + a*x)^(5/2))/
(9*a)

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 x^2 \left (b x^{2/3}+a x\right )^{3/2} \, dx &=\frac{2 x^2 \left (b x^{2/3}+a x\right )^{5/2}}{9 a}-\frac{(22 b) \int x^{5/3} \left (b x^{2/3}+a x\right )^{3/2} \, dx}{27 a}\\ &=-\frac{44 b x^{5/3} \left (b x^{2/3}+a x\right )^{5/2}}{225 a^2}+\frac{2 x^2 \left (b x^{2/3}+a x\right )^{5/2}}{9 a}+\frac{\left (88 b^2\right ) \int x^{4/3} \left (b x^{2/3}+a x\right )^{3/2} \, dx}{135 a^2}\\ &=\frac{176 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{5/2}}{1035 a^3}-\frac{44 b x^{5/3} \left (b x^{2/3}+a x\right )^{5/2}}{225 a^2}+\frac{2 x^2 \left (b x^{2/3}+a x\right )^{5/2}}{9 a}-\frac{\left (176 b^3\right ) \int x \left (b x^{2/3}+a x\right )^{3/2} \, dx}{345 a^3}\\ &=-\frac{352 b^3 x \left (b x^{2/3}+a x\right )^{5/2}}{2415 a^4}+\frac{176 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{5/2}}{1035 a^3}-\frac{44 b x^{5/3} \left (b x^{2/3}+a x\right )^{5/2}}{225 a^2}+\frac{2 x^2 \left (b x^{2/3}+a x\right )^{5/2}}{9 a}+\frac{\left (2816 b^4\right ) \int x^{2/3} \left (b x^{2/3}+a x\right )^{3/2} \, dx}{7245 a^4}\\ &=\frac{5632 b^4 x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{45885 a^5}-\frac{352 b^3 x \left (b x^{2/3}+a x\right )^{5/2}}{2415 a^4}+\frac{176 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{5/2}}{1035 a^3}-\frac{44 b x^{5/3} \left (b x^{2/3}+a x\right )^{5/2}}{225 a^2}+\frac{2 x^2 \left (b x^{2/3}+a x\right )^{5/2}}{9 a}-\frac{\left (5632 b^5\right ) \int \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{3/2} \, dx}{19665 a^5}\\ &=-\frac{11264 b^5 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{5/2}}{111435 a^6}+\frac{5632 b^4 x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{45885 a^5}-\frac{352 b^3 x \left (b x^{2/3}+a x\right )^{5/2}}{2415 a^4}+\frac{176 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{5/2}}{1035 a^3}-\frac{44 b x^{5/3} \left (b x^{2/3}+a x\right )^{5/2}}{225 a^2}+\frac{2 x^2 \left (b x^{2/3}+a x\right )^{5/2}}{9 a}+\frac{\left (22528 b^6\right ) \int \left (b x^{2/3}+a x\right )^{3/2} \, dx}{111435 a^6}\\ &=\frac{45056 b^6 \left (b x^{2/3}+a x\right )^{5/2}}{557175 a^7}-\frac{11264 b^5 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{5/2}}{111435 a^6}+\frac{5632 b^4 x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{45885 a^5}-\frac{352 b^3 x \left (b x^{2/3}+a x\right )^{5/2}}{2415 a^4}+\frac{176 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{5/2}}{1035 a^3}-\frac{44 b x^{5/3} \left (b x^{2/3}+a x\right )^{5/2}}{225 a^2}+\frac{2 x^2 \left (b x^{2/3}+a x\right )^{5/2}}{9 a}-\frac{\left (45056 b^7\right ) \int \frac{\left (b x^{2/3}+a x\right )^{3/2}}{\sqrt [3]{x}} \, dx}{334305 a^7}\\ &=\frac{45056 b^6 \left (b x^{2/3}+a x\right )^{5/2}}{557175 a^7}-\frac{90112 b^7 \left (b x^{2/3}+a x\right )^{5/2}}{1448655 a^8 \sqrt [3]{x}}-\frac{11264 b^5 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{5/2}}{111435 a^6}+\frac{5632 b^4 x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{45885 a^5}-\frac{352 b^3 x \left (b x^{2/3}+a x\right )^{5/2}}{2415 a^4}+\frac{176 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{5/2}}{1035 a^3}-\frac{44 b x^{5/3} \left (b x^{2/3}+a x\right )^{5/2}}{225 a^2}+\frac{2 x^2 \left (b x^{2/3}+a x\right )^{5/2}}{9 