\(\int \frac {x^4}{\sqrt {b x^{2/3}+a x}} \, dx\) [185]

Optimal result

Integrand size = 19, antiderivative size = 401 \[ \int \frac {x^4}{\sqrt {b x^{2/3}+a x}} \, dx=\frac {8388608 b^{12} \sqrt {b x^{2/3}+a x}}{11700675 a^{13}}-\frac {16777216 b^{13} \sqrt {b x^{2/3}+a x}}{11700675 a^{14} \sqrt [3]{x}}-\frac {2097152 b^{11} \sqrt [3]{x} \sqrt {b x^{2/3}+a x}}{3900225 a^{12}}+\frac {1048576 b^{10} x^{2/3} \sqrt {b x^{2/3}+a x}}{2340135 a^{11}}-\frac {131072 b^9 x \sqrt {b x^{2/3}+a x}}{334305 a^{10}}+\frac {65536 b^8 x^{4/3} \sqrt {b x^{2/3}+a x}}{185725 a^9}-\frac {180224 b^7 x^{5/3} \sqrt {b x^{2/3}+a x}}{557175 a^8}+\frac {1171456 b^6 x^2 \sqrt {b x^{2/3}+a x}}{3900225 a^7}-\frac {73216 b^5 x^{7/3} \sqrt {b x^{2/3}+a x}}{260015 a^6}+\frac {36608 b^4 x^{8/3} \sqrt {b x^{2/3}+a x}}{137655 a^5}-\frac {9152 b^3 x^3 \sqrt {b x^{2/3}+a x}}{36225 a^4}+\frac {416 b^2 x^{10/3} \sqrt {b x^{2/3}+a x}}{1725 a^3}-\frac {52 b x^{11/3} \sqrt {b x^{2/3}+a x}}{225 a^2}+\frac {2 x^4 \sqrt {b x^{2/3}+a x}}{9 a} \]

[Out]

8388608/11700675*b^12*(b*x^(2/3)+a*x)^(1/2)/a^13-16777216/11700675*b^13*(b*x^(2/3)+a*x)^(1/2)/a^14/x^(1/3)-209
7152/3900225*b^11*x^(1/3)*(b*x^(2/3)+a*x)^(1/2)/a^12+1048576/2340135*b^10*x^(2/3)*(b*x^(2/3)+a*x)^(1/2)/a^11-1
31072/334305*b^9*x*(b*x^(2/3)+a*x)^(1/2)/a^10+65536/185725*b^8*x^(4/3)*(b*x^(2/3)+a*x)^(1/2)/a^9-180224/557175
*b^7*x^(5/3)*(b*x^(2/3)+a*x)^(1/2)/a^8+1171456/3900225*b^6*x^2*(b*x^(2/3)+a*x)^(1/2)/a^7-73216/260015*b^5*x^(7
/3)*(b*x^(2/3)+a*x)^(1/2)/a^6+36608/137655*b^4*x^(8/3)*(b*x^(2/3)+a*x)^(1/2)/a^5-9152/36225*b^3*x^3*(b*x^(2/3)
+a*x)^(1/2)/a^4+416/1725*b^2*x^(10/3)*(b*x^(2/3)+a*x)^(1/2)/a^3-52/225*b*x^(11/3)*(b*x^(2/3)+a*x)^(1/2)/a^2+2/
9*x^4*(b*x^(2/3)+a*x)^(1/2)/a

