Integrand size = 19, antiderivative size = 343 \[ \int x^2 \left (b x^{2/3}+a x\right )^{3/2} \, dx=\frac {45056 b^6 \left (b x^{2/3}+a x\right )^{5/2}}{557175 a^7}-\frac {1048576 b^{11} \left (b x^{2/3}+a x\right )^{5/2}}{152108775 a^{12} x^{5/3}}+\frac {524288 b^{10} \left (b x^{2/3}+a x\right )^{5/2}}{30421755 a^{11} x^{4/3}}-\frac {131072 b^9 \left (b x^{2/3}+a x\right )^{5/2}}{4345965 a^{10} x}+\frac {65536 b^8 \left (b x^{2/3}+a x\right )^{5/2}}{1448655 a^9 x^{2/3}}-\frac {90112 b^7 \left (b x^{2/3}+a x\right )^{5/2}}{1448655 a^8 \sqrt [3]{x}}-\frac {11264 b^5 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{5/2}}{111435 a^6}+\frac {5632 b^4 x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{45885 a^5}-\frac {352 b^3 x \left (b x^{2/3}+a x\right )^{5/2}}{2415 a^4}+\frac {176 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{5/2}}{1035 a^3}-\frac {44 b x^{5/3} \left (b x^{2/3}+a x\right )^{5/2}}{225 a^2}+\frac {2 x^2 \left (b x^{2/3}+a x\right )^{5/2}}{9 a} \] Output:
45056/557175*b^6*(b*x^(2/3)+a*x)^(5/2)/a^7-1048576/152108775*b^11*(b*x^(2/ 3)+a*x)^(5/2)/a^12/x^(5/3)+524288/30421755*b^10*(b*x^(2/3)+a*x)^(5/2)/a^11 /x^(4/3)-131072/4345965*b^9*(b*x^(2/3)+a*x)^(5/2)/a^10/x+65536/1448655*b^8 *(b*x^(2/3)+a*x)^(5/2)/a^9/x^(2/3)-90112/1448655*b^7*(b*x^(2/3)+a*x)^(5/2) /a^8/x^(1/3)-11264/111435*b^5*x^(1/3)*(b*x^(2/3)+a*x)^(5/2)/a^6+5632/45885 *b^4*x^(2/3)*(b*x^(2/3)+a*x)^(5/2)/a^5-352/2415*b^3*x*(b*x^(2/3)+a*x)^(5/2 )/a^4+176/1035*b^2*x^(4/3)*(b*x^(2/3)+a*x)^(5/2)/a^3-44/225*b*x^(5/3)*(b*x ^(2/3)+a*x)^(5/2)/a^2+2/9*x^2*(b*x^(2/3)+a*x)^(5/2)/a
Time = 5.25 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.49 \[ \int x^2 \left (b x^{2/3}+a x\right )^{3/2} \, dx=\frac {2 \left (b+a \sqrt [3]{x}\right ) \left (b x^{2/3}+a x\right )^{3/2} \left (-524288 b^{11}+1310720 a b^{10} \sqrt [3]{x}-2293760 a^2 b^9 x^{2/3}+3440640 a^3 b^8 x-4730880 a^4 b^7 x^{4/3}+6150144 a^5 b^6 x^{5/3}-7687680 a^6 b^5 x^2+9335040 a^7 b^4 x^{7/3}-11085360 a^8 b^3 x^{8/3}+12932920 a^9 b^2 x^3-14872858 a^{10} b x^{10/3}+16900975 a^{11} x^{11/3}\right )}{152108775 a^{12} x} \] Input:
Integrate[x^2*(b*x^(2/3) + a*x)^(3/2),x]
Output:
(2*(b + a*x^(1/3))*(b*x^(2/3) + a*x)^(3/2)*(-524288*b^11 + 1310720*a*b^10* x^(1/3) - 2293760*a^2*b^9*x^(2/3) + 3440640*a^3*b^8*x - 4730880*a^4*b^7*x^ (4/3) + 6150144*a^5*b^6*x^(5/3) - 7687680*a^6*b^5*x^2 + 9335040*a^7*b^4*x^ (7/3) - 11085360*a^8*b^3*x^(8/3) + 12932920*a^9*b^2*x^3 - 14872858*a^10*b* x^(10/3) + 16900975*a^11*x^(11/3)))/(152108775*a^12*x)
Time = 1.25 (sec) , antiderivative size = 403, 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 = {1922, 1922, 1922, 1922, 1922, 1922, 1908, 1922, 1922, 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.
