Integrand size = 24, antiderivative size = 82 \[ \int \frac {\left (a+b x^2\right )^2}{x^4 \left (c+d x^2\right )^{3/2}} \, dx=\frac {(b c-a d)^2 x}{c^3 \sqrt {c+d x^2}}-\frac {a^2 \sqrt {c+d x^2}}{3 c^2 x^3}-\frac {a (6 b c-5 a d) \sqrt {c+d x^2}}{3 c^3 x} \] Output:
(-a*d+b*c)^2*x/c^3/(d*x^2+c)^(1/2)-1/3*a^2*(d*x^2+c)^(1/2)/c^2/x^3-1/3*a*( -5*a*d+6*b*c)*(d*x^2+c)^(1/2)/c^3/x
Time = 0.12 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b x^2\right )^2}{x^4 \left (c+d x^2\right )^{3/2}} \, dx=\frac {3 b^2 c^2 x^4-6 a b c x^2 \left (c+2 d x^2\right )+a^2 \left (-c^2+4 c d x^2+8 d^2 x^4\right )}{3 c^3 x^3 \sqrt {c+d x^2}} \] Input:
Integrate[(a + b*x^2)^2/(x^4*(c + d*x^2)^(3/2)),x]
Output:
(3*b^2*c^2*x^4 - 6*a*b*c*x^2*(c + 2*d*x^2) + a^2*(-c^2 + 4*c*d*x^2 + 8*d^2 *x^4))/(3*c^3*x^3*Sqrt[c + d*x^2])
Time = 0.21 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.22, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {365, 359, 208}
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.
\(\displaystyle \int \frac {\left (a+b x^2\right )^2}{x^4 \left (c+d x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 365 |
\(\displaystyle \frac {\int \frac {3 b^2 c x^2+2 a (3 b c-2 a d)}{x^2 \left (d x^2+c\right )^{3/2}}dx}{3 c}-\frac {a^2}{3 c x^3 \sqrt {c+d x^2}}\) |
\(\Big \downarrow \) 359 |
\(\displaystyle \frac {\frac {\left (3 b^2 c^2-4 a d (3 b c-2 a d)\right ) \int \frac {1}{\left (d x^2+c\right )^{3/2}}dx}{c}-\frac {2 a (3 b c-2 a d)}{c x \sqrt {c+d x^2}}}{3 c}-\frac {a^2}{3 c x^3 \sqrt {c+d x^2}}\) |
\(\Big \downarrow \) 208 |
\(\displaystyle \frac {\frac {x \left (3 b^2 c^2-4 a d (3 b c-2 a d)\right )}{c^2 \sqrt {c+d x^2}}-\frac {2 a (3 b c-2 a d)}{c x \sqrt {c+d x^2}}}{3 c}-\frac {a^2}{3 c x^3 \sqrt {c+d x^2}}\) |
Input:
Int[(a + b*x^2)^2/(x^4*(c + d*x^2)^(3/2)),x]
Output:
-1/3*a^2/(c*x^3*Sqrt[c + d*x^2]) + ((-2*a*(3*b*c - 2*a*d))/(c*x*Sqrt[c + d *x^2]) + ((3*b^2*c^2 - 4*a*d*(3*b*c - 2*a*d))*x)/(c^2*Sqrt[c + d*x^2]))/(3 *c)
Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)* (a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !ILtQ[p, -1]
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^2, x _Symbol] :> Simp[c^2*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] - Simp[1/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)*(a + b*x^2)^p*Simp[2*b*c^2*(p + 1) + c*(b*c - 2*a*d)*(m + 1) - a*d^2*(m + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1]
Time = 0.50 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.84
method | result | size |
pseudoelliptic | \(-\frac {\left (-3 b^{2} x^{4}+6 a b \,x^{2}+a^{2}\right ) c^{2}-4 a d \,x^{2} \left (-3 b \,x^{2}+a \right ) c -8 a^{2} d^{2} x^{4}}{3 \sqrt {x^{2} d +c}\, x^{3} c^{3}}\) | \(69\) |
risch | \(-\frac {\sqrt {x^{2} d +c}\, a \left (-5 a d \,x^{2}+6 x^{2} b c +a c \right )}{3 c^{3} x^{3}}+\frac {x \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{\sqrt {x^{2} d +c}\, c^{3}}\) | \(73\) |
