Integrand size = 19, antiderivative size = 51 \[ \int \frac {x^2}{\left (b x^2+c x^4\right )^{3/2}} \, dx=\frac {x}{b \sqrt {b x^2+c x^4}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{b^{3/2}} \] Output:
x/b/(c*x^4+b*x^2)^(1/2)-arctanh(b^(1/2)*x/(c*x^4+b*x^2)^(1/2))/b^(3/2)
Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.16 \[ \int \frac {x^2}{\left (b x^2+c x^4\right )^{3/2}} \, dx=\frac {x \left (\sqrt {b}-\sqrt {b+c x^2} \text {arctanh}\left (\frac {\sqrt {b+c x^2}}{\sqrt {b}}\right )\right )}{b^{3/2} \sqrt {x^2 \left (b+c x^2\right )}} \] Input:
Integrate[x^2/(b*x^2 + c*x^4)^(3/2),x]
Output:
(x*(Sqrt[b] - Sqrt[b + c*x^2]*ArcTanh[Sqrt[b + c*x^2]/Sqrt[b]]))/(b^(3/2)* Sqrt[x^2*(b + c*x^2)])
Time = 0.30 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1428, 1400, 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.
\(\displaystyle \int \frac {x^2}{\left (b x^2+c x^4\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 1428 |
\(\displaystyle \frac {\int \frac {1}{\sqrt {c x^4+b x^2}}dx}{b}+\frac {x}{b \sqrt {b x^2+c x^4}}\) |
\(\Big \downarrow \) 1400 |
\(\displaystyle \frac {x}{b \sqrt {b x^2+c x^4}}-\frac {\int \frac {1}{1-\frac {b x^2}{c x^4+b x^2}}d\frac {x}{\sqrt {c x^4+b x^2}}}{b}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {x}{b \sqrt {b x^2+c x^4}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{b^{3/2}}\) |
Input:
Int[x^2/(b*x^2 + c*x^4)^(3/2),x]
Output:
x/(b*Sqrt[b*x^2 + c*x^4]) - ArcTanh[(Sqrt[b]*x)/Sqrt[b*x^2 + c*x^4]]/b^(3/ 2)
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[(b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> -Subst[Int[1/(1 - b*x ^2), x], x, x/Sqrt[b*x^2 + c*x^4]] /; FreeQ[{b, c}, x]
Int[((d_.)*(x_))^(m_)*((b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp [(-d)*(d*x)^(m - 1)*((b*x^2 + c*x^4)^(p + 1)/(2*b*(p + 1))), x] + Simp[d^2* ((m + 4*p + 3)/(2*b*(p + 1))) Int[(d*x)^(m - 2)*(b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{b, c, d, m, p}, x] && !IntegerQ[p] && LtQ[p, -1]
Time = 0.56 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.27
method | result | size |
default | \(\frac {x^{3} \left (c \,x^{2}+b \right ) \left (b^{\frac {3}{2}}-\ln \left (\frac {2 b +2 \sqrt {b}\, \sqrt {c \,x^{2}+b}}{x}\right ) b \sqrt {c \,x^{2}+b}\right )}{\left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} b^{\frac {5}{2}}}\) | \(65\) |
Input:
int(x^2/(c*x^4+b*x^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
x^3*(c*x^2+b)*(b^(3/2)-ln(2*(b^(1/2)*(c*x^2+b)^(1/2)+b)/x)*b*(c*x^2+b)^(1/ 2))/(c*x^4+b*x^2)^(3/2)/b^(5/2)
Time = 0.08 (sec) , antiderivative size = 157, normalized size of antiderivative = 3.08 \[ \int \frac {x^2}{\left (b x^2+c x^4\right )^{3/2}} \, dx=\left [\frac {{\left (c x^{3} + b x\right )} \sqrt {b} \log \left (-\frac {c x^{3} + 2 \, b x - 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {b}}{x^{3}}\right ) + 2 \, \sqrt {c x^{4} + b x^{2}} b}{2 \, {\left (b^{2} c x^{3} + b^{3} x\right )}}, \frac {{\left (c x^{3} + b x\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-b}}{b x}\right ) + \sqrt {c x^{4} + b x^{2}} b}{b^{2} c x^{3} + b^{3} x}\right ] \] Input:
integrate(x^2/(c*x^4+b*x^2)^(3/2),x, algorithm="fricas")
Output:
[1/2*((c*x^3 + b*x)*sqrt(b)*log(-(c*x^3 + 2*b*x - 2*sqrt(c*x^4 + b*x^2)*sq rt(b))/x^3) + 2*sqrt(c*x^4 + b*x^2)*b)/(b^2*c*x^3 + b^3*x), ((c*x^3 + b*x) *sqrt(-b)*arctan(sqrt(c*x^4 + b*x^2)*sqrt(-b)/(b*x)) + sqrt(c*x^4 + b*x^2) *b)/(b^2*c*x^3 + b^3*x)]
\[ \int \frac {x^2}{\left (b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {x^{2}}{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(x**2/(c*x**4+b*x**2)**(3/2),x)
Output:
Integral(x**2/(x**2*(b + c*x**2))**(3/2), x)
\[ \int \frac {x^2}{\left (b x^2+c x^4\right )^{3/2}} \, dx=\int { \frac {x^{2}}{{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^2/(c*x^4+b*x^2)^(3/2),x, algorithm="maxima")
Output:
integrate(x^2/(c*x^4 + b*x^2)^(3/2), x)
Time = 0.12 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.55 \[ \int \frac {x^2}{\left (b x^2+c x^4\right )^{3/2}} \, dx=-\frac {{\left (\sqrt {b} \arctan \left (\frac {\sqrt {b}}{\sqrt {-b}}\right ) + \sqrt {-b}\right )} \mathrm {sgn}\left (x\right )}{\sqrt {-b} b^{\frac {3}{2}}} + \frac {\arctan \left (\frac {\sqrt {c x^{2} + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b \mathrm {sgn}\left (x\right )} + \frac {1}{\sqrt {c x^{2} + b} b \mathrm {sgn}\left (x\right )} \] Input:
integrate(x^2/(c*x^4+b*x^2)^(3/2),x, algorithm="giac")
Output:
-(sqrt(b)*arctan(sqrt(b)/sqrt(-b)) + sqrt(-b))*sgn(x)/(sqrt(-b)*b^(3/2)) + arctan(sqrt(c*x^2 + b)/sqrt(-b))/(sqrt(-b)*b*sgn(x)) + 1/(sqrt(c*x^2 + b) *b*sgn(x))
Timed out. \[ \int \frac {x^2}{\left (b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {x^2}{{\left (c\,x^4+b\,x^2\right )}^{3/2}} \,d x \] Input:
int(x^2/(b*x^2 + c*x^4)^(3/2),x)
Output:
int(x^2/(b*x^2 + c*x^4)^(3/2), x)
Time = 0.18 (sec) , antiderivative size = 136, normalized size of antiderivative = 2.67 \[ \int \frac {x^2}{\left (b x^2+c x^4\right )^{3/2}} \, dx=\frac {\sqrt {c \,x^{2}+b}\, b +\sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+b}-\sqrt {b}+\sqrt {c}\, x}{\sqrt {b}}\right ) b +\sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+b}-\sqrt {b}+\sqrt {c}\, x}{\sqrt {b}}\right ) c \,x^{2}-\sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+b}+\sqrt {b}+\sqrt {c}\, x}{\sqrt {b}}\right ) b -\sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+b}+\sqrt {b}+\sqrt {c}\, x}{\sqrt {b}}\right ) c \,x^{2}}{b^{2} \left (c \,x^{2}+b \right )} \] Input:
int(x^2/(c*x^4+b*x^2)^(3/2),x)
Output:
(sqrt(b + c*x**2)*b + sqrt(b)*log((sqrt(b + c*x**2) - sqrt(b) + sqrt(c)*x) /sqrt(b))*b + sqrt(b)*log((sqrt(b + c*x**2) - sqrt(b) + sqrt(c)*x)/sqrt(b) )*c*x**2 - sqrt(b)*log((sqrt(b + c*x**2) + sqrt(b) + sqrt(c)*x)/sqrt(b))*b - sqrt(b)*log((sqrt(b + c*x**2) + sqrt(b) + sqrt(c)*x)/sqrt(b))*c*x**2)/( b**2*(b + c*x**2))