Integrand size = 19, antiderivative size = 313 \[ \int \frac {x^3}{\sqrt {b x^{2/3}+a x}} \, dx=-\frac {262144 b^9 \sqrt {b x^{2/3}+a x}}{323323 a^{10}}+\frac {524288 b^{10} \sqrt {b x^{2/3}+a x}}{323323 a^{11} \sqrt [3]{x}}+\frac {196608 b^8 \sqrt [3]{x} \sqrt {b x^{2/3}+a x}}{323323 a^9}-\frac {163840 b^7 x^{2/3} \sqrt {b x^{2/3}+a x}}{323323 a^8}+\frac {20480 b^6 x \sqrt {b x^{2/3}+a x}}{46189 a^7}-\frac {18432 b^5 x^{4/3} \sqrt {b x^{2/3}+a x}}{46189 a^6}+\frac {1536 b^4 x^{5/3} \sqrt {b x^{2/3}+a x}}{4199 a^5}-\frac {768 b^3 x^2 \sqrt {b x^{2/3}+a x}}{2261 a^4}+\frac {720 b^2 x^{7/3} \sqrt {b x^{2/3}+a x}}{2261 a^3}-\frac {40 b x^{8/3} \sqrt {b x^{2/3}+a x}}{133 a^2}+\frac {2 x^3 \sqrt {b x^{2/3}+a x}}{7 a} \]
-262144/323323*b^9*(b*x^(2/3)+a*x)^(1/2)/a^10+524288/323323*b^10*(b*x^(2/3 )+a*x)^(1/2)/a^11/x^(1/3)+196608/323323*b^8*x^(1/3)*(b*x^(2/3)+a*x)^(1/2)/ a^9-163840/323323*b^7*x^(2/3)*(b*x^(2/3)+a*x)^(1/2)/a^8+20480/46189*b^6*x* (b*x^(2/3)+a*x)^(1/2)/a^7-18432/46189*b^5*x^(4/3)*(b*x^(2/3)+a*x)^(1/2)/a^ 6+1536/4199*b^4*x^(5/3)*(b*x^(2/3)+a*x)^(1/2)/a^5-768/2261*b^3*x^2*(b*x^(2 /3)+a*x)^(1/2)/a^4+720/2261*b^2*x^(7/3)*(b*x^(2/3)+a*x)^(1/2)/a^3-40/133*b *x^(8/3)*(b*x^(2/3)+a*x)^(1/2)/a^2+2/7*x^3*(b*x^(2/3)+a*x)^(1/2)/a
Time = 0.15 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.47 \[ \int \frac {x^3}{\sqrt {b x^{2/3}+a x}} \, dx=\frac {2 \sqrt {b x^{2/3}+a x} \left (262144 b^{10}-131072 a b^9 \sqrt [3]{x}+98304 a^2 b^8 x^{2/3}-81920 a^3 b^7 x+71680 a^4 b^6 x^{4/3}-64512 a^5 b^5 x^{5/3}+59136 a^6 b^4 x^2-54912 a^7 b^3 x^{7/3}+51480 a^8 b^2 x^{8/3}-48620 a^9 b x^3+46189 a^{10} x^{10/3}\right )}{323323 a^{11} \sqrt [3]{x}} \]
(2*Sqrt[b*x^(2/3) + a*x]*(262144*b^10 - 131072*a*b^9*x^(1/3) + 98304*a^2*b ^8*x^(2/3) - 81920*a^3*b^7*x + 71680*a^4*b^6*x^(4/3) - 64512*a^5*b^5*x^(5/ 3) + 59136*a^6*b^4*x^2 - 54912*a^7*b^3*x^(7/3) + 51480*a^8*b^2*x^(8/3) - 4 8620*a^9*b*x^3 + 46189*a^10*x^(10/3)))/(323323*a^11*x^(1/3))
Time = 0.72 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.16, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.579, Rules used = {1922, 1922, 1922, 1922, 1922, 1922, 1922, 1922, 1922, 1908, 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.