gosper | \(-\frac {-8 a^{2} d^{2} x^{4}+12 a b c d \,x^{4}-3 b^{2} c^{2} x^{4}-4 a^{2} c d \,x^{2}+6 a b \,c^{2} x^{2}+a^{2} c^{2}}{3 x^{3} \sqrt {x^{2} d +c}\, c^{3}}\) | \(77\) |
trager | \(-\frac {-8 a^{2} d^{2} x^{4}+12 a b c d \,x^{4}-3 b^{2} c^{2} x^{4}-4 a^{2} c d \,x^{2}+6 a b \,c^{2} x^{2}+a^{2} c^{2}}{3 x^{3} \sqrt {x^{2} d +c}\, c^{3}}\) | \(77\) |
orering | \(-\frac {-8 a^{2} d^{2} x^{4}+12 a b c d \,x^{4}-3 b^{2} c^{2} x^{4}-4 a^{2} c d \,x^{2}+6 a b \,c^{2} x^{2}+a^{2} c^{2}}{3 x^{3} \sqrt {x^{2} d +c}\, c^{3}}\) | \(77\) |
default | \(\frac {b^{2} x}{c \sqrt {x^{2} d +c}}+a^{2} \left (-\frac {1}{3 c \,x^{3} \sqrt {x^{2} d +c}}-\frac {4 d \left (-\frac {1}{c x \sqrt {x^{2} d +c}}-\frac {2 d x}{c^{2} \sqrt {x^{2} d +c}}\right )}{3 c}\right )+2 a b \left (-\frac {1}{c x \sqrt {x^{2} d +c}}-\frac {2 d x}{c^{2} \sqrt {x^{2} d +c}}\right )\) | \(119\) |
Input:
int((b*x^2+a)^2/x^4/(d*x^2+c)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/3/(d*x^2+c)^(1/2)*((-3*b^2*x^4+6*a*b*x^2+a^2)*c^2-4*a*d*x^2*(-3*b*x^2+a )*c-8*a^2*d^2*x^4)/x^3/c^3
Time = 0.09 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.04 \[ \int \frac {\left (a+b x^2\right )^2}{x^4 \left (c+d x^2\right )^{3/2}} \, dx=\frac {{\left ({\left (3 \, b^{2} c^{2} - 12 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} - a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 2 \, a^{2} c d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{3 \, {\left (c^{3} d x^{5} + c^{4} x^{3}\right )}} \] Input:
integrate((b*x^2+a)^2/x^4/(d*x^2+c)^(3/2),x, algorithm="fricas")
Output:
1/3*((3*b^2*c^2 - 12*a*b*c*d + 8*a^2*d^2)*x^4 - a^2*c^2 - 2*(3*a*b*c^2 - 2 *a^2*c*d)*x^2)*sqrt(d*x^2 + c)/(c^3*d*x^5 + c^4*x^3)
\[ \int \frac {\left (a+b x^2\right )^2}{x^4 \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {\left (a + b x^{2}\right )^{2}}{x^{4} \left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((b*x**2+a)**2/x**4/(d*x**2+c)**(3/2),x)
Output:
Integral((a + b*x**2)**2/(x**4*(c + d*x**2)**(3/2)), x)
Time = 0.04 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.43 \[ \int \frac {\left (a+b x^2\right )^2}{x^4 \left (c+d x^2\right )^{3/2}} \, dx=\frac {b^{2} x}{\sqrt {d x^{2} + c} c} - \frac {4 \, a b d x}{\sqrt {d x^{2} + c} c^{2}} + \frac {8 \, a^{2} d^{2} x}{3 \, \sqrt {d x^{2} + c} c^{3}} - \frac {2 \, a b}{\sqrt {d x^{2} + c} c x} + \frac {4 \, a^{2} d}{3 \, \sqrt {d x^{2} + c} c^{2} x} - \frac {a^{2}}{3 \, \sqrt {d x^{2} + c} c x^{3}} \] Input:
integrate((b*x^2+a)^2/x^4/(d*x^2+c)^(3/2),x, algorithm="maxima")
Output:
b^2*x/(sqrt(d*x^2 + c)*c) - 4*a*b*d*x/(sqrt(d*x^2 + c)*c^2) + 8/3*a^2*d^2* x/(sqrt(d*x^2 + c)*c^3) - 2*a*b/(sqrt(d*x^2 + c)*c*x) + 4/3*a^2*d/(sqrt(d* x^2 + c)*c^2*x) - 1/3*a^2/(sqrt(d*x^2 + c)*c*x^3)
Leaf count of result is larger than twice the leaf count of optimal. 199 vs. \(2 (72) = 144\).
