Integrand size = 28, antiderivative size = 91 \[ \int \sqrt {d x} \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx=\frac {2 a^4 (d x)^{3/2}}{3 d}+\frac {8 a^3 b (d x)^{7/2}}{7 d^3}+\frac {12 a^2 b^2 (d x)^{11/2}}{11 d^5}+\frac {8 a b^3 (d x)^{15/2}}{15 d^7}+\frac {2 b^4 (d x)^{19/2}}{19 d^9} \] Output:
2/3*a^4*(d*x)^(3/2)/d+8/7*a^3*b*(d*x)^(7/2)/d^3+12/11*a^2*b^2*(d*x)^(11/2) /d^5+8/15*a*b^3*(d*x)^(15/2)/d^7+2/19*b^4*(d*x)^(19/2)/d^9
Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.60 \[ \int \sqrt {d x} \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx=\frac {2 x \sqrt {d x} \left (7315 a^4+12540 a^3 b x^2+11970 a^2 b^2 x^4+5852 a b^3 x^6+1155 b^4 x^8\right )}{21945} \] Input:
Integrate[Sqrt[d*x]*(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]
Output:
(2*x*Sqrt[d*x]*(7315*a^4 + 12540*a^3*b*x^2 + 11970*a^2*b^2*x^4 + 5852*a*b^ 3*x^6 + 1155*b^4*x^8))/21945
Time = 0.38 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1380, 27, 244, 2009}
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.
\(\displaystyle \int \sqrt {d x} \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx\) |
\(\Big \downarrow \) 1380 |
\(\displaystyle \frac {\int b^4 \sqrt {d x} \left (b x^2+a\right )^4dx}{b^4}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \sqrt {d x} \left (a+b x^2\right )^4dx\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \int \left (a^4 \sqrt {d x}+\frac {4 a^3 b (d x)^{5/2}}{d^2}+\frac {6 a^2 b^2 (d x)^{9/2}}{d^4}+\frac {4 a b^3 (d x)^{13/2}}{d^6}+\frac {b^4 (d x)^{17/2}}{d^8}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 a^4 (d x)^{3/2}}{3 d}+\frac {8 a^3 b (d x)^{7/2}}{7 d^3}+\frac {12 a^2 b^2 (d x)^{11/2}}{11 d^5}+\frac {8 a b^3 (d x)^{15/2}}{15 d^7}+\frac {2 b^4 (d x)^{19/2}}{19 d^9}\) |
Input:
Int[Sqrt[d*x]*(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]
Output:
(2*a^4*(d*x)^(3/2))/(3*d) + (8*a^3*b*(d*x)^(7/2))/(7*d^3) + (12*a^2*b^2*(d *x)^(11/2))/(11*d^5) + (8*a*b^3*(d*x)^(15/2))/(15*d^7) + (2*b^4*(d*x)^(19/ 2))/(19*d^9)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[(u_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> S imp[1/c^p Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Time = 0.24 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.55
method | result | size |
pseudoelliptic | \(\frac {2 \sqrt {d x}\, \left (\frac {3}{19} b^{4} x^{8}+\frac {4}{5} a \,b^{3} x^{6}+\frac {18}{11} a^{2} b^{2} x^{4}+\frac {12}{7} a^{3} b \,x^{2}+a^{4}\right ) x}{3}\) | \(50\) |
gosper | \(\frac {2 x \left (1155 b^{4} x^{8}+5852 a \,b^{3} x^{6}+11970 a^{2} b^{2} x^{4}+12540 a^{3} b \,x^{2}+7315 a^{4}\right ) \sqrt {d x}}{21945}\) | \(52\) |
trager | \(\frac {2 x \left (1155 b^{4} x^{8}+5852 a \,b^{3} x^{6}+11970 a^{2} b^{2} x^{4}+12540 a^{3} b \,x^{2}+7315 a^{4}\right ) \sqrt {d x}}{21945}\) | \(52\) |
risch | \(\frac {2 d \,x^{2} \left (1155 b^{4} x^{8}+5852 a \,b^{3} x^{6}+11970 a^{2} b^{2} x^{4}+12540 a^{3} b \,x^{2}+7315 a^{4}\right )}{21945 \sqrt {d x}}\) | \(55\) |
derivativedivides | \(\frac {\frac {2 b^{4} \left (d x \right )^{\frac {19}{2}}}{19}+\frac {8 a \,b^{3} d^{2} \left (d x \right )^{\frac {15}{2}}}{15}+\frac {12 a^{2} d^{4} b^{2} \left (d x \right )^{\frac {11}{2}}}{11}+\frac {8 a^{3} b \,d^{6} \left (d x \right )^{\frac {7}{2}}}{7}+\frac {2 a^{4} d^{8} \left (d x \right )^{\frac {3}{2}}}{3}}{d^{9}}\) | \(74\) |
default | \(\frac {\frac {2 b^{4} \left (d x \right )^{\frac {19}{2}}}{19}+\frac {8 a \,b^{3} d^{2} \left (d x \right )^{\frac {15}{2}}}{15}+\frac {12 a^{2} d^{4} b^{2} \left (d x \right )^{\frac {11}{2}}}{11}+\frac {8 a^{3} b \,d^{6} \left (d x \right )^{\frac {7}{2}}}{7}+\frac {2 a^{4} d^{8} \left (d x \right )^{\frac {3}{2}}}{3}}{d^{9}}\) | \(74\) |