a}+\frac{\left (360448 b^8\right ) \int \frac{\left (b x^{2/3}+a x\right )^{3/2}}{x^{2/3}} \, dx}{4345965 a^8}\\ &=\frac{45056 b^6 \left (b x^{2/3}+a x\right )^{5/2}}{557175 a^7}+\frac{65536 b^8 \left (b x^{2/3}+a x\right )^{5/2}}{1448655 a^9 x^{2/3}}-\frac{90112 b^7 \left (b x^{2/3}+a x\right )^{5/2}}{1448655 a^8 \sqrt [3]{x}}-\frac{11264 b^5 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{5/2}}{111435 a^6}+\frac{5632 b^4 x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{45885 a^5}-\frac{352 b^3 x \left (b x^{2/3}+a x\right )^{5/2}}{2415 a^4}+\frac{176 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{5/2}}{1035 a^3}-\frac{44 b x^{5/3} \left (b x^{2/3}+a x\right )^{5/2}}{225 a^2}+\frac{2 x^2 \left (b x^{2/3}+a x\right )^{5/2}}{9 a}-\frac{\left (65536 b^9\right ) \int \frac{\left (b x^{2/3}+a x\right )^{3/2}}{x} \, dx}{1448655 a^9}\\ &=\frac{45056 b^6 \left (b x^{2/3}+a x\right )^{5/2}}{557175 a^7}-\frac{131072 b^9 \left (b x^{2/3}+a x\right )^{5/2}}{4345965 a^{10} x}+\frac{65536 b^8 \left (b x^{2/3}+a x\right )^{5/2}}{1448655 a^9 x^{2/3}}-\frac{90112 b^7 \left (b x^{2/3}+a x\right )^{5/2}}{1448655 a^8 \sqrt [3]{x}}-\frac{11264 b^5 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{5/2}}{111435 a^6}+\frac{5632 b^4 x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{45885 a^5}-\frac{352 b^3 x \left (b x^{2/3}+a x\right )^{5/2}}{2415 a^4}+\frac{176 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{5/2}}{1035 a^3}-\frac{44 b x^{5/3} \left (b x^{2/3}+a x\right )^{5/2}}{225 a^2}+\frac{2 x^2 \left (b x^{2/3}+a x\right )^{5/2}}{9 a}+\frac{\left (262144 b^{10}\right ) \int \frac{\left (b x^{2/3}+a x\right )^{3/2}}{x^{4/3}} \, dx}{13037895 a^{10}}\\ &=\frac{45056 b^6 \left (b x^{2/3}+a x\right )^{5/2}}{557175 a^7}+\frac{524288 b^{10} \left (b x^{2/3}+a x\right )^{5/2}}{30421755 a^{11} x^{4/3}}-\frac{131072 b^9 \left (b x^{2/3}+a x\right )^{5/2}}{4345965 a^{10} x}+\frac{65536 b^8 \left (b x^{2/3}+a x\right )^{5/2}}{1448655 a^9 x^{2/3}}-\frac{90112 b^7 \left (b x^{2/3}+a x\right )^{5/2}}{1448655 a^8 \sqrt [3]{x}}-\frac{11264 b^5 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{5/2}}{111435 a^6}+\frac{5632 b^4 x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{45885 a^5}-\frac{352 b^3 x \left (b x^{2/3}+a x\right )^{5/2}}{2415 a^4}+\frac{176 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{5/2}}{1035 a^3}-\frac{44 b x^{5/3} \left (b x^{2/3}+a x\right )^{5/2}}{225 a^2}+\frac{2 x^2 \left (b x^{2/3}+a x\right )^{5/2}}{9 a}-\frac{\left (524288 b^{11}\right ) \int \frac{\left (b x^{2/3}+a x\right )^{3/2}}{x^{5/3}} \, dx}{91265265 a^{11}}\\ &=\frac{45056 b^6 \left (b x^{2/3}+a x\right )^{5/2}}{557175 a^7}-\frac{1048576 b^{11} \left (b x^{2/3}+a x\right )^{5/2}}{152108775 a^{12} x^{5/3}}+\frac{524288 b^{10} \left (b x^{2/3}+a x\right )^{5/2}}{30421755 a^{11} x^{4/3}}-\frac{131072 b^9 \left (b x^{2/3}+a x\right )^{5/2}}{4345965 a^{10} x}+\frac{65536 b^8 \left (b x^{2/3}+a x\right )^{5/2}}{1448655 a^9 x^{2/3}}-\frac{90112 b^7 \left (b x^{2/3}+a x\right )^{5/2}}{1448655 a^8 \sqrt [3]{x}}-\frac{11264 b^5 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{5/2}}{111435 a^6}+\frac{5632 b^4 x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{45885 a^5}-\frac{352 b^3 x \left (b x^{2/3}+a x\right )^{5/2}}{2415 a^4}+\frac{176 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{5/2}}{1035 a^3}-\frac{44 b x^{5/3} \left (b x^{2/3}+a x\right )^{5/2}}{225 a^2}+\frac{2 x^2 \left (b x^{2/3}+a x\right )^{5/2}}{9 a}\\ \end{align*}

