\(\int x^2 \sqrt {b x^{2/3}+a x} \, dx\) [168]

Optimal result

Integrand size = 19, antiderivative size = 283 \[ \int x^2 \sqrt {b x^{2/3}+a x} \, dx=\frac {8192 b^6 \left (b x^{2/3}+a x\right )^{3/2}}{46189 a^7}-\frac {131072 b^9 \left (b x^{2/3}+a x\right )^{3/2}}{1616615 a^{10} x}+\frac {196608 b^8 \left (b x^{2/3}+a x\right )^{3/2}}{1616615 a^9 x^{2/3}}-\frac {49152 b^7 \left (b x^{2/3}+a x\right )^{3/2}}{323323 a^8 \sqrt [3]{x}}-\frac {9216 b^5 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{3/2}}{46189 a^6}+\frac {4608 b^4 x^{2/3} \left (b x^{2/3}+a x\right )^{3/2}}{20995 a^5}-\frac {384 b^3 x \left (b x^{2/3}+a x\right )^{3/2}}{1615 a^4}+\frac {576 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{3/2}}{2261 a^3}-\frac {36 b x^{5/3} \left (b x^{2/3}+a x\right )^{3/2}}{133 a^2}+\frac {2 x^2 \left (b x^{2/3}+a x\right )^{3/2}}{7 a} \]

[Out]

8192/46189*b^6*(b*x^(2/3)+a*x)^(3/2)/a^7-131072/1616615*b^9*(b*x^(2/3)+a*x)^(3/2)/a^10/x+196608/1616615*b^8*(b
*x^(2/3)+a*x)^(3/2)/a^9/x^(2/3)-49152/323323*b^7*(b*x^(2/3)+a*x)^(3/2)/a^8/x^(1/3)-9216/46189*b^5*x^(1/3)*(b*x
^(2/3)+a*x)^(3/2)/a^6+4608/20995*b^4*x^(2/3)*(b*x^(2/3)+a*x)^(3/2)/a^5-384/1615*b^3*x*(b*x^(2/3)+a*x)^(3/2)/a^
4+576/2261*b^2*x^(4/3)*(b*x^(2/3)+a*x)^(3/2)/a^3-36/133*b*x^(5/3)*(b*x^(2/3)+a*x)^(3/2)/a^2+2/7*x^2*(b*x^(2/3)
+a*x)^(3/2)/a

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2041, 2027, 2039} \[ \int x^2 \sqrt {b x^{2/3}+a x} \, dx=-\frac {131072 b^9 \left (a x+b x^{2/3}\right )^{3/2}}{1616615 a^{10} x}+\frac {196608 b^8 \left (a x+b x^{2/3}\right )^{3/2}}{1616615 a^9 x^{2/3}}-\frac {49152 b^7 \left (a x+b x^{2/3}\right )^{3/2}}{323323 a^8 \sqrt [3]{x}}+\frac {8192 b^6 \left (a x+b x^{2/3}\right )^{3/2}}{46189 a^7}-\frac {9216 b^5 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{3/2}}{46189 a^6}+\frac {4608 b^4 x^{2/3} \left (a x+b x^{2/3}\right )^{3/2}}{20995 a^5}-\frac {384 b^3 x \left (a x+b x^{2/3}\right )^{3/2}}{1615 a^4}+\frac {576 b^2 x^{4/3} \left (a x+b x^{2/3}\right )^{3/2}}{2261 a^3}-\frac {36 b x^{5/3} \left (a x+b x^{2/3}\right )^{3/2}}{133 a^2}+\frac {2 x^2 \left (a x+b x^{2/3}\right )^{3/2}}{7 a} \]

[In]

Int[x^2*Sqrt[b*x^(2/3) + a*x],x]

[Out]

