Integrand size = 24, antiderivative size = 81 \[ \int \frac {\left (a+b x^2\right )^2}{x^2 \left (c+d x^2\right )^{3/2}} \, dx=-\frac {(b c-a d)^2 x}{c^2 d \sqrt {c+d x^2}}-\frac {a^2 \sqrt {c+d x^2}}{c^2 x}+\frac {b^2 \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{d^{3/2}} \] Output:
-(-a*d+b*c)^2*x/c^2/d/(d*x^2+c)^(1/2)-a^2*(d*x^2+c)^(1/2)/c^2/x+b^2*arctan h(d^(1/2)*x/(d*x^2+c)^(1/2))/d^(3/2)
Time = 0.14 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a+b x^2\right )^2}{x^2 \left (c+d x^2\right )^{3/2}} \, dx=\frac {-b^2 c^2 x^2+2 a b c d x^2-a^2 d \left (c+2 d x^2\right )}{c^2 d x \sqrt {c+d x^2}}-\frac {b^2 \log \left (-\sqrt {d} x+\sqrt {c+d x^2}\right )}{d^{3/2}} \] Input:
Integrate[(a + b*x^2)^2/(x^2*(c + d*x^2)^(3/2)),x]
Output:
(-(b^2*c^2*x^2) + 2*a*b*c*d*x^2 - a^2*d*(c + 2*d*x^2))/(c^2*d*x*Sqrt[c + d *x^2]) - (b^2*Log[-(Sqrt[d]*x) + Sqrt[c + d*x^2]])/d^(3/2)
Time = 0.22 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.16, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {365, 298, 224, 219}
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.
\(\displaystyle \int \frac {\left (a+b x^2\right )^2}{x^2 \left (c+d x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 365 |
\(\displaystyle \frac {\int \frac {b^2 c x^2+2 a (b c-a d)}{\left (d x^2+c\right )^{3/2}}dx}{c}-\frac {a^2}{c x \sqrt {c+d x^2}}\) |
\(\Big \downarrow \) 298 |
\(\displaystyle \frac {\frac {b^2 c \int \frac {1}{\sqrt {d x^2+c}}dx}{d}-\frac {x \left (\frac {b^2 c}{d}-\frac {2 a (b c-a d)}{c}\right )}{\sqrt {c+d x^2}}}{c}-\frac {a^2}{c x \sqrt {c+d x^2}}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {\frac {b^2 c \int \frac {1}{1-\frac {d x^2}{d x^2+c}}d\frac {x}{\sqrt {d x^2+c}}}{d}-\frac {x \left (\frac {b^2 c}{d}-\frac {2 a (b c-a d)}{c}\right )}{\sqrt {c+d x^2}}}{c}-\frac {a^2}{c x \sqrt {c+d x^2}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\frac {b^2 c \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{d^{3/2}}-\frac {x \left (\frac {b^2 c}{d}-\frac {2 a (b c-a d)}{c}\right )}{\sqrt {c+d x^2}}}{c}-\frac {a^2}{c x \sqrt {c+d x^2}}\) |
Input:
Int[(a + b*x^2)^2/(x^2*(c + d*x^2)^(3/2)),x]
Output:
-(a^2/(c*x*Sqrt[c + d*x^2])) + (-((((b^2*c)/d - (2*a*(b*c - a*d))/c)*x)/Sq rt[c + d*x^2]) + (b^2*c*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/d^(3/2))/c
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-( b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 2*p + 3))/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 0])
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 = 0.51 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.07
method | result | size |
