\(\int \frac {x^2 (a+b \log (c (4 d-d g x^2)^p))}{\sqrt {4-g x^2}} \, dx\) [683]

Optimal result

Integrand size = 36, antiderivative size = 568 \[ \int \frac {x^2 \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{\sqrt {4-g x^2}} \, dx=\frac {2 b p \sqrt {-\frac {4}{g}+x^2}}{g^2 \sqrt {4-g x^2} \left (x+\sqrt {-\frac {4}{g}+x^2}\right )^2}+\frac {a \sqrt {-\frac {4}{g}+x^2} \left (x+\sqrt {-\frac {4}{g}+x^2}\right )^2}{8 \sqrt {4-g x^2}}-\frac {b p \sqrt {-\frac {4}{g}+x^2} \left (x+\sqrt {-\frac {4}{g}+x^2}\right )^2}{8 \sqrt {4-g x^2}}+\frac {b \sqrt {-\frac {4}{g}+x^2} \left (x+\sqrt {-\frac {4}{g}+x^2}\right )^2 \log \left (c \left (4 d-d g x^2\right )^p\right )}{8 \sqrt {4-g x^2}}-\frac {2 \sqrt {-\frac {4}{g}+x^2} \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{g^2 \sqrt {4-g x^2} \left (x+\sqrt {-\frac {4}{g}+x^2}\right )^2}-\frac {2 b p \sqrt {-\frac {4}{g}+x^2} \log \left (x+\sqrt {-\frac {4}{g}+x^2}\right )}{g \sqrt {4-g x^2}}+\frac {2 \sqrt {-\frac {4}{g}+x^2} \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right ) \log \left (x+\sqrt {-\frac {4}{g}+x^2}\right )}{g \sqrt {4-g x^2}}+\frac {2 b p \sqrt {-\frac {4}{g}+x^2} \log ^2\left (x+\sqrt {-\frac {4}{g}+x^2}\right )}{g \sqrt {4-g x^2}}-\frac {4 b p \sqrt {-\frac {4}{g}+x^2} \log \left (x+\sqrt {-\frac {4}{g}+x^2}\right ) \log \left (1-\frac {1}{4} g \left (x+\sqrt {-\frac {4}{g}+x^2}\right )^2\right )}{g \sqrt {4-g x^2}}-\frac {2 b p \sqrt {-\frac {4}{g}+x^2} \operatorname {PolyLog}\left (2,\frac {1}{4} g \left (x+\sqrt {-\frac {4}{g}+x^2}\right )^2\right )}{g \sqrt {4-g x^2}} \] Output:

2*b*p*(-4/g+x^2)^(1/2)/g^2/(-g*x^2+4)^(1/2)/(x+(-4/g+x^2)^(1/2))^2+1/8*a*( 
-4/g+x^2)^(1/2)*(x+(-4/g+x^2)^(1/2))^2/(-g*x^2+4)^(1/2)-1/8*b*p*(-4/g+x^2) 
^(1/2)*(x+(-4/g+x^2)^(1/2))^2/(-g*x^2+4)^(1/2)+1/8*b*(-4/g+x^2)^(1/2)*(x+( 
-4/g+x^2)^(1/2))^2*ln(c*(-d*g*x^2+4*d)^p)/(-g*x^2+4)^(1/2)-2*(-4/g+x^2)^(1 
/2)*(a+b*ln(c*(-d*g*x^2+4*d)^p))/g^2/(-g*x^2+4)^(1/2)/(x+(-4/g+x^2)^(1/2)) 
^2-2*b*p*(-4/g+x^2)^(1/2)*ln(x+(-4/g+x^2)^(1/2))/g/(-g*x^2+4)^(1/2)+2*(-4/ 
g+x^2)^(1/2)*(a+b*ln(c*(-d*g*x^2+4*d)^p))*ln(x+(-4/g+x^2)^(1/2))/g/(-g*x^2 
+4)^(1/2)+2*b*p*(-4/g+x^2)^(1/2)*ln(x+(-4/g+x^2)^(1/2))^2/g/(-g*x^2+4)^(1/ 
2)-4*b*p*(-4/g+x^2)^(1/2)*ln(x+(-4/g+x^2)^(1/2))*ln(1-1/4*g*(x+(-4/g+x^2)^ 
(1/2))^2)/g/(-g*x^2+4)^(1/2)-2*b*p*(-4/g+x^2)^(1/2)*polylog(2,1/4*g*(x+(-4 
/g+x^2)^(1/2))^2)/g/(-g*x^2+4)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.59 (sec) , antiderivative size = 477, normalized size of antiderivative = 0.84 \[ \int \frac {x^2 \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{\sqrt {4-g x^2}} \, dx=-\frac {a \sqrt {g} x \sqrt {4-g x^2}-b \sqrt {g} p x \sqrt {4-g x^2}-4 a \arcsin \left (\frac {\sqrt {g} x}{2}\right )+4 b p \arcsin \left (\frac {\sqrt {g} x}{2}\right )+8 i b p \pi \arcsin \left (\frac {\sqrt {g} x}{2}\right )-4 i b p \arcsin \left (\frac {\sqrt {g} x}{2}\right )^2+16 b p \pi \log \left (1+e^{-i \arcsin \left (\frac {\sqrt {g} x}{2}\right )}\right )+4 b p \pi \log \left (1-i e^{i \arcsin \left (\frac {\sqrt {g} x}{2}\right )}\right )+8 b p \arcsin \left (\frac {\sqrt {g} x}{2}\right ) \log \left (1-i e^{i \arcsin \left (\frac {\sqrt {g} x}{2}\right )}\right )-4 b p \pi \log \left (1+i e^{i \arcsin \left (\frac {\sqrt {g} x}{2}\right )}\right )+8 b p \arcsin \left (\frac {\sqrt {g} x}{2}\right ) \log \left (1+i e^{i \arcsin \left (\frac {\sqrt {g} x}{2}\right )}\right )+b \sqrt {g} x \sqrt {4-g x^2} \log \left (c \left (d \left (4-g x^2\right )\right )^p\right )-4 b \arcsin \left (\frac {\sqrt {g} x}{2}\right ) \log \left (c \left (d \left (4-g x^2\right )\right )^p\right )-16 b p \pi \log \left (\cos \left (\frac {1}{2} \arcsin \left (\frac {\sqrt {g} x}{2}\right )\right )\right )+4 b p \pi \log \left (-\cos \left (\frac {1}{4} \left (\pi +2 \arcsin \left (\frac {\sqrt {g} x}{2}\right )\right )\right )\right )-4 b p \pi \log \left (\sin \left (\frac {1}{4} \left (\pi +2 \arcsin \left (\frac {\sqrt {g} x}{2}\right )\right )\right )\right )-8 i b p \operatorname {PolyLog}\left (2,-i e^{i \arcsin \left (\frac {\sqrt {g} x}{2}\right )}\right )-8 i b p \operatorname {PolyLog}\left (2,i e^{i \arcsin \left (\frac {\sqrt {g} x}{2}\right )}\right )}{2 g^{3/2}} \] Input:

