Integrand size = 28, antiderivative size = 89 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{\sqrt {d x}} \, dx=\frac {2 a^4 \sqrt {d x}}{d}+\frac {8 a^3 b (d x)^{5/2}}{5 d^3}+\frac {4 a^2 b^2 (d x)^{9/2}}{3 d^5}+\frac {8 a b^3 (d x)^{13/2}}{13 d^7}+\frac {2 b^4 (d x)^{17/2}}{17 d^9} \] Output:
2*a^4*(d*x)^(1/2)/d+8/5*a^3*b*(d*x)^(5/2)/d^3+4/3*a^2*b^2*(d*x)^(9/2)/d^5+ 8/13*a*b^3*(d*x)^(13/2)/d^7+2/17*b^4*(d*x)^(17/2)/d^9
Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.62 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{\sqrt {d x}} \, dx=\frac {2 x \left (3315 a^4+2652 a^3 b x^2+2210 a^2 b^2 x^4+1020 a b^3 x^6+195 b^4 x^8\right )}{3315 \sqrt {d x}} \] Input:
Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^2/Sqrt[d*x],x]
Output:
(2*x*(3315*a^4 + 2652*a^3*b*x^2 + 2210*a^2*b^2*x^4 + 1020*a*b^3*x^6 + 195* b^4*x^8))/(3315*Sqrt[d*x])
Time = 0.37 (sec) , antiderivative size = 89, 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.
\(\displaystyle \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{\sqrt {d x}} \, dx\) |
\(\Big \downarrow \) 1380 |
\(\displaystyle \frac {\int \frac {b^4 \left (b x^2+a\right )^4}{\sqrt {d x}}dx}{b^4}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {\left (a+b x^2\right )^4}{\sqrt {d x}}dx\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \int \left (\frac {a^4}{\sqrt {d x}}+\frac {4 a^3 b (d x)^{3/2}}{d^2}+\frac {6 a^2 b^2 (d x)^{7/2}}{d^4}+\frac {4 a b^3 (d x)^{11/2}}{d^6}+\frac {b^4 (d x)^{15/2}}{d^8}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 a^4 \sqrt {d x}}{d}+\frac {8 a^3 b (d x)^{5/2}}{5 d^3}+\frac {4 a^2 b^2 (d x)^{9/2}}{3 d^5}+\frac {8 a b^3 (d x)^{13/2}}{13 d^7}+\frac {2 b^4 (d x)^{17/2}}{17 d^9}\) |
Input:
Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^2/Sqrt[d*x],x]
Output:
(2*a^4*Sqrt[d*x])/d + (8*a^3*b*(d*x)^(5/2))/(5*d^3) + (4*a^2*b^2*(d*x)^(9/ 2))/(3*d^5) + (8*a*b^3*(d*x)^(13/2))/(13*d^7) + (2*b^4*(d*x)^(17/2))/(17*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 = 0.15 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.58
method | result | size |
gosper | \(\frac {2 \left (195 b^{4} x^{8}+1020 a \,b^{3} x^{6}+2210 a^{2} b^{2} x^{4}+2652 a^{3} b \,x^{2}+3315 a^{4}\right ) x}{3315 \sqrt {d x}}\) | \(52\) |
risch | \(\frac {2 \left (195 b^{4} x^{8}+1020 a \,b^{3} x^{6}+2210 a^{2} b^{2} x^{4}+2652 a^{3} b \,x^{2}+3315 a^{4}\right ) x}{3315 \sqrt {d x}}\) | \(52\) |
