Integrand size = 28, antiderivative size = 87 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{(d x)^{7/2}} \, dx=-\frac {2 a^4}{5 d (d x)^{5/2}}-\frac {8 a^3 b}{d^3 \sqrt {d x}}+\frac {4 a^2 b^2 (d x)^{3/2}}{d^5}+\frac {8 a b^3 (d x)^{7/2}}{7 d^7}+\frac {2 b^4 (d x)^{11/2}}{11 d^9} \] Output:
-2/5*a^4/d/(d*x)^(5/2)-8*a^3*b/d^3/(d*x)^(1/2)+4*a^2*b^2*(d*x)^(3/2)/d^5+8 /7*a*b^3*(d*x)^(7/2)/d^7+2/11*b^4*(d*x)^(11/2)/d^9
Time = 0.03 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.69 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{(d x)^{7/2}} \, dx=-\frac {2 \sqrt {d x} \left (77 a^4+1540 a^3 b x^2-770 a^2 b^2 x^4-220 a b^3 x^6-35 b^4 x^8\right )}{385 d^4 x^3} \] Input:
Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^2/(d*x)^(7/2),x]
Output:
(-2*Sqrt[d*x]*(77*a^4 + 1540*a^3*b*x^2 - 770*a^2*b^2*x^4 - 220*a*b^3*x^6 - 35*b^4*x^8))/(385*d^4*x^3)
Time = 0.37 (sec) , antiderivative size = 87, 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.
\(\displaystyle \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{(d x)^{7/2}} \, dx\) |
\(\Big \downarrow \) 1380 |
\(\displaystyle \frac {\int \frac {b^4 \left (b x^2+a\right )^4}{(d x)^{7/2}}dx}{b^4}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {\left (a+b x^2\right )^4}{(d x)^{7/2}}dx\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \int \left (\frac {a^4}{(d x)^{7/2}}+\frac {4 a^3 b}{d^2 (d x)^{3/2}}+\frac {6 a^2 b^2 \sqrt {d x}}{d^4}+\frac {4 a b^3 (d x)^{5/2}}{d^6}+\frac {b^4 (d x)^{9/2}}{d^8}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 a^4}{5 d (d x)^{5/2}}-\frac {8 a^3 b}{d^3 \sqrt {d x}}+\frac {4 a^2 b^2 (d x)^{3/2}}{d^5}+\frac {8 a b^3 (d x)^{7/2}}{7 d^7}+\frac {2 b^4 (d x)^{11/2}}{11 d^9}\) |
Input:
Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^2/(d*x)^(7/2),x]
Output:
(-2*a^4)/(5*d*(d*x)^(5/2)) - (8*a^3*b)/(d^3*Sqrt[d*x]) + (4*a^2*b^2*(d*x)^ (3/2))/d^5 + (8*a*b^3*(d*x)^(7/2))/(7*d^7) + (2*b^4*(d*x)^(11/2))/(11*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 = 0.18 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.60
method | result | size |
gosper | \(-\frac {2 \left (-35 b^{4} x^{8}-220 a \,b^{3} x^{6}-770 a^{2} b^{2} x^{4}+1540 a^{3} b \,x^{2}+77 a^{4}\right ) x}{385 \left (d x \right )^{\frac {7}{2}}}\) | \(52\) |
pseudoelliptic | \(-\frac {2 \left (-\frac {5}{11} b^{4} x^{8}-\frac {20}{7} a \,b^{3} x^{6}-10 a^{2} b^{2} x^{4}+20 a^{3} b \,x^{2}+a^{4}\right )}{5 \sqrt {d x}\, d^{3} x^{2}}\) | \(55\) |
