Integrand size = 19, antiderivative size = 123 \[ \int \frac {\left (\frac {c}{\left (a+b x^2\right )^2}\right )^{3/2}}{x} \, dx=\frac {c \sqrt {\frac {c}{\left (a+b x^2\right )^2}}}{2 a^2}+\frac {c \sqrt {\frac {c}{\left (a+b x^2\right )^2}}}{4 a \left (a+b x^2\right )}+\frac {c \sqrt {\frac {c}{\left (a+b x^2\right )^2}} \left (a+b x^2\right ) \log (x)}{a^3}-\frac {c \sqrt {\frac {c}{\left (a+b x^2\right )^2}} \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 a^3} \] Output:
1/2*c*(c/(b*x^2+a)^2)^(1/2)/a^2+1/4*c*(c/(b*x^2+a)^2)^(1/2)/a/(b*x^2+a)+c* (c/(b*x^2+a)^2)^(1/2)*(b*x^2+a)*ln(x)/a^3-1/2*c*(c/(b*x^2+a)^2)^(1/2)*(b*x ^2+a)*ln(b*x^2+a)/a^3
Time = 1.03 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.60 \[ \int \frac {\left (\frac {c}{\left (a+b x^2\right )^2}\right )^{3/2}}{x} \, dx=\frac {\left (\frac {c}{\left (a+b x^2\right )^2}\right )^{3/2} \left (a+b x^2\right ) \left (a \left (3 a+2 b x^2\right )+4 \left (a+b x^2\right )^2 \log (x)-2 \left (a+b x^2\right )^2 \log \left (a+b x^2\right )\right )}{4 a^3} \] Input:
Integrate[(c/(a + b*x^2)^2)^(3/2)/x,x]
Output:
((c/(a + b*x^2)^2)^(3/2)*(a + b*x^2)*(a*(3*a + 2*b*x^2) + 4*(a + b*x^2)^2* Log[x] - 2*(a + b*x^2)^2*Log[a + b*x^2]))/(4*a^3)
Time = 0.41 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.63, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2045, 27, 243, 54, 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.
\(\displaystyle \int \frac {\left (\frac {c}{\left (a+b x^2\right )^2}\right )^{3/2}}{x} \, dx\) |
\(\Big \downarrow \) 2045 |
\(\displaystyle \frac {c \left (a+b x^2\right ) \sqrt {\frac {c}{\left (a+b x^2\right )^2}} \int \frac {a^3}{x \left (b x^2+a\right )^3}dx}{a^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle c \left (a+b x^2\right ) \sqrt {\frac {c}{\left (a+b x^2\right )^2}} \int \frac {1}{x \left (b x^2+a\right )^3}dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} c \left (a+b x^2\right ) \sqrt {\frac {c}{\left (a+b x^2\right )^2}} \int \frac {1}{x^2 \left (b x^2+a\right )^3}dx^2\) |
\(\Big \downarrow \) 54 |
\(\displaystyle \frac {1}{2} c \left (a+b x^2\right ) \sqrt {\frac {c}{\left (a+b x^2\right )^2}} \int \left (-\frac {b}{a^3 \left (b x^2+a\right )}-\frac {b}{a^2 \left (b x^2+a\right )^2}-\frac {b}{a \left (b x^2+a\right )^3}+\frac {1}{a^3 x^2}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} c \left (a+b x^2\right ) \sqrt {\frac {c}{\left (a+b x^2\right )^2}} \left (-\frac {\log \left (a+b x^2\right )}{a^3}+\frac {\log \left (x^2\right )}{a^3}+\frac {1}{a^2 \left (a+b x^2\right )}+\frac {1}{2 a \left (a+b x^2\right )^2}\right )\) |
