Integrand size = 19, antiderivative size = 283 \[ \int x^2 \sqrt {b x^{2/3}+a x} \, dx=\frac {8192 b^6 \left (b x^{2/3}+a x\right )^{3/2}}{46189 a^7}-\frac {131072 b^9 \left (b x^{2/3}+a x\right )^{3/2}}{1616615 a^{10} x}+\frac {196608 b^8 \left (b x^{2/3}+a x\right )^{3/2}}{1616615 a^9 x^{2/3}}-\frac {49152 b^7 \left (b x^{2/3}+a x\right )^{3/2}}{323323 a^8 \sqrt [3]{x}}-\frac {9216 b^5 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{3/2}}{46189 a^6}+\frac {4608 b^4 x^{2/3} \left (b x^{2/3}+a x\right )^{3/2}}{20995 a^5}-\frac {384 b^3 x \left (b x^{2/3}+a x\right )^{3/2}}{1615 a^4}+\frac {576 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{3/2}}{2261 a^3}-\frac {36 b x^{5/3} \left (b x^{2/3}+a x\right )^{3/2}}{133 a^2}+\frac {2 x^2 \left (b x^{2/3}+a x\right )^{3/2}}{7 a} \] Output:
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^5-384/1 615*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
Time = 0.10 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.47 \[ \int x^2 \sqrt {b x^{2/3}+a x} \, dx=\frac {2 \left (b x^{2/3}+a x\right )^{3/2} \left (-65536 b^9+98304 a b^8 \sqrt [3]{x}-122880 a^2 b^7 x^{2/3}+143360 a^3 b^6 x-161280 a^4 b^5 x^{4/3}+177408 a^5 b^4 x^{5/3}-192192 a^6 b^3 x^2+205920 a^7 b^2 x^{7/3}-218790 a^8 b x^{8/3}+230945 a^9 x^3\right )}{1616615 a^{10} x} \] Input:
Integrate[x^2*Sqrt[b*x^(2/3) + a*x],x]
Output:
(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 - 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))/(1616615*a^10*x)
Time = 1.03 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.17, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {1922, 1922, 1922, 1922, 1922, 1922, 1908, 1922, 1922, 1920}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^2 \sqrt {a x+b x^{2/3}} \, dx\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^2 \left (a x+b x^{2/3}\right )^{3/2}}{7 a}-\frac {6 b \int x^{5/3} \sqrt {x^{2/3} b+a x}dx}{7 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^2 \left (a x+b x^{2/3}\right )^{3/2}}{7 a}-\frac {6 b \left (\frac {6 x^{5/3} \left (a x+b x^{2/3}\right )^{3/2}}{19 a}-\frac {16 b \int x^{4/3} \sqrt {x^{2/3} b+a x}dx}{19 a}\right )}{7 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^2 \left (a x+b x^{2/3}\right )^{3/2}}{7 a}-\frac {6 b \left (\frac {6 x^{5/3} \left (a x+b x^{2/3}\right )^{3/2}}{19 a}-\frac {16 b \left (\frac {6 x^{4/3} \left (a x+b x^{2/3}\right )^{3/2}}{17 a}-\frac {14 b \int x \sqrt {x^{2/3} b+a x}dx}{17 a}\right )}{19 a}\right )}{7 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^2 \left (a x+b x^{2/3}\right )^{3/2}}{7 a}-\frac {6 b \left (\frac {6 x^{5/3} \left (a x+b x^{2/3}\right )^{3/2}}{19 a}-\frac {16 b \left (\frac {6 x^{4/3} \left (a x+b x^{2/3}\right )^{3/2}}{17 a}-\frac {14 b \left (\frac {2 x \left (a x+b x^{2/3}\right )^{3/2}}{5 a}-\frac {4 b \int x^{2/3} \sqrt {x^{2/3} b+a x}dx}{5 a}\right )}{17 a}\right )}{19 a}\right )}{7 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^2 \left (a x+b x^{2/3}\right )^{3/2}}{7 a}-\frac {6 b \left (\frac {6 x^{5/3} \left (a x+b x^{2/3}\right )^{3/2}}{19 a}-\frac {16 b \left (\frac {6 x^{4/3} \left (a x+b x^{2/3}\right )^{3/2}}{17 a}-\frac {14 b \left (\frac {2 x \left (a x+b x^{2/3}\right )^{3/2}}{5 a}-\frac {4 b \left (\frac {6 x^{2/3} \left (a x+b x^{2/3}\right )^{3/2}}{13 a}-\frac {10 b \int \sqrt [3]{x} \sqrt {x^{2/3} b+a x}dx}{13 a}\right )}{5 a}\right )}{17 a}\right )}{19 a}\right )}{7 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^2 \left (a x+b x^{2/3}\right )^{3/2}}{7 a}-\frac {6 b \left (\frac {6 x^{5/3} \left (a x+b x^{2/3}\right )^{3/2}}{19 a}-\frac {16 b \left (\frac {6 x^{4/3} \left (a x+b x^{2/3}\right )^{3/2}}{17 a}-\frac {14 b \left (\frac {2 x \left (a x+b x^{2/3}\right )^{3/2}}{5 a}-\frac {4 b \left (\frac {6 x^{2/3} \left (a x+b x^{2/3}\right )^{3/2}}{13 a}-\frac {10 b \left (\frac {6 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{3/2}}{11 a}-\frac {8 b \int \sqrt {x^{2/3} b+a x}dx}{11 a}\right )}{13 a}\right )}{5 a}\right )}{17 a}\right )}{19 a}\right )}{7 a}\) |
| \(\Big \downarrow \) 1908 |
| \(\displaystyle \frac {2 x^2 \left (a x+b x^{2/3}\right )^{3/2}}{7 a}-\frac {6 b \left (\frac {6 x^{5/3} \left (a x+b x^{2/3}\right )^{3/2}}{19 a}-\frac {16 b \left (\frac {6 x^{4/3} \left (a x+b x^{2/3}\right )^{3/2}}{17 a}-\frac {14 b \left (\frac {2 x \left (a x+b x^{2/3}\right )^{3/2}}{5 a}-\frac {4 b \left (\frac {6 x^{2/3} \left (a x+b x^{2/3}\right )^{3/2}}{13 a}-\frac {10 b \left (\frac {6 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{3/2}}{11 a}-\frac {8 b \left (\frac {2 \left (a x+b x^{2/3}\right )^{3/2}}{3 a}-\frac {2 b \int \frac {\sqrt {x^{2/3} b+a x}}{\sqrt [3]{x}}dx}{3 a}\right )}{11 a}\right )}{13 a}\right )}{5 a}\right )}{17 a}\right )}{19 a}\right )}{7 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^2 \left (a x+b x^{2/3}\right )^{3/2}}{7 a}-\frac {6 b \left (\frac {6 x^{5/3} \left (a x+b x^{2/3}\right )^{3/2}}{19 a}-\frac {16 b \left (\frac {6 x^{4/3} \left (a x+b x^{2/3}\right )^{3/2}}{17 a}-\frac {14 b \left (\frac {2 x \left (a x+b x^{2/3}\right )^{3/2}}{5 a}-\frac {4 b \left (\frac {6 x^{2/3} \left (a x+b x^{2/3}\right )^{3/2}}{13 a}-\frac {10 b \left (\frac {6 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{3/2}}{11 a}-\frac {8 b \left (\frac {2 \left (a x+b x^{2/3}\right )^{3/2}}{3 a}-\frac {2 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{3/2}}{7 