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2041, 2027, 2039} \[ \int \frac {x^4}{\sqrt {b x^{2/3}+a x}} \, dx=-\frac {16777216 b^{13} \sqrt {a x+b x^{2/3}}}{11700675 a^{14} \sqrt [3]{x}}+\frac {8388608 b^{12} \sqrt {a x+b x^{2/3}}}{11700675 a^{13}}-\frac {2097152 b^{11} \sqrt [3]{x} \sqrt {a x+b x^{2/3}}}{3900225 a^{12}}+\frac {1048576 b^{10} x^{2/3} \sqrt {a x+b x^{2/3}}}{2340135 a^{11}}-\frac {131072 b^9 x \sqrt {a x+b x^{2/3}}}{334305 a^{10}}+\frac {65536 b^8 x^{4/3} \sqrt {a x+b x^{2/3}}}{185725 a^9}-\frac {180224 b^7 x^{5/3} \sqrt {a x+b x^{2/3}}}{557175 a^8}+\frac {1171456 b^6 x^2 \sqrt {a x+b x^{2/3}}}{3900225 a^7}-\frac {73216 b^5 x^{7/3} \sqrt {a x+b x^{2/3}}}{260015 a^6}+\frac {36608 b^4 x^{8/3} \sqrt {a x+b x^{2/3}}}{137655 a^5}-\frac {9152 b^3 x^3 \sqrt {a x+b x^{2/3}}}{36225 a^4}+\frac {416 b^2 x^{10/3} \sqrt {a x+b x^{2/3}}}{1725 a^3}-\frac {52 b x^{11/3} \sqrt {a x+b x^{2/3}}}{225 a^2}+\frac {2 x^4 \sqrt {a x+b x^{2/3}}}{9 a} \]

[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 2027

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 2039

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] &&
NeQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

Rule 2041

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])

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 0.19 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.46 \[ \int \frac {x^4}{\sqrt {b x^{2/3}+a x}} \, dx=\frac {2 \sqrt {b x^{2/3}+a x} \left (-8388608 b^{13}+4194304 a b^{12} \sqrt [3]{x}-3145728 a^2 b^{11} x^{2/3}+2621440 a^3 b^{10} x-2293760 a^4 b^9 x^{4/3}+2064384 a^5 b^8 x^{5/3}-1892352 a^6 b^7 x^2+1757184 a^7 b^6 x^{7/3}-1647360 a^8 b^5 x^{8/3}+1555840 a^9 b^4 x^3-1478048 a^{10} b^3 x^{10/3}+1410864 a^{11} b^2 x^{11/3}-1352078 a^{12} b x^4+1300075 a^{13} x^{13/3}\right )}{11700675 a^{14} \sqrt [3]{x}} \]

[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))

Maple [A] (verified)

Time = 2.07 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.42

method	result	size
derivativedivides	\(\frac {2 x^{\frac {1}{3}} \left (b +a \,x^{\frac {1}{3}}\right ) \left (1300075 a^{13} x^{\frac {13}{3}}-1352078 a^{12} b \,x^{4}+1410864 a^{11} b^{2} x^{\frac {11}{3}}-1478048 a^{10} b^{3} x^{\frac {10}{3}}+1555840 a^{9} b^{4} x^{3}-1647360 a^{8} b^{5} x^{\frac {8}{3}}+1757184 a^{7} b^{6} x^{\frac {7}{3}}-1892352 a^{6} b^{7} x^{2}+2064384 a^{5} b^{8} x^{\frac {5}{3}}-2293760 a^{4} b^{9} x^{\frac {4}{3}}+2621440 a^{3} b^{10} x -3145728 a^{2} b^{11} x^{\frac {2}{3}}+4194304 a \,b^{12} x^{\frac {1}{3}}-8388608 b^{13}\right )}{11700675 \sqrt {b \,x^{\frac {2}{3}}+a x}\, a^{14}}\)	\(167\)
default	\(\frac {2 x^{\frac {1}{3}} \left (b +a \,x^{\frac {1}{3}}\right ) \left (1300075 a^{13} x^{\frac {13}{3}}-1352078 a^{12} b \,x^{4}+1410864 a^{11} b^{2} x^{\frac {11}{3}}-1478048 a^{10} b^{3} x^{\frac {10}{3}}+1555840 a^{9} b^{4} x^{3}-1647360 a^{8} b^{5} x^{\frac {8}{3}}+1757184 a^{7} b^{6} x^{\frac {7}{3}}-1892352 a^{6} b^{7} x^{2}+2064384 a^{5} b^{8} x^{\frac {5}{3}}-2293760 a^{4} b^{9} x^{\frac {4}{3}}+2621440 a^{3} b^{10} x -3145728 a^{2} b^{11} x^{\frac {2}{3}}+4194304 a \,b^{12} x^{\frac {1}{3}}-8388608 b^{13}\right )}{11700675 \sqrt {b \,x^{\frac {2}{3}}+a x}\, a^{14}}\)	\(167\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1294 vs. \(2 (299) = 598\).