| \(\displaystyle \int x^2 \left (a x+b x^{2/3}\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^2 \left (a x+b x^{2/3}\right )^{5/2}}{9 a}-\frac {22 b \int x^{5/3} \left (x^{2/3} b+a x\right )^{3/2}dx}{27 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^2 \left (a x+b x^{2/3}\right )^{5/2}}{9 a}-\frac {22 b \left (\frac {6 x^{5/3} \left (a x+b x^{2/3}\right )^{5/2}}{25 a}-\frac {4 b \int x^{4/3} \left (x^{2/3} b+a x\right )^{3/2}dx}{5 a}\right )}{27 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^2 \left (a x+b x^{2/3}\right )^{5/2}}{9 a}-\frac {22 b \left (\frac {6 x^{5/3} \left (a x+b x^{2/3}\right )^{5/2}}{25 a}-\frac {4 b \left (\frac {6 x^{4/3} \left (a x+b x^{2/3}\right )^{5/2}}{23 a}-\frac {18 b \int x \left (x^{2/3} b+a x\right )^{3/2}dx}{23 a}\right )}{5 a}\right )}{27 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^2 \left (a x+b x^{2/3}\right )^{5/2}}{9 a}-\frac {22 b \left (\frac {6 x^{5/3} \left (a x+b x^{2/3}\right )^{5/2}}{25 a}-\frac {4 b \left (\frac {6 x^{4/3} \left (a x+b x^{2/3}\right )^{5/2}}{23 a}-\frac {18 b \left (\frac {2 x \left (a x+b x^{2/3}\right )^{5/2}}{7 a}-\frac {16 b \int x^{2/3} \left (x^{2/3} b+a x\right )^{3/2}dx}{21 a}\right )}{23 a}\right )}{5 a}\right )}{27 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^2 \left (a x+b x^{2/3}\right )^{5/2}}{9 a}-\frac {22 b \left (\frac {6 x^{5/3} \left (a x+b x^{2/3}\right )^{5/2}}{25 a}-\frac {4 b \left (\frac {6 x^{4/3} \left (a x+b x^{2/3}\right )^{5/2}}{23 a}-\frac {18 b \left (\frac {2 x \left (a x+b x^{2/3}\right )^{5/2}}{7 a}-\frac {16 b \left (\frac {6 x^{2/3} \left (a x+b x^{2/3}\right )^{5/2}}{19 a}-\frac {14 b \int \sqrt [3]{x} \left (x^{2/3} b+a x\right )^{3/2}dx}{19 a}\right )}{21 a}\right )}{23 a}\right )}{5 a}\right )}{27 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^2 \left (a x+b x^{2/3}\right )^{5/2}}{9 a}-\frac {22 b \left (\frac {6 x^{5/3} \left (a x+b x^{2/3}\right )^{5/2}}{25 a}-\frac {4 b \left (\frac {6 x^{4/3} \left (a x+b x^{2/3}\right )^{5/2}}{23 a}-\frac {18 b \left (\frac {2 x \left (a x+b x^{2/3}\right )^{5/2}}{7 a}-\frac {16 b \left (\frac {6 x^{2/3} \left (a x+b x^{2/3}\right )^{5/2}}{19 a}-\frac {14 b \left (\frac {6 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{5/2}}{17 a}-\frac {12 b \int \left (x^{2/3} b+a x\right )^{3/2}dx}{17 a}\right )}{19 a}\right )}{21 a}\right )}{23 a}\right )}{5 a}\right )}{27 a}\) |
| \(\Big \downarrow \) 1908 |