a \sqrt [3]{x}}-\frac {4 b \int \frac {\sqrt {x^{2/3} b+a x}}{x^{2/3}}dx}{7 a}\right )}{3 a}\right )}{11 a}\right )}{13 a}\right )}{5 a}\right )}{17 a}\right )}{19 a}\right )}{7 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^2 \left (a x+b x^{2/3}\right )^{3/2}}{7 a}-\frac {6 b \left (\frac {6 x^{5/3} \left (a x+b x^{2/3}\right )^{3/2}}{19 a}-\frac {16 b \left (\frac {6 x^{4/3} \left (a x+b x^{2/3}\right )^{3/2}}{17 a}-\frac {14 b \left (\frac {2 x \left (a x+b x^{2/3}\right )^{3/2}}{5 a}-\frac {4 b \left (\frac {6 x^{2/3} \left (a x+b x^{2/3}\right )^{3/2}}{13 a}-\frac {10 b \left (\frac {6 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{3/2}}{11 a}-\frac {8 b \left (\frac {2 \left (a x+b x^{2/3}\right )^{3/2}}{3 a}-\frac {2 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{3/2}}{7 a \sqrt [3]{x}}-\frac {4 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{3/2}}{5 a x^{2/3}}-\frac {2 b \int \frac {\sqrt {x^{2/3} b+a x}}{x}dx}{5 a}\right )}{7 a}\right )}{3 a}\right )}{11 a}\right )}{13 a}\right )}{5 a}\right )}{17 a}\right )}{19 a}\right )}{7 a}\) |
| \(\Big \downarrow \) 1920 |
| \(\displaystyle \frac {2 x^2 \left (a x+b x^{2/3}\right )^{3/2}}{7 a}-\frac {6 b \left (\frac {6 x^{5/3} \left (a x+b x^{2/3}\right )^{3/2}}{19 a}-\frac {16 b \left (\frac {6 x^{4/3} \left (a x+b x^{2/3}\right )^{3/2}}{17 a}-\frac {14 b \left (\frac {2 x \left (a x+b x^{2/3}\right )^{3/2}}{5 a}-\frac {4 b \left (\frac {6 x^{2/3} \left (a x+b x^{2/3}\right )^{3/2}}{13 a}-\frac {10 b \left (\frac {6 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{3/2}}{11 a}-\frac {8 b \left (\frac {2 \left (a x+b x^{2/3}\right )^{3/2}}{3 a}-\frac {2 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{3/2}}{7 a \sqrt [3]{x}}-\frac {4 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{3/2}}{5 a x^{2/3}}-\frac {4 b \left (a x+b x^{2/3}\right )^{3/2}}{5 a^2 x}\right )}{7 a}\right )}{3 a}\right )}{11 a}\right )}{13 a}\right )}{5 a}\right )}{17 a}\right )}{19 a}\right )}{7 a}\) |
Input:
Int[x^2*Sqrt[b*x^(2/3) + a*x],x]
Output:
(2*x^2*(b*x^(2/3) + a*x)^(3/2))/(7*a) - (6*b*((6*x^(5/3)*(b*x^(2/3) + a*x) ^(3/2))/(19*a) - (16*b*((6*x^(4/3)*(b*x^(2/3) + a*x)^(3/2))/(17*a) - (14*b *((2*x*(b*x^(2/3) + a*x)^(3/2))/(5*a) - (4*b*((6*x^(2/3)*(b*x^(2/3) + a*x) ^(3/2))/(13*a) - (10*b*((6*x^(1/3)*(b*x^(2/3) + a*x)^(3/2))/(11*a) - (8*b* ((2*(b*x^(2/3) + a*x)^(3/2))/(3*a) - (2*b*((6*(b*x^(2/3) + a*x)^(3/2))/(7* a*x^(1/3)) - (4*b*((-4*b*(b*x^(2/3) + a*x)^(3/2))/(5*a^2*x) + (6*(b*x^(2/3 ) + a*x)^(3/2))/(5*a*x^(2/3))))/(7*a)))/(3*a)))/(11*a)))/(13*a)))/(5*a)))/ (17*a)))/(19*a)))/(7*a)
