Integrand size = 28, antiderivative size = 127 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{(d x)^{5/2}} \, dx=-\frac {2 a^6}{3 d (d x)^{3/2}}+\frac {12 a^5 b \sqrt {d x}}{d^3}+\frac {6 a^4 b^2 (d x)^{5/2}}{d^5}+\frac {40 a^3 b^3 (d x)^{9/2}}{9 d^7}+\frac {30 a^2 b^4 (d x)^{13/2}}{13 d^9}+\frac {12 a b^5 (d x)^{17/2}}{17 d^{11}}+\frac {2 b^6 (d x)^{21/2}}{21 d^{13}} \] Output:
-2/3*a^6/d/(d*x)^(3/2)+12*a^5*b*(d*x)^(1/2)/d^3+6*a^4*b^2*(d*x)^(5/2)/d^5+ 40/9*a^3*b^3*(d*x)^(9/2)/d^7+30/13*a^2*b^4*(d*x)^(13/2)/d^9+12/17*a*b^5*(d *x)^(17/2)/d^11+2/21*b^6*(d*x)^(21/2)/d^13
Time = 0.03 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.61 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{(d x)^{5/2}} \, dx=-\frac {2 x \left (4641 a^6-83538 a^5 b x^2-41769 a^4 b^2 x^4-30940 a^3 b^3 x^6-16065 a^2 b^4 x^8-4914 a b^5 x^{10}-663 b^6 x^{12}\right )}{13923 (d x)^{5/2}} \] Input:
Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/(d*x)^(5/2),x]
Output:
(-2*x*(4641*a^6 - 83538*a^5*b*x^2 - 41769*a^4*b^2*x^4 - 30940*a^3*b^3*x^6 - 16065*a^2*b^4*x^8 - 4914*a*b^5*x^10 - 663*b^6*x^12))/(13923*(d*x)^(5/2))
Time = 0.41 (sec) , antiderivative size = 127, 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.
\(\displaystyle \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{(d x)^{5/2}} \, dx\) |
\(\Big \downarrow \) 1380 |
\(\displaystyle \frac {\int \frac {b^6 \left (b x^2+a\right )^6}{(d x)^{5/2}}dx}{b^6}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {\left (a+b x^2\right )^6}{(d x)^{5/2}}dx\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \int \left (\frac {a^6}{(d x)^{5/2}}+\frac {6 a^5 b}{d^2 \sqrt {d x}}+\frac {15 a^4 b^2 (d x)^{3/2}}{d^4}+\frac {20 a^3 b^3 (d x)^{7/2}}{d^6}+\frac {15 a^2 b^4 (d x)^{11/2}}{d^8}+\frac {6 a b^5 (d x)^{15/2}}{d^{10}}+\frac {b^6 (d x)^{19/2}}{d^{12}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 a^6}{3 d (d x)^{3/2}}+\frac {12 a^5 b \sqrt {d x}}{d^3}+\frac {6 a^4 b^2 (d x)^{5/2}}{d^5}+\frac {40 a^3 b^3 (d x)^{9/2}}{9 d^7}+\frac {30 a^2 b^4 (d x)^{13/2}}{13 d^9}+\frac {12 a b^5 (d x)^{17/2}}{17 d^{11}}+\frac {2 b^6 (d x)^{21/2}}{21 d^{13}}\) |
Input:
Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/(d*x)^(5/2),x]
Output:
(-2*a^6)/(3*d*(d*x)^(3/2)) + (12*a^5*b*Sqrt[d*x])/d^3 + (6*a^4*b^2*(d*x)^( 5/2))/d^5 + (40*a^3*b^3*(d*x)^(9/2))/(9*d^7) + (30*a^2*b^4*(d*x)^(13/2))/( 13*d^9) + (12*a*b^5*(d*x)^(17/2))/(17*d^11) + (2*b^6*(d*x)^(21/2))/(21*d^1 3)