| \(\displaystyle \frac {2 x^2 \left (a x+b x^{2/3}\right )^{5/2}}{9 a}-\frac {22 b \left (\frac {6 x^{5/3} \left (a x+b x^{2/3}\right )^{5/2}}{25 a}-\frac {4 b \left (\frac {6 x^{4/3} \left (a x+b x^{2/3}\right )^{5/2}}{23 a}-\frac {18 b \left (\frac {2 x \left (a x+b x^{2/3}\right )^{5/2}}{7 a}-\frac {16 b \left (\frac {6 x^{2/3} \left (a x+b x^{2/3}\right )^{5/2}}{19 a}-\frac {14 b \left (\frac {6 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{5/2}}{17 a}-\frac {12 b \left (\frac {2 \left (a x+b x^{2/3}\right )^{5/2}}{5 a}-\frac {2 b \int \frac {\left (x^{2/3} b+a x\right )^{3/2}}{\sqrt [3]{x}}dx}{3 a}\right )}{17 a}\right )}{19 a}\right )}{21 a}\right )}{23 a}\right )}{5 a}\right )}{27 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^2 \left (a x+b x^{2/3}\right )^{5/2}}{9 a}-\frac {22 b \left (\frac {6 x^{5/3} \left (a x+b x^{2/3}\right )^{5/2}}{25 a}-\frac {4 b \left (\frac {6 x^{4/3} \left (a x+b x^{2/3}\right )^{5/2}}{23 a}-\frac {18 b \left (\frac {2 x \left (a x+b x^{2/3}\right )^{5/2}}{7 a}-\frac {16 b \left (\frac {6 x^{2/3} \left (a x+b x^{2/3}\right )^{5/2}}{19 a}-\frac {14 b \left (\frac {6 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{5/2}}{17 a}-\frac {12 b \left (\frac {2 \left (a x+b x^{2/3}\right )^{5/2}}{5 a}-\frac {2 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{5/2}}{13 a \sqrt [3]{x}}-\frac {8 b \int \frac {\left (x^{2/3} b+a x\right )^{3/2}}{x^{2/3}}dx}{13 a}\right )}{3 a}\right )}{17 a}\right )}{19 a}\right )}{21 a}\right )}{23 a}\right )}{5 a}\right )}{27 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^2 \left (a x+b x^{2/3}\right )^{5/2}}{9 a}-\frac {22 b \left (\frac {6 x^{5/3} \left (a x+b x^{2/3}\right )^{5/2}}{25 a}-\frac {4 b \left (\frac {6 x^{4/3} \left (a x+b x^{2/3}\right )^{5/2}}{23 a}-\frac {18 b \left (\frac {2 x \left (a x+b x^{2/3}\right )^{5/2}}{7 a}-\frac {16 b \left (\frac {6 x^{2/3} \left (a x+b x^{2/3}\right )^{5/2}}{19 a}-\frac {14 b \left (\frac {6 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{5/2}}{17 a}-\frac {12 b \left (\frac {2 \left (a x+b x^{2/3}\right )^{5/2}}{5 a}-\frac {2 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{5/2}}{13 a \sqrt [3]{x}}-\frac {8 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{5/2}}{11 a x^{2/3}}-\frac {6 b \int \frac {\left (x^{2/3} b+a x\right )^{3/2}}{x}dx}{11 a}\right )}{13 a}\right )}{3 a}\right )}{17 a}\right )}{19 a}\right )}{21 a}\right )}{23 a}\right )}{5 a}\right )}{27 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^2 \left (a x+b x^{2/3}\right )^{5/2}}{9 a}-\frac {22 b \left (\frac {6 x^{5/3} \left (a x+b x^{2/3}\right )^{5/2}}{25 a}-\frac {4 b \left (\frac {6 x^{4/3} \left (a x+b x^{2/3}\right )^{5/2}}{23 a}-\frac {18 b \left (\frac {2 x \left (a x+b x^{2/3}\right )^{5/2}}{7 a}-\frac {16 b \left (\frac {6 x^{2/3} \left (a x+b x^{2/3}\right )^{5/2}}{19 a}-\frac {14 b \left (\frac {6 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{5/2}}{17 a}-\frac {12 b \left (\frac {2 \left (a x+b x^{2/3}\right )^{5/2}}{5 a}-\frac {2 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{5/2}}{13 a \sqrt [3]{x}}-\frac {8 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{5/2}}{11 a x^{2/3}}-\frac {6 b \left (\frac {2 \left (a x+b x^{2/3}\right )^{5/2}}{3 a x}-\frac {4 b \int \frac {\left (x^{2/3} b+a x\right )^{3/2}}{x^{4/3}}dx}{9 a}\right )}{11 a}\right )}{13 a}\right )}{3 a}\right )}{17 a}\right )}{19 a}\right )}{21 a}\right )}{23 a}\right )}{5 a}\right )}{27 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^2 \left (a x+b x^{2/3}\right )^{5/2}}{9 a}-\frac {22 b \left (\frac {6 x^{5/3} \left (a x+b x^{2/3}\right )^{5/2}}{25 a}-\frac {4 b \left (\frac {6 x^{4/3} \left (a x+b x^{2/3}\right )^{5/2}}{23 a}-\frac {18 b \left (\frac {2 x \left (a x+b x^{2/3}\right )^{5/2}}{7 a}-\frac {16 b \left (\frac {6 x^{2/3} \left (a x+b x^{2/3}\right )^{5/2}}{19 a}-\frac {14 b \left (\frac {6 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{5/2}}{17 a}-\frac {12 b \left (\frac {2 \left (a x+b x^{2/3}\right )^{5/2}}{5 a}-\frac {2 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{5/2}}{13 a \sqrt [3]{x}}-\frac {8 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{5/2}}{11 a x^{2/3}}-\frac {6 b \left (\frac {2 \left (a x+b x^{2/3}\right )^{5/2}}{3 a x}-\frac {4 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{5/2}}{7 a x^{4/3}}-\frac {2 b \int \frac {\left (x^{2/3} b+a x\right )^{3/2}}{x^{5/3}}dx}{7 a}\right )}{9 a}\right )}{11 a}\right )}{13 a}\right )}{3 a}\right )}{17 a}\right )}{19 a}\right )}{21 a}\right )}{23 a}\right )}{5 a}\right )}{27 a}\) |
| \(\Big \downarrow \) 1920 |
| \(\displaystyle \frac {2 x^2 \left (a x+b x^{2/3}\right )^{5/2}}{9 a}-\frac {22 b \left (\frac {6 x^{5/3} \left (a x+b x^{2/3}\right )^{5/2}}{25 a}-\frac {4 b \left (\frac {6 x^{4/3} \left (a x+b x^{2/3}\right )^{5/2}}{23 a}-\frac {18 b \left (\frac {2 x \left (a x+b x^{2/3}\right )^{5/2}}{7 a}-\frac {16 b \left (\frac {6 x^{2/3} \left (a x+b x^{2/3}\right )^{5/2}}{19 a}-\frac {14 b \left (\frac {6 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{5/2}}{17 a}-\frac {12 b \left (\frac {2 \left (a x+b x^{2/3}\right )^{5/2}}{5 a}-\frac {2 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{5/2}}{13 a \sqrt [3]{x}}-\frac {8 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{5/2}}{11 a x^{2/3}}-\frac {6 b \left (\frac {2 \left (a x+b x^{2/3}\right )^{5/2}}{3 a x}-\frac {4 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{5/2}}{7 a x^{4/3}}-\frac {12 b \left (a x+b x^{2/3}\right )^{5/2}}{35 a^2 x^{5/3}}\right )}{9 a}\right )}{11 a}\right )}{13 a}\right )}{3 a}\right )}{17 a}\right )}{19 a}\right )}{21 a}\right )}{23 a}\right )}{5 a}\right )}{27 a}\) |
