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

Optimal. Leaf size=401 \[ -\frac{16777216 b^{13} \sqrt{a x+b x^{2/3}}}{11700675 a^{14} \sqrt [3]{x}}+\frac{8388608 b^{12} \sqrt{a x+b x^{2/3}}}{11700675 a^{13}}-\frac{2097152 b^{11} \sqrt [3]{x} \sqrt{a x+b x^{2/3}}}{3900225 a^{12}}+\frac{1048576 b^{10} x^{2/3} \sqrt{a x+b x^{2/3}}}{2340135 a^{11}}-\frac{131072 b^9 x \sqrt{a x+b x^{2/3}}}{334305 a^{10}}+\frac{65536 b^8 x^{4/3} \sqrt{a x+b x^{2/3}}}{185725 a^9}-\frac{180224 b^7 x^{5/3} \sqrt{a x+b x^{2/3}}}{557175 a^8}+\frac{1171456 b^6 x^2 \sqrt{a x+b x^{2/3}}}{3900225 a^7}-\frac{73216 b^5 x^{7/3} \sqrt{a x+b x^{2/3}}}{260015 a^6}+\frac{36608 b^4 x^{8/3} \sqrt{a x+b x^{2/3}}}{137655 a^5}-\frac{9152 b^3 x^3 \sqrt{a x+b x^{2/3}}}{36225 a^4}+\frac{416 b^2 x^{10/3} \sqrt{a x+b x^{2/3}}}{1725 a^3}-\frac{52 b x^{11/3} \sqrt{a x+b x^{2/3}}}{225 a^2}+\frac{2 x^4 \sqrt{a x+b x^{2/3}}}{9 a} \]

[Out]

(8388608*b^12*Sqrt[b*x^(2/3) + a*x])/(11700675*a^13) - (16777216*b^13*Sqrt[b*x^(2/3) + a*x])/(11700675*a^14*x^
(1/3)) - (2097152*b^11*x^(1/3)*Sqrt[b*x^(2/3) + a*x])/(3900225*a^12) + (1048576*b^10*x^(2/3)*Sqrt[b*x^(2/3) +
a*x])/(2340135*a^11) - (131072*b^9*x*Sqrt[b*x^(2/3) + a*x])/(334305*a^10) + (65536*b^8*x^(4/3)*Sqrt[b*x^(2/3)
+ a*x])/(185725*a^9) - (180224*b^7*x^(5/3)*Sqrt[b*x^(2/3) + a*x])/(557175*a^8) + (1171456*b^6*x^2*Sqrt[b*x^(2/
3) + a*x])/(3900225*a^7) - (73216*b^5*x^(7/3)*Sqrt[b*x^(2/3) + a*x])/(260015*a^6) + (36608*b^4*x^(8/3)*Sqrt[b*
x^(2/3) + a*x])/(137655*a^5) - (9152*b^3*x^3*Sqrt[b*x^(2/3) + a*x])/(36225*a^4) + (416*b^2*x^(10/3)*Sqrt[b*x^(
2/3) + a*x])/(1725*a^3) - (52*b*x^(11/3)*Sqrt[b*x^(2/3) + a*x])/(225*a^2) + (2*x^4*Sqrt[b*x^(2/3) + a*x])/(9*a
)

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Rubi [A]  time = 0.728113, antiderivative size = 401, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2016, 2002, 2014} \[ -\frac{16777216 b^{13} \sqrt{a x+b x^{2/3}}}{11700675 a^{14} \sqrt [3]{x}}+\frac{8388608 b^{12} \sqrt{a x+b x^{2/3}}}{11700675 a^{13}}-\frac{2097152 b^{11} \sqrt [3]{x} \sqrt{a x+b x^{2/3}}}{3900225 a^{12}}+\frac{1048576 b^{10} x^{2/3} \sqrt{a x+b x^{2/3}}}{2340135 a^{11}}-\frac{131072 b^9 x \sqrt{a x+b x^{2/3}}}{334305 a^{10}}+\frac{65536 b^8 x^{4/3} \sqrt{a x+b x^{2/3}}}{185725 a^9}-\frac{180224 b^7 x^{5/3} \sqrt{a x+b x^{2/3}}}{557175 a^8}+\frac{1171456 b^6 x^2 \sqrt{a x+b x^{2/3}}}{3900225 a^7}-\frac{73216 b^5 x^{7/3} \sqrt{a x+b x^{2/3}}}{260015 a^6}+\frac{36608 b^4 x^{8/3} \sqrt{a x+b x^{2/3}}}{137655 a^5}-\frac{9152 b^3 x^3 \sqrt{a x+b x^{2/3}}}{36225 a^4}+\frac{416 b^2 x^{10/3} \sqrt{a x+b x^{2/3}}}{1725 a^3}-\frac{52 b x^{11/3} \sqrt{a x+b x^{2/3}}}{225 a^2}+\frac{2 x^4 \sqrt{a x+b x^{2/3}}}{9 a} \]

Antiderivative was successfully verified.

