Integrand size = 34, antiderivative size = 70 \[ \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^2 \sqrt {f+g x^2}} \, dx=\frac {2 b \sqrt {g} p \text {arctanh}\left (\frac {\sqrt {g} x}{\sqrt {f+g x^2}}\right )}{f}-\frac {\sqrt {f+g x^2} \left (a+b \log \left (c \left (d f+d g x^2\right )^p\right )\right )}{f x} \] Output:
2*b*g^(1/2)*p*arctanh(g^(1/2)*x/(g*x^2+f)^(1/2))/f-(g*x^2+f)^(1/2)*(a+b*ln (c*(d*g*x^2+d*f)^p))/f/x
Time = 0.15 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.16 \[ \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^2 \sqrt {f+g x^2}} \, dx=-\frac {a \sqrt {f+g x^2}+b \sqrt {f+g x^2} \log \left (c \left (d \left (f+g x^2\right )\right )^p\right )-2 b \sqrt {g} p x \log \left (g x+\sqrt {g} \sqrt {f+g x^2}\right )}{f x} \] Input:
Integrate[(a + b*Log[c*(d*f + d*g*x^2)^p])/(x^2*Sqrt[f + g*x^2]),x]
Output:
-((a*Sqrt[f + g*x^2] + b*Sqrt[f + g*x^2]*Log[c*(d*(f + g*x^2))^p] - 2*b*Sq rt[g]*p*x*Log[g*x + Sqrt[g]*Sqrt[f + g*x^2]])/(f*x))
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.
\(\displaystyle \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^2 \sqrt {f+g x^2}} \, dx\) |
\(\Big \downarrow \) 2929 |
\(\displaystyle \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^2 \sqrt {f+g x^2}}dx\) |
Input:
Int[(a + b*Log[c*(d*f + d*g*x^2)^p])/(x^2*Sqrt[f + g*x^2]),x]
Output:
$Aborted
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*((h_.)* (x_))^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol] :> Unintegrable[(h*x) ^m*(f + g*x^s)^r*(a + b*Log[c*(d + e*x^n)^p])^q, x] /; FreeQ[{a, b, c, d, e , f, g, h, m, n, p, q, r, s}, x]
\[\int \frac {a +b \ln \left (c \left (d g \,x^{2}+d f \right )^{p}\right )}{x^{2} \sqrt {g \,x^{2}+f}}d x\]
Input:
int((a+b*ln(c*(d*g*x^2+d*f)^p))/x^2/(g*x^2+f)^(1/2),x)
Output:
int((a+b*ln(c*(d*g*x^2+d*f)^p))/x^2/(g*x^2+f)^(1/2),x)
Time = 0.09 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.99 \[ \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^2 \sqrt {f+g x^2}} \, dx=\left [\frac {b \sqrt {g} p x \log \left (-2 \, g x^{2} - 2 \, \sqrt {g x^{2} + f} \sqrt {g} x - f\right ) - \sqrt {g x^{2} + f} {\left (b p \log \left (d g x^{2} + d f\right ) + b \log \left (c\right ) + a\right )}}{f x}, -\frac {2 \, b \sqrt {-g} p x \arctan \left (\frac {\sqrt {-g} x}{\sqrt {g x^{2} + f}}\right ) + \sqrt {g x^{2} + f} {\left (b p \log \left (d g x^{2} + d f\right ) + b \log \left (c\right ) + a\right )}}{f x}\right ] \] Input:
integrate((a+b*log(c*(d*g*x^2+d*f)^p))/x^2/(g*x^2+f)^(1/2),x, algorithm="f ricas")
Output:
[(b*sqrt(g)*p*x*log(-2*g*x^2 - 2*sqrt(g*x^2 + f)*sqrt(g)*x - f) - sqrt(g*x ^2 + f)*(b*p*log(d*g*x^2 + d*f) + b*log(c) + a))/(f*x), -(2*b*sqrt(-g)*p*x *arctan(sqrt(-g)*x/sqrt(g*x^2 + f)) + sqrt(g*x^2 + f)*(b*p*log(d*g*x^2 + d *f) + b*log(c) + a))/(f*x)]
\[ \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^2 \sqrt {f+g x^2}} \, dx=\int \frac {a + b \log {\left (c \left (d f + d g x^{2}\right )^{p} \right )}}{x^{2} \sqrt {f + g x^{2}}}\, dx \] Input:
integrate((a+b*ln(c*(d*g*x**2+d*f)**p))/x**2/(g*x**2+f)**(1/2),x)
Output:
Integral((a + b*log(c*(d*f + d*g*x**2)**p))/(x**2*sqrt(f + g*x**2)), x)
\[ \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^2 \sqrt {f+g x^2}} \, dx=\int { \frac {b \log \left ({\left (d g x^{2} + d f\right )}^{p} c\right ) + a}{\sqrt {g x^{2} + f} x^{2}} \,d x } \] Input:
integrate((a+b*log(c*(d*g*x^2+d*f)^p))/x^2/(g*x^2+f)^(1/2),x, algorithm="m axima")
Output:
(2*g^2*p*integrate(x^2/((f*g*x^2 + f^2)*sqrt(g*x^2 + f)), x) - ((g*p*log(d ) - 2*g*p + g*log(c))*x^2 + f*p*log(d) + (g*p*x^2 + f*p)*log(g*x^2 + f) + f*log(c))/(sqrt(g*x^2 + f)*f*x))*b - sqrt(g*x^2 + f)*a/(f*x)
Leaf count of result is larger than twice the leaf count of optimal. 152 vs. \(2 (62) = 124\).