pseudoelliptic | \(\frac {2 \sqrt {d x}\, \left (\frac {1}{17} b^{4} x^{8}+\frac {4}{13} a \,b^{3} x^{6}+\frac {2}{3} a^{2} b^{2} x^{4}+\frac {4}{5} a^{3} b \,x^{2}+a^{4}\right )}{d}\) | \(52\) |
trager | \(\frac {\left (\frac {2}{17} b^{4} x^{8}+\frac {8}{13} a \,b^{3} x^{6}+\frac {4}{3} a^{2} b^{2} x^{4}+\frac {8}{5} a^{3} b \,x^{2}+2 a^{4}\right ) \sqrt {d x}}{d}\) | \(53\) |
derivativedivides | \(\frac {\frac {2 b^{4} \left (d x \right )^{\frac {17}{2}}}{17}+\frac {8 a \,b^{3} d^{2} \left (d x \right )^{\frac {13}{2}}}{13}+\frac {4 a^{2} d^{4} b^{2} \left (d x \right )^{\frac {9}{2}}}{3}+\frac {8 a^{3} d^{6} b \left (d x \right )^{\frac {5}{2}}}{5}+2 \sqrt {d x}\, a^{4} d^{8}}{d^{9}}\) | \(73\) |
default | \(\frac {\frac {2 b^{4} \left (d x \right )^{\frac {17}{2}}}{17}+\frac {8 a \,b^{3} d^{2} \left (d x \right )^{\frac {13}{2}}}{13}+\frac {4 a^{2} d^{4} b^{2} \left (d x \right )^{\frac {9}{2}}}{3}+\frac {8 a^{3} d^{6} b \left (d x \right )^{\frac {5}{2}}}{5}+2 \sqrt {d x}\, a^{4} d^{8}}{d^{9}}\) | \(73\) |
orering | \(\frac {2 \left (195 b^{4} x^{8}+1020 a \,b^{3} x^{6}+2210 a^{2} b^{2} x^{4}+2652 a^{3} b \,x^{2}+3315 a^{4}\right ) x \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{2}}{3315 \left (b \,x^{2}+a \right )^{4} \sqrt {d x}}\) | \(81\) |
Input:
int((b^2*x^4+2*a*b*x^2+a^2)^2/(d*x)^(1/2),x,method=_RETURNVERBOSE)
Output:
2/3315*(195*b^4*x^8+1020*a*b^3*x^6+2210*a^2*b^2*x^4+2652*a^3*b*x^2+3315*a^ 4)*x/(d*x)^(1/2)
Time = 0.13 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.60 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{\sqrt {d x}} \, dx=\frac {2 \, {\left (195 \, b^{4} x^{8} + 1020 \, a b^{3} x^{6} + 2210 \, a^{2} b^{2} x^{4} + 2652 \, a^{3} b x^{2} + 3315 \, a^{4}\right )} \sqrt {d x}}{3315 \, d} \] Input:
integrate((b^2*x^4+2*a*b*x^2+a^2)^2/(d*x)^(1/2),x, algorithm="fricas")
Output:
2/3315*(195*b^4*x^8 + 1020*a*b^3*x^6 + 2210*a^2*b^2*x^4 + 2652*a^3*b*x^2 + 3315*a^4)*sqrt(d*x)/d
Time = 0.29 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{\sqrt {d x}} \, dx=\frac {2 a^{4} x}{\sqrt {d x}} + \frac {8 a^{3} b x^{3}}{5 \sqrt {d x}} + \frac {4 a^{2} b^{2} x^{5}}{3 \sqrt {d x}} + \frac {8 a b^{3} x^{7}}{13 \sqrt {d x}} + \frac {2 b^{4} x^{9}}{17 \sqrt {d x}} \] Input:
integrate((b**2*x**4+2*a*b*x**2+a**2)**2/(d*x)**(1/2),x)
Output:
2*a**4*x/sqrt(d*x) + 8*a**3*b*x**3/(5*sqrt(d*x)) + 4*a**2*b**2*x**5/(3*sqr t(d*x)) + 8*a*b**3*x**7/(13*sqrt(d*x)) + 2*b**4*x**9/(17*sqrt(d*x))