pseudoelliptic | \(\frac {-\frac {\sqrt {x^{2} d +c}\, a^{2}}{x}+\frac {\left (-a^{2} d^{2}+2 a b c d -b^{2} c^{2}\right ) x}{d \sqrt {x^{2} d +c}}+\frac {b^{2} c^{2} \operatorname {arctanh}\left (\frac {\sqrt {x^{2} d +c}}{x \sqrt {d}}\right )}{d^{\frac {3}{2}}}}{c^{2}}\) | \(87\) |
default | \(a^{2} \left (-\frac {1}{c x \sqrt {x^{2} d +c}}-\frac {2 d x}{c^{2} \sqrt {x^{2} d +c}}\right )+b^{2} \left (-\frac {x}{d \sqrt {x^{2} d +c}}+\frac {\ln \left (\sqrt {d}\, x +\sqrt {x^{2} d +c}\right )}{d^{\frac {3}{2}}}\right )+\frac {2 a b x}{c \sqrt {x^{2} d +c}}\) | \(97\) |
risch | \(-\frac {a^{2} \sqrt {x^{2} d +c}}{c^{2} x}-\frac {b^{2} x}{d \sqrt {x^{2} d +c}}+\frac {b^{2} \ln \left (\sqrt {d}\, x +\sqrt {x^{2} d +c}\right )}{d^{\frac {3}{2}}}-\frac {d \,a^{2} x}{c^{2} \sqrt {x^{2} d +c}}+\frac {2 a b x}{c \sqrt {x^{2} d +c}}\) | \(99\) |
Input:
int((b*x^2+a)^2/x^2/(d*x^2+c)^(3/2),x,method=_RETURNVERBOSE)
Output:
(-(d*x^2+c)^(1/2)/x*a^2+(-a^2*d^2+2*a*b*c*d-b^2*c^2)/d/(d*x^2+c)^(1/2)*x+b ^2*c^2/d^(3/2)*arctanh((d*x^2+c)^(1/2)/x/d^(1/2)))/c^2
Time = 0.09 (sec) , antiderivative size = 239, normalized size of antiderivative = 2.95 \[ \int \frac {\left (a+b x^2\right )^2}{x^2 \left (c+d x^2\right )^{3/2}} \, dx=\left [\frac {{\left (b^{2} c^{2} d x^{3} + b^{2} c^{3} x\right )} \sqrt {d} \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) - 2 \, {\left (a^{2} c d^{2} + {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + 2 \, a^{2} d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{2 \, {\left (c^{2} d^{3} x^{3} + c^{3} d^{2} x\right )}}, -\frac {{\left (b^{2} c^{2} d x^{3} + b^{2} c^{3} x\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) + {\left (a^{2} c d^{2} + {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + 2 \, a^{2} d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{c^{2} d^{3} x^{3} + c^{3} d^{2} x}\right ] \] Input:
integrate((b*x^2+a)^2/x^2/(d*x^2+c)^(3/2),x, algorithm="fricas")
Output:
[1/2*((b^2*c^2*d*x^3 + b^2*c^3*x)*sqrt(d)*log(-2*d*x^2 - 2*sqrt(d*x^2 + c) *sqrt(d)*x - c) - 2*(a^2*c*d^2 + (b^2*c^2*d - 2*a*b*c*d^2 + 2*a^2*d^3)*x^2 )*sqrt(d*x^2 + c))/(c^2*d^3*x^3 + c^3*d^2*x), -((b^2*c^2*d*x^3 + b^2*c^3*x )*sqrt(-d)*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)) + (a^2*c*d^2 + (b^2*c^2*d - 2*a*b*c*d^2 + 2*a^2*d^3)*x^2)*sqrt(d*x^2 + c))/(c^2*d^3*x^3 + c^3*d^2*x)]
\[ \int \frac {\left (a+b x^2\right )^2}{x^2 \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {\left (a + b x^{2}\right )^{2}}{x^{2} \left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((b*x**2+a)**2/x**2/(d*x**2+c)**(3/2),x)
Output:
Integral((a + b*x**2)**2/(x**2*(c + d*x**2)**(3/2)), x)
Time = 0.05 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.12 \[ \int \frac {\left (a+b x^2\right )^2}{x^2 \left (c+d x^2\right )^{3/2}} \, dx=\frac {2 \, a b x}{\sqrt {d x^{2} + c} c} - \frac {b^{2} x}{\sqrt {d x^{2} + c} d} - \frac {2 \, a^{2} d x}{\sqrt {d x^{2} + c} c^{2}} + \frac {b^{2} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{d^{\frac {3}{2}}} - \frac {a^{2}}{\sqrt {d x^{2} + c} c x} \] Input:
