Integrand size = 28, antiderivative size = 91 \[ \int (d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx=\frac {2 a^4 (d x)^{5/2}}{5 d}+\frac {8 a^3 b (d x)^{9/2}}{9 d^3}+\frac {12 a^2 b^2 (d x)^{13/2}}{13 d^5}+\frac {8 a b^3 (d x)^{17/2}}{17 d^7}+\frac {2 b^4 (d x)^{21/2}}{21 d^9} \] Output:
2/5*a^4*(d*x)^(5/2)/d+8/9*a^3*b*(d*x)^(9/2)/d^3+12/13*a^2*b^2*(d*x)^(13/2) /d^5+8/17*a*b^3*(d*x)^(17/2)/d^7+2/21*b^4*(d*x)^(21/2)/d^9
Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.60 \[ \int (d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx=\frac {2 x (d x)^{3/2} \left (13923 a^4+30940 a^3 b x^2+32130 a^2 b^2 x^4+16380 a b^3 x^6+3315 b^4 x^8\right )}{69615} \] Input:
Integrate[(d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]
Output:
(2*x*(d*x)^(3/2)*(13923*a^4 + 30940*a^3*b*x^2 + 32130*a^2*b^2*x^4 + 16380* a*b^3*x^6 + 3315*b^4*x^8))/69615
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.
\(\displaystyle \int (d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx\) |
\(\Big \downarrow \) 1380 |
\(\displaystyle \frac {\int b^4 (d x)^{3/2} \left (b x^2+a\right )^4dx}{b^4}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int (d x)^{3/2} \left (a+b x^2\right )^4dx\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \int \left (a^4 (d x)^{3/2}+\frac {4 a^3 b (d x)^{7/2}}{d^2}+\frac {6 a^2 b^2 (d x)^{11/2}}{d^4}+\frac {4 a b^3 (d x)^{15/2}}{d^6}+\frac {b^4 (d x)^{19/2}}{d^8}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 a^4 (d x)^{5/2}}{5 d}+\frac {8 a^3 b (d x)^{9/2}}{9 d^3}+\frac {12 a^2 b^2 (d x)^{13/2}}{13 d^5}+\frac {8 a b^3 (d x)^{17/2}}{17 d^7}+\frac {2 b^4 (d x)^{21/2}}{21 d^9}\) |
Input:
Int[(d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]
Output:
(2*a^4*(d*x)^(5/2))/(5*d) + (8*a^3*b*(d*x)^(9/2))/(9*d^3) + (12*a^2*b^2*(d *x)^(13/2))/(13*d^5) + (8*a*b^3*(d*x)^(17/2))/(17*d^7) + (2*b^4*(d*x)^(21/ 2))/(21*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 = 0.22 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.57
method | result | size |
gosper | \(\frac {2 x \left (3315 b^{4} x^{8}+16380 a \,b^{3} x^{6}+32130 a^{2} b^{2} x^{4}+30940 a^{3} b \,x^{2}+13923 a^{4}\right ) \left (d x \right )^{\frac {3}{2}}}{69615}\) | \(52\) |
pseudoelliptic | \(\frac {2 \sqrt {d x}\, d \left (\frac {5}{21} b^{4} x^{8}+\frac {20}{17} a \,b^{3} x^{6}+\frac {30}{13} a^{2} b^{2} x^{4}+\frac {20}{9} a^{3} b \,x^{2}+a^{4}\right ) x^{2}}{5}\) | \(53\) |
trager | \(\frac {2 d \,x^{2} \left (3315 b^{4} x^{8}+16380 a \,b^{3} x^{6}+32130 a^{2} b^{2} x^{4}+30940 a^{3} b \,x^{2}+13923 a^{4}\right ) \sqrt {d x}}{69615}\) | \(55\) |
risch | \(\frac {2 d^{2} x^{3} \left (3315 b^{4} x^{8}+16380 a \,b^{3} x^{6}+32130 a^{2} b^{2} x^{4}+30940 a^{3} b \,x^{2}+13923 a^{4}\right )}{69615 \sqrt {d x}}\) | \(57\) |
derivativedivides | \(\frac {\frac {2 b^{4} \left (d x \right )^{\frac {21}{2}}}{21}+\frac {8 a \,b^{3} d^{2} \left (d x \right )^{\frac {17}{2}}}{17}+\frac {12 a^{2} d^{4} b^{2} \left (d x \right )^{\frac {13}{2}}}{13}+\frac {8 a^{3} d^{6} b \left (d x \right )^{\frac {9}{2}}}{9}+\frac {2 a^{4} d^{8} \left (d x \right )^{\frac {5}{2}}}{5}}{d^{9}}\) | \(74\) |
default | \(\frac {\frac {2 b^{4} \left (d x \right )^{\frac {21}{2}}}{21}+\frac {8 a \,b^{3} d^{2} \left (d x \right )^{\frac {17}{2}}}{17}+\frac {12 a^{2} d^{4} b^{2} \left (d x \right )^{\frac {13}{2}}}{13}+\frac {8 a^{3} d^{6} b \left (d x \right )^{\frac {9}{2}}}{9}+\frac {2 a^{4} d^{8} \left (d x \right )^{\frac {5}{2}}}{5}}{d^{9}}\) | \(74\) |
orering | \(\frac {2 x \left (3315 b^{4} x^{8}+16380 a \,b^{3} x^{6}+32130 a^{2} b^{2} x^{4}+30940 a^{3} b \,x^{2}+13923 a^{4}\right ) \left (d x \right )^{\frac {3}{2}} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{2}}{69615 \left (b \,x^{2}+a \right )^{4}}\) | \(81\) |
Input:
int((d*x)^(3/2)*(b^2*x^4+2*a*b*x^2+a^2)^2,x,method=_RETURNVERBOSE)