Time = 148.43 (sec) , antiderivative size = 1294, normalized size of antiderivative = 3.23 \[ \int \frac {x^4}{\sqrt {b x^{2/3}+a x}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/11700675*((211106232532992*b^19 + 43980465111040*b^18 + 206158430208*(64*a^3 - 3)*b^16 - 4123168604160*b^17
- 1073741824*(11264*a^3 - 53)*b^15 + 15143273600*a^15 - 402653184*(5504*a^3 + 1)*b^14 + 12582912*(3194880*a^6
- 114688*a^3 - 3)*b^13 + 469762048*(18816*a^6 + 103*a^3)*b^12 - 50331648*(48816*a^6 + 23*a^3)*b^11 - 786432*(4
5731840*a^9 - 495872*a^6 - 15*a^3)*b^10 - 7340032*(1349120*a^9 + 3439*a^6)*b^9 + 250478592*(5600*a^9 + 3*a^6)*
b^8 + 12288*(2616979456*a^12 - 21542400*a^9 - 693*a^6)*b^7 + 212992*(43743616*a^12 + 89111*a^9)*b^6 - 638976*(
1652476*a^12 + 935*a^9)*b^5 + 3264*(3608543232*a^15 + 64599808*a^12 + 2145*a^9)*b^4 + 578816*(13049856*a^15 -
27313*a^12)*b^3 + 217056*(6211584*a^15 + 2353*a^12)*b^2 - 156009*(2547712*a^15 + 39*a^12)*b)*x + 2*(1300075*(1
6777216*a^13*b^6 + 6291456*a^13*b^5 + 196608*a^13*b^4 - 262144*a^16 - 114688*a^13*b^3 - 2304*a^13*b^2 + 864*a^
13*b - 27*a^13)*x^5 - 1478048*(16777216*a^10*b^9 + 6291456*a^10*b^8 + 196608*a^10*b^7 - 114688*a^10*b^6 - 2304
*a^10*b^5 + 864*a^10*b^4 - (262144*a^13 + 27*a^10)*b^3)*x^4 + 1757184*(16777216*a^7*b^12 + 6291456*a^7*b^11 +
196608*a^7*b^10 - 114688*a^7*b^9 - 2304*a^7*b^8 + 864*a^7*b^7 - (262144*a^10 + 27*a^7)*b^6)*x^3 - 2293760*(167
77216*a^4*b^15 + 6291456*a^4*b^14 + 196608*a^4*b^13 - 114688*a^4*b^12 - 2304*a^4*b^11 + 864*a^4*b^10 - (262144
*a^7 + 27*a^4)*b^9)*x^2 + 4194304*(16777216*a*b^18 + 6291456*a*b^17 + 196608*a*b^16 - 114688*a*b^15 - 2304*a*b
^14 + 864*a*b^13 - (262144*a^4 + 27*a)*b^12)*x - 2*(70368744177664*b^19 + 26388279066624*b^18 + 824633720832*b
^17 - 481036337152*b^16 - 9663676416*b^15 - 4194304*(262144*a^3 + 27)*b^13 + 3623878656*b^14 + 676039*(1677721
6*a^12*b^7 + 6291456*a^12*b^6 + 196608*a^12*b^5 - 114688*a^12*b^4 - 2304*a^12*b^3 + 864*a^12*b^2 - (262144*a^1
5 + 27*a^12)*b)*x^4 - 777920*(16777216*a^9*b^10 + 6291456*a^9*b^9 + 196608*a^9*b^8 - 114688*a^9*b^7 - 2304*a^9
*b^6 + 864*a^9*b^5 - (262144*a^12 + 27*a^9)*b^4)*x^3 + 946176*(16777216*a^6*b^13 + 6291456*a^6*b^12 + 196608*a
^6*b^11 - 114688*a^6*b^10 - 2304*a^6*b^9 + 864*a^6*b^8 - (262144*a^9 + 27*a^6)*b^7)*x^2 - 1310720*(16777216*a^
3*b^16 + 6291456*a^3*b^15 + 196608*a^3*b^14 - 114688*a^3*b^13 - 2304*a^3*b^12 + 864*a^3*b^11 - (262144*a^6 + 2
7*a^3)*b^10)*x)*x^(2/3) + 48*(29393*(16777216*a^11*b^8 + 6291456*a^11*b^7 + 196608*a^11*b^6 - 114688*a^11*b^5
- 2304*a^11*b^4 + 864*a^11*b^3 - (262144*a^14 + 27*a^11)*b^2)*x^4 - 34320*(16777216*a^8*b^11 + 6291456*a^8*b^1
0 + 196608*a^8*b^9 - 114688*a^8*b^8 - 2304*a^8*b^7 + 864*a^8*b^6 - (262144*a^11 + 27*a^8)*b^5)*x^3 + 43008*(16
777216*a^5*b^14 + 6291456*a^5*b^13 + 196608*a^5*b^12 - 114688*a^5*b^11 - 2304*a^5*b^10 + 864*a^5*b^9 - (262144
*a^8 + 27*a^5)*b^8)*x^2 - 65536*(16777216*a^2*b^17 + 6291456*a^2*b^16 + 196608*a^2*b^15 - 114688*a^2*b^14 - 23
04*a^2*b^13 + 864*a^2*b^12 - (262144*a^5 + 27*a^2)*b^11)*x)*x^(1/3))*sqrt(a*x + b*x^(2/3)))/((16777216*a^14*b^
6 + 6291456*a^14*b^5 + 196608*a^14*b^4 - 262144*a^17 - 114688*a^14*b^3 - 2304*a^14*b^2 + 864*a^14*b - 27*a^14)
*x)