Input:
Int[(c/(a + b*x^2)^2)^(3/2)/x,x]
Output:
(c*Sqrt[c/(a + b*x^2)^2]*(a + b*x^2)*(1/(2*a*(a + b*x^2)^2) + 1/(a^2*(a + b*x^2)) + Log[x^2]/a^3 - Log[a + b*x^2]/a^3))/2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[(u_.)*((c_.)*((a_) + (b_.)*(x_)^(n_.))^(q_))^(p_), x_Symbol] :> Simp[Si mp[(c*(a + b*x^n)^q)^p/(1 + b*(x^n/a))^(p*q)] Int[u*(1 + b*(x^n/a))^(p*q) , x], x] /; FreeQ[{a, b, c, n, p, q}, x] && !GeQ[a, 0]
Time = 0.17 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.83
method | result | size |
risch | \(\frac {c \sqrt {\frac {c}{\left (b \,x^{2}+a \right )^{2}}}\, \left (\frac {b \,x^{2}}{2 a^{2}}+\frac {3}{4 a}\right )}{b \,x^{2}+a}+\frac {c \sqrt {\frac {c}{\left (b \,x^{2}+a \right )^{2}}}\, \left (b \,x^{2}+a \right ) \ln \left (x \right )}{a^{3}}-\frac {c \sqrt {\frac {c}{\left (b \,x^{2}+a \right )^{2}}}\, \left (b \,x^{2}+a \right ) \ln \left (b \,x^{2}+a \right )}{2 a^{3}}\) | \(102\) |
default | \(-\frac {\left (2 \ln \left (b \,x^{2}+a \right ) b^{2} x^{4}-4 \ln \left (x \right ) b^{2} x^{4}+4 \ln \left (b \,x^{2}+a \right ) a b \,x^{2}-8 \ln \left (x \right ) a b \,x^{2}-2 a b \,x^{2}+2 \ln \left (b \,x^{2}+a \right ) a^{2}-4 a^{2} \ln \left (x \right )-3 a^{2}\right ) \left (b \,x^{2}+a \right ) {\left (\frac {c}{\left (b \,x^{2}+a \right )^{2}}\right )}^{\frac {3}{2}}}{4 a^{3}}\) | \(109\) |
Input:
int((c/(b*x^2+a)^2)^(3/2)/x,x,method=_RETURNVERBOSE)
Output:
c/(b*x^2+a)*(c/(b*x^2+a)^2)^(1/2)*(1/2*b/a^2*x^2+3/4/a)+c*(c/(b*x^2+a)^2)^ (1/2)*(b*x^2+a)*ln(x)/a^3-1/2*c*(c/(b*x^2+a)^2)^(1/2)*(b*x^2+a)*ln(b*x^2+a )/a^3
Time = 0.08 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.92 \[ \int \frac {\left (\frac {c}{\left (a+b x^2\right )^2}\right )^{3/2}}{x} \, dx=\frac {{\left (2 \, a b c x^{2} + 3 \, a^{2} c - 2 \, {\left (b^{2} c x^{4} + 2 \, a b c x^{2} + a^{2} c\right )} \log \left (b x^{2} + a\right ) + 4 \, {\left (b^{2} c x^{4} + 2 \, a b c x^{2} + a^{2} c\right )} \log \left (x\right )\right )} \sqrt {\frac {c}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}}}{4 \, {\left (a^{3} b x^{2} + a^{4}\right )}} \] Input:
integrate((c/(b*x^2+a)^2)^(3/2)/x,x, algorithm="fricas")
Output:
1/4*(2*a*b*c*x^2 + 3*a^2*c - 2*(b^2*c*x^4 + 2*a*b*c*x^2 + a^2*c)*log(b*x^2 + a) + 4*(b^2*c*x^4 + 2*a*b*c*x^2 + a^2*c)*log(x))*sqrt(c/(b^2*x^4 + 2*a* b*x^2 + a^2))/(a^3*b*x^2 + a^4)
\[ \int \frac {\left (\frac {c}{\left (a+b x^2\right )^2}\right )^{3/2}}{x} \, dx=\int \frac {\left (\frac {c}{a^{2} + 2 a b x^{2} + b^{2} x^{4}}\right )^{\frac {3}{2}}}{x}\, dx \] Input:
integrate((c/(b*x**2+a)**2)**(3/2)/x,x)
Output:
Integral((c/(a**2 + 2*a*b*x**2 + b**2*x**4))**(3/2)/x, x)
Time = 0.04 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.59 \[ \int \frac {\left (\frac {c}{\left (a+b x^2\right )^2}\right )^{3/2}}{x} \, dx=\frac {2 \, b c^{\frac {3}{2}} x^{2} + 3 \, a c^{\frac {3}{2}}}{4 \, {\left (a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}\right )}} - \frac {c^{\frac {3}{2}} \log \left (b x^{2} + a\right )}{2 \, a^{3}} + \frac {c^{\frac {3}{2}} \log \left (x^{2}\right )}{2 \, a^{3}} \] Input:
integrate((c/(b*x^2+a)^2)^(3/2)/x,x, algorithm="maxima")
Output:
1/4*(2*b*c^(3/2)*x^2 + 3*a*c^(3/2))/(a^2*b^2*x^4 + 2*a^3*b*x^2 + a^4) - 1/ 2*c^(3/2)*log(b*x^2 + a)/a^3 + 1/2*c^(3/2)*log(x^2)/a^3
Time = 0.13 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.58 \[ \int \frac {\left (\frac {c}{\left (a+b x^2\right )^2}\right )^{3/2}}{x} \, dx=\frac {1}{4} \, c^{\frac {3}{2}} {\left (\frac {2 \, \log \left (x^{2}\right )}{a^{3}} - \frac {2 \, \log \left ({\left | b x^{2} + a \right |}\right )}{a^{3}} + \frac {3 \, b^{2} x^{4} + 8 \, a b x^{2} + 6 \, a^{2}}{{\left (b x^{2} + a\right )}^{2} a^{3}}\right )} \mathrm {sgn}\left (b x^{2} + a\right ) \] Input:
integrate((c/(b*x^2+a)^2)^(3/2)/x,x, algorithm="giac")
Output:
1/4*c^(3/2)*(2*log(x^2)/a^3 - 2*log(abs(b*x^2 + a))/a^3 + (3*b^2*x^4 + 8*a *b*x^2 + 6*a^2)/((b*x^2 + a)^2*a^3))*sgn(b*x^2 + a)
Timed out. \[ \int \frac {\left (\frac {c}{\left (a+b x^2\right )^2}\right )^{3/2}}{x} \, dx=\int \frac {{\left (\frac {c}{{\left (b\,x^2+a\right )}^2}\right )}^{3/2}}{x} \,d x \] Input:
int((c/(a + b*x^2)^2)^(3/2)/x,x)
Output:
int((c/(a + b*x^2)^2)^(3/2)/x, x)
Time = 0.21 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.91 \[ \int \frac {\left (\frac {c}{\left (a+b x^2\right )^2}\right )^{3/2}}{x} \, dx=\frac {\sqrt {c}\, c \left (-2 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{2}-4 \,\mathrm {log}\left (b \,x^{2}+a \right ) a b \,x^{2}-2 \,\mathrm {log}\left (b \,x^{2}+a \right ) b^{2} x^{4}+4 \,\mathrm {log}\left (x \right ) a^{2}+8 \,\mathrm {log}\left (x \right ) a b \,x^{2}+4 \,\mathrm {log}\left (x \right ) b^{2} x^{4}+2 a^{2}-b^{2} x^{4}\right )}{4 a^{3} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )} \] Input:
int((c/(b*x^2+a)^2)^(3/2)/x,x)
Output:
(sqrt(c)*c*( - 2*log(a + b*x**2)*a**2 - 4*log(a + b*x**2)*a*b*x**2 - 2*log (a + b*x**2)*b**2*x**4 + 4*log(x)*a**2 + 8*log(x)*a*b*x**2 + 4*log(x)*b**2 *x**4 + 2*a**2 - b**2*x**4))/(4*a**3*(a**2 + 2*a*b*x**2 + b**2*x**4))