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 = 0.27 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.58
method | result | size |
gosper | \(-\frac {2 \left (-663 b^{6} x^{12}-4914 a \,b^{5} x^{10}-16065 a^{2} b^{4} x^{8}-30940 a^{3} b^{3} x^{6}-41769 a^{4} b^{2} x^{4}-83538 a^{5} b \,x^{2}+4641 a^{6}\right ) x}{13923 \left (d x \right )^{\frac {5}{2}}}\) | \(74\) |
pseudoelliptic | \(-\frac {2 \left (-\frac {1}{7} b^{6} x^{12}-\frac {18}{17} a \,b^{5} x^{10}-\frac {45}{13} a^{2} b^{4} x^{8}-\frac {20}{3} a^{3} b^{3} x^{6}-9 a^{4} b^{2} x^{4}-18 a^{5} b \,x^{2}+a^{6}\right )}{3 \sqrt {d x}\, d^{2} x}\) | \(77\) |
trager | \(-\frac {2 \left (-663 b^{6} x^{12}-4914 a \,b^{5} x^{10}-16065 a^{2} b^{4} x^{8}-30940 a^{3} b^{3} x^{6}-41769 a^{4} b^{2} x^{4}-83538 a^{5} b \,x^{2}+4641 a^{6}\right ) \sqrt {d x}}{13923 d^{3} x^{2}}\) | \(79\) |
risch | \(-\frac {2 \left (-663 b^{6} x^{12}-4914 a \,b^{5} x^{10}-16065 a^{2} b^{4} x^{8}-30940 a^{3} b^{3} x^{6}-41769 a^{4} b^{2} x^{4}-83538 a^{5} b \,x^{2}+4641 a^{6}\right )}{13923 d^{2} x \sqrt {d x}}\) | \(79\) |
orering | \(-\frac {2 \left (-663 b^{6} x^{12}-4914 a \,b^{5} x^{10}-16065 a^{2} b^{4} x^{8}-30940 a^{3} b^{3} x^{6}-41769 a^{4} b^{2} x^{4}-83538 a^{5} b \,x^{2}+4641 a^{6}\right ) x \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{3}}{13923 \left (b \,x^{2}+a \right )^{6} \left (d x \right )^{\frac {5}{2}}}\) | \(103\) |
derivativedivides | \(\frac {\frac {2 b^{6} \left (d x \right )^{\frac {21}{2}}}{21}+\frac {12 a \,b^{5} d^{2} \left (d x \right )^{\frac {17}{2}}}{17}+\frac {30 a^{2} b^{4} d^{4} \left (d x \right )^{\frac {13}{2}}}{13}+\frac {40 a^{3} b^{3} d^{6} \left (d x \right )^{\frac {9}{2}}}{9}+6 a^{4} b^{2} d^{8} \left (d x \right )^{\frac {5}{2}}+12 \sqrt {d x}\, a^{5} b \,d^{10}-\frac {2 a^{6} d^{12}}{3 \left (d x \right )^{\frac {3}{2}}}}{d^{13}}\) | \(106\) |
default | \(\frac {\frac {2 b^{6} \left (d x \right )^{\frac {21}{2}}}{21}+\frac {12 a \,b^{5} d^{2} \left (d x \right )^{\frac {17}{2}}}{17}+\frac {30 a^{2} b^{4} d^{4} \left (d x \right )^{\frac {13}{2}}}{13}+\frac {40 a^{3} b^{3} d^{6} \left (d x \right )^{\frac {9}{2}}}{9}+6 a^{4} b^{2} d^{8} \left (d x \right )^{\frac {5}{2}}+12 \sqrt {d x}\, a^{5} b \,d^{10}-\frac {2 a^{6} d^{12}}{3 \left (d x \right )^{\frac {3}{2}}}}{d^{13}}\) | \(106\) |
