Integrand size = 19, antiderivative size = 336 \[ \int \frac {x^4}{\left (b x^{2/3}+a x\right )^{3/2}} \, dx=-\frac {6 x^4}{a \sqrt {b x^{2/3}+a x}}-\frac {524288 b^9 \sqrt {b x^{2/3}+a x}}{29393 a^{11}}+\frac {1048576 b^{10} \sqrt {b x^{2/3}+a x}}{29393 a^{12} \sqrt [3]{x}}+\frac {393216 b^8 \sqrt [3]{x} \sqrt {b x^{2/3}+a x}}{29393 a^{10}}-\frac {327680 b^7 x^{2/3} \sqrt {b x^{2/3}+a x}}{29393 a^9}+\frac {40960 b^6 x \sqrt {b x^{2/3}+a x}}{4199 a^8}-\frac {36864 b^5 x^{4/3} \sqrt {b x^{2/3}+a x}}{4199 a^7}+\frac {33792 b^4 x^{5/3} \sqrt {b x^{2/3}+a x}}{4199 a^6}-\frac {16896 b^3 x^2 \sqrt {b x^{2/3}+a x}}{2261 a^5}+\frac {15840 b^2 x^{7/3} \sqrt {b x^{2/3}+a x}}{2261 a^4}-\frac {880 b x^{8/3} \sqrt {b x^{2/3}+a x}}{133 a^3}+\frac {44 x^3 \sqrt {b x^{2/3}+a x}}{7 a^2} \] Output:
-6*x^4/a/(b*x^(2/3)+a*x)^(1/2)-524288/29393*b^9*(b*x^(2/3)+a*x)^(1/2)/a^11 +1048576/29393*b^10*(b*x^(2/3)+a*x)^(1/2)/a^12/x^(1/3)+393216/29393*b^8*x^ (1/3)*(b*x^(2/3)+a*x)^(1/2)/a^10-327680/29393*b^7*x^(2/3)*(b*x^(2/3)+a*x)^ (1/2)/a^9+40960/4199*b^6*x*(b*x^(2/3)+a*x)^(1/2)/a^8-36864/4199*b^5*x^(4/3 )*(b*x^(2/3)+a*x)^(1/2)/a^7+33792/4199*b^4*x^(5/3)*(b*x^(2/3)+a*x)^(1/2)/a ^6-16896/2261*b^3*x^2*(b*x^(2/3)+a*x)^(1/2)/a^5+15840/2261*b^2*x^(7/3)*(b* x^(2/3)+a*x)^(1/2)/a^4-880/133*b*x^(8/3)*(b*x^(2/3)+a*x)^(1/2)/a^3+44/7*x^ 3*(b*x^(2/3)+a*x)^(1/2)/a^2
Time = 4.51 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.48 \[ \int \frac {x^4}{\left (b x^{2/3}+a x\right )^{3/2}} \, dx=\frac {2 \sqrt [3]{x} \left (524288 b^{11}+262144 a b^{10} \sqrt [3]{x}-65536 a^2 b^9 x^{2/3}+32768 a^3 b^8 x-20480 a^4 b^7 x^{4/3}+14336 a^5 b^6 x^{5/3}-10752 a^6 b^5 x^2+8448 a^7 b^4 x^{7/3}-6864 a^8 b^3 x^{8/3}+5720 a^9 b^2 x^3-4862 a^{10} b x^{10/3}+4199 a^{11} x^{11/3}\right )}{29393 a^{12} \sqrt {b x^{2/3}+a x}} \] Input:
Integrate[x^4/(b*x^(2/3) + a*x)^(3/2),x]
Output:
(2*x^(1/3)*(524288*b^11 + 262144*a*b^10*x^(1/3) - 65536*a^2*b^9*x^(2/3) + 32768*a^3*b^8*x - 20480*a^4*b^7*x^(4/3) + 14336*a^5*b^6*x^(5/3) - 10752*a^ 6*b^5*x^2 + 8448*a^7*b^4*x^(7/3) - 6864*a^8*b^3*x^(8/3) + 5720*a^9*b^2*x^3 - 4862*a^10*b*x^(10/3) + 4199*a^11*x^(11/3)))/(29393*a^12*Sqrt[b*x^(2/3) + a*x])
Time = 1.23 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.17, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.632, Rules used = {1921, 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.