(8192*b^6*(b*x^(2/3) + a*x)^(3/2))/(46189*a^7) - (131072*b^9*(b*x^(2/3) + a*x)^(3/2))/(1616615*a^10*x) + (1966
08*b^8*(b*x^(2/3) + a*x)^(3/2))/(1616615*a^9*x^(2/3)) - (49152*b^7*(b*x^(2/3) + a*x)^(3/2))/(323323*a^8*x^(1/3
)) - (9216*b^5*x^(1/3)*(b*x^(2/3) + a*x)^(3/2))/(46189*a^6) + (4608*b^4*x^(2/3)*(b*x^(2/3) + a*x)^(3/2))/(2099
5*a^5) - (384*b^3*x*(b*x^(2/3) + a*x)^(3/2))/(1615*a^4) + (576*b^2*x^(4/3)*(b*x^(2/3) + a*x)^(3/2))/(2261*a^3)
- (36*b*x^(5/3)*(b*x^(2/3) + a*x)^(3/2))/(133*a^2) + (2*x^2*(b*x^(2/3) + a*x)^(3/2))/(7*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^2 \left (b x^{2/3}+a x\right )^{3/2}}{7 a}-\frac {(6 b) \int x^{5/3} \sqrt {b x^{2/3}+a x} \, dx}{7 a} \\ & = -\frac {36 b x^{5/3} \left (b x^{2/3}+a x\right )^{3/2}}{133 a^2}+\frac {2 x^2 \left (b x^{2/3}+a x\right )^{3/2}}{7 a}+\frac {\left (96 b^2\right ) \int x^{4/3} \sqrt {b x^{2/3}+a x} \, dx}{133 a^2} \\ & = \frac {576 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{3/2}}{2261 a^3}-\frac {36 b x^{5/3} \left (b x^{2/3}+a x\right )^{3/2}}{133 a^2}+\frac {2 x^2 \left (b x^{2/3}+a x\right )^{3/2}}{7 a}-\frac {\left (192 b^3\right ) \int x \sqrt {b x^{2/3}+a x} \, dx}{323 a^3} \\ & = -\frac {384 b^3 x \left (b x^{2/3}+a x\right )^{3/2}}{1615 a^4}+\frac {576 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{3/2}}{2261 a^3}-\frac {36 b x^{5/3} \left (b x^{2/3}+a x\right )^{3/2}}{133 a^2}+\frac {2 x^2 \left (b x^{2/3}+a x\right )^{3/2}}{7 a}+\frac {\left (768 b^4\right ) \int x^{2/3} \sqrt {b x^{2/3}+a x} \, dx}{1615 a^4} \\ & = \frac {4608 b^4 x^{2/3} \left (b x^{2/3}+a x\right )^{3/2}}{20995 a^5}-\frac {384 b^3 x \left (b x^{2/3}+a x\right )^{3/2}}{1615 a^4}+\frac {576 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{3/2}}{2261 a^3}-\frac {36 b x^{5/3} \left (b x^{2/3}+a x\right )^{3/2}}{133 a^2}+\frac {2 x^2 \left (b x^{2/3}+a x\right )^{3/2}}{7 a}-\frac {\left (1536 b^5\right ) \int \sqrt [3]{x} \sqrt {b x^{2/3}+a x} \, dx}{4199 a^5} \\ & = -\frac {9216 b^5 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{3/2}}{46189 a^6}+\frac {4608 b^4 x^{2/3} \left (b x^{2/3}+a x\right )^{3/2}}{20995 a^5}-\frac {384 b^3 x \left (b x^{2/3}+a x\right )^{3/2}}{1615 a^4}+\frac {576 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{3/2}}{2261 a^3}-\frac {36 b x^{5/3} \left (b x^{2/3}+a x\right )^{3/2}}{133 a^2}+\frac {2 x^2 \left (b x^{2/3}+a x\right )^{3/2}}{7 a}+\frac {\left (12288 b^6\right ) \int \sqrt {b x^{2/3}+a x} \, dx}{46189 a^6} \\ & = \frac {8192 b^6 \left (b x^{2/3}+a x\right )^{3/2}}{46189 a^7}-\frac {9216 b^5 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{3/2}}{46189 a^6}+\frac {4608 b^4 x^{2/3} \left (b x^{2/3}+a x\right )^{3/2}}{20995 a^5}-\frac {384 b^3 x \left (b x^{2/3}+a x\right )^{3/2}}{1615 a^4}+\frac {576 