Input:
int((b^2*x^4+2*a*b*x^2+a^2)^3/(d*x)^(5/2),x,method=_RETURNVERBOSE)
Output:
-2/13923*(-663*b^6*x^12-4914*a*b^5*x^10-16065*a^2*b^4*x^8-30940*a^3*b^3*x^ 6-41769*a^4*b^2*x^4-83538*a^5*b*x^2+4641*a^6)*x/(d*x)^(5/2)
Time = 0.08 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.61 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{(d x)^{5/2}} \, dx=\frac {2 \, {\left (663 \, b^{6} x^{12} + 4914 \, a b^{5} x^{10} + 16065 \, a^{2} b^{4} x^{8} + 30940 \, a^{3} b^{3} x^{6} + 41769 \, a^{4} b^{2} x^{4} + 83538 \, a^{5} b x^{2} - 4641 \, a^{6}\right )} \sqrt {d x}}{13923 \, d^{3} x^{2}} \] Input:
integrate((b^2*x^4+2*a*b*x^2+a^2)^3/(d*x)^(5/2),x, algorithm="fricas")
Output:
2/13923*(663*b^6*x^12 + 4914*a*b^5*x^10 + 16065*a^2*b^4*x^8 + 30940*a^3*b^ 3*x^6 + 41769*a^4*b^2*x^4 + 83538*a^5*b*x^2 - 4641*a^6)*sqrt(d*x)/(d^3*x^2 )
Time = 0.45 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{(d x)^{5/2}} \, dx=- \frac {2 a^{6} x}{3 \left (d x\right )^{\frac {5}{2}}} + \frac {12 a^{5} b x^{3}}{\left (d x\right )^{\frac {5}{2}}} + \frac {6 a^{4} b^{2} x^{5}}{\left (d x\right )^{\frac {5}{2}}} + \frac {40 a^{3} b^{3} x^{7}}{9 \left (d x\right )^{\frac {5}{2}}} + \frac {30 a^{2} b^{4} x^{9}}{13 \left (d x\right )^{\frac {5}{2}}} + \frac {12 a b^{5} x^{11}}{17 \left (d x\right )^{\frac {5}{2}}} + \frac {2 b^{6} x^{13}}{21 \left (d x\right )^{\frac {5}{2}}} \] Input:
integrate((b**2*x**4+2*a*b*x**2+a**2)**3/(d*x)**(5/2),x)
Output:
-2*a**6*x/(3*(d*x)**(5/2)) + 12*a**5*b*x**3/(d*x)**(5/2) + 6*a**4*b**2*x** 5/(d*x)**(5/2) + 40*a**3*b**3*x**7/(9*(d*x)**(5/2)) + 30*a**2*b**4*x**9/(1 3*(d*x)**(5/2)) + 12*a*b**5*x**11/(17*(d*x)**(5/2)) + 2*b**6*x**13/(21*(d* x)**(5/2))
Time = 0.03 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{(d x)^{5/2}} \, dx=-\frac {2 \, {\left (\frac {4641 \, a^{6}}{\left (d x\right )^{\frac {3}{2}}} - \frac {663 \, \left (d x\right )^{\frac {21}{2}} b^{6} + 4914 \, \left (d x\right )^{\frac {17}{2}} a b^{5} d^{2} + 16065 \, \left (d x\right )^{\frac {13}{2}} a^{2} b^{4} d^{4} + 30940 \, \left (d x\right )^{\frac {9}{2}} a^{3} b^{3} d^{6} + 41769 \, \left (d x\right )^{\frac {5}{2}} a^{4} b^{2} d^{8} + 83538 \, \sqrt {d x} a^{5} b d^{10}}{d^{12}}\right )}}{13923 \, d} \] Input:
integrate((b^2*x^4+2*a*b*x^2+a^2)^3/(d*x)^(5/2),x, algorithm="maxima")
Output:
-2/13923*(4641*a^6/(d*x)^(3/2) - (663*(d*x)^(21/2)*b^6 + 4914*(d*x)^(17/2) *a*b^5*d^2 + 16065*(d*x)^(13/2)*a^2*b^4*d^4 + 30940*(d*x)^(9/2)*a^3*b^3*d^ 6 + 41769*(d*x)^(5/2)*a^4*b^2*d^8 + 83538*sqrt(d*x)*a^5*b*d^10)/d^12)/d
Time = 0.12 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.04 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{(d x)^{5/2}} \, dx=-\frac {2 \, {\left (\frac {4641 \, a^{6}}{\sqrt {d x} d x} - \frac {663 \, \sqrt {d x} b^{6} d^{250} x^{10} + 4914 \, \sqrt {d x} a b^{5} d^{250} x^{8} + 16065 \, \sqrt {d x} a^{2} b^{4} d^{250} x^{6} + 30940 \, \sqrt {d x} a^{3} b^{3} d^{250} x^{4} + 41769 \, \sqrt {d x} a^{4} b^{2} d^{250} x^{2} + 83538 \, \sqrt {d x} a^{5} b d^{250}}{d^{252}}\right )}}{13923 \, d} \] Input:
integrate((b^2*x^4+2*a*b*x^2+a^2)^3/(d*x)^(5/2),x, algorithm="giac")
Output:
-2/13923*(4641*a^6/(sqrt(d*x)*d*x) - (663*sqrt(d*x)*b^6*d^250*x^10 + 4914* sqrt(d*x)*a*b^5*d^250*x^8 + 16065*sqrt(d*x)*a^2*b^4*d^250*x^6 + 30940*sqrt (d*x)*a^3*b^3*d^250*x^4 + 41769*sqrt(d*x)*a^4*b^2*d^250*x^2 + 83538*sqrt(d *x)*a^5*b*d^250)/d^252)/d
Time = 0.04 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{(d x)^{5/2}} \, dx=\frac {2\,b^6\,{\left (d\,x\right )}^{21/2}}{21\,d^{13}}-\frac {2\,a^6}{3\,d\,{\left (d\,x\right )}^{3/2}}+\frac {6\,a^4\,b^2\,{\left (d\,x\right )}^{5/2}}{d^5}+\frac {40\,a^3\,b^3\,{\left (d\,x\right )}^{9/2}}{9\,d^7}+\frac {30\,a^2\,b^4\,{\left (d\,x\right )}^{13/2}}{13\,d^9}+\frac {12\,a^5\,b\,\sqrt {d\,x}}{d^3}+\frac {12\,a\,b^5\,{\left (d\,x\right )}^{17/2}}{17\,d^{11}} \] Input:
int((a^2 + b^2*x^4 + 2*a*b*x^2)^3/(d*x)^(5/2),x)
Output:
(2*b^6*(d*x)^(21/2))/(21*d^13) - (2*a^6)/(3*d*(d*x)^(3/2)) + (6*a^4*b^2*(d *x)^(5/2))/d^5 + (40*a^3*b^3*(d*x)^(9/2))/(9*d^7) + (30*a^2*b^4*(d*x)^(13/ 2))/(13*d^9) + (12*a^5*b*(d*x)^(1/2))/d^3 + (12*a*b^5*(d*x)^(17/2))/(17*d^ 11)
Time = 0.22 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.62 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{(d x)^{5/2}} \, dx=\frac {2 \sqrt {d}\, \left (663 b^{6} x^{12}+4914 a \,b^{5} x^{10}+16065 a^{2} b^{4} x^{8}+30940 a^{3} b^{3} x^{6}+41769 a^{4} b^{2} x^{4}+83538 a^{5} b \,x^{2}-4641 a^{6}\right )}{13923 \sqrt {x}\, d^{3} x} \] Input:
int((b^2*x^4+2*a*b*x^2+a^2)^3/(d*x)^(5/2),x)
Output:
(2*sqrt(d)*( - 4641*a**6 + 83538*a**5*b*x**2 + 41769*a**4*b**2*x**4 + 3094 0*a**3*b**3*x**6 + 16065*a**2*b**4*x**8 + 4914*a*b**5*x**10 + 663*b**6*x** 12))/(13923*sqrt(x)*d**3*x)