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] - Simp[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]
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])
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] - Simp[b*((m + n*p + n - j + 1)/(a*c^(n - j)*(m + j*p + 1))) I nt[(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])
Time = 0.35 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.43
| method | result | size |
| derivativedivides | \(\frac {2 \sqrt {b \,x^{\frac {2}{3}}+a x}\, \left (x^{\frac {1}{3}} a +b \right ) \left (230945 a^{9} x^{3}-218790 a^{8} b \,x^{\frac {8}{3}}+205920 a^{7} b^{2} x^{\frac {7}{3}}-192192 a^{6} b^{3} x^{2}+177408 a^{5} b^{4} x^{\frac {5}{3}}-161280 a^{4} b^{5} x^{\frac {4}{3}}+143360 a^{3} b^{6} x -122880 a^{2} b^{7} x^{\frac {2}{3}}+98304 a \,b^{8} x^{\frac {1}{3}}-65536 b^{9}\right )}{1616615 x^{\frac {1}{3}} a^{10}}\) | \(123\) |
| default | \(-\frac {2 \sqrt {b \,x^{\frac {2}{3}}+a x}\, \left (x^{\frac {1}{3}} a +b \right ) \left (218790 a^{8} b \,x^{\frac {8}{3}}-205920 a^{7} b^{2} x^{\frac {7}{3}}-177408 a^{5} b^{4} x^{\frac {5}{3}}+161280 a^{4} b^{5} x^{\frac {4}{3}}-230945 a^{9} x^{3}+122880 a^{2} b^{7} x^{\frac {2}{3}}+192192 a^{6} b^{3} x^{2}-98304 a \,b^{8} x^{\frac {1}{3}}-143360 a^{3} b^{6} x +65536 b^{9}\right )}{1616615 x^{\frac {1}{3}} a^{10}}\) | \(123\) |
Input:
int(x^2*(b*x^(2/3)+a*x)^(1/2),x,method=_RETURNVERBOSE)
Output:
2/1616615*(b*x^(2/3)+a*x)^(1/2)*(x^(1/3)*a+b)*(230945*a^9*x^3-218790*a^8*b *x^(8/3)+205920*a^7*b^2*x^(7/3)-192192*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
Leaf count of result is larger than twice the leaf count of optimal. 1031 vs. \(2 (211) = 422\).
Time = 123.05 (sec) , antiderivative size = 1031, normalized size of antiderivative = 3.64 \[ \int x^2 \sqrt {b x^{2/3}+a x} \, dx=\text {Too large to display} \] Input:
integrate(x^2*(b*x^(2/3)+a*x)^(1/2),x, algorithm="fricas")
Output:
1/3233230*((3298534883328*b^16 + 687194767360*b^15 + 3221225472*(64*a^3 - 3)*b^13 - 64424509440*b^14 - 16777216*(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 + 1 958400*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 - 109395*(516096*a^12 - a^9)*b)*x + 4*(230945*(1 6777216*a^10*b^6 + 6291456*a^10*b^5 + 196608*a^10*b^4 - 262144*a^13 - 1146 88*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 + 196608*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^1 2 + 6291456*a^4*b^11 + 196608*a^4*b^10 - 114688*a^4*b^9 - 2304*a^4*b^8 + 8 64*a^4*b^7 - (262144*a^7 + 27*a^4)*b^6)*x^2 + 32768*(16777216*a*b^15 + 629 1456*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 + 1288 4901888*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*(16777216*a^6*b^10 + 6291456*a^6*b^9 + 196608*a^6*b...