| \(\displaystyle \int \frac {x^3}{\sqrt {a x+b x^{2/3}}} \, dx\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^3 \sqrt {a x+b x^{2/3}}}{7 a}-\frac {20 b \int \frac {x^{8/3}}{\sqrt {x^{2/3} b+a x}}dx}{21 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^3 \sqrt {a x+b x^{2/3}}}{7 a}-\frac {20 b \left (\frac {6 x^{8/3} \sqrt {a x+b x^{2/3}}}{19 a}-\frac {18 b \int \frac {x^{7/3}}{\sqrt {x^{2/3} b+a x}}dx}{19 a}\right )}{21 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^3 \sqrt {a x+b x^{2/3}}}{7 a}-\frac {20 b \left (\frac {6 x^{8/3} \sqrt {a x+b x^{2/3}}}{19 a}-\frac {18 b \left (\frac {6 x^{7/3} \sqrt {a x+b x^{2/3}}}{17 a}-\frac {16 b \int \frac {x^2}{\sqrt {x^{2/3} b+a x}}dx}{17 a}\right )}{19 a}\right )}{21 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^3 \sqrt {a x+b x^{2/3}}}{7 a}-\frac {20 b \left (\frac {6 x^{8/3} \sqrt {a x+b x^{2/3}}}{19 a}-\frac {18 b \left (\frac {6 x^{7/3} \sqrt {a x+b x^{2/3}}}{17 a}-\frac {16 b \left (\frac {2 x^2 \sqrt {a x+b x^{2/3}}}{5 a}-\frac {14 b \int \frac {x^{5/3}}{\sqrt {x^{2/3} b+a x}}dx}{15 a}\right )}{17 a}\right )}{19 a}\right )}{21 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^3 \sqrt {a x+b x^{2/3}}}{7 a}-\frac {20 b \left (\frac {6 x^{8/3} \sqrt {a x+b x^{2/3}}}{19 a}-\frac {18 b \left (\frac {6 x^{7/3} \sqrt {a x+b x^{2/3}}}{17 a}-\frac {16 b \left (\frac {2 x^2 \sqrt {a x+b x^{2/3}}}{5 a}-\frac {14 b \left (\frac {6 x^{5/3} \sqrt {a x+b x^{2/3}}}{13 a}-\frac {12 b \int \frac {x^{4/3}}{\sqrt {x^{2/3} b+a x}}dx}{13 a}\right )}{15 a}\right )}{17 a}\right )}{19 a}\right )}{21 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^3 \sqrt {a x+b x^{2/3}}}{7 a}-\frac {20 b \left (\frac {6 x^{8/3} \sqrt {a x+b x^{2/3}}}{19 a}-\frac {18 b \left (\frac {6 x^{7/3} \sqrt {a x+b x^{2/3}}}{17 a}-\frac {16 b \left (\frac {2 x^2 \sqrt {a x+b x^{2/3}}}{5 a}-\frac {14 b \left (\frac {6 x^{5/3} \sqrt {a x+b x^{2/3}}}{13 a}-\frac {12 b \left (\frac {6 x^{4/3} \sqrt {a x+b x^{2/3}}}{11 a}-\frac {10 b \int \frac {x}{\sqrt {x^{2/3} b+a x}}dx}{11 a}\right )}{13 a}\right )}{15 a}\right )}{17 a}\right )}{19 a}\right )}{21 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^3 \sqrt {a x+b x^{2/3}}}{7 a}-\frac {20 b \left (\frac {6 x^{8/3} \sqrt {a x+b x^{2/3}}}{19 a}-\frac {18 b \left (\frac {6 x^{7/3} \sqrt {a x+b x^{2/3}}}{17 a}-\frac {16 b \left (\frac {2 x^2 \sqrt {a x+b x^{2/3}}}{5 a}-\frac {14 b \left (\frac {6 x^{5/3} \sqrt {a x+b x^{2/3}}}{13 a}-\frac {12 b \left (\frac {6 x^{4/3} \sqrt {a x+b x^{2/3}}}{11 a}-\frac {10 b \left (\frac {2 x \sqrt {a x+b x^{2/3}}}{3 a}-\frac {8 b \int \frac {x^{2/3}}{\sqrt {x^{2/3} b+a x}}dx}{9 a}\right )}{11 a}\right )}{13 