Integrand size = 30, antiderivative size = 293 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{\sqrt {d x}} \, dx=\frac {2 a^5 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d \left (a+b x^2\right )}+\frac {2 a^4 b (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^3 \left (a+b x^2\right )}+\frac {20 a^3 b^2 (d x)^{9/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 d^5 \left (a+b x^2\right )}+\frac {20 a^2 b^3 (d x)^{13/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{13 d^7 \left (a+b x^2\right )}+\frac {10 a b^4 (d x)^{17/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{17 d^9 \left (a+b x^2\right )}+\frac {2 b^5 (d x)^{21/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{21 d^{11} \left (a+b x^2\right )} \] Output:
2*a^5*(d*x)^(1/2)*((b*x^2+a)^2)^(1/2)/d/(b*x^2+a)+2*a^4*b*(d*x)^(5/2)*((b* x^2+a)^2)^(1/2)/d^3/(b*x^2+a)+20/9*a^3*b^2*(d*x)^(9/2)*((b*x^2+a)^2)^(1/2) /d^5/(b*x^2+a)+20/13*a^2*b^3*(d*x)^(13/2)*((b*x^2+a)^2)^(1/2)/d^7/(b*x^2+a )+10/17*a*b^4*(d*x)^(17/2)*((b*x^2+a)^2)^(1/2)/d^9/(b*x^2+a)+2/21*b^5*(d*x )^(21/2)*((b*x^2+a)^2)^(1/2)/d^11/(b*x^2+a)
Time = 0.03 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.30 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{\sqrt {d x}} \, dx=\frac {2 x \sqrt {\left (a+b x^2\right )^2} \left (13923 a^5+13923 a^4 b x^2+15470 a^3 b^2 x^4+10710 a^2 b^3 x^6+4095 a b^4 x^8+663 b^5 x^{10}\right )}{13923 \sqrt {d x} \left (a+b x^2\right )} \] Input:
Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/Sqrt[d*x],x]
Output:
(2*x*Sqrt[(a + b*x^2)^2]*(13923*a^5 + 13923*a^4*b*x^2 + 15470*a^3*b^2*x^4 + 10710*a^2*b^3*x^6 + 4095*a*b^4*x^8 + 663*b^5*x^10))/(13923*Sqrt[d*x]*(a + b*x^2))
Time = 0.45 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.47, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1384, 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.
\(\displaystyle \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{\sqrt {d x}} \, dx\) |
\(\Big \downarrow \) 1384 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \frac {b^5 \left (b x^2+a\right )^5}{\sqrt {d x}}dx}{b^5 \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \frac {\left (b x^2+a\right )^5}{\sqrt {d x}}dx}{a+b x^2}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \left (\frac {b^5 (d x)^{19/2}}{d^{10}}+\frac {5 a b^4 (d x)^{15/2}}{d^8}+\frac {10 a^2 b^3 (d x)^{11/2}}{d^6}+\frac {10 a^3 b^2 (d x)^{7/2}}{d^4}+\frac {5 a^4 b (d x)^{3/2}}{d^2}+\frac {a^5}{\sqrt {d x}}\right )dx}{a+b x^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \left (\frac {2 a^5 \sqrt {d x}}{d}+\frac {2 a^4 b (d x)^{5/2}}{d^3}+\frac {20 a^3 b^2 (d x)^{9/2}}{9 d^5}+\frac {20 a^2 b^3 (d x)^{13/2}}{13 d^7}+\frac {10 a b^4 (d x)^{17/2}}{17 d^9}+\frac {2 b^5 (d x)^{21/2}}{21 d^{11}}\right )}{a+b x^2}\) |
Input:
Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/Sqrt[d*x],x]
Output:
(Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*((2*a^5*Sqrt[d*x])/d + (2*a^4*b*(d*x)^(5/ 2))/d^3 + (20*a^3*b^2*(d*x)^(9/2))/(9*d^5) + (20*a^2*b^3*(d*x)^(13/2))/(13 *d^7) + (10*a*b^4*(d*x)^(17/2))/(17*d^9) + (2*b^5*(d*x)^(21/2))/(21*d^11)) )/(a + b*x^2)