Integrate[(x^2*(a + b*Log[c*(4*d - d*g*x^2)^p]))/Sqrt[4 - g*x^2],x]
 

Output:

-1/2*(a*Sqrt[g]*x*Sqrt[4 - g*x^2] - b*Sqrt[g]*p*x*Sqrt[4 - g*x^2] - 4*a*Ar 
cSin[(Sqrt[g]*x)/2] + 4*b*p*ArcSin[(Sqrt[g]*x)/2] + (8*I)*b*p*Pi*ArcSin[(S 
qrt[g]*x)/2] - (4*I)*b*p*ArcSin[(Sqrt[g]*x)/2]^2 + 16*b*p*Pi*Log[1 + E^((- 
I)*ArcSin[(Sqrt[g]*x)/2])] + 4*b*p*Pi*Log[1 - I*E^(I*ArcSin[(Sqrt[g]*x)/2] 
)] + 8*b*p*ArcSin[(Sqrt[g]*x)/2]*Log[1 - I*E^(I*ArcSin[(Sqrt[g]*x)/2])] - 
4*b*p*Pi*Log[1 + I*E^(I*ArcSin[(Sqrt[g]*x)/2])] + 8*b*p*ArcSin[(Sqrt[g]*x) 
/2]*Log[1 + I*E^(I*ArcSin[(Sqrt[g]*x)/2])] + b*Sqrt[g]*x*Sqrt[4 - g*x^2]*L 
og[c*(d*(4 - g*x^2))^p] - 4*b*ArcSin[(Sqrt[g]*x)/2]*Log[c*(d*(4 - g*x^2))^ 
p] - 16*b*p*Pi*Log[Cos[ArcSin[(Sqrt[g]*x)/2]/2]] + 4*b*p*Pi*Log[-Cos[(Pi + 
 2*ArcSin[(Sqrt[g]*x)/2])/4]] - 4*b*p*Pi*Log[Sin[(Pi + 2*ArcSin[(Sqrt[g]*x 
)/2])/4]] - (8*I)*b*p*PolyLog[2, (-I)*E^(I*ArcSin[(Sqrt[g]*x)/2])] - (8*I) 
*b*p*PolyLog[2, I*E^(I*ArcSin[(Sqrt[g]*x)/2])])/g^(3/2)
 

Rubi [F]

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.