[In]

Int[x^4/Sqrt[b*x^(2/3) + a*x],x]

[Out]

(8388608*b^12*Sqrt[b*x^(2/3) + a*x])/(11700675*a^13) - (16777216*b^13*Sqrt[b*x^(2/3) + a*x])/(11700675*a^14*x^
(1/3)) - (2097152*b^11*x^(1/3)*Sqrt[b*x^(2/3) + a*x])/(3900225*a^12) + (1048576*b^10*x^(2/3)*Sqrt[b*x^(2/3) +
a*x])/(2340135*a^11) - (131072*b^9*x*Sqrt[b*x^(2/3) + a*x])/(334305*a^10) + (65536*b^8*x^(4/3)*Sqrt[b*x^(2/3)
+ a*x])/(185725*a^9) - (180224*b^7*x^(5/3)*Sqrt[b*x^(2/3) + a*x])/(557175*a^8) + (1171456*b^6*x^2*Sqrt[b*x^(2/
3) + a*x])/(3900225*a^7) - (73216*b^5*x^(7/3)*Sqrt[b*x^(2/3) + a*x])/(260015*a^6) + (36608*b^4*x^(8/3)*Sqrt[b*
x^(2/3) + a*x])/(137655*a^5) - (9152*b^3*x^3*Sqrt[b*x^(2/3) + a*x])/(36225*a^4) + (416*b^2*x^(10/3)*Sqrt[b*x^(
2/3) + a*x])/(1725*a^3) - (52*b*x^(11/3)*Sqrt[b*x^(2/3) + a*x])/(225*a^2) + (2*x^4*Sqrt[b*x^(2/3) + a*x])/(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 \frac{x^4}{\sqrt{b x^{2/3}+a x}} \, dx &=\frac{2 x^4 \sqrt{b x^{2/3}+a x}}{9 a}-\frac{(26 b) \int \frac{x^{11/3}}{\sqrt{b x^{2/3}+a x}} \, dx}{27 a}\\ &=-\frac{52 b x^{11/3} \sqrt{b x^{2/3}+a x}}{225 a^2}+\frac{2 x^4 \sqrt{b x^{2/3}+a x}}{9 a}+\frac{\left (208 b^2\right ) \int \frac{x^{10/3}}{\sqrt{b x^{2/3}+a x}} \, dx}{225 a^2}\\ &=\frac{416 b^2 x^{10/3} \sqrt{b x^{2/3}+a x}}{1725 a^3}-\frac{52 b x^{11/3} \sqrt{b x^{2/3}+a x}}{225 a^2}+\frac{2 x^4 \sqrt{b x^{2/3}+a x}}{9 a}-\frac{\left (4576 b^3\right ) \int \frac{x^3}{\sqrt{b x^{2/3}+a x}} \, dx}{5175 a^3}\\ &=-\frac{9152 b^3 x^3 \sqrt{b x^{2/3}+a x}}{36225 a^4}+\frac{416 b^2 x^{10/3} \sqrt{b x^{2/3}+a x}}{1725 a^3}-\frac{52 b x^{11/3} \sqrt{b x^{2/3}+a x}}{225 a^2}+\frac{2 x^4 \sqrt{b x^{2/3}+a x}}{9 a}+\frac{\left (18304 b^4\right ) \int \frac{x^{8/3}}{\sqrt{b x^{2/3}+a x}} \, dx}{21735 a^4}\\ &=\frac{36608 b^4 x^{8/3} \sqrt{b x^{2/3}+a x}}{137655 a^5}-\frac{9152 b^3 x^3 \sqrt{b x^{2/3}+a x}}{36225 a^4}+\frac{416 b^2 x^{10/3} \sqrt{b x^{2/3}+a x}}{1725 a^3}-\frac{52 b x^{11/3} \sqrt{b x^{2/3}+a x}}{225 a^2}+\frac{2 x^4 \sqrt{b x^{2/3}+a x}}{9 a}-\frac{\left (36608 b^5\right ) \int \frac{x^{7/3}}{\sqrt{b x^{2/3}+a x}} \, dx}{45885 a^5}\\ &=-\frac{73216 b^5 x^{7/3} \sqrt{b x^{2/3}+a x}}{260015 a^6}+\frac{36608 b^4 x^{8/3} \sqrt{b x^{2/3}+a x}}{137655 a^5}-\frac{9152 b^3 x^3 \sqrt{b x^{2/3}+a x}}{36225 a^4}+\frac{416 b^2 x^{10/3} \sqrt{b x^{2/3}+a x}}{1725 a^3}-\frac{52 b x^{11/3} \sqrt{b x^{2/3}+a x}}{225 a^2}+\frac{2 x^4 \sqrt{b x^{2/3}+a x}}{9 a}+\frac{\left (585728 b^6\right ) \int \frac{x^2}{\sqrt{b x^{2/3}+a x}} \, dx}{780045 a^6}\\ &=\frac{1171456 b^6 x^2 \sqrt{b x^{2/3}+a x}}{3900225 a^7}-\frac{73216 b^5 x^{7/3} \sqrt{b x^{2/3}+a x}}{260015 a^6}+\frac{36608 b^4 x^{8/3} \sqrt{b x^{2/3}+a x}}{137655 a^5}-\frac{9152 b^3 x^3 \sqrt{b x^{2/3}+a x}}{36225 a^4}+\frac{416 b^2 x^{10/3} \sqrt{b x^{2/3}+a x}}{1725 a^3}-\frac{52 b x^{11/3} \sqrt{b x^{2/3}+a x}}{225 a^2}+\frac{2 x^4 \sqrt{b x^{2/3}+a x}}{9 a}-\frac{\left (1171456 b^7\right ) \int \frac{x^{5/3}}{\sqrt{b x^{2/3}+a x}} \, dx}{1671525 a^7}\\ &=-\frac{180224 b^7 x^{5/3} \sqrt{b x^{2/3}+a x}}{557175 a^8}+\frac{1171456 b^6 x^2 \sqrt{b x^{2/3}+a x}}{3900225 a^7}-\frac{73216 b^5 x^{7/3} \sqrt{b x^{2/3}+a x}}{260015 a^6}+\frac{36608 b^4 x^{8/3} \sqrt{b x^{2/3}+a x}}{137655 a^5}-\frac{9152 b^3 x^3 \sqrt{b x^{2/3}+a x}}{36225 a^4}+\frac{416 b^2 x^{10/3} \sqrt{b x^{2/3}+a x}}{1725 a^3}-\frac{52 b x^{11/3} \sqrt{b x^{2/3}+a x}}{225 a^2}+\frac{2 x^4 \sqrt{b x^{2/3}+a x}}{9 a}+\frac{\left (360448 b^8\right ) \int \frac{x^{4/3}}{\sqrt{b x^{2/3}+a x}} \, dx}{557175 a^8}\\ &=\frac{65536 b^8 x^{4/3} \sqrt{b x^{2/3}+a x}}{185725 a^9}-\frac{180224 b^7 x^{5/3} \sqrt{b x^{2/3}+a x}}{557175 a^8}+\frac{1171456 b^6 x^2 \sqrt{b x^{2/3}+a x}}{3900225 a^7}-\frac{73216 b^5 x^{7/3} \sqrt{b x^{2/3}+a x}}{260015 a^6}+\frac{36608 b^4 x^{8/3} \sqrt{b x^{2/3}+a x}}{137655 a^5}-\frac{9152 b^3 x^3 \sqrt{b x^{2/3}+a x}}{36225 a^4}+\frac{416 b^2 x^{10/3} \sqrt{b x^{2/3}+a x}}{1725 a^3}-\frac{52 b x^{11/3} \sqrt{b x^{2/3}+a x}}{225 a^2}+\frac{2 x^4 \sqrt{b x^{2/3}+a x}}{9 a}-\frac{\left (65536 b^9\right ) \int \frac{x}{\sqrt{b x^{2/3}+a x}} \, dx}{111435 a^9}\\ &=-\frac{131072 b^9 x \sqrt{b x^{2/3}+a x}}{334305 a^{10}}+\frac{65536 b^8 x^{4/3} \sqrt{b x^{2/3}+a x}}{185725 a^9}-\frac{180224 b^7 x^{5/3} \sqrt{b x^{2/3}+a x}}{557175 a^8}+\frac{1171456 b^6 x^2 \sqrt{b x^{2/3}+a x}}{3900225 a^7}-\frac{73216 b^5 x^{7/3} \sqrt{b x^{2/3}+a x}}{260015 a^6}+\frac{36608 b^4 x^{8/3} \sqrt{b x^{2/3}+a x}}{137655 a^5}-\frac{9152 b^3 x^3 \sqrt{b x^{2/3}+a x}}{36225 a^4}+\frac{416 b^2 x^{10/3} \sqrt{b x^{2/3}+a x}}{1725 a^3}-\frac{52 b x^{11/3} \sqrt{b x^{2/3}+a x}}{225 a^2}+\frac{2 x^4 \sqrt{b x^{2/3}+a x}}{9 a}+\frac{\left (524288 b^{10}\right ) \int \frac{x^{2/3}}{\sqrt{b x^{2/3}+a x}} \, dx}{1002915 a^{10}}\\ &=\frac{1048576 b^{10} x^{2/3} \sqrt{b x^{2/3}+a x}}{2340135 a^{11}}-\frac{131072 b^9 x \sqrt{b x^{2/3}+a x}}{334305 a^{10}}+\frac{65536 b^8 x^{4/3} \sqrt{b x^{2/3}+a x}}{185725 a^9}-\frac{180224 b^7 x^{5/3} \sqrt{b x^{2/3}+a x}}{557175 a^8}+\frac{1171456 b^6 x^2 \sqrt{b x^{2/3}+a x}}{3900225 a^7}-\frac{73216 b^5 x^{7/3} \sqrt{b x^{2/3}+a x}}{260015 a^6}+\frac{36608 b^4 x^{8/3} \sqrt{b x^{2/3}+a x}}{137655 a^5}-\frac{9152 b^3 x^3 \sqrt{b x^{2/3}+a x}}{36225 a^4}+\frac{416 b^2 x^{10/3} \sqrt{b x^{2/3}+a x}}{1725 a^3}-\frac{52 b x^{11/3} \sqrt{b x^{2/3}+a x}}{225 a^2}+\frac{2 x^4 \sqrt{b x^{2/3}+a x}}{9 a}-\frac{\left (1048576 b^{11}\right ) \int \frac{\sqrt [3]{x}}{\sqrt{b x^{2/3}+a x}} \, dx}{2340135 a^{11}}\\ &=-\frac{2097152 b^{11} \sqrt [3]{x} \sqrt{b x^{2/3}+a x}}{3900225 a^{12}}+\frac{1048576 b^{10} x^{2/3} \sqrt{b x^{2/3}+a x}}{2340135 a^{11}}-\frac{131072 b^9 x \sqrt{b x^{2/3}+a x}}{334305 a^{10}}+\frac{65536 b^8 x^{4/3} \sqrt{b x^{2/3}+a x}}{185725 a^9}-\frac{180224 b^7 x^{5/3} \sqrt{b x^{2/3}+a x}}{557175 a^8}+\frac{1171456 b^6 x^2 \sqrt{b x^{2/3}+a x}}{3900225 a^7}-\frac{73216 b^5 x^{7/3} \sqrt{b x^{2/3}+a x}}{260015 a^6}+\frac{36608 b^4 x^{8/3} \sqrt{b x^{2/3}+a x}}{137655 a^5}-\frac{9152 b^3 x^3 \sqrt{b x^{2/3}+a x}}{36225 a^4}+\frac{416 b^2 x^{10/3} \sqrt{b x^{2/3}+a x}}{1725 a^3}-\frac{52 b x^{11/3} \sqrt{b x^{2/3}+a x}}{225 a^2}+\frac{2 x^4 \sqrt{b x^{2/3}+a x}}{9 a}+\frac{\left (4194304 b^{12}\right ) \int \frac{1}{\sqrt{b x^{2/3}+a x}} \, dx}{11700675 a^{12}}\\ &=\frac{8388608 b^{12} \sqrt{b x^{2/3}+a x}}{11700675 a^{13}}-\frac{2097152 b^{11} \sqrt [3]{x} \sqrt{b x^{2/3}+a x}}{3900225 a^{12}}+\frac{1048576 b^{10} x^{2/3} \sqrt{b x^{2/3}+a x}}{2340135 a^{11}}-\frac{131072 b^9 x \sqrt{b x^{2/3}+a x}}{334305 a^{10}}+\frac{65536 b^8 x^{4/3} \sqrt{b x^{2/3}+a x}}{185725 a^9}-\frac{180224 b^7 x^{5/3} \sqrt{b x^{2/3}+a x}}{557175 a^8}+\frac{1171456 b^6 x^2 \sqrt{b x^{2/3}+a x}}{3900225 a^7}-\frac{73216 b^5 x^{7/3} \sqrt{b x^{2/3}+a x}}{260015 a^6}+\frac{36608 b^4 x^{8/3} \sqrt{b x^{2/3}+a x}}{137655 a^5}-\frac{9152 b^3 x^3 \sqrt{b x^{2/3}+a x}}{36225 a^4}+\frac{416 b^2 x^{10/3} \sqrt{b x^{2/3}+a x}}{1725 a^3}-\frac{52 b x^{11/3} \sqrt{b x^{2/3}+a x}}{225 a^2}+\frac{2 x^4 \sqrt{b x^{2/3}+a x}}{9 a}-\frac{\left (8388608 b^{13}\right ) \int \frac{1}{\sqrt [3]{x} \sqrt{b x^{2/3}+a x}} \, dx}{35102025 a^{13}}\\ &=\frac{8388608 b^{12} \sqrt{b x^{2/3}+a x}}{11700675 a^{13}}-\frac{16777216 b^{13} \sqrt{b x^{2/3}+a x}}{11700675 a^{14} \sqrt [3]{x}}-\frac{2097152 b^{11} \sqrt [3]{x} \sqrt{b x^{2/3}+a x}}{3900225 a^{12}}+\frac{1048576 b^{10} x^{2/3} \sqrt{b x^{2/3}+a x}}{2340135 a^{11}}-\frac{131072 b^9 x \sqrt{b x^{2/3}+a x}}{334305 a^{10}}+\frac{65536 b^8 x^{4/3} \sqrt{b x^{2/3}+a x}}{185725 a^9}-\frac{180224 b^7 x^{5/3} \sqrt{b x^{2/3}+a x}}{557175 a^8}+\frac{1171456 b^6 x^2 \sqrt{b x^{2/3}+a x}}{3900225 a^7}-\frac{73216 b^5 x^{7/3} \sqrt{b x^{2/3}+a x}}{260015 a^6}+\frac{36608 b^4 x^{8/3} \sqrt{b x^{2/3}+a x}}{137655 a^5}-\frac{9152 b^3 x^3 \sqrt{b x^{2/3}+a x}}{36225 a^4}+\frac{416 b^2 x^{10/3} \sqrt{b x^{2/3}+a x}}{1725 a^3}-\frac{52 b x^{11/3} \sqrt{b x^{2/3}+a x}}{225 a^2}+\frac{2 x^4 \sqrt{b x^{2/3}+a x}}{9 a}\\ \end{align*}