Output:
2/69615*x*(3315*b^4*x^8+16380*a*b^3*x^6+32130*a^2*b^2*x^4+30940*a^3*b*x^2+ 13923*a^4)*(d*x)^(3/2)
Time = 0.09 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.64 \[ \int (d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx=\frac {2}{69615} \, {\left (3315 \, b^{4} d x^{10} + 16380 \, a b^{3} d x^{8} + 32130 \, a^{2} b^{2} d x^{6} + 30940 \, a^{3} b d x^{4} + 13923 \, a^{4} d x^{2}\right )} \sqrt {d x} \] Input:
integrate((d*x)^(3/2)*(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="fricas")
Output:
2/69615*(3315*b^4*d*x^10 + 16380*a*b^3*d*x^8 + 32130*a^2*b^2*d*x^6 + 30940 *a^3*b*d*x^4 + 13923*a^4*d*x^2)*sqrt(d*x)
Time = 0.33 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.97 \[ \int (d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx=\frac {2 a^{4} x \left (d x\right )^{\frac {3}{2}}}{5} + \frac {8 a^{3} b x^{3} \left (d x\right )^{\frac {3}{2}}}{9} + \frac {12 a^{2} b^{2} x^{5} \left (d x\right )^{\frac {3}{2}}}{13} + \frac {8 a b^{3} x^{7} \left (d x\right )^{\frac {3}{2}}}{17} + \frac {2 b^{4} x^{9} \left (d x\right )^{\frac {3}{2}}}{21} \] Input:
integrate((d*x)**(3/2)*(b**2*x**4+2*a*b*x**2+a**2)**2,x)
Output:
2*a**4*x*(d*x)**(3/2)/5 + 8*a**3*b*x**3*(d*x)**(3/2)/9 + 12*a**2*b**2*x**5 *(d*x)**(3/2)/13 + 8*a*b**3*x**7*(d*x)**(3/2)/17 + 2*b**4*x**9*(d*x)**(3/2 )/21
Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.80 \[ \int (d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx=\frac {2 \, {\left (3315 \, \left (d x\right )^{\frac {21}{2}} b^{4} + 16380 \, \left (d x\right )^{\frac {17}{2}} a b^{3} d^{2} + 32130 \, \left (d x\right )^{\frac {13}{2}} a^{2} b^{2} d^{4} + 30940 \, \left (d x\right )^{\frac {9}{2}} a^{3} b d^{6} + 13923 \, \left (d x\right )^{\frac {5}{2}} a^{4} d^{8}\right )}}{69615 \, d^{9}} \] Input:
integrate((d*x)^(3/2)*(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="maxima")
Output:
2/69615*(3315*(d*x)^(21/2)*b^4 + 16380*(d*x)^(17/2)*a*b^3*d^2 + 32130*(d*x )^(13/2)*a^2*b^2*d^4 + 30940*(d*x)^(9/2)*a^3*b*d^6 + 13923*(d*x)^(5/2)*a^4 *d^8)/d^9
Time = 0.12 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.81 \[ \int (d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx=\frac {2}{69615} \, {\left (3315 \, \sqrt {d x} b^{4} x^{10} + 16380 \, \sqrt {d x} a b^{3} x^{8} + 32130 \, \sqrt {d x} a^{2} b^{2} x^{6} + 30940 \, \sqrt {d x} a^{3} b x^{4} + 13923 \, \sqrt {d x} a^{4} x^{2}\right )} d \] Input:
integrate((d*x)^(3/2)*(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="giac")
Output:
2/69615*(3315*sqrt(d*x)*b^4*x^10 + 16380*sqrt(d*x)*a*b^3*x^8 + 32130*sqrt( d*x)*a^2*b^2*x^6 + 30940*sqrt(d*x)*a^3*b*x^4 + 13923*sqrt(d*x)*a^4*x^2)*d
Time = 0.03 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.78 \[ \int (d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx=\frac {2\,a^4\,{\left (d\,x\right )}^{5/2}}{5\,d}+\frac {2\,b^4\,{\left (d\,x\right )}^{21/2}}{21\,d^9}+\frac {12\,a^2\,b^2\,{\left (d\,x\right )}^{13/2}}{13\,d^5}+\frac {8\,a^3\,b\,{\left (d\,x\right )}^{9/2}}{9\,d^3}+\frac {8\,a\,b^3\,{\left (d\,x\right )}^{17/2}}{17\,d^7} \] Input:
int((d*x)^(3/2)*(a^2 + b^2*x^4 + 2*a*b*x^2)^2,x)
Output:
(2*a^4*(d*x)^(5/2))/(5*d) + (2*b^4*(d*x)^(21/2))/(21*d^9) + (12*a^2*b^2*(d *x)^(13/2))/(13*d^5) + (8*a^3*b*(d*x)^(9/2))/(9*d^3) + (8*a*b^3*(d*x)^(17/ 2))/(17*d^7)
Time = 0.17 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.58 \[ \int (d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx=\frac {2 \sqrt {x}\, \sqrt {d}\, d \,x^{2} \left (3315 b^{4} x^{8}+16380 a \,b^{3} x^{6}+32130 a^{2} b^{2} x^{4}+30940 a^{3} b \,x^{2}+13923 a^{4}\right )}{69615} \] Input:
int((d*x)^(3/2)*(b^2*x^4+2*a*b*x^2+a^2)^2,x)
Output:
(2*sqrt(x)*sqrt(d)*d*x**2*(13923*a**4 + 30940*a**3*b*x**2 + 32130*a**2*b** 2*x**4 + 16380*a*b**3*x**6 + 3315*b**4*x**8))/69615