\(\displaystyle \int \frac {x^2 \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{\sqrt {4-g x^2}} \, dx\)

\(\Big \downarrow \) 2929

\(\displaystyle \int \frac {x^2 \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{\sqrt {4-g x^2}}dx\)

Input:

Int[(x^2*(a + b*Log[c*(4*d - d*g*x^2)^p]))/Sqrt[4 - g*x^2],x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 2929

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]
 

Maple [F]

\[\int \frac {x^{2} \left (a +b \ln \left (c \left (-d g \,x^{2}+4 d \right )^{p}\right )\right )}{\sqrt {-g \,x^{2}+4}}d x\]

Input:

int(x^2*(a+b*ln(c*(-d*g*x^2+4*d)^p))/(-g*x^2+4)^(1/2),x)
 

Output:

int(x^2*(a+b*ln(c*(-d*g*x^2+4*d)^p))/(-g*x^2+4)^(1/2),x)
 

Fricas [F]

\[ \int \frac {x^2 \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{\sqrt {4-g x^2}} \, dx=\int { \frac {{\left (b \log \left ({\left (-d g x^{2} + 4 \, d\right )}^{p} c\right ) + a\right )} x^{2}}{\sqrt {-g x^{2} + 4}} \,d x } \] Input:

integrate(x^2*(a+b*log(c*(-d*g*x^2+4*d)^p))/(-g*x^2+4)^(1/2),x, algorithm= 
"fricas")
 

Output:

integral(-(sqrt(-g*x^2 + 4)*b*x^2*log((-d*g*x^2 + 4*d)^p*c) + sqrt(-g*x^2 
+ 4)*a*x^2)/(g*x^2 - 4), x)
 

Sympy [F]

\[ \int \frac {x^2 \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{\sqrt {4-g x^2}} \, dx=\int \frac {x^{2} \left (a + b \log {\left (c \left (- d g x^{2} + 4 d\right )^{p} \right )}\right )}{\sqrt {- g x^{2} + 4}}\, dx \] Input:

integrate(x**2*(a+b*ln(c*(-d*g*x**2+4*d)**p))/(-g*x**2+4)**(1/2),x)
 

Output:

Integral(x**2*(a + b*log(c*(-d*g*x**2 + 4*d)**p))/sqrt(-g*x**2 + 4), x)
 

Maxima [F]

\[ \int \frac {x^2 \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{\sqrt {4-g x^2}} \, dx=\int { \frac {{\left (b \log \left ({\left (-d g x^{2} + 4 \, d\right )}^{p} c\right ) + a\right )} x^{2}}{\sqrt {-g x^{2} + 4}} \,d x } \] Input:

integrate(x^2*(a+b*log(c*(-d*g*x^2+4*d)^p))/(-g*x^2+4)^(1/2),x, algorithm= 
"maxima")
 

Output:

-1/2*a*(sqrt(-g*x^2 + 4)*x/g - 4*arcsin(1/2*sqrt(g)*x)/g^(3/2)) + b*integr 
ate(((p*log(d) + log(c))*x^2 + x^2*log((-g*x^2 + 4)^p))/sqrt(-g*x^2 + 4), 
x)
 

Giac [F]

\[ \int \frac {x^2 \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{\sqrt {4-g x^2}} \, dx=\int { \frac {{\left (b \log \left ({\left (-d g x^{2} + 4 \, d\right )}^{p} c\right ) + a\right )} x^{2}}{\sqrt {-g x^{2} + 4}} \,d x } \] Input:

integrate(x^2*(a+b*log(c*(-d*g*x^2+4*d)^p))/(-g*x^2+4)^(1/2),x, algorithm= 
"giac")
 

Output:

integrate((b*log((-d*g*x^2 + 4*d)^p*c) + a)*x^2/sqrt(-g*x^2 + 4), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{\sqrt {4-g x^2}} \, dx=\int \frac {x^2\,\left (a+b\,\ln \left (c\,{\left (4\,d-d\,g\,x^2\right )}^p\right )\right )}{\sqrt {4-g\,x^2}} \,d x \] Input:

int((x^2*(a + b*log(c*(4*d - d*g*x^2)^p)))/(4 - g*x^2)^(1/2),x)
 

Output:

int((x^2*(a + b*log(c*(4*d - d*g*x^2)^p)))/(4 - g*x^2)^(1/2), x)
 

Reduce [F]

\[ \int \frac {x^2 \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{\sqrt {4-g x^2}} \, dx=\frac {4 \sqrt {g}\, \mathit {asin} \left (\frac {\sqrt {g}\, x}{2}\right ) a -\sqrt {-g \,x^{2}+4}\, a g x +2 \left (\int \frac {\mathrm {log}\left (\left (-d g \,x^{2}+4 d \right )^{p} c \right ) x^{2}}{\sqrt {-g \,x^{2}+4}}d x \right ) b \,g^{2}}{2 g^{2}} \] Input:

int(x^2*(a+b*log(c*(-d*g*x^2+4*d)^p))/(-g*x^2+4)^(1/2),x)
 

Output:

(4*sqrt(g)*asin((sqrt(g)*x)/2)*a - sqrt( - g*x**2 + 4)*a*g*x + 2*int((log( 
( - d*g*x**2 + 4*d)**p*c)*x**2)/sqrt( - g*x**2 + 4),x)*b*g**2)/(2*g**2)