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{3/2}}{2261 a^3}-\frac {36 b x^{5/3} \left (b x^{2/3}+a x\right )^{3/2}}{133 a^2}+\frac {2 x^2 \left (b x^{2/3}+a x\right )^{3/2}}{7 a}-\frac {\left (8192 b^7\right ) \int \frac {\sqrt {b x^{2/3}+a x}}{\sqrt [3]{x}} \, dx}{46189 a^7} \\ & = \frac {8192 b^6 \left (b x^{2/3}+a x\right )^{3/2}}{46189 a^7}-\frac {49152 b^7 \left (b x^{2/3}+a x\right )^{3/2}}{323323 a^8 \sqrt [3]{x}}-\frac {9216 b^5 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{3/2}}{46189 a^6}+\frac {4608 b^4 x^{2/3} \left (b x^{2/3}+a x\right )^{3/2}}{20995 a^5}-\frac {384 b^3 x \left (b x^{2/3}+a x\right )^{3/2}}{1615 a^4}+\frac {576 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{3/2}}{2261 a^3}-\frac {36 b x^{5/3} \left (b x^{2/3}+a x\right )^{3/2}}{133 a^2}+\frac {2 x^2 \left (b x^{2/3}+a x\right )^{3/2}}{7 a}+\frac {\left (32768 b^8\right ) \int \frac {\sqrt {b x^{2/3}+a x}}{x^{2/3}} \, dx}{323323 a^8} \\ & = \frac {8192 b^6 \left (b x^{2/3}+a x\right )^{3/2}}{46189 a^7}+\frac {196608 b^8 \left (b x^{2/3}+a x\right )^{3/2}}{1616615 a^9 x^{2/3}}-\frac {49152 b^7 \left (b x^{2/3}+a x\right )^{3/2}}{323323 a^8 \sqrt [3]{x}}-\frac {9216 b^5 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{3/2}}{46189 a^6}+\frac {4608 b^4 x^{2/3} \left (b x^{2/3}+a x\right )^{3/2}}{20995 a^5}-\frac {384 b^3 x \left (b x^{2/3}+a x\right )^{3/2}}{1615 a^4}+\frac {576 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{3/2}}{2261 a^3}-\frac {36 b x^{5/3} \left (b x^{2/3}+a x\right )^{3/2}}{133 a^2}+\frac {2 x^2 \left (b x^{2/3}+a x\right )^{3/2}}{7 a}-\frac {\left (65536 b^9\right ) \int \frac {\sqrt {b x^{2/3}+a x}}{x} \, dx}{1616615 a^9} \\ & = \frac {8192 b^6 \left (b x^{2/3}+a x\right )^{3/2}}{46189 a^7}-\frac {131072 b^9 \left (b x^{2/3}+a x\right )^{3/2}}{1616615 a^{10} x}+\frac {196608 b^8 \left (b x^{2/3}+a x\right )^{3/2}}{1616615 a^9 x^{2/3}}-\frac {49152 b^7 \left (b x^{2/3}+a x\right )^{3/2}}{323323 a^8 \sqrt [3]{x}}-\frac {9216 b^5 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{3/2}}{46189 a^6}+\frac {4608 b^4 x^{2/3} \left (b x^{2/3}+a x\right )^{3/2}}{20995 a^5}-\frac {384 b^3 x \left (b x^{2/3}+a x\right )^{3/2}}{1615 a^4}+\frac {576 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{3/2}}{2261 a^3}-\frac {36 b x^{5/3} \left (b x^{2/3}+a x\right )^{3/2}}{133 a^2}+\frac {2 x^2 \left (b x^{2/3}+a x\right )^{3/2}}{7 a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.47 \[ \int x^2 \sqrt {b x^{2/3}+a x} \, dx=\frac {2 \left (b x^{2/3}+a x\right )^{3/2} \left (-65536 b^9+98304 a b^8 \sqrt [3]{x}-122880 a^2 b^7 x^{2/3}+143360 a^3 b^6 x-161280 a^4 b^5 x^{4/3}+177408 a^5 b^4 x^{5/3}-192192 a^6 b^3 x^2+205920 a^7 b^2 x^{7/3}-218790 a^8 b x^{8/3}+230945 a^9 x^3\right )}{1616615 a^{10} x} \]