orering | \(\frac {2 x \left (1155 b^{4} x^{8}+5852 a \,b^{3} x^{6}+11970 a^{2} b^{2} x^{4}+12540 a^{3} b \,x^{2}+7315 a^{4}\right ) \sqrt {d x}\, \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{2}}{21945 \left (b \,x^{2}+a \right )^{4}}\) | \(81\) |
Input:
int((d*x)^(1/2)*(b^2*x^4+2*a*b*x^2+a^2)^2,x,method=_RETURNVERBOSE)
Output:
2/3*(d*x)^(1/2)*(3/19*b^4*x^8+4/5*a*b^3*x^6+18/11*a^2*b^2*x^4+12/7*a^3*b*x ^2+a^4)*x
Time = 0.08 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.56 \[ \int \sqrt {d x} \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx=\frac {2}{21945} \, {\left (1155 \, b^{4} x^{9} + 5852 \, a b^{3} x^{7} + 11970 \, a^{2} b^{2} x^{5} + 12540 \, a^{3} b x^{3} + 7315 \, a^{4} x\right )} \sqrt {d x} \] Input:
integrate((d*x)^(1/2)*(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="fricas")
Output:
2/21945*(1155*b^4*x^9 + 5852*a*b^3*x^7 + 11970*a^2*b^2*x^5 + 12540*a^3*b*x ^3 + 7315*a^4*x)*sqrt(d*x)
Time = 0.21 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.97 \[ \int \sqrt {d x} \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx=\frac {2 a^{4} x \sqrt {d x}}{3} + \frac {8 a^{3} b x^{3} \sqrt {d x}}{7} + \frac {12 a^{2} b^{2} x^{5} \sqrt {d x}}{11} + \frac {8 a b^{3} x^{7} \sqrt {d x}}{15} + \frac {2 b^{4} x^{9} \sqrt {d x}}{19} \] Input:
integrate((d*x)**(1/2)*(b**2*x**4+2*a*b*x**2+a**2)**2,x)
Output:
2*a**4*x*sqrt(d*x)/3 + 8*a**3*b*x**3*sqrt(d*x)/7 + 12*a**2*b**2*x**5*sqrt( d*x)/11 + 8*a*b**3*x**7*sqrt(d*x)/15 + 2*b**4*x**9*sqrt(d*x)/19
Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.80 \[ \int \sqrt {d x} \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx=\frac {2 \, {\left (1155 \, \left (d x\right )^{\frac {19}{2}} b^{4} + 5852 \, \left (d x\right )^{\frac {15}{2}} a b^{3} d^{2} + 11970 \, \left (d x\right )^{\frac {11}{2}} a^{2} b^{2} d^{4} + 12540 \, \left (d x\right )^{\frac {7}{2}} a^{3} b d^{6} + 7315 \, \left (d x\right )^{\frac {3}{2}} a^{4} d^{8}\right )}}{21945 \, d^{9}} \] Input:
integrate((d*x)^(1/2)*(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="maxima")
Output:
2/21945*(1155*(d*x)^(19/2)*b^4 + 5852*(d*x)^(15/2)*a*b^3*d^2 + 11970*(d*x) ^(11/2)*a^2*b^2*d^4 + 12540*(d*x)^(7/2)*a^3*b*d^6 + 7315*(d*x)^(3/2)*a^4*d ^8)/d^9
Time = 0.13 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.76 \[ \int \sqrt {d x} \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx=\frac {2}{19} \, \sqrt {d x} b^{4} x^{9} + \frac {8}{15} \, \sqrt {d x} a b^{3} x^{7} + \frac {12}{11} \, \sqrt {d x} a^{2} b^{2} x^{5} + \frac {8}{7} \, \sqrt {d x} a^{3} b x^{3} + \frac {2}{3} \, \sqrt {d x} a^{4} x \] Input:
integrate((d*x)^(1/2)*(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="giac")
Output:
2/19*sqrt(d*x)*b^4*x^9 + 8/15*sqrt(d*x)*a*b^3*x^7 + 12/11*sqrt(d*x)*a^2*b^ 2*x^5 + 8/7*sqrt(d*x)*a^3*b*x^3 + 2/3*sqrt(d*x)*a^4*x
Time = 0.03 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.78 \[ \int \sqrt {d x} \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx=\frac {2\,a^4\,{\left (d\,x\right )}^{3/2}}{3\,d}+\frac {2\,b^4\,{\left (d\,x\right )}^{19/2}}{19\,d^9}+\frac {12\,a^2\,b^2\,{\left (d\,x\right )}^{11/2}}{11\,d^5}+\frac {8\,a^3\,b\,{\left (d\,x\right )}^{7/2}}{7\,d^3}+\frac {8\,a\,b^3\,{\left (d\,x\right )}^{15/2}}{15\,d^7} \] Input:
int((d*x)^(1/2)*(a^2 + b^2*x^4 + 2*a*b*x^2)^2,x)
Output:
(2*a^4*(d*x)^(3/2))/(3*d) + (2*b^4*(d*x)^(19/2))/(19*d^9) + (12*a^2*b^2*(d *x)^(11/2))/(11*d^5) + (8*a^3*b*(d*x)^(7/2))/(7*d^3) + (8*a*b^3*(d*x)^(15/ 2))/(15*d^7)
Time = 0.17 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.55 \[ \int \sqrt {d x} \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx=\frac {2 \sqrt {x}\, \sqrt {d}\, x \left (1155 b^{4} x^{8}+5852 a \,b^{3} x^{6}+11970 a^{2} b^{2} x^{4}+12540 a^{3} b \,x^{2}+7315 a^{4}\right )}{21945} \] Input:
int((d*x)^(1/2)*(b^2*x^4+2*a*b*x^2+a^2)^2,x)
Output:
(2*sqrt(x)*sqrt(d)*x*(7315*a**4 + 12540*a**3*b*x**2 + 11970*a**2*b**2*x**4 + 5852*a*b**3*x**6 + 1155*b**4*x**8))/21945