a}\right )}{15 a}\right )}{17 a}\right )}{19 a}\right )}{21 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^3 \sqrt {a x+b x^{2/3}}}{7 a}-\frac {20 b \left (\frac {6 x^{8/3} \sqrt {a x+b x^{2/3}}}{19 a}-\frac {18 b \left (\frac {6 x^{7/3} \sqrt {a x+b x^{2/3}}}{17 a}-\frac {16 b \left (\frac {2 x^2 \sqrt {a x+b x^{2/3}}}{5 a}-\frac {14 b \left (\frac {6 x^{5/3} \sqrt {a x+b x^{2/3}}}{13 a}-\frac {12 b \left (\frac {6 x^{4/3} \sqrt {a x+b x^{2/3}}}{11 a}-\frac {10 b \left (\frac {2 x \sqrt {a x+b x^{2/3}}}{3 a}-\frac {8 b \left (\frac {6 x^{2/3} \sqrt {a x+b x^{2/3}}}{7 a}-\frac {6 b \int \frac {\sqrt [3]{x}}{\sqrt {x^{2/3} b+a x}}dx}{7 a}\right )}{9 a}\right )}{11 a}\right )}{13 a}\right )}{15 a}\right )}{17 a}\right )}{19 a}\right )}{21 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^3 \sqrt {a x+b x^{2/3}}}{7 a}-\frac {20 b \left (\frac {6 x^{8/3} \sqrt {a x+b x^{2/3}}}{19 a}-\frac {18 b \left (\frac {6 x^{7/3} \sqrt {a x+b x^{2/3}}}{17 a}-\frac {16 b \left (\frac {2 x^2 \sqrt {a x+b x^{2/3}}}{5 a}-\frac {14 b \left (\frac {6 x^{5/3} \sqrt {a x+b x^{2/3}}}{13 a}-\frac {12 b \left (\frac {6 x^{4/3} \sqrt {a x+b x^{2/3}}}{11 a}-\frac {10 b \left (\frac {2 x \sqrt {a x+b x^{2/3}}}{3 a}-\frac {8 b \left (\frac {6 x^{2/3} \sqrt {a x+b x^{2/3}}}{7 a}-\frac {6 b \left (\frac {6 \sqrt [3]{x} \sqrt {a x+b x^{2/3}}}{5 a}-\frac {4 b \int \frac {1}{\sqrt {x^{2/3} b+a x}}dx}{5 a}\right )}{7 a}\right )}{9 a}\right )}{11 a}\right )}{13 a}\right )}{15 a}\right )}{17 a}\right )}{19 a}\right )}{21 a}\) |
| \(\Big \downarrow \) 1908 |
| \(\displaystyle \frac {2 x^3 \sqrt {a x+b x^{2/3}}}{7 a}-\frac {20 b \left (\frac {6 x^{8/3} \sqrt {a x+b x^{2/3}}}{19 a}-\frac {18 b \left (\frac {6 x^{7/3} \sqrt {a x+b x^{2/3}}}{17 a}-\frac {16 b \left (\frac {2 x^2 \sqrt {a x+b x^{2/3}}}{5 a}-\frac {14 b \left (\frac {6 x^{5/3} \sqrt {a x+b x^{2/3}}}{13 a}-\frac {12 b \left (\frac {6 x^{4/3} \sqrt {a x+b x^{2/3}}}{11 a}-\frac {10 b \left (\frac {2 x \sqrt {a x+b x^{2/3}}}{3 a}-\frac {8 b \left (\frac {6 x^{2/3} \sqrt {a x+b x^{2/3}}}{7 a}-\frac {6 b \left (\frac {6 \sqrt [3]{x} \sqrt {a x+b x^{2/3}}}{5 a}-\frac {4 b \left (\frac {2 \sqrt {a x+b x^{2/3}}}{a}-\frac {2 b \int \frac {1}{\sqrt [3]{x} \sqrt {x^{2/3} b+a x}}dx}{3 a}\right )}{5 a}\right )}{7 a}\right )}{9 a}\right )}{11 a}\right )}{13 a}\right )}{15 a}\right )}{17 a}\right )}{19 a}\right )}{21 a}\) |
| \(\Big \downarrow \) 1920 |
