Integrand size = 30, antiderivative size = 195 \[ \int (d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx=\frac {2 a^3 (d x)^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 d \left (a+b x^2\right )}+\frac {6 a^2 b (d x)^{11/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{11 d^3 \left (a+b x^2\right )}+\frac {2 a b^2 (d x)^{15/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{5 d^5 \left (a+b x^2\right )}+\frac {2 b^3 (d x)^{19/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{19 d^7 \left (a+b x^2\right )} \] Output:
2/7*a^3*(d*x)^(7/2)*((b*x^2+a)^2)^(1/2)/d/(b*x^2+a)+6/11*a^2*b*(d*x)^(11/2 )*((b*x^2+a)^2)^(1/2)/d^3/(b*x^2+a)+2/5*a*b^2*(d*x)^(15/2)*((b*x^2+a)^2)^( 1/2)/d^5/(b*x^2+a)+2/19*b^3*(d*x)^(19/2)*((b*x^2+a)^2)^(1/2)/d^7/(b*x^2+a)
Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.34 \[ \int (d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx=\frac {2 x (d x)^{5/2} \sqrt {\left (a+b x^2\right )^2} \left (1045 a^3+1995 a^2 b x^2+1463 a b^2 x^4+385 b^3 x^6\right )}{7315 \left (a+b x^2\right )} \] Input:
Integrate[(d*x)^(5/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]
Output:
(2*x*(d*x)^(5/2)*Sqrt[(a + b*x^2)^2]*(1045*a^3 + 1995*a^2*b*x^2 + 1463*a*b ^2*x^4 + 385*b^3*x^6))/(7315*(a + b*x^2))
Time = 0.40 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.53, 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.
\(\displaystyle \int (d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 1384 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int b^3 (d x)^{5/2} \left (b x^2+a\right )^3dx}{b^3 \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int (d x)^{5/2} \left (b x^2+a\right )^3dx}{a+b x^2}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \left (\frac {b^3 (d x)^{17/2}}{d^6}+\frac {3 a b^2 (d x)^{13/2}}{d^4}+\frac {3 a^2 b (d x)^{9/2}}{d^2}+a^3 (d x)^{5/2}\right )dx}{a+b x^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \left (\frac {2 a^3 (d x)^{7/2}}{7 d}+\frac {6 a^2 b (d x)^{11/2}}{11 d^3}+\frac {2 a b^2 (d x)^{15/2}}{5 d^5}+\frac {2 b^3 (d x)^{19/2}}{19 d^7}\right )}{a+b x^2}\) |
Input:
Int[(d*x)^(5/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]
Output:
(Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*((2*a^3*(d*x)^(7/2))/(7*d) + (6*a^2*b*(d* x)^(11/2))/(11*d^3) + (2*a*b^2*(d*x)^(15/2))/(5*d^5) + (2*b^3*(d*x)^(19/2) )/(19*d^7)))/(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 = 0.12 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.31
method | result | size |
gosper | \(\frac {2 x \left (385 b^{3} x^{6}+1463 b^{2} x^{4} a +1995 a^{2} b \,x^{2}+1045 a^{3}\right ) \left (d x \right )^{\frac {5}{2}} {\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {3}{2}}}{7315 \left (b \,x^{2}+a \right )^{3}}\) | \(61\) |
default | \(\frac {2 {\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {3}{2}} \left (d x \right )^{\frac {7}{2}} \left (385 b^{3} x^{6}+1463 b^{2} x^{4} a +1995 a^{2} b \,x^{2}+1045 a^{3}\right )}{7315 d \left (b \,x^{2}+a \right )^{3}}\) | \(63\) |
risch | \(\frac {2 d^{3} \sqrt {\left (b \,x^{2}+a \right )^{2}}\, x^{4} \left (385 b^{3} x^{6}+1463 b^{2} x^{4} a +1995 a^{2} b \,x^{2}+1045 a^{3}\right )}{7315 \left (b \,x^{2}+a \right ) \sqrt {d x}}\) | \(66\) |
orering | \(\frac {2 x \left (385 b^{3} x^{6}+1463 b^{2} x^{4} a +1995 a^{2} b \,x^{2}+1045 a^{3}\right ) \left (d x \right )^{\frac {5}{2}} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{\frac {3}{2}}}{7315 \left (b \,x^{2}+a \right )^{3}}\) | \(70\) |