integrate((b*x^2+a)^2/x^2/(d*x^2+c)^(3/2),x, algorithm="maxima")
Output:
2*a*b*x/(sqrt(d*x^2 + c)*c) - b^2*x/(sqrt(d*x^2 + c)*d) - 2*a^2*d*x/(sqrt( d*x^2 + c)*c^2) + b^2*arcsinh(d*x/sqrt(c*d))/d^(3/2) - a^2/(sqrt(d*x^2 + c )*c*x)
Time = 0.14 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.28 \[ \int \frac {\left (a+b x^2\right )^2}{x^2 \left (c+d x^2\right )^{3/2}} \, dx=-\frac {b^{2} \log \left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2}\right )}{2 \, d^{\frac {3}{2}}} + \frac {2 \, a^{2} \sqrt {d}}{{\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c\right )} c} - \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x}{\sqrt {d x^{2} + c} c^{2} d} \] Input:
integrate((b*x^2+a)^2/x^2/(d*x^2+c)^(3/2),x, algorithm="giac")
Output:
-1/2*b^2*log((sqrt(d)*x - sqrt(d*x^2 + c))^2)/d^(3/2) + 2*a^2*sqrt(d)/(((s qrt(d)*x - sqrt(d*x^2 + c))^2 - c)*c) - (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x/ (sqrt(d*x^2 + c)*c^2*d)
Timed out. \[ \int \frac {\left (a+b x^2\right )^2}{x^2 \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^2}{x^2\,{\left (d\,x^2+c\right )}^{3/2}} \,d x \] Input:
int((a + b*x^2)^2/(x^2*(c + d*x^2)^(3/2)),x)
Output:
int((a + b*x^2)^2/(x^2*(c + d*x^2)^(3/2)), x)
Time = 0.22 (sec) , antiderivative size = 230, normalized size of antiderivative = 2.84 \[ \int \frac {\left (a+b x^2\right )^2}{x^2 \left (c+d x^2\right )^{3/2}} \, dx=\frac {-\sqrt {d \,x^{2}+c}\, a^{2} c \,d^{2}-2 \sqrt {d \,x^{2}+c}\, a^{2} d^{3} x^{2}+2 \sqrt {d \,x^{2}+c}\, a b c \,d^{2} x^{2}-\sqrt {d \,x^{2}+c}\, b^{2} c^{2} d \,x^{2}+\sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d \,x^{2}+c}+\sqrt {d}\, x}{\sqrt {c}}\right ) b^{2} c^{3} x +\sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d \,x^{2}+c}+\sqrt {d}\, x}{\sqrt {c}}\right ) b^{2} c^{2} d \,x^{3}-2 \sqrt {d}\, a^{2} c \,d^{2} x -2 \sqrt {d}\, a^{2} d^{3} x^{3}+2 \sqrt {d}\, a b \,c^{2} d x +2 \sqrt {d}\, a b c \,d^{2} x^{3}-\sqrt {d}\, b^{2} c^{3} x -\sqrt {d}\, b^{2} c^{2} d \,x^{3}}{c^{2} d^{2} x \left (d \,x^{2}+c \right )} \] Input:
int((b*x^2+a)^2/x^2/(d*x^2+c)^(3/2),x)
Output:
( - sqrt(c + d*x**2)*a**2*c*d**2 - 2*sqrt(c + d*x**2)*a**2*d**3*x**2 + 2*s qrt(c + d*x**2)*a*b*c*d**2*x**2 - sqrt(c + d*x**2)*b**2*c**2*d*x**2 + sqrt (d)*log((sqrt(c + d*x**2) + sqrt(d)*x)/sqrt(c))*b**2*c**3*x + sqrt(d)*log( (sqrt(c + d*x**2) + sqrt(d)*x)/sqrt(c))*b**2*c**2*d*x**3 - 2*sqrt(d)*a**2* c*d**2*x - 2*sqrt(d)*a**2*d**3*x**3 + 2*sqrt(d)*a*b*c**2*d*x + 2*sqrt(d)*a *b*c*d**2*x**3 - sqrt(d)*b**2*c**3*x - sqrt(d)*b**2*c**2*d*x**3)/(c**2*d** 2*x*(c + d*x**2))