| \(\displaystyle \int \frac {x^4}{\left (a x+b x^{2/3}\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1921 |
| \(\displaystyle \frac {22 \int \frac {x^3}{\sqrt {x^{2/3} b+a x}}dx}{a}-\frac {6 x^4}{a \sqrt {a x+b x^{2/3}}}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {22 \left (\frac {2 x^3 \sqrt {a x+b x^{2/3}}}{7 a}-\frac {20 b \int \frac {x^{8/3}}{\sqrt {x^{2/3} b+a x}}dx}{21 a}\right )}{a}-\frac {6 x^4}{a \sqrt {a x+b x^{2/3}}}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {22 \left (\frac {2 x^3 \sqrt {a x+b x^{2/3}}}{7 a}-\frac {20 b \left (\frac {6 x^{8/3} \sqrt {a x+b x^{2/3}}}{19 a}-\frac {18 b \int \frac {x^{7/3}}{\sqrt {x^{2/3} b+a x}}dx}{19 a}\right )}{21 a}\right )}{a}-\frac {6 x^4}{a \sqrt {a x+b x^{2/3}}}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {22 \left (\frac {2 x^3 \sqrt {a x+b x^{2/3}}}{7 a}-\frac {20 b \left (\frac {6 x^{8/3} \sqrt {a x+b x^{2/3}}}{19 a}-\frac {18 b \left (\frac {6 x^{7/3} \sqrt {a x+b x^{2/3}}}{17 a}-\frac {16 b \int \frac {x^2}{\sqrt {x^{2/3} b+a x}}dx}{17 a}\right )}{19 a}\right )}{21 a}\right )}{a}-\frac {6 x^4}{a \sqrt {a x+b x^{2/3}}}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {22 \left (\frac {2 x^3 \sqrt {a x+b x^{2/3}}}{7 a}-\frac {20 b \left (\frac {6 x^{8/3} \sqrt {a x+b x^{2/3}}}{19 a}-\frac {18 b \left (\frac {6 x^{7/3} \sqrt {a x+b x^{2/3}}}{17 a}-\frac {16 b \left (\frac {2 x^2 \sqrt {a x+b x^{2/3}}}{5 a}-\frac {14 b \int \frac {x^{5/3}}{\sqrt {x^{2/3} b+a x}}dx}{15 a}\right )}{17 a}\right )}{19 a}\right )}{21 a}\right )}{a}-\frac {6 x^4}{a \sqrt {a x+b x^{2/3}}}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {22 \left (\frac {2 x^3 \sqrt {a x+b x^{2/3}}}{7 a}-\frac {20 b \left (\frac {6 x^{8/3} \sqrt {a x+b x^{2/3}}}{19 a}-\frac {18 b \left (\frac {6 x^{7/3} \sqrt {a x+b x^{2/3}}}{17 a}-\frac {16 b \left (\frac {2 x^2 \sqrt {a x+b x^{2/3}}}{5 a}-\frac {14 b \left (\frac {6 x^{5/3} \sqrt {a x+b x^{2/3}}}{13 a}-\frac {12 b \int \frac {x^{4/3}}{\sqrt {x^{2/3} b+a x}}dx}{13 a}\right )}{15 a}\right )}{17 a}\right )}{19 a}\right )}{21 a}\right )}{a}-\frac {6 x^4}{a \sqrt {a x+b x^{2/3}}}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {22 \left (\frac {2 x^3 \sqrt {a x+b x^{2/3}}}{7 a}-\frac {20 b \left (\frac {6 x^{8/3} \sqrt {a x+b