\[ \int x^2 \sqrt {b x^{2/3}+a x} \, dx=\int x^{2} \sqrt {a x + b x^{\frac {2}{3}}}\, dx \] Input:
integrate(x**2*(b*x**(2/3)+a*x)**(1/2),x)
Output:
Integral(x**2*sqrt(a*x + b*x**(2/3)), x)
\[ \int x^2 \sqrt {b x^{2/3}+a x} \, dx=\int { \sqrt {a x + b x^{\frac {2}{3}}} x^{2} \,d x } \] Input:
integrate(x^2*(b*x^(2/3)+a*x)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(a*x + b*x^(2/3))*x^2, x)
Time = 0.29 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.10 \[ \int x^2 \sqrt {b x^{2/3}+a x} \, dx=\frac {131072 \, b^{\frac {21}{2}}}{1616615 \, a^{10}} + \frac {2 \, {\left (\frac {21 \, {\left (12155 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {19}{2}} - 122265 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} b + 554268 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} b^{2} - 1492260 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} b^{3} + 2645370 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} b^{4} - 3233230 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b^{5} + 2771340 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{6} - 1662804 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{7} + 692835 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{8} - 230945 \, \sqrt {a x^{\frac {1}{3}} + b} b^{9}\right )} b}{a^{9}} + \frac {5 \, {\left (46189 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {21}{2}} - 510510 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {19}{2}} b + 2567565 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} b^{2} - 7759752 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} b^{3} + 15668730 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} b^{4} - 22221108 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} b^{5} + 22632610 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b^{6} - 16628040 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{7} + 8729721 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{8} - 3233230 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{9} + 969969 \, \sqrt {a x^{\frac {1}{3}} + b} b^{10}\right )}}{a^{9}}\right )}}{1616615 \, a} \] Input:
integrate(x^2*(b*x^(2/3)+a*x)^(1/2),x, algorithm="giac")
Output:
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 - 1 492260*(a*x^(1/3) + b)^(13/2)*b^3 + 2645370*(a*x^(1/3) + b)^(11/2)*b^4 - 3 233230*(a*x^(1/3) + b)^(9/2)*b^5 + 2771340*(a*x^(1/3) + b)^(7/2)*b^6 - 166 2804*(a*x^(1/3) + b)^(5/2)*b^7 + 692835*(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) - 510510 *(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 + 15668730*(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 - 1662804 0*(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 + 969969*sqrt(a*x^(1/3) + b)*b^10)/a^9)/a
Timed out. \[ \int x^2 \sqrt {b x^{2/3}+a x} \, dx=\int x^2\,\sqrt {a\,x+b\,x^{2/3}} \,d x \] Input:
int(x^2*(a*x + b*x^(2/3))^(1/2),x)
Output:
int(x^2*(a*x + b*x^(2/3))^(1/2), x)
Time = 0.22 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.42 \[ \int x^2 \sqrt {b x^{2/3}+a x} \, dx=\frac {2 \sqrt {x^{\frac {1}{3}} a +b}\, \left (-12870 x^{\frac {8}{3}} a^{8} b^{2}+16128 x^{\frac {5}{3}} a^{5} b^{5}-24576 x^{\frac {2}{3}} a^{2} b^{8}+230945 x^{\frac {10}{3}} a^{10}+13728 x^{\frac {7}{3}} a^{7} b^{3}-17920 x^{\frac {4}{3}} a^{4} b^{6}+32768 x^{\frac {1}{3}} a \,b^{9}+12155 a^{9} b \,x^{3}-14784 a^{6} b^{4} x^{2}+20480 a^{3} b^{7} x -65536 b^{10}\right )}{1616615 a^{10}} \] Input:
int(x^2*(b*x^(2/3)+a*x)^(1/2),x)
Output:
(2*sqrt(x**(1/3)*a + b)*( - 12870*x**(2/3)*a**8*b**2*x**2 + 16128*x**(2/3) *a**5*b**5*x - 24576*x**(2/3)*a**2*b**8 + 230945*x**(1/3)*a**10*x**3 + 137 28*x**(1/3)*a**7*b**3*x**2 - 17920*x**(1/3)*a**4*b**6*x + 32768*x**(1/3)*a *b**9 + 12155*a**9*b*x**3 - 14784*a**6*b**4*x**2 + 20480*a**3*b**7*x - 655 36*b**10))/(1616615*a**10)