Sympy [F]

\[ \int \frac {x^4}{\sqrt {b x^{2/3}+a x}} \, dx=\int \frac {x^{4}}{\sqrt {a x + b x^{\frac {2}{3}}}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {x^4}{\sqrt {b x^{2/3}+a x}} \, dx=\int { \frac {x^{4}}{\sqrt {a x + b x^{\frac {2}{3}}}} \,d x } \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.65 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.51 \[ \int \frac {x^4}{\sqrt {b x^{2/3}+a x}} \, dx=\frac {16777216 \, b^{\frac {27}{2}}}{11700675 \, a^{14}} + \frac {2 \, {\left (1300075 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {27}{2}} - 18253053 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {25}{2}} b + 119041650 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {23}{2}} b^{2} - 478056150 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {21}{2}} b^{3} + 1320944625 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {19}{2}} b^{4} - 2657429775 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} b^{5} + 4015671660 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} b^{6} - 4633467300 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} b^{7} + 4106936925 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} b^{8} - 2788660875 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b^{9} + 1434168450 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{10} - 547591590 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{11} + 152108775 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{12} - 35102025 \, \sqrt {a x^{\frac {1}{3}} + b} b^{13}\right )}}{11700675 \, a^{14}} \]

[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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4}{\sqrt {b x^{2/3}+a x}} \, dx=\int \frac {x^4}{\sqrt {a\,x+b\,x^{2/3}}} \,d x \]

[In]

int(x^4/(a*x + b*x^(2/3))^(1/2),x)

[Out]

int(x^4/(a*x + b*x^(2/3))^(1/2), x)