Integrand size = 36, antiderivative size = 73 \[ \int \frac {a+b \log \left (c \left (d f-d g x^2\right )^p\right )}{x^2 \sqrt {f-g x^2}} \, dx=-\frac {2 b \sqrt {g} p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f-g x^2}}\right )}{f}-\frac {\sqrt {f-g x^2} \left (a+b \log \left (c \left (d f-d g x^2\right )^p\right )\right )}{f x} \] Output:
-2*b*g^(1/2)*p*arctan(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 = 0.16 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.97 \[ \int \frac {a+b \log \left (c \left (d f-d g x^2\right )^p\right )}{x^2 \sqrt {f-g x^2}} \, dx=-\frac {2 b \sqrt {g} p x \arctan \left (\frac {\sqrt {g} x}{\sqrt {f-g x^2}}\right )+\sqrt {f-g x^2} \left (a+b \log \left (c \left (d \left (f-g x^2\right )\right )^p\right )\right )}{f x} \] Input:
Integrate[(a + b*Log[c*(d*f - d*g*x^2)^p])/(x^2*Sqrt[f - g*x^2]),x]
Output:
-((2*b*Sqrt[g]*p*x*ArcTan[(Sqrt[g]*x)/Sqrt[f - g*x^2]] + Sqrt[f - g*x^2]*( a + b*Log[c*(d*(f - g*x^2))^p]))/(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 = 0.14 (sec) , antiderivative size = 156, normalized size of antiderivative = 2.14 \[ \int \frac {a+b \log \left (c \left (d f-d g x^2\right )^p\right )}{x^2 \sqrt {f-g x^2}} \, dx=\left [\frac {b \sqrt {-g} p x \log \left (2 \, g x^{2} - 2 \, \sqrt {-g x^{2} + f} \sqrt {-g} x - f\right ) - \sqrt {-g x^{2} + f} {\left (b p \log \left (-d g x^{2} + d f\right ) + b \log \left (c\right ) + a\right )}}{f x}, \frac {2 \, b \sqrt {g} p x \arctan \left (\frac {\sqrt {-g x^{2} + f} \sqrt {g} x}{g x^{2} - f}\right ) - \sqrt {-g x^{2} + f} {\left (b p \log \left (-d g x^{2} + d f\right ) + b \log \left (c\right ) + a\right )}}{f x}\right ] \] Input:
integrate((a+b*log(c*(-d*g*x^2+d*f)^p))/x^2/(-g*x^2+f)^(1/2),x, algorithm= "fricas")
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^2 + f)*sqrt(g)*x/(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)
Time = 0.14 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.03 \[ \int \frac {a+b \log \left (c \left (d f-d g x^2\right )^p\right )}{x^2 \sqrt {f-g x^2}} \, dx=-\frac {2 \, b \sqrt {g} p \arcsin \left (\frac {x}{\sqrt {\frac {f}{g}}}\right )}{f} - \frac {\sqrt {-g x^{2} + f} b \log \left ({\left (-d g x^{2} + d f\right )}^{p} c\right )}{f x} - \frac {\sqrt {-g x^{2} + f} a}{f x} \] Input:
integrate((a+b*log(c*(-d*g*x^2+d*f)^p))/x^2/(-g*x^2+f)^(1/2),x, algorithm= "maxima")
Output:
-2*b*sqrt(g)*p*arcsin(x/sqrt(f/g))/f - sqrt(-g*x^2 + f)*b*log((-d*g*x^2 + d*f)^p*c)/(f*x) - sqrt(-g*x^2 + f)*a/(f*x)
Leaf count of result is larger than twice the leaf count of optimal. 178 vs. \(2 (65) = 130\).
Time = 0.47 (sec) , antiderivative size = 178, normalized size of antiderivative = 2.44 \[ \int \frac {a+b \log \left (c \left (d f-d g x^2\right )^p\right )}{x^2 \sqrt {f-g x^2}} \, dx=b p {\left (\frac {2 \, \sqrt {-g} \log \left (-d g x^{2} + d f\right )}{{\left (\sqrt {-g} x - \sqrt {-g x^{2} + f}\right )}^{2} - f} + \frac {\sqrt {-g} \log \left ({\left | g x^{2} - f \right |}\right )}{f} + \frac {2 \, g \log \left ({\left | -\sqrt {-g} x + \sqrt {-g x^{2} + f} \right |}\right )}{f \sqrt {-g}}\right )} + \frac {2 \, b \sqrt {-g} \log \left (c\right )}{{\left (\sqrt {-g} x - \sqrt {-g x^{2} + f}\right )}^{2} - f} + \frac {2 \, a \sqrt {-g}}{{\left (\sqrt {-g} x - \sqrt {-g x^{2} + f}\right )}^{2} - f} \] Input:
integrate((a+b*log(c*(-d*g*x^2+d*f)^p))/x^2/(-g*x^2+f)^(1/2),x, algorithm= "giac")
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*g*log(abs(-sqrt(-g)*x + sqrt(-g*x^2 + f)))/(f*sqrt(-g))) + 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. \[ \int \frac {a+b \log \left (c \left (d f-d g x^2\right )^p\right )}{x^2 \sqrt {f-g x^2}} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d\,f-d\,g\,x^2\right )}^p\right )}{x^2\,\sqrt {f-g\,x^2}} \,d x \] 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 = 101, normalized size of antiderivative = 1.38 \[ \int \frac {a+b \log \left (c \left (d f-d g x^2\right )^p\right )}{x^2 \sqrt {f-g x^2}} \, dx=\frac {-2 \sqrt {g}\, \mathit {asin} \left (\frac {\sqrt {g}\, x}{\sqrt {f}}\right ) b p x -\sqrt {-g \,x^{2}+f}\, \mathrm {log}\left (\frac {f^{p} d^{p} \left (-\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {g}\, x}{\sqrt {f}}\right )}{2}\right )^{2}+1\right )^{2 p} c}{\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {g}\, x}{\sqrt {f}}\right )}{2}\right )^{2}+1\right )^{2 p}}\right ) b -\sqrt {-g \,x^{2}+f}\, a}{f x} \] Input:
int((a+b*log(c*(-d*g*x^2+d*f)^p))/x^2/(-g*x^2+f)^(1/2),x)
Output:
( - 2*sqrt(g)*asin((sqrt(g)*x)/sqrt(f))*b*p*x - sqrt(f - g*x**2)*log((f**p *d**p*( - tan(asin((sqrt(g)*x)/sqrt(f))/2)**2 + 1)**(2*p)*c)/(tan(asin((sq rt(g)*x)/sqrt(f))/2)**2 + 1)**(2*p))*b - sqrt(f - g*x**2)*a)/(f*x)