x^{2/3}}}{19 a}-\frac {18 b \left (\frac {6 x^{7/3} \sqrt {a x+b x^{2/3}}}{17 a}-\frac {16 b \left (\frac {2 x^2 \sqrt {a x+b x^{2/3}}}{5 a}-\frac {14 b \left (\frac {6 x^{5/3} \sqrt {a x+b x^{2/3}}}{13 a}-\frac {12 b \left (\frac {6 x^{4/3} \sqrt {a x+b x^{2/3}}}{11 a}-\frac {10 b \int \frac {x}{\sqrt {x^{2/3} b+a x}}dx}{11 a}\right )}{13 a}\right )}{15 a}\right )}{17 a}\right )}{19 a}\right )}{21 a}\right )}{a}-\frac {6 x^4}{a \sqrt {a x+b x^{2/3}}}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {22 \left (\frac {2 x^3 \sqrt {a x+b x^{2/3}}}{7 a}-\frac {20 b \left (\frac {6 x^{8/3} \sqrt {a x+b x^{2/3}}}{19 a}-\frac {18 b \left (\frac {6 x^{7/3} \sqrt {a x+b x^{2/3}}}{17 a}-\frac {16 b \left (\frac {2 x^2 \sqrt {a x+b x^{2/3}}}{5 a}-\frac {14 b \left (\frac {6 x^{5/3} \sqrt {a x+b x^{2/3}}}{13 a}-\frac {12 b \left (\frac {6 x^{4/3} \sqrt {a x+b x^{2/3}}}{11 a}-\frac {10 b \left (\frac {2 x \sqrt {a x+b x^{2/3}}}{3 a}-\frac {8 b \int \frac {x^{2/3}}{\sqrt {x^{2/3} b+a x}}dx}{9 a}\right )}{11 a}\right )}{13 a}\right )}{15 a}\right )}{17 a}\right )}{19 a}\right )}{21 a}\right )}{a}-\frac {6 x^4}{a \sqrt {a x+b x^{2/3}}}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {22 \left (\frac {2 x^3 \sqrt {a x+b x^{2/3}}}{7 a}-\frac {20 b \left (\frac {6 x^{8/3} \sqrt {a x+b x^{2/3}}}{19 a}-\frac {18 b \left (\frac {6 x^{7/3} \sqrt {a x+b x^{2/3}}}{17 a}-\frac {16 b \left (\frac {2 x^2 \sqrt {a x+b x^{2/3}}}{5 a}-\frac {14 b \left (\frac {6 x^{5/3} \sqrt {a x+b x^{2/3}}}{13 a}-\frac {12 b \left (\frac {6 x^{4/3} \sqrt {a x+b x^{2/3}}}{11 a}-\frac {10 b \left (\frac {2 x \sqrt {a x+b x^{2/3}}}{3 a}-\frac {8 b \left (\frac {6 x^{2/3} \sqrt {a x+b x^{2/3}}}{7 a}-\frac {6 b \int \frac {\sqrt [3]{x}}{\sqrt {x^{2/3} b+a x}}dx}{7 a}\right )}{9 a}\right )}{11 a}\right )}{13 a}\right )}{15 a}\right )}{17 a}\right )}{19 a}\right )}{21 a}\right )}{a}-\frac {6 x^4}{a \sqrt {a x+b x^{2/3}}}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {22 \left (\frac {2 x^3 \sqrt {a x+b x^{2/3}}}{7 a}-\frac {20 b \left (\frac {6 x^{8/3} \sqrt {a x+b x^{2/3}}}{19 a}-\frac {18 b \left (\frac {6 x^{7/3} \sqrt {a x+b x^{2/3}}}{17 a}-\frac {16 b \left (\frac {2 x^2 \sqrt {a x+b x^{2/3}}}{5 a}-\frac {14 b \left (\frac {6 x^{5/3} \sqrt {a x+b x^{2/3}}}{13 a}-\frac {12 b \left (\frac {6 x^{4/3} \sqrt {a x+b x^{2/3}}}{11 a}-\frac {10 b \left (\frac {2 x \sqrt {a x+b x^{2/3}}}{3 a}-\frac {8 b \left (\frac {6 x^{2/3} \sqrt {a x+b x^{2/3}}}{7 a}-\frac {6 b \left (\frac {6 \sqrt [3]{x} \sqrt {a x+b x^{2/3}}}{5 a}-\frac {4 b \int \frac {1}{\sqrt {x^{2/3} b+a x}}dx}{5 a}\right )}{7 a}\right )}{9 a}\right )}{11 a}\right )}{13 a}\right )}{15 a}\right )}{17 a}\right )}{19 a}\right )}{21 a}\right )}{a}-\frac {6 x^4}{a \sqrt {a x+b x^{2/3}}}\) |
| \(\Big \downarrow \) 1908 |
| \(\displaystyle \frac {22 \left (\frac {2 x^3 \sqrt {a x+b x^{2/3}}}{7 a}-\frac {20 b \left (\frac {6 x^{8/3} \sqrt {a x+b x^{2/3}}}{19 a}-\frac {18 b \left (\frac {6 x^{7/3} \sqrt {a x+b x^{2/3}}}{17 a}-\frac {16 b \left (\frac {2 x^2 \sqrt {a x+b x^{2/3}}}{5 a}-\frac {14 b \left (\frac {6 x^{5/3} \sqrt {a x+b x^{2/3}}}{13 a}-\frac {12 b \left (\frac {6 x^{4/3} \sqrt {a x+b x^{2/3}}}{11 a}-\frac {10 b \left (\frac {2 x \sqrt {a x+b x^{2/3}}}{3 a}-\frac {8 b \left (\frac {6 x^{2/3} \sqrt {a x+b x^{2/3}}}{7 a}-\frac {6 b \left (\frac {6 \sqrt [3]{x} \sqrt {a x+b x^{2/3}}}{5 a}-\frac {4 b \left (\frac {2 \sqrt {a x+b x^{2/3}}}{a}-\frac {2 b \int \frac {1}{\sqrt [3]{x} \sqrt {x^{2/3} b+a x}}dx}{3 a}\right )}{5 a}\right )}{7 a}\right )}{9 a}\right )}{11 a}\right )}{13 a}\right )}{15 a}\right )}{17 a}\right )}{19 a}\right )}{21 a}\right )}{a}-\frac {6 x^4}{a \sqrt {a x+b x^{2/3}}}\) |
| \(\Big \downarrow \) 1920 |
| \(\displaystyle \frac {22 \left (\frac {2 x^3 \sqrt {a x+b x^{2/3}}}{7 a}-\frac {20 b \left (\frac {6 x^{8/3} \sqrt {a x+b x^{2/3}}}{19 a}-\frac {18 b \left (\frac {6 x^{7/3} \sqrt {a x+b x^{2/3}}}{17 a}-\frac {16 b \left (\frac {2 x^2 \sqrt {a x+b x^{2/3}}}{5 a}-\frac {14 b \left (\frac {6 x^{5/3} \sqrt {a x+b x^{2/3}}}{13 a}-\frac {12 b \left (\frac {6 x^{4/3} \sqrt {a x+b x^{2/3}}}{11 a}-\frac {10 b \left (\frac {2 x \sqrt {a x+b x^{2/3}}}{3 a}-\frac {8 b \left (\frac {6 x^{2/3} \sqrt {a x+b x^{2/3}}}{7 a}-\frac {6 b \left (\frac {6 \sqrt [3]{x} \sqrt {a x+b x^{2/3}}}{5 a}-\frac {4 b \left (\frac {2 \sqrt {a x+b x^{2/3}}}{a}-\frac {4 b \sqrt {a x+b x^{2/3}}}{a^2 \sqrt [3]{x}}\right )}{5 a}\right )}{7 a}\right )}{9 a}\right )}{11 a}\right )}{13 a}\right )}{15 a}\right )}{17 a}\right )}{19 a}\right )}{21 a}\right )}{a}-\frac {6 x^4}{a \sqrt {a x+b x^{2/3}}}\) |