Input:
Int[x^2*(b*x^(2/3) + a*x)^(3/2),x]
Output:
(2*x^2*(b*x^(2/3) + a*x)^(5/2))/(9*a) - (22*b*((6*x^(5/3)*(b*x^(2/3) + a*x )^(5/2))/(25*a) - (4*b*((6*x^(4/3)*(b*x^(2/3) + a*x)^(5/2))/(23*a) - (18*b *((2*x*(b*x^(2/3) + a*x)^(5/2))/(7*a) - (16*b*((6*x^(2/3)*(b*x^(2/3) + a*x )^(5/2))/(19*a) - (14*b*((6*x^(1/3)*(b*x^(2/3) + a*x)^(5/2))/(17*a) - (12* b*((2*(b*x^(2/3) + a*x)^(5/2))/(5*a) - (2*b*((6*(b*x^(2/3) + a*x)^(5/2))/( 13*a*x^(1/3)) - (8*b*((6*(b*x^(2/3) + a*x)^(5/2))/(11*a*x^(2/3)) - (6*b*(( 2*(b*x^(2/3) + a*x)^(5/2))/(3*a*x) - (4*b*((-12*b*(b*x^(2/3) + a*x)^(5/2)) /(35*a^2*x^(5/3)) + (6*(b*x^(2/3) + a*x)^(5/2))/(7*a*x^(4/3))))/(9*a)))/(1 1*a)))/(13*a)))/(3*a)))/(17*a)))/(19*a)))/(21*a)))/(23*a)))/(5*a)))/(27*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 = 0.39 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.42
| method | result | size |
| derivativedivides | \(\frac {2 \left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {3}{2}} \left (x^{\frac {1}{3}} a +b \right ) \left (16900975 a^{11} x^{\frac {11}{3}}-14872858 a^{10} b \,x^{\frac {10}{3}}+12932920 a^{9} b^{2} x^{3}-11085360 a^{8} b^{3} x^{\frac {8}{3}}+9335040 a^{7} b^{4} x^{\frac {7}{3}}-7687680 a^{6} b^{5} x^{2}+6150144 a^{5} b^{6} x^{\frac {5}{3}}-4730880 a^{4} b^{7} x^{\frac {4}{3}}+3440640 a^{3} b^{8} x -2293760 a^{2} b^{9} x^{\frac {2}{3}}+1310720 a \,b^{10} x^{\frac {1}{3}}-524288 b^{11}\right )}{152108775 x \,a^{12}}\) | \(145\) |
| default | \(\frac {2 \left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {3}{2}} \left (x^{\frac {1}{3}} a +b \right ) \left (16900975 a^{11} x^{\frac {11}{3}}-14872858 a^{10} b \,x^{\frac {10}{3}}+12932920 a^{9} b^{2} x^{3}-11085360 a^{8} b^{3} x^{\frac {8}{3}}+9335040 a^{7} b^{4} x^{\frac {7}{3}}-7687680 a^{6} b^{5} x^{2}+6150144 a^{5} b^{6} x^{\frac {5}{3}}-4730880 a^{4} b^{7} x^{\frac {4}{3}}+3440640 a^{3} b^{8} x -2293760 a^{2} b^{9} x^{\frac {2}{3}}+1310720 a \,b^{10} x^{\frac {1}{3}}-524288 b^{11}\right )}{152108775 x \,a^{12}}\) | \(145\) |
Input:
int(x^2*(b*x^(2/3)+a*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
2/152108775*(b*x^(2/3)+a*x)^(3/2)*(x^(1/3)*a+b)*(16900975*a^11*x^(11/3)-14 872858*a^10*b*x^(10/3)+12932920*a^9*b^2*x^3-11085360*a^8*b^3*x^(8/3)+93350 40*a^7*b^4*x^(7/3)-7687680*a^6*b^5*x^2+6150144*a^5*b^6*x^(5/3)-4730880*a^4 *b^7*x^(4/3)+3440640*a^3*b^8*x-2293760*a^2*b^9*x^(2/3)+1310720*a*b^10*x^(1 /3)-524288*b^11)/x/a^12
Leaf count of result is larger than twice the leaf count of optimal. 1293 vs. \(2 (255) = 510\).