[In]

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

[Out]

(2*(b*x^(2/3) + a*x)^(3/2)*(-65536*b^9 + 98304*a*b^8*x^(1/3) - 122880*a^2*b^7*x^(2/3) + 143360*a^3*b^6*x - 161
280*a^4*b^5*x^(4/3) + 177408*a^5*b^4*x^(5/3) - 192192*a^6*b^3*x^2 + 205920*a^7*b^2*x^(7/3) - 218790*a^8*b*x^(8
/3) + 230945*a^9*x^3))/(1616615*a^10*x)

Maple [A] (verified)

Time = 2.07 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.43

method	result	size
derivativedivides	\(\frac {2 \sqrt {b \,x^{\frac {2}{3}}+a x}\, \left (b +a \,x^{\frac {1}{3}}\right ) \left (230945 a^{9} x^{3}-218790 a^{8} b \,x^{\frac {8}{3}}+205920 a^{7} b^{2} x^{\frac {7}{3}}-192192 a^{6} b^{3} x^{2}+177408 a^{5} b^{4} x^{\frac {5}{3}}-161280 a^{4} b^{5} x^{\frac {4}{3}}+143360 a^{3} b^{6} x -122880 a^{2} b^{7} x^{\frac {2}{3}}+98304 a \,b^{8} x^{\frac {1}{3}}-65536 b^{9}\right )}{1616615 x^{\frac {1}{3}} a^{10}}\)	\(123\)
default	\(-\frac {2 \sqrt {b \,x^{\frac {2}{3}}+a x}\, \left (b +a \,x^{\frac {1}{3}}\right ) \left (218790 a^{8} b \,x^{\frac {8}{3}}-205920 a^{7} b^{2} x^{\frac {7}{3}}-177408 a^{5} b^{4} x^{\frac {5}{3}}+161280 a^{4} b^{5} x^{\frac {4}{3}}-230945 a^{9} x^{3}+122880 a^{2} b^{7} x^{\frac {2}{3}}+192192 a^{6} b^{3} x^{2}-98304 a \,b^{8} x^{\frac {1}{3}}-143360 a^{3} b^{6} x +65536 b^{9}\right )}{1616615 x^{\frac {1}{3}} a^{10}}\)	\(123\)

[In]

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

[Out]

2/1616615*(b*x^(2/3)+a*x)^(1/2)*(b+a*x^(1/3))*(230945*a^9*x^3-218790*a^8*b*x^(8/3)+205920*a^7*b^2*x^(7/3)-1921
92*a^6*b^3*x^2+177408*a^5*b^4*x^(5/3)-161280*a^4*b^5*x^(4/3)+143360*a^3*b^6*x-122880*a^2*b^7*x^(2/3)+98304*a*b
^8*x^(1/3)-65536*b^9)/x^(1/3)/a^10

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1031 vs. \(2 (211) = 422\).

Time = 150.03 (sec) , antiderivative size = 1031, normalized size of antiderivative = 3.64 \[ \int x^2 \sqrt {b x^{2/3}+a x} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/3233230*((3298534883328*b^16 + 687194767360*b^15 + 3221225472*(64*a^3 - 3)*b^13 - 64424509440*b^14 - 1677721
6*(11264*a^3 - 53)*b^12 + 5380094720*a^12 - 6291456*(5504*a^3 + 1)*b^11 + 196608*(3194880*a^6 - 114688*a^3 - 3
)*b^10 + 7340032*(18816*a^6 + 103*a^3)*b^9 - 786432*(48816*a^6 + 23*a^3)*b^8 - 12288*(45731840*a^9 - 495872*a^
6 - 15*a^3)*b^7 - 114688*(1349120*a^9 + 3439*a^6)*b^6 + 3913728*(5600*a^9 + 3*a^6)*b^5 - 2112*(2027683840*a^12
+ 1958400*a^9 + 63*a^6)*b^4 - 36608*(59351040*a^12 - 8101*a^9)*b^3 - 549120*(566272*a^12 + 17*a^9)*b^2 - 1093
95*(516096*a^12 - a^9)*b)*x + 4*(230945*(16777216*a^10*b^6 + 6291456*a^10*b^5 + 196608*a^10*b^4 - 262144*a^13
- 114688*a^10*b^3 - 2304*a^10*b^2 + 864*a^10*b - 27*a^10)*x^4 + 13728*(16777216*a^7*b^9 + 6291456*a^7*b^8 + 19
6608*a^7*b^7 - 114688*a^7*b^6 - 2304*a^7*b^5 + 864*a^7*b^4 - (262144*a^10 + 27*a^7)*b^3)*x^3 - 17920*(16777216
*a^4*b^12 + 6291456*a^4*b^11 + 196608*a^4*b^10 - 114688*a^4*b^9 - 2304*a^4*b^8 + 864*a^4*b^7 - (262144*a^7 + 2
7*a^4)*b^6)*x^2 + 32768*(16777216*a*b^15 + 6291456*a*b^14 + 196608*a*b^13 - 114688*a*b^12 - 2304*a*b^11 + 864*
a*b^10 - (262144*a^4 + 27*a)*b^9)*x - (1099511627776*b^16 + 412316860416*b^15 + 12884901888*b^14 - 7516192768*
b^13 - 150994944*b^12 - 65536*(262144*a^3 + 27)*b^10 + 56623104*b^11 - 12155*(16777216*a^9*b^7 + 6291456*a^9*b
^6 + 196608*a^9*b^5 - 114688*a^9*b^4 - 2304*a^9*b^3 + 864*a^9*b^2 - (262144*a^12 + 27*a^9)*b)*x^3 + 14784*(167
77216*a^6*b^10 + 6291456*a^6*b^9 + 196608*a^6*b^8 - 114688*a^6*b^7 - 2304*a^6*b^6 + 864*a^6*b^5 - (262144*a^9
+ 27*a^6)*b^4)*x^2 - 20480*(16777216*a^3*b^13 + 6291456*a^3*b^12 + 196608*a^3*b^11 - 114688*a^3*b^10 - 2304*a^
3*b^9 + 864*a^3*b^8 - (262144*a^6 + 27*a^3)*b^7)*x)*x^(2/3) - 6*(2145*(16777216*a^8*b^8 + 6291456*a^8*b^7 + 19
6608*a^8*b^6 - 114688*a^8*b^5 - 2304*a^8*b^4 + 864*a^8*b^3 - (262144*a^11 + 27*a^8)*b^2)*x^3 - 2688*(16777216*
a^5*b^11 + 6291456*a^5*b^10 + 196608*a^5*b^9 - 114688*a^5*b^8 - 2304*a^5*b^7 + 864*a^5*b^6 - (262144*a^8 + 27*
a^5)*b^5)*x^2 + 4096*(16777216*a^2*b^14 + 6291456*a^2*b^13 + 196608*a^2*b^12 - 114688*a^2*b^11 - 2304*a^2*b^10
+ 864*a^2*b^9 - (262144*a^5 + 27*a^2)*b^8)*x)*x^(1/3))*sqrt(a*x + b*x^(2/3)))/((16777216*a^10*b^6 + 6291456*a
^10*b^5 + 196608*a^10*b^4 - 262144*a^13 - 114688*a^10*b^3 - 2304*a^10*b^2 + 864*a^10*b - 27*a^10)*x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.10 \[ \int x^2 \sqrt {b x^{2/3}+a x} \, dx=\frac {131072 \, b^{\frac {21}{2}}}{1616615 \, a^{10}} + \frac {2 \, {\left (\frac {21 \, {\left (12155 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {19}{2}} - 122265 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} b + 554268 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} b^{2} - 1492260 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} b^{3} + 2645370 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} b^{4} - 3233230 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b^{5} + 2771340 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{6} - 1662804 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{7} + 692835 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{8} - 230945 \, \sqrt {a x^{\frac {1}{3}} + b} b^{9}\right )} b}{a^{9}} + \frac {5 \, {\left (46189 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {21}{2}} - 510510 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {19}{2}} b + 2567565 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} b^{2} - 7759752 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} b^{3} + 15668730 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} b^{4} - 22221108 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} b^{5} + 22632610 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b^{6} - 16628040 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{7} + 8729721 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{8} - 3233230 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{9} + 969969 \, \sqrt {a x^{\frac {1}{3}} + b} b^{10}\right )}}{a^{9}}\right )}}{1616615 \, a} \]