Input:
Int[x^4/(b*x^(2/3) + a*x)^(3/2),x]
Output:
(-6*x^4)/(a*Sqrt[b*x^(2/3) + a*x]) + (22*((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)*Sqr t[b*x^(2/3) + a*x])/(11*a) - (10*b*((2*x*Sqrt[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)))/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*(n - j )*(p + 1))), x] + Simp[c^j*((m + n*p + n - j + 1)/(a*(n - j)*(p + 1))) In t[(c*x)^(m - j)*(a*x^j + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, j, m, n} , x] && !IntegerQ[p] && NeQ[n, j] && ILtQ[Simplify[(m + n*p + n - j + 1)/( n - j)], 0] && LtQ[p, -1] && (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 = 143, normalized size of antiderivative = 0.43
| method | result | size |
| derivativedivides | \(\frac {2 x \left (x^{\frac {1}{3}} a +b \right ) \left (4199 a^{11} x^{\frac {11}{3}}-4862 a^{10} b \,x^{\frac {10}{3}}+5720 a^{9} b^{2} x^{3}-6864 a^{8} b^{3} x^{\frac {8}{3}}+8448 a^{7} b^{4} x^{\frac {7}{3}}-10752 a^{6} b^{5} x^{2}+14336 a^{5} b^{6} x^{\frac {5}{3}}-20480 a^{4} b^{7} x^{\frac {4}{3}}+32768 a^{3} b^{8} x -65536 a^{2} b^{9} x^{\frac {2}{3}}+262144 a \,b^{10} x^{\frac {1}{3}}+524288 b^{11}\right )}{29393 \left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {3}{2}} a^{12}}\) | \(143\) |
| default | \(\frac {2 x \left (x^{\frac {1}{3}} a +b \right ) \left (4199 a^{11} x^{\frac {11}{3}}-4862 a^{10} b \,x^{\frac {10}{3}}+5720 a^{9} b^{2} x^{3}-6864 a^{8} b^{3} x^{\frac {8}{3}}+8448 a^{7} b^{4} x^{\frac {7}{3}}-10752 a^{6} b^{5} x^{2}+14336 a^{5} b^{6} x^{\frac {5}{3}}-20480 a^{4} b^{7} x^{\frac {4}{3}}+32768 a^{3} b^{8} x -65536 a^{2} b^{9} x^{\frac {2}{3}}+262144 a \,b^{10} x^{\frac {1}{3}}+524288 b^{11}\right )}{29393 \left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {3}{2}} a^{12}}\) | \(143\) |
Input:
int(x^4/(b*x^(2/3)+a*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
2/29393*x*(x^(1/3)*a+b)*(4199*a^11*x^(11/3)-4862*a^10*b*x^(10/3)+5720*a^9* b^2*x^3-6864*a^8*b^3*x^(8/3)+8448*a^7*b^4*x^(7/3)-10752*a^6*b^5*x^2+14336* a^5*b^6*x^(5/3)-20480*a^4*b^7*x^(4/3)+32768*a^3*b^8*x-65536*a^2*b^9*x^(2/3 )+262144*a*b^10*x^(1/3)+524288*b^11)/(b*x^(2/3)+a*x)^(3/2)/a^12
Leaf count of result is larger than twice the leaf count of optimal. 2566 vs. \(2 (252) = 504\).