Time = 0.03 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{\sqrt {d x}} \, dx=\frac {2 \, {\left (9945 \, \sqrt {d x} a^{4} + \frac {585 \, \left (d x\right )^{\frac {17}{2}} b^{4}}{d^{8}} + \frac {3060 \, \left (d x\right )^{\frac {13}{2}} a b^{3}}{d^{6}} + \frac {4420 \, \left (d x\right )^{\frac {9}{2}} a^{2} b^{2}}{d^{4}} + 442 \, {\left (\frac {5 \, \left (d x\right )^{\frac {9}{2}} b^{2}}{d^{4}} + \frac {18 \, \left (d x\right )^{\frac {5}{2}} a b}{d^{2}}\right )} a^{2}\right )}}{9945 \, d} \] Input:
integrate((b^2*x^4+2*a*b*x^2+a^2)^2/(d*x)^(1/2),x, algorithm="maxima")
Output:
2/9945*(9945*sqrt(d*x)*a^4 + 585*(d*x)^(17/2)*b^4/d^8 + 3060*(d*x)^(13/2)* a*b^3/d^6 + 4420*(d*x)^(9/2)*a^2*b^2/d^4 + 442*(5*(d*x)^(9/2)*b^2/d^4 + 18 *(d*x)^(5/2)*a*b/d^2)*a^2)/d
Time = 0.11 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{\sqrt {d x}} \, dx=\frac {2 \, {\left (195 \, \sqrt {d x} b^{4} x^{8} + 1020 \, \sqrt {d x} a b^{3} x^{6} + 2210 \, \sqrt {d x} a^{2} b^{2} x^{4} + 2652 \, \sqrt {d x} a^{3} b x^{2} + 3315 \, \sqrt {d x} a^{4}\right )}}{3315 \, d} \] Input:
integrate((b^2*x^4+2*a*b*x^2+a^2)^2/(d*x)^(1/2),x, algorithm="giac")
Output:
2/3315*(195*sqrt(d*x)*b^4*x^8 + 1020*sqrt(d*x)*a*b^3*x^6 + 2210*sqrt(d*x)* a^2*b^2*x^4 + 2652*sqrt(d*x)*a^3*b*x^2 + 3315*sqrt(d*x)*a^4)/d
Time = 0.03 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{\sqrt {d x}} \, dx=\frac {2\,a^4\,\sqrt {d\,x}}{d}+\frac {2\,b^4\,{\left (d\,x\right )}^{17/2}}{17\,d^9}+\frac {4\,a^2\,b^2\,{\left (d\,x\right )}^{9/2}}{3\,d^5}+\frac {8\,a^3\,b\,{\left (d\,x\right )}^{5/2}}{5\,d^3}+\frac {8\,a\,b^3\,{\left (d\,x\right )}^{13/2}}{13\,d^7} \] Input:
int((a^2 + b^2*x^4 + 2*a*b*x^2)^2/(d*x)^(1/2),x)
Output:
(2*a^4*(d*x)^(1/2))/d + (2*b^4*(d*x)^(17/2))/(17*d^9) + (4*a^2*b^2*(d*x)^( 9/2))/(3*d^5) + (8*a^3*b*(d*x)^(5/2))/(5*d^3) + (8*a*b^3*(d*x)^(13/2))/(13 *d^7)
Time = 0.16 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.58 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{\sqrt {d x}} \, dx=\frac {2 \sqrt {x}\, \sqrt {d}\, \left (195 b^{4} x^{8}+1020 a \,b^{3} x^{6}+2210 a^{2} b^{2} x^{4}+2652 a^{3} b \,x^{2}+3315 a^{4}\right )}{3315 d} \] Input:
int((b^2*x^4+2*a*b*x^2+a^2)^2/(d*x)^(1/2),x)
Output:
(2*sqrt(x)*sqrt(d)*(3315*a**4 + 2652*a**3*b*x**2 + 2210*a**2*b**2*x**4 + 1 020*a*b**3*x**6 + 195*b**4*x**8))/(3315*d)