Integrand size = 28, antiderivative size = 125 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{(d x)^{3/2}} \, dx=-\frac {2 a^6}{d \sqrt {d x}}+\frac {4 a^5 b (d x)^{3/2}}{d^3}+\frac {30 a^4 b^2 (d x)^{7/2}}{7 d^5}+\frac {40 a^3 b^3 (d x)^{11/2}}{11 d^7}+\frac {2 a^2 b^4 (d x)^{15/2}}{d^9}+\frac {12 a b^5 (d x)^{19/2}}{19 d^{11}}+\frac {2 b^6 (d x)^{23/2}}{23 d^{13}} \]
4*a^5*b*(d*x)^(3/2)/d^3+30/7*a^4*b^2*(d*x)^(7/2)/d^5+40/11*a^3*b^3*(d*x)^( 11/2)/d^7+2*a^2*b^4*(d*x)^(15/2)/d^9+12/19*a*b^5*(d*x)^(19/2)/d^11+2/23*b^ 6*(d*x)^(23/2)/d^13-2*a^6/d/(d*x)^(1/2)
Time = 0.03 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.62 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{(d x)^{3/2}} \, dx=-\frac {2 x \left (33649 a^6-67298 a^5 b x^2-72105 a^4 b^2 x^4-61180 a^3 b^3 x^6-33649 a^2 b^4 x^8-10626 a b^5 x^{10}-1463 b^6 x^{12}\right )}{33649 (d x)^{3/2}} \]
(-2*x*(33649*a^6 - 67298*a^5*b*x^2 - 72105*a^4*b^2*x^4 - 61180*a^3*b^3*x^6 - 33649*a^2*b^4*x^8 - 10626*a*b^5*x^10 - 1463*b^6*x^12))/(33649*(d*x)^(3/ 2))
Time = 0.26 (sec) , antiderivative size = 125, 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.
\(\displaystyle \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{(d x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 1380 |
\(\displaystyle \frac {\int \frac {b^6 \left (b x^2+a\right )^6}{(d x)^{3/2}}dx}{b^6}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {\left (a+b x^2\right )^6}{(d x)^{3/2}}dx\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \int \left (\frac {a^6}{(d x)^{3/2}}+\frac {6 a^5 b \sqrt {d x}}{d^2}+\frac {15 a^4 b^2 (d x)^{5/2}}{d^4}+\frac {20 a^3 b^3 (d x)^{9/2}}{d^6}+\frac {15 a^2 b^4 (d x)^{13/2}}{d^8}+\frac {6 a b^5 (d x)^{17/2}}{d^{10}}+\frac {b^6 (d x)^{21/2}}{d^{12}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 a^6}{d \sqrt {d x}}+\frac {4 a^5 b (d x)^{3/2}}{d^3}+\frac {30 a^4 b^2 (d x)^{7/2}}{7 d^5}+\frac {40 a^3 b^3 (d x)^{11/2}}{11 d^7}+\frac {2 a^2 b^4 (d x)^{15/2}}{d^9}+\frac {12 a b^5 (d x)^{19/2}}{19 d^{11}}+\frac {2 b^6 (d x)^{23/2}}{23 d^{13}}\) |
(-2*a^6)/(d*Sqrt[d*x]) + (4*a^5*b*(d*x)^(3/2))/d^3 + (30*a^4*b^2*(d*x)^(7/ 2))/(7*d^5) + (40*a^3*b^3*(d*x)^(11/2))/(11*d^7) + (2*a^2*b^4*(d*x)^(15/2) )/d^9 + (12*a*b^5*(d*x)^(19/2))/(19*d^11) + (2*b^6*(d*x)^(23/2))/(23*d^13)
3.7.83.3.1 Defintions of rubi rules used
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 = 74, normalized size of antiderivative = 0.59
method | result | size |