| \(\displaystyle \frac {2 x^3 \sqrt {a x+b x^{2/3}}}{7 a}-\frac {20 b \left (\frac {6 x^{8/3} \sqrt {a x+b x^{2/3}}}{19 a}-\frac {18 b \left (\frac {6 x^{7/3} \sqrt {a x+b x^{2/3}}}{17 a}-\frac {16 b \left (\frac {2 x^2 \sqrt {a x+b x^{2/3}}}{5 a}-\frac {14 b \left (\frac {6 x^{5/3} \sqrt {a x+b x^{2/3}}}{13 a}-\frac {12 b \left (\frac {6 x^{4/3} \sqrt {a x+b x^{2/3}}}{11 a}-\frac {10 b \left (\frac {2 x \sqrt {a x+b x^{2/3}}}{3 a}-\frac {8 b \left (\frac {6 x^{2/3} \sqrt {a x+b x^{2/3}}}{7 a}-\frac {6 b \left (\frac {6 \sqrt [3]{x} \sqrt {a x+b x^{2/3}}}{5 a}-\frac {4 b \left (\frac {2 \sqrt {a x+b x^{2/3}}}{a}-\frac {4 b \sqrt {a x+b x^{2/3}}}{a^2 \sqrt [3]{x}}\right )}{5 a}\right )}{7 a}\right )}{9 a}\right )}{11 a}\right )}{13 a}\right )}{15 a}\right )}{17 a}\right )}{19 a}\right )}{21 a}\) |
(2*x^3*Sqrt[b*x^(2/3) + a*x])/(7*a) - (20*b*((6*x^(8/3)*Sqrt[b*x^(2/3) + a *x])/(19*a) - (18*b*((6*x^(7/3)*Sqrt[b*x^(2/3) + a*x])/(17*a) - (16*b*((2* x^2*Sqrt[b*x^(2/3) + a*x])/(5*a) - (14*b*((6*x^(5/3)*Sqrt[b*x^(2/3) + a*x] )/(13*a) - (12*b*((6*x^(4/3)*Sqrt[b*x^(2/3) + a*x])/(11*a) - (10*b*((2*x*S qrt[b*x^(2/3) + a*x])/(3*a) - (8*b*((6*x^(2/3)*Sqrt[b*x^(2/3) + a*x])/(7*a ) - (6*b*((6*x^(1/3)*Sqrt[b*x^(2/3) + a*x])/(5*a) - (4*b*((2*Sqrt[b*x^(2/3 ) + a*x])/a - (4*b*Sqrt[b*x^(2/3) + a*x])/(a^2*x^(1/3))))/(5*a)))/(7*a)))/ (9*a)))/(11*a)))/(13*a)))/(15*a)))/(17*a)))/(19*a)))/(21*a)
3.2.86.3.1 Defintions of rubi rules used
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 = 1.76 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.43
| method | result | size |
| derivativedivides | \(\frac {2 x^{\frac {1}{3}} \left (b +a \,x^{\frac {1}{3}}\right ) \left (46189 a^{10} x^{\frac {10}{3}}-48620 a^{9} b \,x^{3}+51480 a^{8} b^{2} x^{\frac {8}{3}}-54912 a^{7} b^{3} x^{\frac {7}{3}}+59136 x^{2} a^{6} b^{4}-64512 a^{5} b^{5} x^{\frac {5}{3}}+71680 a^{4} x^{\frac {4}{3}} b^{6}-81920 a^{3} b^{7} x +98304 a^{2} b^{8} x^{\frac {2}{3}}-131072 a \,b^{9} x^{\frac {1}{3}}+262144 b^{10}\right )}{323323 \sqrt {b \,x^{\frac {2}{3}}+a x}\, a^{11}}\) | \(134\) |
| default | \(\frac {2 x^{\frac {1}{3}} \left (b +a \,x^{\frac {1}{3}}\right ) \left (46189 a^{10} x^{\frac {10}{3}}-48620 a^{9} b \,x^{3}+51480 a^{8} b^{2} x^{\frac {8}{3}}-54912 a^{7} b^{3} x^{\frac {7}{3}}+59136 x^{2} a^{6} b^{4}-64512 a^{5} b^{5} x^{\frac {5}{3}}+71680 a^{4} x^{\frac {4}{3}} b^{6}-81920 a^{3} b^{7} x +98304 a^{2} b^{8} x^{\frac {2}{3}}-131072 a \,b^{9} x^{\frac {1}{3}}+262144 b^{10}\right )}{323323 \sqrt {b \,x^{\frac {2}{3}}+a x}\, a^{11}}\) | \(134\) |
2/323323*x^(1/3)*(b+a*x^(1/3))*(46189*a^10*x^(10/3)-48620*a^9*b*x^3+51480* a^8*b^2*x^(8/3)-54912*a^7*b^3*x^(7/3)+59136*x^2*a^6*b^4-64512*a^5*b^5*x^(5 /3)+71680*a^4*x^(4/3)*b^6-81920*a^3*b^7*x+98304*a^2*b^8*x^(2/3)-131072*a*b ^9*x^(1/3)+262144*b^10)/(b*x^(2/3)+a*x)^(1/2)/a^11
Leaf count of result is larger than twice the leaf count of optimal. 1031 vs. \(2 (233) = 466\).