Mathematica [A]  time = 0.153033, size = 185, normalized size = 0.46 \[ \frac{2 \sqrt{a x+b x^{2/3}} \left (1410864 a^{11} b^2 x^{11/3}-1478048 a^{10} b^3 x^{10/3}+1555840 a^9 b^4 x^3-1647360 a^8 b^5 x^{8/3}+1757184 a^7 b^6 x^{7/3}-1892352 a^6 b^7 x^2+2064384 a^5 b^8 x^{5/3}-2293760 a^4 b^9 x^{4/3}-3145728 a^2 b^{11} x^{2/3}+2621440 a^3 b^{10} x-1352078 a^{12} b x^4+1300075 a^{13} x^{13/3}+4194304 a b^{12} \sqrt [3]{x}-8388608 b^{13}\right )}{11700675 a^{14} \sqrt [3]{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[b*x^(2/3) + a*x]*(-8388608*b^13 + 4194304*a*b^12*x^(1/3) - 3145728*a^2*b^11*x^(2/3) + 2621440*a^3*b^10
*x - 2293760*a^4*b^9*x^(4/3) + 2064384*a^5*b^8*x^(5/3) - 1892352*a^6*b^7*x^2 + 1757184*a^7*b^6*x^(7/3) - 16473
60*a^8*b^5*x^(8/3) + 1555840*a^9*b^4*x^3 - 1478048*a^10*b^3*x^(10/3) + 1410864*a^11*b^2*x^(11/3) - 1352078*a^1
2*b*x^4 + 1300075*a^13*x^(13/3)))/(11700675*a^14*x^(1/3))