gosper | \(-\frac {2 \left (-1463 b^{6} x^{12}-10626 a \,b^{5} x^{10}-33649 a^{2} b^{4} x^{8}-61180 a^{3} b^{3} x^{6}-72105 a^{4} b^{2} x^{4}-67298 a^{5} b \,x^{2}+33649 a^{6}\right ) x}{33649 \left (d x \right )^{\frac {3}{2}}}\) | \(74\) |
pseudoelliptic | \(-\frac {2 \left (-\frac {1}{23} b^{6} x^{12}-\frac {6}{19} a \,b^{5} x^{10}-a^{2} b^{4} x^{8}-\frac {20}{11} a^{3} b^{3} x^{6}-\frac {15}{7} a^{4} b^{2} x^{4}-2 a^{5} b \,x^{2}+a^{6}\right )}{\sqrt {d x}\, d}\) | \(74\) |
risch | \(-\frac {2 \left (-1463 b^{6} x^{12}-10626 a \,b^{5} x^{10}-33649 a^{2} b^{4} x^{8}-61180 a^{3} b^{3} x^{6}-72105 a^{4} b^{2} x^{4}-67298 a^{5} b \,x^{2}+33649 a^{6}\right )}{33649 d \sqrt {d x}}\) | \(76\) |
trager | \(-\frac {2 \left (-1463 b^{6} x^{12}-10626 a \,b^{5} x^{10}-33649 a^{2} b^{4} x^{8}-61180 a^{3} b^{3} x^{6}-72105 a^{4} b^{2} x^{4}-67298 a^{5} b \,x^{2}+33649 a^{6}\right ) \sqrt {d x}}{33649 d^{2} x}\) | \(79\) |
derivativedivides | \(\frac {\frac {2 b^{6} \left (d x \right )^{\frac {23}{2}}}{23}+\frac {12 a \,b^{5} d^{2} \left (d x \right )^{\frac {19}{2}}}{19}+2 a^{2} b^{4} d^{4} \left (d x \right )^{\frac {15}{2}}+\frac {40 a^{3} b^{3} d^{6} \left (d x \right )^{\frac {11}{2}}}{11}+\frac {30 a^{4} b^{2} d^{8} \left (d x \right )^{\frac {7}{2}}}{7}+4 a^{5} b \,d^{10} \left (d x \right )^{\frac {3}{2}}-\frac {2 a^{6} d^{12}}{\sqrt {d x}}}{d^{13}}\) | \(105\) |
default | \(\frac {\frac {2 b^{6} \left (d x \right )^{\frac {23}{2}}}{23}+\frac {12 a \,b^{5} d^{2} \left (d x \right )^{\frac {19}{2}}}{19}+2 a^{2} b^{4} d^{4} \left (d x \right )^{\frac {15}{2}}+\frac {40 a^{3} b^{3} d^{6} \left (d x \right )^{\frac {11}{2}}}{11}+\frac {30 a^{4} b^{2} d^{8} \left (d x \right )^{\frac {7}{2}}}{7}+4 a^{5} b \,d^{10} \left (d x \right )^{\frac {3}{2}}-\frac {2 a^{6} d^{12}}{\sqrt {d x}}}{d^{13}}\) | \(105\) |
-2/33649*(-1463*b^6*x^12-10626*a*b^5*x^10-33649*a^2*b^4*x^8-61180*a^3*b^3* x^6-72105*a^4*b^2*x^4-67298*a^5*b*x^2+33649*a^6)*x/(d*x)^(3/2)
Time = 0.26 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.62 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{(d x)^{3/2}} \, dx=\frac {2 \, {\left (1463 \, b^{6} x^{12} + 10626 \, a b^{5} x^{10} + 33649 \, a^{2} b^{4} x^{8} + 61180 \, a^{3} b^{3} x^{6} + 72105 \, a^{4} b^{2} x^{4} + 67298 \, a^{5} b x^{2} - 33649 \, a^{6}\right )} \sqrt {d x}}{33649 \, d^{2} x} \]
2/33649*(1463*b^6*x^12 + 10626*a*b^5*x^10 + 33649*a^2*b^4*x^8 + 61180*a^3* b^3*x^6 + 72105*a^4*b^2*x^4 + 67298*a^5*b*x^2 - 33649*a^6)*sqrt(d*x)/(d^2* x)
