Integrand size = 36, antiderivative size = 62 \[ \int \frac {a+b \log \left (c \left (4 d-d g x^2\right )^p\right )}{x^2 \sqrt {4-g x^2}} \, dx=-\frac {1}{2} b \sqrt {g} p \arcsin \left (\frac {\sqrt {g} x}{2}\right )-\frac {\sqrt {4-g x^2} \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{4 x} \] Output:
-1/2*b*g^(1/2)*p*arcsin(1/2*g^(1/2)*x)-1/4*(-g*x^2+4)^(1/2)*(a+b*ln(c*(-d* g*x^2+4*d)^p))/x
Time = 0.14 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.98 \[ \int \frac {a+b \log \left (c \left (4 d-d g x^2\right )^p\right )}{x^2 \sqrt {4-g x^2}} \, dx=-\frac {1}{2} b \sqrt {g} p \arcsin \left (\frac {\sqrt {g} x}{2}\right )-\frac {\sqrt {4-g x^2} \left (a+b \log \left (c \left (d \left (4-g x^2\right )\right )^p\right )\right )}{4 x} \] Input:
Integrate[(a + b*Log[c*(4*d - d*g*x^2)^p])/(x^2*Sqrt[4 - g*x^2]),x]
Output:
-1/2*(b*Sqrt[g]*p*ArcSin[(Sqrt[g]*x)/2]) - (Sqrt[4 - g*x^2]*(a + b*Log[c*( d*(4 - g*x^2))^p]))/(4*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.
\(\displaystyle \int \frac {a+b \log \left (c \left (4 d-d g x^2\right )^p\right )}{x^2 \sqrt {4-g x^2}} \, dx\) |
\(\Big \downarrow \) 2929 |
\(\displaystyle \int \frac {a+b \log \left (c \left (4 d-d g x^2\right )^p\right )}{x^2 \sqrt {4-g x^2}}dx\) |
Input:
Int[(a + b*Log[c*(4*d - d*g*x^2)^p])/(x^2*Sqrt[4 - 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]
\[\int \frac {a +b \ln \left (c \left (-d g \,x^{2}+4 d \right )^{p}\right )}{x^{2} \sqrt {-g \,x^{2}+4}}d x\]
Input:
int((a+b*ln(c*(-d*g*x^2+4*d)^p))/x^2/(-g*x^2+4)^(1/2),x)
Output:
int((a+b*ln(c*(-d*g*x^2+4*d)^p))/x^2/(-g*x^2+4)^(1/2),x)
Time = 0.11 (sec) , antiderivative size = 139, normalized size of antiderivative = 2.24 \[ \int \frac {a+b \log \left (c \left (4 d-d g x^2\right )^p\right )}{x^2 \sqrt {4-g x^2}} \, dx=\left [\frac {2 \, b \sqrt {-g} p x \log \left (-\sqrt {-g} x - \sqrt {-g x^{2} + 4}\right ) - \sqrt {-g x^{2} + 4} {\left (b p \log \left (-d g x^{2} + 4 \, d\right ) + b \log \left (c\right ) + a\right )}}{4 \, x}, \frac {4 \, b \sqrt {g} p x \arctan \left (\frac {\sqrt {-g x^{2} + 4} - 2}{\sqrt {g} x}\right ) - \sqrt {-g x^{2} + 4} {\left (b p \log \left (-d g x^{2} + 4 \, d\right ) + b \log \left (c\right ) + a\right )}}{4 \, x}\right ] \] Input:
integrate((a+b*log(c*(-d*g*x^2+4*d)^p))/x^2/(-g*x^2+4)^(1/2),x, algorithm= "fricas")
Output:
[1/4*(2*b*sqrt(-g)*p*x*log(-sqrt(-g)*x - sqrt(-g*x^2 + 4)) - sqrt(-g*x^2 + 4)*(b*p*log(-d*g*x^2 + 4*d) + b*log(c) + a))/x, 1/4*(4*b*sqrt(g)*p*x*arct an((sqrt(-g*x^2 + 4) - 2)/(sqrt(g)*x)) - sqrt(-g*x^2 + 4)*(b*p*log(-d*g*x^ 2 + 4*d) + b*log(c) + a))/x]