Time = 0.46 (sec) , antiderivative size = 152, normalized size of antiderivative = 2.17 \[ \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^2 \sqrt {f+g x^2}} \, dx=b p {\left (\frac {2 \, \sqrt {g} \log \left (d g x^{2} + d f\right )}{{\left (\sqrt {g} x - \sqrt {g x^{2} + f}\right )}^{2} - f} + \frac {\sqrt {g} \log \left ({\left | g x^{2} + f \right |}\right )}{f} - \frac {2 \, \sqrt {g} \log \left ({\left | -\sqrt {g} x + \sqrt {g x^{2} + f} \right |}\right )}{f}\right )} + \frac {2 \, b \sqrt {g} \log \left (c\right )}{{\left (\sqrt {g} x - \sqrt {g x^{2} + f}\right )}^{2} - f} + \frac {2 \, a \sqrt {g}}{{\left (\sqrt {g} x - \sqrt {g x^{2} + f}\right )}^{2} - f} \] Input:
integrate((a+b*log(c*(d*g*x^2+d*f)^p))/x^2/(g*x^2+f)^(1/2),x, algorithm="g iac")
Output:
b*p*(2*sqrt(g)*log(d*g*x^2 + d*f)/((sqrt(g)*x - sqrt(g*x^2 + f))^2 - f) + sqrt(g)*log(abs(g*x^2 + f))/f - 2*sqrt(g)*log(abs(-sqrt(g)*x + sqrt(g*x^2 + f)))/f) + 2*b*sqrt(g)*log(c)/((sqrt(g)*x - sqrt(g*x^2 + f))^2 - f) + 2*a *sqrt(g)/((sqrt(g)*x - sqrt(g*x^2 + f))^2 - f)
Timed out. \[ \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^2 \sqrt {f+g x^2}} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d\,g\,x^2+d\,f\right )}^p\right )}{x^2\,\sqrt {g\,x^2+f}} \,d x \] Input:
int((a + b*log(c*(d*f + d*g*x^2)^p))/(x^2*(f + g*x^2)^(1/2)),x)
Output:
int((a + b*log(c*(d*f + d*g*x^2)^p))/(x^2*(f + g*x^2)^(1/2)), x)
Time = 0.17 (sec) , antiderivative size = 195, normalized size of antiderivative = 2.79 \[ \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^2 \sqrt {f+g x^2}} \, dx=\frac {-\sqrt {g \,x^{2}+f}\, \mathrm {log}\left (\frac {d^{p} \left (2 \sqrt {g}\, \sqrt {g \,x^{2}+f}\, x +2 f +2 g \,x^{2}\right )^{2 p} c}{\left (\sqrt {g \,x^{2}+f}+\sqrt {g}\, x \right )^{2 p} 2^{2 p}}\right ) b -\sqrt {g \,x^{2}+f}\, a +2 \sqrt {g}\, \mathrm {log}\left (\frac {2 \sqrt {g}\, \sqrt {g \,x^{2}+f}\, x +2 f +2 g \,x^{2}}{f}\right ) b p x -\sqrt {g}\, \mathrm {log}\left (\frac {d^{p} \left (2 \sqrt {g}\, \sqrt {g \,x^{2}+f}\, x +2 f +2 g \,x^{2}\right )^{2 p} c}{\left (\sqrt {g \,x^{2}+f}+\sqrt {g}\, x \right )^{2 p} 2^{2 p}}\right ) b x -\sqrt {g}\, a x}{f x} \] Input:
int((a+b*log(c*(d*g*x^2+d*f)^p))/x^2/(g*x^2+f)^(1/2),x)
Output:
( - sqrt(f + g*x**2)*log((d**p*(2*sqrt(g)*sqrt(f + g*x**2)*x + 2*f + 2*g*x **2)**(2*p)*c)/((sqrt(f + g*x**2) + sqrt(g)*x)**(2*p)*2**(2*p)))*b - sqrt( f + g*x**2)*a + 2*sqrt(g)*log((2*sqrt(g)*sqrt(f + g*x**2)*x + 2*f + 2*g*x* *2)/f)*b*p*x - sqrt(g)*log((d**p*(2*sqrt(g)*sqrt(f + g*x**2)*x + 2*f + 2*g *x**2)**(2*p)*c)/((sqrt(f + g*x**2) + sqrt(g)*x)**(2*p)*2**(2*p)))*b*x - s qrt(g)*a*x)/(f*x)