Time = 0.41 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{(d x)^{3/2}} \, dx=- \frac {2 a^{6} x}{\left (d x\right )^{\frac {3}{2}}} + \frac {4 a^{5} b x^{3}}{\left (d x\right )^{\frac {3}{2}}} + \frac {30 a^{4} b^{2} x^{5}}{7 \left (d x\right )^{\frac {3}{2}}} + \frac {40 a^{3} b^{3} x^{7}}{11 \left (d x\right )^{\frac {3}{2}}} + \frac {2 a^{2} b^{4} x^{9}}{\left (d x\right )^{\frac {3}{2}}} + \frac {12 a b^{5} x^{11}}{19 \left (d x\right )^{\frac {3}{2}}} + \frac {2 b^{6} x^{13}}{23 \left (d x\right )^{\frac {3}{2}}} \]
-2*a**6*x/(d*x)**(3/2) + 4*a**5*b*x**3/(d*x)**(3/2) + 30*a**4*b**2*x**5/(7 *(d*x)**(3/2)) + 40*a**3*b**3*x**7/(11*(d*x)**(3/2)) + 2*a**2*b**4*x**9/(d *x)**(3/2) + 12*a*b**5*x**11/(19*(d*x)**(3/2)) + 2*b**6*x**13/(23*(d*x)**( 3/2))
Time = 0.20 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{(d x)^{3/2}} \, dx=-\frac {2 \, {\left (\frac {33649 \, a^{6}}{\sqrt {d x}} - \frac {1463 \, \left (d x\right )^{\frac {23}{2}} b^{6} + 10626 \, \left (d x\right )^{\frac {19}{2}} a b^{5} d^{2} + 33649 \, \left (d x\right )^{\frac {15}{2}} a^{2} b^{4} d^{4} + 61180 \, \left (d x\right )^{\frac {11}{2}} a^{3} b^{3} d^{6} + 72105 \, \left (d x\right )^{\frac {7}{2}} a^{4} b^{2} d^{8} + 67298 \, \left (d x\right )^{\frac {3}{2}} a^{5} b d^{10}}{d^{12}}\right )}}{33649 \, d} \]
-2/33649*(33649*a^6/sqrt(d*x) - (1463*(d*x)^(23/2)*b^6 + 10626*(d*x)^(19/2 )*a*b^5*d^2 + 33649*(d*x)^(15/2)*a^2*b^4*d^4 + 61180*(d*x)^(11/2)*a^3*b^3* d^6 + 72105*(d*x)^(7/2)*a^4*b^2*d^8 + 67298*(d*x)^(3/2)*a^5*b*d^10)/d^12)/ d
Time = 0.28 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{(d x)^{3/2}} \, dx=-\frac {2 \, {\left (\frac {33649 \, a^{6}}{\sqrt {d x}} - \frac {1463 \, \sqrt {d x} b^{6} d^{275} x^{11} + 10626 \, \sqrt {d x} a b^{5} d^{275} x^{9} + 33649 \, \sqrt {d x} a^{2} b^{4} d^{275} x^{7} + 61180 \, \sqrt {d x} a^{3} b^{3} d^{275} x^{5} + 72105 \, \sqrt {d x} a^{4} b^{2} d^{275} x^{3} + 67298 \, \sqrt {d x} a^{5} b d^{275} x}{d^{276}}\right )}}{33649 \, d} \]
-2/33649*(33649*a^6/sqrt(d*x) - (1463*sqrt(d*x)*b^6*d^275*x^11 + 10626*sqr t(d*x)*a*b^5*d^275*x^9 + 33649*sqrt(d*x)*a^2*b^4*d^275*x^7 + 61180*sqrt(d* x)*a^3*b^3*d^275*x^5 + 72105*sqrt(d*x)*a^4*b^2*d^275*x^3 + 67298*sqrt(d*x) *a^5*b*d^275*x)/d^276)/d
Time = 0.04 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{(d x)^{3/2}} \, dx=\frac {2\,b^6\,{\left (d\,x\right )}^{23/2}}{23\,d^{13}}-\frac {2\,a^6}{d\,\sqrt {d\,x}}+\frac {30\,a^4\,b^2\,{\left (d\,x\right )}^{7/2}}{7\,d^5}+\frac {40\,a^3\,b^3\,{\left (d\,x\right )}^{11/2}}{11\,d^7}+\frac {2\,a^2\,b^4\,{\left (d\,x\right )}^{15/2}}{d^9}+\frac {4\,a^5\,b\,{\left (d\,x\right )}^{3/2}}{d^3}+\frac {12\,a\,b^5\,{\left (d\,x\right )}^{19/2}}{19\,d^{11}} \]