[In]

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

[Out]

131072/1616615*b^(21/2)/a^10 + 2/1616615*(21*(12155*(a*x^(1/3) + b)^(19/2) - 122265*(a*x^(1/3) + b)^(17/2)*b +
554268*(a*x^(1/3) + b)^(15/2)*b^2 - 1492260*(a*x^(1/3) + b)^(13/2)*b^3 + 2645370*(a*x^(1/3) + b)^(11/2)*b^4 -
3233230*(a*x^(1/3) + b)^(9/2)*b^5 + 2771340*(a*x^(1/3) + b)^(7/2)*b^6 - 1662804*(a*x^(1/3) + b)^(5/2)*b^7 + 6
92835*(a*x^(1/3) + b)^(3/2)*b^8 - 230945*sqrt(a*x^(1/3) + b)*b^9)*b/a^9 + 5*(46189*(a*x^(1/3) + b)^(21/2) - 51
0510*(a*x^(1/3) + b)^(19/2)*b + 2567565*(a*x^(1/3) + b)^(17/2)*b^2 - 7759752*(a*x^(1/3) + b)^(15/2)*b^3 + 1566
8730*(a*x^(1/3) + b)^(13/2)*b^4 - 22221108*(a*x^(1/3) + b)^(11/2)*b^5 + 22632610*(a*x^(1/3) + b)^(9/2)*b^6 - 1
6628040*(a*x^(1/3) + b)^(7/2)*b^7 + 8729721*(a*x^(1/3) + b)^(5/2)*b^8 - 3233230*(a*x^(1/3) + b)^(3/2)*b^9 + 96
9969*sqrt(a*x^(1/3) + b)*b^10)/a^9)/a

Mupad [F(-1)]

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

[In]

int(x^2*(a*x + b*x^(2/3))^(1/2),x)

[Out]

int(x^2*(a*x + b*x^(2/3))^(1/2), x)