\[ \int \frac {a+b \log \left (c \left (4 d-d g x^2\right )^p\right )}{x^2 \sqrt {4-g x^2}} \, dx=\int \frac {a + b \log {\left (c \left (- d g x^{2} + 4 d\right )^{p} \right )}}{x^{2} \sqrt {- g x^{2} + 4}}\, dx \] Input:
integrate((a+b*ln(c*(-d*g*x**2+4*d)**p))/x**2/(-g*x**2+4)**(1/2),x)
Output:
Integral((a + b*log(c*(-d*g*x**2 + 4*d)**p))/(x**2*sqrt(-g*x**2 + 4)), x)
Time = 0.12 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.02 \[ \int \frac {a+b \log \left (c \left (4 d-d g x^2\right )^p\right )}{x^2 \sqrt {4-g x^2}} \, dx=-\frac {1}{2} \, b \sqrt {g} p \arcsin \left (\frac {1}{2} \, \sqrt {g} x\right ) - \frac {\sqrt {-g x^{2} + 4} b \log \left ({\left (-d g x^{2} + 4 \, d\right )}^{p} c\right )}{4 \, x} - \frac {\sqrt {-g x^{2} + 4} a}{4 \, x} \] Input:
integrate((a+b*log(c*(-d*g*x^2+4*d)^p))/x^2/(-g*x^2+4)^(1/2),x, algorithm= "maxima")
Output:
-1/2*b*sqrt(g)*p*arcsin(1/2*sqrt(g)*x) - 1/4*sqrt(-g*x^2 + 4)*b*log((-d*g* x^2 + 4*d)^p*c)/x - 1/4*sqrt(-g*x^2 + 4)*a/x
Exception generated. \[ \int \frac {a+b \log \left (c \left (4 d-d g x^2\right )^p\right )}{x^2 \sqrt {4-g x^2}} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((a+b*log(c*(-d*g*x^2+4*d)^p))/x^2/(-g*x^2+4)^(1/2),x, algorithm= "giac")
Output:
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {a+b \log \left (c \left (4 d-d g x^2\right )^p\right )}{x^2 \sqrt {4-g x^2}} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (4\,d-d\,g\,x^2\right )}^p\right )}{x^2\,\sqrt {4-g\,x^2}} \,d x \] Input:
int((a + b*log(c*(4*d - d*g*x^2)^p))/(x^2*(4 - g*x^2)^(1/2)),x)
Output:
int((a + b*log(c*(4*d - d*g*x^2)^p))/(x^2*(4 - g*x^2)^(1/2)), x)
Time = 0.16 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.45 \[ \int \frac {a+b \log \left (c \left (4 d-d g x^2\right )^p\right )}{x^2 \sqrt {4-g x^2}} \, dx=\frac {-2 \sqrt {g}\, \mathit {asin} \left (\frac {\sqrt {g}\, x}{2}\right ) b p x -\sqrt {-g \,x^{2}+4}\, \mathrm {log}\left (\frac {d^{p} \left (-\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {g}\, x}{2}\right )}{2}\right )^{2}+1\right )^{2 p} 4^{p} c}{\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {g}\, x}{2}\right )}{2}\right )^{2}+1\right )^{2 p}}\right ) b -\sqrt {-g \,x^{2}+4}\, a}{4 x} \] Input:
int((a+b*log(c*(-d*g*x^2+4*d)^p))/x^2/(-g*x^2+4)^(1/2),x)
Output:
( - 2*sqrt(g)*asin((sqrt(g)*x)/2)*b*p*x - sqrt( - g*x**2 + 4)*log((d**p*( - tan(asin((sqrt(g)*x)/2)/2)**2 + 1)**(2*p)*4**p*c)/(tan(asin((sqrt(g)*x)/ 2)/2)**2 + 1)**(2*p))*b - sqrt( - g*x**2 + 4)*a)/(4*x)