trager | \(-\frac {2 \left (-35 b^{4} x^{8}-220 a \,b^{3} x^{6}-770 a^{2} b^{2} x^{4}+1540 a^{3} b \,x^{2}+77 a^{4}\right ) \sqrt {d x}}{385 d^{4} x^{3}}\) | \(57\) |
risch | \(-\frac {2 \left (-35 b^{4} x^{8}-220 a \,b^{3} x^{6}-770 a^{2} b^{2} x^{4}+1540 a^{3} b \,x^{2}+77 a^{4}\right )}{385 d^{3} x^{2} \sqrt {d x}}\) | \(57\) |
derivativedivides | \(\frac {\frac {2 b^{4} \left (d x \right )^{\frac {11}{2}}}{11}+\frac {8 a \,b^{3} d^{2} \left (d x \right )^{\frac {7}{2}}}{7}+4 a^{2} b^{2} d^{4} \left (d x \right )^{\frac {3}{2}}-\frac {8 a^{3} b \,d^{6}}{\sqrt {d x}}-\frac {2 a^{4} d^{8}}{5 \left (d x \right )^{\frac {5}{2}}}}{d^{9}}\) | \(74\) |
default | \(\frac {\frac {2 b^{4} \left (d x \right )^{\frac {11}{2}}}{11}+\frac {8 a \,b^{3} d^{2} \left (d x \right )^{\frac {7}{2}}}{7}+4 a^{2} b^{2} d^{4} \left (d x \right )^{\frac {3}{2}}-\frac {8 a^{3} b \,d^{6}}{\sqrt {d x}}-\frac {2 a^{4} d^{8}}{5 \left (d x \right )^{\frac {5}{2}}}}{d^{9}}\) | \(74\) |
orering | \(-\frac {2 \left (-35 b^{4} x^{8}-220 a \,b^{3} x^{6}-770 a^{2} b^{2} x^{4}+1540 a^{3} b \,x^{2}+77 a^{4}\right ) x \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{2}}{385 \left (b \,x^{2}+a \right )^{4} \left (d x \right )^{\frac {7}{2}}}\) | \(81\) |
Input:
int((b^2*x^4+2*a*b*x^2+a^2)^2/(d*x)^(7/2),x,method=_RETURNVERBOSE)
Output:
-2/385*(-35*b^4*x^8-220*a*b^3*x^6-770*a^2*b^2*x^4+1540*a^3*b*x^2+77*a^4)*x /(d*x)^(7/2)
Time = 0.09 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.64 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{(d x)^{7/2}} \, dx=\frac {2 \, {\left (35 \, b^{4} x^{8} + 220 \, a b^{3} x^{6} + 770 \, a^{2} b^{2} x^{4} - 1540 \, a^{3} b x^{2} - 77 \, a^{4}\right )} \sqrt {d x}}{385 \, d^{4} x^{3}} \] Input:
integrate((b^2*x^4+2*a*b*x^2+a^2)^2/(d*x)^(7/2),x, algorithm="fricas")
Output:
2/385*(35*b^4*x^8 + 220*a*b^3*x^6 + 770*a^2*b^2*x^4 - 1540*a^3*b*x^2 - 77* a^4)*sqrt(d*x)/(d^4*x^3)
Time = 0.37 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{(d x)^{7/2}} \, dx=- \frac {2 a^{4} x}{5 \left (d x\right )^{\frac {7}{2}}} - \frac {8 a^{3} b x^{3}}{\left (d x\right )^{\frac {7}{2}}} + \frac {4 a^{2} b^{2} x^{5}}{\left (d x\right )^{\frac {7}{2}}} + \frac {8 a b^{3} x^{7}}{7 \left (d x\right )^{\frac {7}{2}}} + \frac {2 b^{4} x^{9}}{11 \left (d x\right )^{\frac {7}{2}}} \] Input:
integrate((b**2*x**4+2*a*b*x**2+a**2)**2/(d*x)**(7/2),x)
Output:
-2*a**4*x/(5*(d*x)**(7/2)) - 8*a**3*b*x**3/(d*x)**(7/2) + 4*a**2*b**2*x**5 /(d*x)**(7/2) + 8*a*b**3*x**7/(7*(d*x)**(7/2)) + 2*b**4*x**9/(11*(d*x)**(7 /2))
Time = 0.03 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{(d x)^{7/2}} \, dx=-\frac {2 \, {\left (\frac {77 \, {\left (20 \, a^{3} b d^{2} x^{2} + a^{4} d^{2}\right )}}{\left (d x\right )^{\frac {5}{2}} d^{2}} - \frac {5 \, {\left (7 \, \left (d x\right )^{\frac {11}{2}} b^{4} + 44 \, \left (d x\right )^{\frac {7}{2}} a b^{3} d^{2} + 154 \, \left (d x\right )^{\frac {3}{2}} a^{2} b^{2} d^{4}\right )}}{d^{8}}\right )}}{385 \, d} \] Input:
integrate((b^2*x^4+2*a*b*x^2+a^2)^2/(d*x)^(7/2),x, algorithm="maxima")
Output:
-2/385*(77*(20*a^3*b*d^2*x^2 + a^4*d^2)/((d*x)^(5/2)*d^2) - 5*(7*(d*x)^(11 /2)*b^4 + 44*(d*x)^(7/2)*a*b^3*d^2 + 154*(d*x)^(3/2)*a^2*b^2*d^4)/d^8)/d
Time = 0.15 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{(d x)^{7/2}} \, dx=-\frac {2 \, {\left (\frac {77 \, {\left (20 \, a^{3} b d^{3} x^{2} + a^{4} d^{3}\right )}}{\sqrt {d x} d^{2} x^{2}} - \frac {5 \, {\left (7 \, \sqrt {d x} b^{4} d^{55} x^{5} + 44 \, \sqrt {d x} a b^{3} d^{55} x^{3} + 154 \, \sqrt {d x} a^{2} b^{2} d^{55} x\right )}}{d^{55}}\right )}}{385 \, d^{4}} \] Input:
integrate((b^2*x^4+2*a*b*x^2+a^2)^2/(d*x)^(7/2),x, algorithm="giac")
Output:
-2/385*(77*(20*a^3*b*d^3*x^2 + a^4*d^3)/(sqrt(d*x)*d^2*x^2) - 5*(7*sqrt(d* x)*b^4*d^55*x^5 + 44*sqrt(d*x)*a*b^3*d^55*x^3 + 154*sqrt(d*x)*a^2*b^2*d^55 *x)/d^55)/d^4
Time = 0.06 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{(d x)^{7/2}} \, dx=\frac {2\,b^4\,{\left (d\,x\right )}^{11/2}}{11\,d^9}-\frac {\frac {2\,a^4\,d^2}{5}+8\,b\,a^3\,d^2\,x^2}{d^3\,{\left (d\,x\right )}^{5/2}}+\frac {4\,a^2\,b^2\,{\left (d\,x\right )}^{3/2}}{d^5}+\frac {8\,a\,b^3\,{\left (d\,x\right )}^{7/2}}{7\,d^7} \] Input:
int((a^2 + b^2*x^4 + 2*a*b*x^2)^2/(d*x)^(7/2),x)
Output:
(2*b^4*(d*x)^(11/2))/(11*d^9) - ((2*a^4*d^2)/5 + 8*a^3*b*d^2*x^2)/(d^3*(d* x)^(5/2)) + (4*a^2*b^2*(d*x)^(3/2))/d^5 + (8*a*b^3*(d*x)^(7/2))/(7*d^7)
Time = 0.16 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.66 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{(d x)^{7/2}} \, dx=\frac {2 \sqrt {d}\, \left (35 b^{4} x^{8}+220 a \,b^{3} x^{6}+770 a^{2} b^{2} x^{4}-1540 a^{3} b \,x^{2}-77 a^{4}\right )}{385 \sqrt {x}\, d^{4} x^{2}} \] Input:
int((b^2*x^4+2*a*b*x^2+a^2)^2/(d*x)^(7/2),x)
Output:
(2*sqrt(d)*( - 77*a**4 - 1540*a**3*b*x**2 + 770*a**2*b**2*x**4 + 220*a*b** 3*x**6 + 35*b**4*x**8))/(385*sqrt(x)*d**4*x**2)