Time = 176.92 (sec) , antiderivative size = 1031, normalized size of antiderivative = 3.29 \[ \int \frac {x^3}{\sqrt {b x^{2/3}+a x}} \, dx=\text {Too large to display} \]
-2/323323*((3298534883328*b^16 + 687194767360*b^15 + 3221225472*(64*a^3 - 3)*b^13 - 64424509440*b^14 - 16777216*(11264*a^3 - 53)*b^12 - 269004736*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 + 3 439*a^6)*b^6 + 3913728*(5600*a^9 + 3*a^6)*b^5 - 2112*(101384192*a^12 + 195 8400*a^9 + 63*a^6)*b^4 - 36608*(3784704*a^12 - 8101*a^9)*b^3 - 109824*(226 688*a^12 + 85*a^9)*b^2 + 7293*(974848*a^12 + 15*a^9)*b)*x - (46189*(167772 16*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 - 54912*(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 + 86 4*a^7*b^4 - (262144*a^10 + 27*a^7)*b^3)*x^3 + 71680*(16777216*a^4*b^12 + 6 291456*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 + 27*a^4)*b^6)*x^2 - 131072*(16777216*a*b^15 + 6291456 *a*b^14 + 196608*a*b^13 - 114688*a*b^12 - 2304*a*b^11 + 864*a*b^10 - (2621 44*a^4 + 27*a)*b^9)*x + 4*(1099511627776*b^16 + 412316860416*b^15 + 128849 01888*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^8...
\[ \int \frac {x^3}{\sqrt {b x^{2/3}+a x}} \, dx=\int \frac {x^{3}}{\sqrt {a x + b x^{\frac {2}{3}}}}\, dx \]
\[ \int \frac {x^3}{\sqrt {b x^{2/3}+a x}} \, dx=\int { \frac {x^{3}}{\sqrt {a x + b x^{\frac {2}{3}}}} \,d x } \]
Time = 0.28 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.52 \[ \int \frac {x^3}{\sqrt {b x^{2/3}+a x}} \, dx=-\frac {524288 \, b^{\frac {21}{2}}}{323323 \, a^{11}} + \frac {2 \, {\left (46189 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {21}{2}} - 510510 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {19}{2}} b + 2567565 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} b^{2} - 7759752 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} b^{3} + 15668730 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} b^{4} - 22221108 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} b^{5} + 22632610 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b^{6} - 16628040 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{7} + 8729721 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{8} - 3233230 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{9} + 969969 \, \sqrt {a x^{\frac {1}{3}} + b} b^{10}\right )}}{323323 \, a^{11}} \]
-524288/323323*b^(21/2)/a^11 + 2/323323*(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 - 77597 52*(a*x^(1/3) + b)^(15/2)*b^3 + 15668730*(a*x^(1/3) + b)^(13/2)*b^4 - 2222 1108*(a*x^(1/3) + b)^(11/2)*b^5 + 22632610*(a*x^(1/3) + b)^(9/2)*b^6 - 166 28040*(a*x^(1/3) + b)^(7/2)*b^7 + 8729721*(a*x^(1/3) + b)^(5/2)*b^8 - 3233 230*(a*x^(1/3) + b)^(3/2)*b^9 + 969969*sqrt(a*x^(1/3) + b)*b^10)/a^11
Timed out. \[ \int \frac {x^3}{\sqrt {b x^{2/3}+a x}} \, dx=\int \frac {x^3}{\sqrt {a\,x+b\,x^{2/3}}} \,d x \]