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[(a + b*x^n + c*x^(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*Frac Part[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 - 1/2] && NeQ[u, x^(n - 1)] && NeQ[u, x^(2*n - 1)] && !(EqQ[p, 1/2] && EqQ[u, x^(-2*n - 1)])
Time = 0.08 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.28
method | result | size |
gosper | \(\frac {2 x \left (663 x^{10} b^{5}+4095 a \,x^{8} b^{4}+10710 a^{2} x^{6} b^{3}+15470 a^{3} x^{4} b^{2}+13923 x^{2} a^{4} b +13923 a^{5}\right ) {\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {5}{2}}}{13923 \left (b \,x^{2}+a \right )^{5} \sqrt {d x}}\) | \(83\) |
risch | \(\frac {2 \sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (663 x^{10} b^{5}+4095 a \,x^{8} b^{4}+10710 a^{2} x^{6} b^{3}+15470 a^{3} x^{4} b^{2}+13923 x^{2} a^{4} b +13923 a^{5}\right ) x}{13923 \left (b \,x^{2}+a \right ) \sqrt {d x}}\) | \(83\) |
default | \(\frac {2 {\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {5}{2}} \sqrt {d x}\, \left (663 x^{10} b^{5}+4095 a \,x^{8} b^{4}+10710 a^{2} x^{6} b^{3}+15470 a^{3} x^{4} b^{2}+13923 x^{2} a^{4} b +13923 a^{5}\right )}{13923 d \left (b \,x^{2}+a \right )^{5}}\) | \(85\) |
orering | \(\frac {2 \left (663 x^{10} b^{5}+4095 a \,x^{8} b^{4}+10710 a^{2} x^{6} b^{3}+15470 a^{3} x^{4} b^{2}+13923 x^{2} a^{4} b +13923 a^{5}\right ) x \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{\frac {5}{2}}}{13923 \left (b \,x^{2}+a \right )^{5} \sqrt {d x}}\) | \(92\) |
Input:
int((b^2*x^4+2*a*b*x^2+a^2)^(5/2)/(d*x)^(1/2),x,method=_RETURNVERBOSE)
Output:
2/13923*x*(663*b^5*x^10+4095*a*b^4*x^8+10710*a^2*b^3*x^6+15470*a^3*b^2*x^4 +13923*a^4*b*x^2+13923*a^5)*((b*x^2+a)^2)^(5/2)/(b*x^2+a)^5/(d*x)^(1/2)
Time = 0.07 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.22 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{\sqrt {d x}} \, dx=\frac {2 \, {\left (663 \, b^{5} x^{10} + 4095 \, a b^{4} x^{8} + 10710 \, a^{2} b^{3} x^{6} + 15470 \, a^{3} b^{2} x^{4} + 13923 \, a^{4} b x^{2} + 13923 \, a^{5}\right )} \sqrt {d x}}{13923 \, d} \] Input:
integrate((b^2*x^4+2*a*b*x^2+a^2)^(5/2)/(d*x)^(1/2),x, algorithm="fricas")
Output:
2/13923*(663*b^5*x^10 + 4095*a*b^4*x^8 + 10710*a^2*b^3*x^6 + 15470*a^3*b^2 *x^4 + 13923*a^4*b*x^2 + 13923*a^5)*sqrt(d*x)/d
\[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{\sqrt {d x}} \, dx=\int \frac {\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}}{\sqrt {d x}}\, dx \] Input:
integrate((b**2*x**4+2*a*b*x**2+a**2)**(5/2)/(d*x)**(1/2),x)
Output:
Integral(((a + b*x**2)**2)**(5/2)/sqrt(d*x), x)
Time = 0.04 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.52 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{\sqrt {d x}} \, dx=\frac {2 \, {\left (195 \, {\left (17 \, b^{5} \sqrt {d} x^{3} + 21 \, a b^{4} \sqrt {d} x\right )} x^{\frac {15}{2}} + 1260 \, {\left (13 \, a b^{4} \sqrt {d} x^{3} + 17 \, a^{2} b^{3} \sqrt {d} x\right )} x^{\frac {11}{2}} + 3570 \, {\left (9 \, a^{2} b^{3} \sqrt {d} x^{3} + 13 \, a^{3} b^{2} \sqrt {d} x\right )} x^{\frac {7}{2}} + 6188 \, {\left (5 \, a^{3} b^{2} \sqrt {d} x^{3} + 9 \, a^{4} b \sqrt {d} x\right )} x^{\frac {3}{2}} + \frac {13923 \, {\left (a^{4} b \sqrt {d} x^{3} + 5 \, a^{5} \sqrt {d} x\right )}}{\sqrt {x}}\right )}}{69615 \, d} \] Input:
integrate((b^2*x^4+2*a*b*x^2+a^2)^(5/2)/(d*x)^(1/2),x, algorithm="maxima")
Output:
2/69615*(195*(17*b^5*sqrt(d)*x^3 + 21*a*b^4*sqrt(d)*x)*x^(15/2) + 1260*(13 *a*b^4*sqrt(d)*x^3 + 17*a^2*b^3*sqrt(d)*x)*x^(11/2) + 3570*(9*a^2*b^3*sqrt (d)*x^3 + 13*a^3*b^2*sqrt(d)*x)*x^(7/2) + 6188*(5*a^3*b^2*sqrt(d)*x^3 + 9* a^4*b*sqrt(d)*x)*x^(3/2) + 13923*(a^4*b*sqrt(d)*x^3 + 5*a^5*sqrt(d)*x)/sqr t(x))/d
Time = 0.15 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.47 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{\sqrt {d x}} \, dx=\frac {2 \, {\left (663 \, \sqrt {d x} b^{5} x^{10} \mathrm {sgn}\left (b x^{2} + a\right ) + 4095 \, \sqrt {d x} a b^{4} x^{8} \mathrm {sgn}\left (b x^{2} + a\right ) + 10710 \, \sqrt {d x} a^{2} b^{3} x^{6} \mathrm {sgn}\left (b x^{2} + a\right ) + 15470 \, \sqrt {d x} a^{3} b^{2} x^{4} \mathrm {sgn}\left (b x^{2} + a\right ) + 13923 \, \sqrt {d x} a^{4} b x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 13923 \, \sqrt {d x} a^{5} \mathrm {sgn}\left (b x^{2} + a\right )\right )}}{13923 \, d} \] Input:
integrate((b^2*x^4+2*a*b*x^2+a^2)^(5/2)/(d*x)^(1/2),x, algorithm="giac")
Output:
2/13923*(663*sqrt(d*x)*b^5*x^10*sgn(b*x^2 + a) + 4095*sqrt(d*x)*a*b^4*x^8* sgn(b*x^2 + a) + 10710*sqrt(d*x)*a^2*b^3*x^6*sgn(b*x^2 + a) + 15470*sqrt(d *x)*a^3*b^2*x^4*sgn(b*x^2 + a) + 13923*sqrt(d*x)*a^4*b*x^2*sgn(b*x^2 + a) + 13923*sqrt(d*x)*a^5*sgn(b*x^2 + a))/d
Time = 17.95 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.38 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{\sqrt {d x}} \, dx=\frac {2\,x\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}\,\left (5731\,a^4+8192\,a^3\,b\,x^2+7278\,a^2\,b^2\,x^4+3432\,a\,b^3\,x^6+663\,b^4\,x^8\right )}{13923\,\sqrt {d\,x}}+\frac {16384\,a^5\,x\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{13923\,\sqrt {d\,x}\,\left (b\,x^2+a\right )} \] Input:
int((a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2)/(d*x)^(1/2),x)
Output:
(2*x*(a^2 + b^2*x^4 + 2*a*b*x^2)^(1/2)*(5731*a^4 + 663*b^4*x^8 + 8192*a^3* b*x^2 + 3432*a*b^3*x^6 + 7278*a^2*b^2*x^4))/(13923*(d*x)^(1/2)) + (16384*a ^5*x*(a^2 + b^2*x^4 + 2*a*b*x^2)^(1/2))/(13923*(d*x)^(1/2)*(a + b*x^2))
Time = 0.16 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.22 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{\sqrt {d x}} \, dx=\frac {2 \sqrt {x}\, \sqrt {d}\, \left (663 b^{5} x^{10}+4095 a \,b^{4} x^{8}+10710 a^{2} b^{3} x^{6}+15470 a^{3} b^{2} x^{4}+13923 a^{4} b \,x^{2}+13923 a^{5}\right )}{13923 d} \] Input:
int((b^2*x^4+2*a*b*x^2+a^2)^(5/2)/(d*x)^(1/2),x)
Output:
(2*sqrt(x)*sqrt(d)*(13923*a**5 + 13923*a**4*b*x**2 + 15470*a**3*b**2*x**4 + 10710*a**2*b**3*x**6 + 4095*a*b**4*x**8 + 663*b**5*x**10))/(13923*d)