Input:
int((d*x)^(5/2)*(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
2/7315*x*(385*b^3*x^6+1463*a*b^2*x^4+1995*a^2*b*x^2+1045*a^3)*(d*x)^(5/2)* ((b*x^2+a)^2)^(3/2)/(b*x^2+a)^3
Time = 0.06 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.28 \[ \int (d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx=\frac {2}{7315} \, {\left (385 \, b^{3} d^{2} x^{9} + 1463 \, a b^{2} d^{2} x^{7} + 1995 \, a^{2} b d^{2} x^{5} + 1045 \, a^{3} d^{2} x^{3}\right )} \sqrt {d x} \] Input:
integrate((d*x)^(5/2)*(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x, algorithm="fricas")
Output:
2/7315*(385*b^3*d^2*x^9 + 1463*a*b^2*d^2*x^7 + 1995*a^2*b*d^2*x^5 + 1045*a ^3*d^2*x^3)*sqrt(d*x)
\[ \int (d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx=\int \left (d x\right )^{\frac {5}{2}} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}\, dx \] Input:
integrate((d*x)**(5/2)*(b**2*x**4+2*a*b*x**2+a**2)**(3/2),x)
Output:
Integral((d*x)**(5/2)*((a + b*x**2)**2)**(3/2), x)
Time = 0.04 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.43 \[ \int (d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx=\frac {2}{285} \, {\left (15 \, b^{3} d^{\frac {5}{2}} x^{3} + 19 \, a b^{2} d^{\frac {5}{2}} x\right )} x^{\frac {13}{2}} + \frac {4}{165} \, {\left (11 \, a b^{2} d^{\frac {5}{2}} x^{3} + 15 \, a^{2} b d^{\frac {5}{2}} x\right )} x^{\frac {9}{2}} + \frac {2}{77} \, {\left (7 \, a^{2} b d^{\frac {5}{2}} x^{3} + 11 \, a^{3} d^{\frac {5}{2}} x\right )} x^{\frac {5}{2}} \] Input:
integrate((d*x)^(5/2)*(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x, algorithm="maxima")
Output:
2/285*(15*b^3*d^(5/2)*x^3 + 19*a*b^2*d^(5/2)*x)*x^(13/2) + 4/165*(11*a*b^2 *d^(5/2)*x^3 + 15*a^2*b*d^(5/2)*x)*x^(9/2) + 2/77*(7*a^2*b*d^(5/2)*x^3 + 1 1*a^3*d^(5/2)*x)*x^(5/2)
Time = 0.15 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.51 \[ \int (d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx=\frac {2}{19} \, \sqrt {d x} b^{3} d^{2} x^{9} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {2}{5} \, \sqrt {d x} a b^{2} d^{2} x^{7} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {6}{11} \, \sqrt {d x} a^{2} b d^{2} x^{5} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {2}{7} \, \sqrt {d x} a^{3} d^{2} x^{3} \mathrm {sgn}\left (b x^{2} + a\right ) \] Input:
integrate((d*x)^(5/2)*(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x, algorithm="giac")
Output:
2/19*sqrt(d*x)*b^3*d^2*x^9*sgn(b*x^2 + a) + 2/5*sqrt(d*x)*a*b^2*d^2*x^7*sg n(b*x^2 + a) + 6/11*sqrt(d*x)*a^2*b*d^2*x^5*sgn(b*x^2 + a) + 2/7*sqrt(d*x) *a^3*d^2*x^3*sgn(b*x^2 + a)
Timed out. \[ \int (d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx=\int {\left (d\,x\right )}^{5/2}\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{3/2} \,d x \] Input:
int((d*x)^(5/2)*(a^2 + b^2*x^4 + 2*a*b*x^2)^(3/2),x)
Output:
int((d*x)^(5/2)*(a^2 + b^2*x^4 + 2*a*b*x^2)^(3/2), x)
Time = 0.17 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.23 \[ \int (d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx=\frac {2 \sqrt {x}\, \sqrt {d}\, d^{2} x^{3} \left (385 b^{3} x^{6}+1463 a \,b^{2} x^{4}+1995 a^{2} b \,x^{2}+1045 a^{3}\right )}{7315} \] Input:
int((d*x)^(5/2)*(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x)
Output:
(2*sqrt(x)*sqrt(d)*d**2*x**3*(1045*a**3 + 1995*a**2*b*x**2 + 1463*a*b**2*x **4 + 385*b**3*x**6))/7315