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Maple [A]  time = 0.004, size = 167, normalized size = 0.4 \begin{align*}{\frac{2}{11700675\,{a}^{14}}\sqrt [3]{x} \left ( b+a\sqrt [3]{x} \right ) \left ( 1300075\,{x}^{13/3}{a}^{13}-1352078\,{x}^{4}{a}^{12}b+1410864\,{x}^{11/3}{a}^{11}{b}^{2}-1478048\,{x}^{10/3}{a}^{10}{b}^{3}+1555840\,{x}^{3}{a}^{9}{b}^{4}-1647360\,{x}^{8/3}{a}^{8}{b}^{5}+1757184\,{x}^{7/3}{a}^{7}{b}^{6}-1892352\,{x}^{2}{a}^{6}{b}^{7}+2064384\,{x}^{5/3}{a}^{5}{b}^{8}-2293760\,{x}^{4/3}{a}^{4}{b}^{9}+2621440\,x{a}^{3}{b}^{10}-3145728\,{x}^{2/3}{a}^{2}{b}^{11}+4194304\,\sqrt [3]{x}a{b}^{12}-8388608\,{b}^{13} \right ){\frac{1}{\sqrt{b{x}^{{\frac{2}{3}}}+ax}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/11700675*x^(1/3)*(b+a*x^(1/3))*(1300075*x^(13/3)*a^13-1352078*x^4*a^12*b+1410864*x^(11/3)*a^11*b^2-1478048*x
^(10/3)*a^10*b^3+1555840*x^3*a^9*b^4-1647360*x^(8/3)*a^8*b^5+1757184*x^(7/3)*a^7*b^6-1892352*x^2*a^6*b^7+20643
84*x^(5/3)*a^5*b^8-2293760*x^(4/3)*a^4*b^9+2621440*x*a^3*b^10-3145728*x^(2/3)*a^2*b^11+4194304*x^(1/3)*a*b^12-
8388608*b^13)/(b*x^(2/3)+a*x)^(1/2)/a^14