Mathematica [A]  time = 0.131135, size = 172, normalized size = 0.5 \[ \frac{2 \left (a \sqrt [3]{x}+b\right )^2 \sqrt{a x+b x^{2/3}} \left (12932920 a^9 b^2 x^3-11085360 a^8 b^3 x^{8/3}+9335040 a^7 b^4 x^{7/3}-7687680 a^6 b^5 x^2+6150144 a^5 b^6 x^{5/3}-4730880 a^4 b^7 x^{4/3}-2293760 a^2 b^9 x^{2/3}+3440640 a^3 b^8 x-14872858 a^{10} b x^{10/3}[phone] a^{11} x^{11/3}+1310720 a b^{10} \sqrt [3]{x}-524288 b^{11}\right )}{152108775 a^{12} \sqrt [3]{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b + a*x^(1/3))^2*Sqrt[b*x^(2/3) + a*x]*(-524288*b^11 + 1310720*a*b^10*x^(1/3) - 2293760*a^2*b^9*x^(2/3) +
3440640*a^3*b^8*x - 4730880*a^4*b^7*x^(4/3) + 6150144*a^5*b^6*x^(5/3) - 7687680*a^6*b^5*x^2 + 9335040*a^7*b^4*
x^(7/3) - 11085360*a^8*b^3*x^(8/3) + 12932920*a^9*b^2*x^3 - 14872858*a^10*b*x^(10/3) + 16900975*a^11*x^(11/3))
)/(152108775*a^12*x^(1/3))

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 145, normalized size = 0.4 \begin{align*}{\frac{2}{152108775\,x{a}^{12}} \left ( b{x}^{{\frac{2}{3}}}+ax \right ) ^{{\frac{3}{2}}} \left ( b+a\sqrt [3]{x} \right ) \left ( 16900975\,{x}^{11/3}{a}^{11}-14872858\,{x}^{10/3}{a}^{10}b+12932920\,{x}^{3}{a}^{9}{b}^{2}-11085360\,{x}^{8/3}{a}^{8}{b}^{3}+9335040\,{x}^{7/3}{a}^{7}{b}^{4}-7687680\,{x}^{2}{a}^{6}{b}^{5}+6150144\,{x}^{5/3}{a}^{5}{b}^{6}-4730880\,{x}^{4/3}{a}^{4}{b}^{7}+3440640\,x{a}^{3}{b}^{8}-2293760\,{x}^{2/3}{a}^{2}{b}^{9}+1310720\,\sqrt [3]{x}a{b}^{10}-524288\,{b}^{11} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/152108775*(b*x^(2/3)+a*x)^(3/2)*(b+a*x^(1/3))*(16900975*x^(11/3)*a^11-14872858*x^(10/3)*a^10*b+12932920*x^3*
a^9*b^2-11085360*x^(8/3)*a^8*b^3+9335040*x^(7/3)*a^7*b^4-7687680*x^2*a^6*b^5+6150144*x^(5/3)*a^5*b^6-4730880*x
^(4/3)*a^4*b^7+3440640*x*a^3*b^8-2293760*x^(2/3)*a^2*b^9+1310720*x^(1/3)*a*b^10-524288*b^11)/x/a^12

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.18264, size = 508, normalized size = 1.48 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/16900975*b*(524288*b^(25/2)/a^12 + (2028117*(a*x^(1/3) + b)^(25/2) - 24249225*(a*x^(1/3) + b)^(23/2)*b + 132
793375*(a*x^(1/3) + b)^(21/2)*b^2 - 440314875*(a*x^(1/3) + b)^(19/2)*b^3 + 984233250*(a*x^(1/3) + b)^(17/2)*b^
4 - 1561650090*(a*x^(1/3) + b)^(15/2)*b^5 + 1801903950*(a*x^(1/3) + b)^(13/2)*b^6 - 1521087750*(a*x^(1/3) + b)
^(11/2)*b^7 + 929553625*(a*x^(1/3) + b)^(9/2)*b^8 - 398380125*(a*x^(1/3) + b)^(7/2)*b^9 + 111546435*(a*x^(1/3)
+ b)^(5/2)*b^10 - 16900975*(a*x^(1/3) + b)^(3/2)*b^11)/a^12) - 2/152108775*a*(4194304*b^(27/2)/a^13 - (169009
75*(a*x^(1/3) + b)^(27/2) - 219036636*(a*x^(1/3) + b)^(25/2)*b + 1309458150*(a*x^(1/3) + b)^(23/2)*b^2 - 47805
61500*(a*x^(1/3) + b)^(21/2)*b^3 + 11888501625*(a*x^(1/3) + b)^(19/2)*b^4 - 21259438200*(a*x^(1/3) + b)^(17/2)
*b^5 + 28109701620*(a*x^(1/3) + b)^(15/2)*b^6 - 27800803800*(a*x^(1/3) + b)^(13/2)*b^7 + 20534684625*(a*x^(1/3
) + b)^(11/2)*b^8 - 11154643500*(a*x^(1/3) + b)^(9/2)*b^9 + 4302505350*(a*x^(1/3) + b)^(7/2)*b^10 - 1095183180
*(a*x^(1/3) + b)^(5/2)*b^11 + 152108775*(a*x^(1/3) + b)^(3/2)*b^12)/a^13)