Time = 117.47 (sec) , antiderivative size = 2566, normalized size of antiderivative = 7.64 \[ \int \frac {x^4}{\left (b x^{2/3}+a x\right )^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate(x^4/(b*x^(2/3)+a*x)^(3/2),x, algorithm="fricas")
Output:
-1/29393*((6442450944*a^3*b^19 + 5368709120*a^3*b^18 - 2013265920*a^3*b^17 - 6113744*a^18 + 402653184*(17*a^6 - 3*a^3)*b^16 + 8388608*(464*a^6 + 53* a^3)*b^15 - 12582912*(246*a^6 + a^3)*b^14 + 1572864*(1036*a^9 - 2560*a^6 - 3*a^3)*b^13 - 524288*(758*a^9 - 1569*a^6)*b^12 - 393216*(5803*a^9 + 124*a ^6)*b^11 + 98304*(1315*a^12 - 20924*a^9 - 33*a^6)*b^10 - 57344*(2264*a^12 - 3153*a^9)*b^9 - 6144*(83789*a^12 + 2066*a^9)*b^8 - 1536*(46256*a^15 - 15 9272*a^12 - 267*a^9)*b^7 - 128*(264488*a^15 + 382229*a^12)*b^6 + 9984*(155 47*a^15 + 482*a^12)*b^5 - 24*(2376192*a^18 + 4735792*a^15 + 7887*a^12)*b^4 - 1664*(107856*a^18 - 16759*a^15)*b^3 - 156*(935424*a^18 + 17935*a^15)*b^ 2 + 663*(97664*a^18 + 123*a^15)*b)*x^2 + (6442450944*b^22 + 5368709120*b^2 1 + 402653184*(17*a^3 - 3)*b^19 - 2013265920*b^20 + 8388608*(464*a^3 + 53) *b^18 - 6113744*a^15*b^3 - 12582912*(246*a^3 + 1)*b^17 + 1572864*(1036*a^6 - 2560*a^3 - 3)*b^16 - 524288*(758*a^6 - 1569*a^3)*b^15 - 393216*(5803*a^ 6 + 124*a^3)*b^14 + 98304*(1315*a^9 - 20924*a^6 - 33*a^3)*b^13 - 57344*(22 64*a^9 - 3153*a^6)*b^12 - 6144*(83789*a^9 + 2066*a^6)*b^11 - 1536*(46256*a ^12 - 159272*a^9 - 267*a^6)*b^10 - 128*(264488*a^12 + 382229*a^9)*b^9 + 99 84*(15547*a^12 + 482*a^9)*b^8 - 24*(2376192*a^15 + 4735792*a^12 + 7887*a^9 )*b^7 - 1664*(107856*a^15 - 16759*a^12)*b^6 - 156*(935424*a^15 + 17935*a^1 2)*b^5 + 663*(97664*a^15 + 123*a^12)*b^4)*x - 2*(4199*(4096*a^13*b^9 + 614 4*a^13*b^8 + 768*a^13*b^7 - 4096*a^19 - 144*a^16*b^2 + 216*a^16*b - 27*...
\[ \int \frac {x^4}{\left (b x^{2/3}+a x\right )^{3/2}} \, dx=\int \frac {x^{4}}{\left (a x + b x^{\frac {2}{3}}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(x**4/(b*x**(2/3)+a*x)**(3/2),x)
Output:
Integral(x**4/(a*x + b*x**(2/3))**(3/2), x)
\[ \int \frac {x^4}{\left (b x^{2/3}+a x\right )^{3/2}} \, dx=\int { \frac {x^{4}}{{\left (a x + b x^{\frac {2}{3}}\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^4/(b*x^(2/3)+a*x)^(3/2),x, algorithm="maxima")
Output:
integrate(x^4/(a*x + b*x^(2/3))^(3/2), x)
Time = 0.44 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.64 \[ \int \frac {x^4}{\left (b x^{2/3}+a x\right )^{3/2}} \, dx=-\frac {1048576 \, b^{\frac {21}{2}}}{29393 \, a^{12}} + \frac {6 \, b^{11}}{\sqrt {a x^{\frac {1}{3}} + b} a^{12}} + \frac {2 \, {\left (4199 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {21}{2}} a^{240} - 51051 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {19}{2}} a^{240} b + 285285 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} a^{240} b^{2} - 969969 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} a^{240} b^{3} + 2238390 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} a^{240} b^{4} - 3703518 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} a^{240} b^{5} + 4526522 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} a^{240} b^{6} - 4157010 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} a^{240} b^{7} + 2909907 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} a^{240} b^{8} - 1616615 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} a^{240} b^{9} + 969969 \, \sqrt {a x^{\frac {1}{3}} + b} a^{240} b^{10}\right )}}{29393 \, a^{252}} \] Input:
integrate(x^4/(b*x^(2/3)+a*x)^(3/2),x, algorithm="giac")
Output:
-1048576/29393*b^(21/2)/a^12 + 6*b^11/(sqrt(a*x^(1/3) + b)*a^12) + 2/29393 *(4199*(a*x^(1/3) + b)^(21/2)*a^240 - 51051*(a*x^(1/3) + b)^(19/2)*a^240*b + 285285*(a*x^(1/3) + b)^(17/2)*a^240*b^2 - 969969*(a*x^(1/3) + b)^(15/2) *a^240*b^3 + 2238390*(a*x^(1/3) + b)^(13/2)*a^240*b^4 - 3703518*(a*x^(1/3) + b)^(11/2)*a^240*b^5 + 4526522*(a*x^(1/3) + b)^(9/2)*a^240*b^6 - 4157010 *(a*x^(1/3) + b)^(7/2)*a^240*b^7 + 2909907*(a*x^(1/3) + b)^(5/2)*a^240*b^8 - 1616615*(a*x^(1/3) + b)^(3/2)*a^240*b^9 + 969969*sqrt(a*x^(1/3) + b)*a^ 240*b^10)/a^252
Timed out. \[ \int \frac {x^4}{\left (b x^{2/3}+a x\right )^{3/2}} \, dx=\int \frac {x^4}{{\left (a\,x+b\,x^{2/3}\right )}^{3/2}} \,d x \] Input:
int(x^4/(a*x + b*x^(2/3))^(3/2),x)
Output:
int(x^4/(a*x + b*x^(2/3))^(3/2), x)
Time = 0.20 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.40 \[ \int \frac {x^4}{\left (b x^{2/3}+a x\right )^{3/2}} \, dx=\frac {\frac {2 x^{\frac {11}{3}} a^{11}}{7}-\frac {1056 x^{\frac {8}{3}} a^{8} b^{3}}{2261}+\frac {4096 x^{\frac {5}{3}} a^{5} b^{6}}{4199}-\frac {131072 x^{\frac {2}{3}} a^{2} b^{9}}{29393}-\frac {44 x^{\frac {10}{3}} a^{10} b}{133}+\frac {16896 x^{\frac {7}{3}} a^{7} b^{4}}{29393}-\frac {40960 x^{\frac {4}{3}} a^{4} b^{7}}{29393}+\frac {524288 x^{\frac {1}{3}} a \,b^{10}}{29393}+\frac {880 a^{9} b^{2} x^{3}}{2261}-\frac {3072 a^{6} b^{5} x^{2}}{4199}+\frac {65536 a^{3} b^{8} x}{29393}+\frac {1048576 b^{11}}{29393}}{\sqrt {x^{\frac {1}{3}} a +b}\, a^{12}} \] Input:
int(x^4/(b*x^(2/3)+a*x)^(3/2),x)
Output:
(2*(4199*x**(2/3)*a**11*x**3 - 6864*x**(2/3)*a**8*b**3*x**2 + 14336*x**(2/ 3)*a**5*b**6*x - 65536*x**(2/3)*a**2*b**9 - 4862*x**(1/3)*a**10*b*x**3 + 8 448*x**(1/3)*a**7*b**4*x**2 - 20480*x**(1/3)*a**4*b**7*x + 262144*x**(1/3) *a*b**10 + 5720*a**9*b**2*x**3 - 10752*a**6*b**5*x**2 + 32768*a**3*b**8*x + 524288*b**11))/(29393*sqrt(x**(1/3)*a + b)*a**12)