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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Giac [A]  time = 1.14363, size = 278, normalized size = 0.69 \begin{align*} \frac{16777216 \, b^{\frac{27}{2}}}{11700675 \, a^{14}} + \frac{2 \,{\left (1300075 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{27}{2}} - 18253053 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{25}{2}} b + 119041650 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{23}{2}} b^{2} - 478056150 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{21}{2}} b^{3} + 1320944625 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{19}{2}} b^{4} - 2657429775 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{17}{2}} b^{5} + 4015671660 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{15}{2}} b^{6} - 4633467300 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{13}{2}} b^{7} + 4106936925 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{11}{2}} b^{8} - 2788660875 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{9}{2}} b^{9} + 1434168450 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{7}{2}} b^{10} - 547591590 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} b^{11} + 152108775 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} b^{12} - 35102025 \, \sqrt{a x^{\frac{1}{3}} + b} b^{13}\right )}}{11700675 \, a^{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

16777216/11700675*b^(27/2)/a^14 + 2/11700675*(1300075*(a*x^(1/3) + b)^(27/2) - 18253053*(a*x^(1/3) + b)^(25/2)
*b + 119041650*(a*x^(1/3) + b)^(23/2)*b^2 - 478056150*(a*x^(1/3) + b)^(21/2)*b^3 + 1320944625*(a*x^(1/3) + b)^
(19/2)*b^4 - 2657429775*(a*x^(1/3) + b)^(17/2)*b^5 + 4015671660*(a*x^(1/3) + b)^(15/2)*b^6 - 4633467300*(a*x^(
1/3) + b)^(13/2)*b^7 + 4106936925*(a*x^(1/3) + b)^(11/2)*b^8 - 2788660875*(a*x^(1/3) + b)^(9/2)*b^9 + 14341684
50*(a*x^(1/3) + b)^(7/2)*b^10 - 547591590*(a*x^(1/3) + b)^(5/2)*b^11 + 152108775*(a*x^(1/3) + b)^(3/2)*b^12 -
35102025*sqrt(a*x^(1/3) + b)*b^13)/a^14