Time = 0.14 (sec) , antiderivative size = 199, normalized size of antiderivative = 2.43 \[ \int \frac {\left (a+b x^2\right )^2}{x^4 \left (c+d x^2\right )^{3/2}} \, dx=\frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x}{\sqrt {d x^{2} + c} c^{3}} + \frac {2 \, {\left (6 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a b c \sqrt {d} - 3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a^{2} d^{\frac {3}{2}} - 12 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b c^{2} \sqrt {d} + 12 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{2} c d^{\frac {3}{2}} + 6 \, a b c^{3} \sqrt {d} - 5 \, a^{2} c^{2} d^{\frac {3}{2}}\right )}}{3 \, {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c\right )}^{3} c^{2}} \] Input:
integrate((b*x^2+a)^2/x^4/(d*x^2+c)^(3/2),x, algorithm="giac")
Output:
(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x/(sqrt(d*x^2 + c)*c^3) + 2/3*(6*(sqrt(d)* x - sqrt(d*x^2 + c))^4*a*b*c*sqrt(d) - 3*(sqrt(d)*x - sqrt(d*x^2 + c))^4*a ^2*d^(3/2) - 12*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a*b*c^2*sqrt(d) + 12*(sqrt (d)*x - sqrt(d*x^2 + c))^2*a^2*c*d^(3/2) + 6*a*b*c^3*sqrt(d) - 5*a^2*c^2*d ^(3/2))/(((sqrt(d)*x - sqrt(d*x^2 + c))^2 - c)^3*c^2)
Time = 1.13 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+b x^2\right )^2}{x^4 \left (c+d x^2\right )^{3/2}} \, dx=-\frac {a^2\,c^2-4\,a^2\,c\,d\,x^2-8\,a^2\,d^2\,x^4+6\,a\,b\,c^2\,x^2+12\,a\,b\,c\,d\,x^4-3\,b^2\,c^2\,x^4}{3\,c^3\,x^3\,\sqrt {d\,x^2+c}} \] Input:
int((a + b*x^2)^2/(x^4*(c + d*x^2)^(3/2)),x)
Output:
-(a^2*c^2 - 8*a^2*d^2*x^4 - 3*b^2*c^2*x^4 + 6*a*b*c^2*x^2 - 4*a^2*c*d*x^2 + 12*a*b*c*d*x^4)/(3*c^3*x^3*(c + d*x^2)^(1/2))
Time = 0.21 (sec) , antiderivative size = 215, normalized size of antiderivative = 2.62 \[ \int \frac {\left (a+b x^2\right )^2}{x^4 \left (c+d x^2\right )^{3/2}} \, dx=\frac {-\sqrt {d \,x^{2}+c}\, a^{2} c^{2} d +4 \sqrt {d \,x^{2}+c}\, a^{2} c \,d^{2} x^{2}+8 \sqrt {d \,x^{2}+c}\, a^{2} d^{3} x^{4}-6 \sqrt {d \,x^{2}+c}\, a b \,c^{2} d \,x^{2}-12 \sqrt {d \,x^{2}+c}\, a b c \,d^{2} x^{4}+3 \sqrt {d \,x^{2}+c}\, b^{2} c^{2} d \,x^{4}-8 \sqrt {d}\, a^{2} c \,d^{2} x^{3}-8 \sqrt {d}\, a^{2} d^{3} x^{5}+12 \sqrt {d}\, a b \,c^{2} d \,x^{3}+12 \sqrt {d}\, a b c \,d^{2} x^{5}-6 \sqrt {d}\, b^{2} c^{3} x^{3}-6 \sqrt {d}\, b^{2} c^{2} d \,x^{5}}{3 c^{3} d \,x^{3} \left (d \,x^{2}+c \right )} \] Input:
int((b*x^2+a)^2/x^4/(d*x^2+c)^(3/2),x)
Output:
( - sqrt(c + d*x**2)*a**2*c**2*d + 4*sqrt(c + d*x**2)*a**2*c*d**2*x**2 + 8 *sqrt(c + d*x**2)*a**2*d**3*x**4 - 6*sqrt(c + d*x**2)*a*b*c**2*d*x**2 - 12 *sqrt(c + d*x**2)*a*b*c*d**2*x**4 + 3*sqrt(c + d*x**2)*b**2*c**2*d*x**4 - 8*sqrt(d)*a**2*c*d**2*x**3 - 8*sqrt(d)*a**2*d**3*x**5 + 12*sqrt(d)*a*b*c** 2*d*x**3 + 12*sqrt(d)*a*b*c*d**2*x**5 - 6*sqrt(d)*b**2*c**3*x**3 - 6*sqrt( d)*b**2*c**2*d*x**5)/(3*c**3*d*x**3*(c + d*x**2))