Time = 132.79 (sec) , antiderivative size = 1293, normalized size of antiderivative = 3.77 \[ \int x^2 \left (b x^{2/3}+a x\right )^{3/2} \, dx=\text {Too large to display} \] Input:
integrate(x^2*(b*x^(2/3)+a*x)^(3/2),x, algorithm="fricas")
Output:
2/152108775*((6597069766656*b^19 + 1374389534720*b^18 + 6442450944*(64*a^3 - 3)*b^16 - 128849018880*b^17 - 33554432*(11264*a^3 - 53)*b^15 + 98431278 400*a^15 - 12582912*(5504*a^3 + 1)*b^14 + 393216*(3194880*a^6 - 114688*a^3 - 3)*b^13 + 14680064*(18816*a^6 + 103*a^3)*b^12 - 1572864*(48816*a^6 + 23 *a^3)*b^11 - 24576*(45731840*a^9 - 495872*a^6 - 15*a^3)*b^10 - 229376*(134 9120*a^9 + 3439*a^6)*b^9 + 7827456*(5600*a^9 + 3*a^6)*b^8 - 384*(620420562 944*a^12 + 21542400*a^9 + 693*a^6)*b^7 - 6656*(7444688384*a^12 - 89111*a^9 )*b^6 + 19968*(232361024*a^12 - 935*a^9)*b^5 - 1326*(173210075136*a^15 - 5 33564416*a^12 - 165*a^9)*b^4 - 1881152*(45121536*a^15 + 34547*a^12)*b^3 - 352716*(19243008*a^15 - 1339*a^12)*b^2 + 2028117*(237568*a^15 + 21*a^12)*b )*x + (16900975*(16777216*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 - 92378*(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 + 109824*(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 - 1 43360*(16777216*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 + 26 2144*(16777216*a*b^18 + 6291456*a*b^17 + 196608*a*b^16 - 114688*a*b^15 - 2 304*a*b^14 + 864*a*b^13 - (262144*a^4 + 27*a)*b^12)*x - 4*(219902325555...
\[ \int x^2 \left (b x^{2/3}+a x\right )^{3/2} \, dx=\int x^{2} \left (a x + b x^{\frac {2}{3}}\right )^{\frac {3}{2}}\, dx \] Input:
integrate(x**2*(b*x**(2/3)+a*x)**(3/2),x)
Output:
Integral(x**2*(a*x + b*x**(2/3))**(3/2), x)
\[ \int x^2 \left (b x^{2/3}+a x\right )^{3/2} \, dx=\int { {\left (a x + b x^{\frac {2}{3}}\right )}^{\frac {3}{2}} x^{2} \,d x } \] Input:
integrate(x^2*(b*x^(2/3)+a*x)^(3/2),x, algorithm="maxima")
Output:
integrate((a*x + b*x^(2/3))^(3/2)*x^2, x)
Leaf count of result is larger than twice the leaf count of optimal. 770 vs. \(2 (255) = 510\).
Time = 0.31 (sec) , antiderivative size = 770, normalized size of antiderivative = 2.24 \[ \int x^2 \left (b x^{2/3}+a x\right )^{3/2} \, dx=\text {Too large to display} \] Input:
integrate(x^2*(b*x^(2/3)+a*x)^(3/2),x, algorithm="giac")
Output:
2/16900975*b*(524288*b^(25/2)/a^12 + (25*(88179*(a*x^(1/3) + b)^(23/2) - 1 062347*(a*x^(1/3) + b)^(21/2)*b + 5870865*(a*x^(1/3) + b)^(19/2)*b^2 - 196 84665*(a*x^(1/3) + b)^(17/2)*b^3 + 44618574*(a*x^(1/3) + b)^(15/2)*b^4 - 7 2076158*(a*x^(1/3) + b)^(13/2)*b^5 + 85180914*(a*x^(1/3) + b)^(11/2)*b^6 - 74364290*(a*x^(1/3) + b)^(9/2)*b^7 + 47805615*(a*x^(1/3) + b)^(7/2)*b^8 - 22309287*(a*x^(1/3) + b)^(5/2)*b^9 + 7436429*(a*x^(1/3) + b)^(3/2)*b^10 - 2028117*sqrt(a*x^(1/3) + b)*b^11)*b/a^11 + 3*(676039*(a*x^(1/3) + b)^(25/ 2) - 8817900*(a*x^(1/3) + b)^(23/2)*b + 53117350*(a*x^(1/3) + b)^(21/2)*b^ 2 - 195695500*(a*x^(1/3) + b)^(19/2)*b^3 + 492116625*(a*x^(1/3) + b)^(17/2 )*b^4 - 892371480*(a*x^(1/3) + b)^(15/2)*b^5 + 1201269300*(a*x^(1/3) + b)^ (13/2)*b^6 - 1216870200*(a*x^(1/3) + b)^(11/2)*b^7 + 929553625*(a*x^(1/3) + b)^(9/2)*b^8 - 531173500*(a*x^(1/3) + b)^(7/2)*b^9 + 223092870*(a*x^(1/3 ) + b)^(5/2)*b^10 - 67603900*(a*x^(1/3) + b)^(3/2)*b^11 + 16900975*sqrt(a* x^(1/3) + b)*b^12)/a^11)/a) - 2/152108775*a*(4194304*b^(27/2)/a^13 - (27*( 676039*(a*x^(1/3) + b)^(25/2) - 8817900*(a*x^(1/3) + b)^(23/2)*b + 5311735 0*(a*x^(1/3) + b)^(21/2)*b^2 - 195695500*(a*x^(1/3) + b)^(19/2)*b^3 + 4921 16625*(a*x^(1/3) + b)^(17/2)*b^4 - 892371480*(a*x^(1/3) + b)^(15/2)*b^5 + 1201269300*(a*x^(1/3) + b)^(13/2)*b^6 - 1216870200*(a*x^(1/3) + b)^(11/2)* b^7 + 929553625*(a*x^(1/3) + b)^(9/2)*b^8 - 531173500*(a*x^(1/3) + b)^(7/2 )*b^9 + 223092870*(a*x^(1/3) + b)^(5/2)*b^10 - 67603900*(a*x^(1/3) + b)...
Timed out. \[ \int x^2 \left (b x^{2/3}+a x\right )^{3/2} \, dx=\int x^2\,{\left (a\,x+b\,x^{2/3}\right )}^{3/2} \,d x \] Input:
int(x^2*(a*x + b*x^(2/3))^(3/2),x)
Output:
int(x^2*(a*x + b*x^(2/3))^(3/2), x)
Time = 0.27 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.45 \[ \int x^2 \left (b x^{2/3}+a x\right )^{3/2} \, dx=\frac {2 \sqrt {x^{\frac {1}{3}} a +b}\, \left (88179 x^{\frac {11}{3}} a^{11} b^{2}-102960 x^{\frac {8}{3}} a^{8} b^{5}+129024 x^{\frac {5}{3}} a^{5} b^{8}-196608 x^{\frac {2}{3}} a^{2} b^{11}+16900975 x^{\frac {13}{3}} a^{13}-92378 x^{\frac {10}{3}} a^{10} b^{3}+109824 x^{\frac {7}{3}} a^{7} b^{6}-143360 x^{\frac {4}{3}} a^{4} b^{9}+262144 x^{\frac {1}{3}} a \,b^{12}+18929092 a^{12} b \,x^{4}+97240 a^{9} b^{4} x^{3}-118272 a^{6} b^{7} x^{2}+163840 a^{3} b^{10} x -524288 b^{13}\right )}{152108775 a^{12}} \] Input:
int(x^2*(b*x^(2/3)+a*x)^(3/2),x)
Output:
(2*sqrt(x**(1/3)*a + b)*(88179*x**(2/3)*a**11*b**2*x**3 - 102960*x**(2/3)* a**8*b**5*x**2 + 129024*x**(2/3)*a**5*b**8*x - 196608*x**(2/3)*a**2*b**11 + 16900975*x**(1/3)*a**13*x**4 - 92378*x**(1/3)*a**10*b**3*x**3 + 109824*x **(1/3)*a**7*b**6*x**2 - 143360*x**(1/3)*a**4*b**9*x + 262144*x**(1/3)*a*b **12 + 18929092*a**12*b*x**4 + 97240*a**9*b**4*x**3 - 118272*a**6*b**7*x** 2 + 163840*a**3*b**10*x - 524288*b**13))/(152108775*a**12)