Integrand size = 36, antiderivative size = 124 \[ \int \frac {a+b \log \left (c \left (4 d-d g x^2\right )^p\right )}{x^4 \sqrt {4-g x^2}} \, dx=\frac {b g p \sqrt {4-g x^2}}{24 x}-\frac {1}{12} b g^{3/2} p \arcsin \left (\frac {\sqrt {g} x}{2}\right )-\frac {\sqrt {4-g x^2} \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{12 x^3}-\frac {g \sqrt {4-g x^2} \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{24 x} \] Output:
1/24*b*g*p*(-g*x^2+4)^(1/2)/x-1/12*b*g^(3/2)*p*arcsin(1/2*g^(1/2)*x)-1/12* (-g*x^2+4)^(1/2)*(a+b*ln(c*(-d*g*x^2+4*d)^p))/x^3-1/24*g*(-g*x^2+4)^(1/2)* (a+b*ln(c*(-d*g*x^2+4*d)^p))/x
Time = 0.21 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.68 \[ \int \frac {a+b \log \left (c \left (4 d-d g x^2\right )^p\right )}{x^4 \sqrt {4-g x^2}} \, dx=\frac {1}{24} \left (-2 b g^{3/2} p \arcsin \left (\frac {\sqrt {g} x}{2}\right )-\frac {\sqrt {4-g x^2} \left (2 a+a g x^2-b g p x^2+b \left (2+g x^2\right ) \log \left (c \left (d \left (4-g x^2\right )\right )^p\right )\right )}{x^3}\right ) \] Input:
Integrate[(a + b*Log[c*(4*d - d*g*x^2)^p])/(x^4*Sqrt[4 - g*x^2]),x]
Output:
(-2*b*g^(3/2)*p*ArcSin[(Sqrt[g]*x)/2] - (Sqrt[4 - g*x^2]*(2*a + a*g*x^2 - b*g*p*x^2 + b*(2 + g*x^2)*Log[c*(d*(4 - g*x^2))^p]))/x^3)/24
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.
\(\displaystyle \int \frac {a+b \log \left (c \left (4 d-d g x^2\right )^p\right )}{x^4 \sqrt {4-g x^2}} \, dx\) |
\(\Big \downarrow \) 2929 |
\(\displaystyle \int \frac {a+b \log \left (c \left (4 d-d g x^2\right )^p\right )}{x^4 \sqrt {4-g x^2}}dx\) |
Input:
Int[(a + b*Log[c*(4*d - d*g*x^2)^p])/(x^4*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]
\[\int \frac {a +b \ln \left (c \left (-d g \,x^{2}+4 d \right )^{p}\right )}{x^{4} \sqrt {-g \,x^{2}+4}}d x\]
Input:
int((a+b*ln(c*(-d*g*x^2+4*d)^p))/x^4/(-g*x^2+4)^(1/2),x)
Output:
int((a+b*ln(c*(-d*g*x^2+4*d)^p))/x^4/(-g*x^2+4)^(1/2),x)
Time = 0.10 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.73 \[ \int \frac {a+b \log \left (c \left (4 d-d g x^2\right )^p\right )}{x^4 \sqrt {4-g x^2}} \, dx=\left [\frac {2 \, b \sqrt {-g} g p x^{3} \log \left (-\sqrt {-g} x - \sqrt {-g x^{2} + 4}\right ) + {\left ({\left (b g p - a g\right )} x^{2} - {\left (b g p x^{2} + 2 \, b p\right )} \log \left (-d g x^{2} + 4 \, d\right ) - {\left (b g x^{2} + 2 \, b\right )} \log \left (c\right ) - 2 \, a\right )} \sqrt {-g x^{2} + 4}}{24 \, x^{3}}, \frac {4 \, b g^{\frac {3}{2}} p x^{3} \arctan \left (\frac {\sqrt {-g x^{2} + 4} - 2}{\sqrt {g} x}\right ) + {\left ({\left (b g p - a g\right )} x^{2} - {\left (b g p x^{2} + 2 \, b p\right )} \log \left (-d g x^{2} + 4 \, d\right ) - {\left (b g x^{2} + 2 \, b\right )} \log \left (c\right ) - 2 \, a\right )} \sqrt {-g x^{2} + 4}}{24 \, x^{3}}\right ] \] Input:
integrate((a+b*log(c*(-d*g*x^2+4*d)^p))/x^4/(-g*x^2+4)^(1/2),x, algorithm= "fricas")
Output:
[1/24*(2*b*sqrt(-g)*g*p*x^3*log(-sqrt(-g)*x - sqrt(-g*x^2 + 4)) + ((b*g*p - a*g)*x^2 - (b*g*p*x^2 + 2*b*p)*log(-d*g*x^2 + 4*d) - (b*g*x^2 + 2*b)*log (c) - 2*a)*sqrt(-g*x^2 + 4))/x^3, 1/24*(4*b*g^(3/2)*p*x^3*arctan((sqrt(-g* x^2 + 4) - 2)/(sqrt(g)*x)) + ((b*g*p - a*g)*x^2 - (b*g*p*x^2 + 2*b*p)*log( -d*g*x^2 + 4*d) - (b*g*x^2 + 2*b)*log(c) - 2*a)*sqrt(-g*x^2 + 4))/x^3]
\[ \int \frac {a+b \log \left (c \left (4 d-d g x^2\right )^p\right )}{x^4 \sqrt {4-g x^2}} \, dx=\int \frac {a + b \log {\left (c \left (- d g x^{2} + 4 d\right )^{p} \right )}}{x^{4} \sqrt {- g x^{2} + 4}}\, dx \] Input:
integrate((a+b*ln(c*(-d*g*x**2+4*d)**p))/x**4/(-g*x**2+4)**(1/2),x)
Output:
Integral((a + b*log(c*(-d*g*x**2 + 4*d)**p))/(x**4*sqrt(-g*x**2 + 4)), x)
\[ \int \frac {a+b \log \left (c \left (4 d-d g x^2\right )^p\right )}{x^4 \sqrt {4-g x^2}} \, dx=\int { \frac {b \log \left ({\left (-d g x^{2} + 4 \, d\right )}^{p} c\right ) + a}{\sqrt {-g x^{2} + 4} x^{4}} \,d x } \] Input:
integrate((a+b*log(c*(-d*g*x^2+4*d)^p))/x^4/(-g*x^2+4)^(1/2),x, algorithm= "maxima")
Output:
-1/24*a*(sqrt(-g*x^2 + 4)*g/x + 2*sqrt(-g*x^2 + 4)/x^3) + 1/24*b*((g^2*x^4 - 2*g*x^2 - 8)*log((-g*x^2 + 4)^p)/(sqrt(-g*x^2 + 4)*x^3) - 2*integrate(( g^2*p*x^4 + 2*g*p*x^2 - 12*p*log(d) - 12*log(c))/(sqrt(-g*x^2 + 4)*x^4), x ))
Exception generated. \[ \int \frac {a+b \log \left (c \left (4 d-d g x^2\right )^p\right )}{x^4 \sqrt {4-g x^2}} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((a+b*log(c*(-d*g*x^2+4*d)^p))/x^4/(-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. \[ \int \frac {a+b \log \left (c \left (4 d-d g x^2\right )^p\right )}{x^4 \sqrt {4-g x^2}} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (4\,d-d\,g\,x^2\right )}^p\right )}{x^4\,\sqrt {4-g\,x^2}} \,d x \] Input:
int((a + b*log(c*(4*d - d*g*x^2)^p))/(x^4*(4 - g*x^2)^(1/2)),x)
Output:
int((a + b*log(c*(4*d - d*g*x^2)^p))/(x^4*(4 - g*x^2)^(1/2)), x)
Time = 0.17 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.52 \[ \int \frac {a+b \log \left (c \left (4 d-d g x^2\right )^p\right )}{x^4 \sqrt {4-g x^2}} \, dx=\frac {-2 \sqrt {g}\, \mathit {asin} \left (\frac {\sqrt {g}\, x}{2}\right ) b g p \,x^{3}-\sqrt {-g \,x^{2}+4}\, \mathrm {log}\left (\frac {d^{p} \left (-\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {g}\, x}{2}\right )}{2}\right )^{2}+1\right )^{2 p} 4^{p} c}{\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {g}\, x}{2}\right )}{2}\right )^{2}+1\right )^{2 p}}\right ) b g \,x^{2}-2 \sqrt {-g \,x^{2}+4}\, \mathrm {log}\left (\frac {d^{p} \left (-\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {g}\, x}{2}\right )}{2}\right )^{2}+1\right )^{2 p} 4^{p} c}{\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {g}\, x}{2}\right )}{2}\right )^{2}+1\right )^{2 p}}\right ) b -\sqrt {-g \,x^{2}+4}\, a g \,x^{2}-2 \sqrt {-g \,x^{2}+4}\, a +\sqrt {-g \,x^{2}+4}\, b g p \,x^{2}}{24 x^{3}} \] Input:
int((a+b*log(c*(-d*g*x^2+4*d)^p))/x^4/(-g*x^2+4)^(1/2),x)
Output:
( - 2*sqrt(g)*asin((sqrt(g)*x)/2)*b*g*p*x**3 - 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*g*x**2 - 2*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*g*x**2 - 2*sqrt( - g*x**2 + 4)*a + sqrt( - g*x**2 + 4)*b*g*p*x**2)/(24*x**3)