Integrand size = 36, antiderivative size = 117 \[ \int \frac {x^3 \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{\sqrt {4-g x^2}} \, dx=\frac {8 b p \sqrt {4-g x^2}}{g^2}-\frac {2 b p \left (4-g x^2\right )^{3/2}}{9 g^2}-\frac {4 \sqrt {4-g x^2} \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{g^2}+\frac {\left (4-g x^2\right )^{3/2} \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{3 g^2} \] Output:
8*b*p*(-g*x^2+4)^(1/2)/g^2-2/9*b*p*(-g*x^2+4)^(3/2)/g^2-4*(-g*x^2+4)^(1/2) *(a+b*ln(c*(-d*g*x^2+4*d)^p))/g^2+1/3*(-g*x^2+4)^(3/2)*(a+b*ln(c*(-d*g*x^2 +4*d)^p))/g^2
Time = 0.09 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.56 \[ \int \frac {x^3 \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{\sqrt {4-g x^2}} \, dx=\frac {\sqrt {4-g x^2} \left (-3 a \left (8+g x^2\right )+2 b p \left (32+g x^2\right )-3 b \left (8+g x^2\right ) \log \left (c \left (d \left (4-g x^2\right )\right )^p\right )\right )}{9 g^2} \] Input:
Integrate[(x^3*(a + b*Log[c*(4*d - d*g*x^2)^p]))/Sqrt[4 - g*x^2],x]
Output:
(Sqrt[4 - g*x^2]*(-3*a*(8 + g*x^2) + 2*b*p*(32 + g*x^2) - 3*b*(8 + g*x^2)* Log[c*(d*(4 - g*x^2))^p]))/(9*g^2)
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.
\(\displaystyle \int \frac {x^3 \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{\sqrt {4-g x^2}} \, dx\) |
\(\Big \downarrow \) 2929 |
\(\displaystyle \int \frac {x^3 \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{\sqrt {4-g x^2}}dx\) |
Input:
Int[(x^3*(a + b*Log[c*(4*d - d*g*x^2)^p]))/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]
\[\int \frac {x^{3} \left (a +b \ln \left (c \left (-d g \,x^{2}+4 d \right )^{p}\right )\right )}{\sqrt {-g \,x^{2}+4}}d x\]
Input:
int(x^3*(a+b*ln(c*(-d*g*x^2+4*d)^p))/(-g*x^2+4)^(1/2),x)
Output:
int(x^3*(a+b*ln(c*(-d*g*x^2+4*d)^p))/(-g*x^2+4)^(1/2),x)
Time = 0.11 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.66 \[ \int \frac {x^3 \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{\sqrt {4-g x^2}} \, dx=\frac {{\left ({\left (2 \, b g p - 3 \, a g\right )} x^{2} + 64 \, b p - 3 \, {\left (b g p x^{2} + 8 \, b p\right )} \log \left (-d g x^{2} + 4 \, d\right ) - 3 \, {\left (b g x^{2} + 8 \, b\right )} \log \left (c\right ) - 24 \, a\right )} \sqrt {-g x^{2} + 4}}{9 \, g^{2}} \] Input:
integrate(x^3*(a+b*log(c*(-d*g*x^2+4*d)^p))/(-g*x^2+4)^(1/2),x, algorithm= "fricas")
Output:
1/9*((2*b*g*p - 3*a*g)*x^2 + 64*b*p - 3*(b*g*p*x^2 + 8*b*p)*log(-d*g*x^2 + 4*d) - 3*(b*g*x^2 + 8*b)*log(c) - 24*a)*sqrt(-g*x^2 + 4)/g^2
Timed out. \[ \int \frac {x^3 \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{\sqrt {4-g x^2}} \, dx=\text {Timed out} \] Input:
integrate(x**3*(a+b*ln(c*(-d*g*x**2+4*d)**p))/(-g*x**2+4)**(1/2),x)
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.02 \[ \int \frac {x^3 \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{\sqrt {4-g x^2}} \, dx=-\frac {1}{3} \, {\left (\frac {\sqrt {-g x^{2} + 4} x^{2}}{g} + \frac {8 \, \sqrt {-g x^{2} + 4}}{g^{2}}\right )} b \log \left ({\left (-d g x^{2} + 4 \, d\right )}^{p} c\right ) - \frac {1}{3} \, {\left (\frac {\sqrt {-g x^{2} + 4} x^{2}}{g} + \frac {8 \, \sqrt {-g x^{2} + 4}}{g^{2}}\right )} a - \frac {2 \, {\left ({\left (-g x^{2} + 4\right )}^{\frac {3}{2}} - 36 \, \sqrt {-g x^{2} + 4}\right )} b p}{9 \, g^{2}} \] Input:
integrate(x^3*(a+b*log(c*(-d*g*x^2+4*d)^p))/(-g*x^2+4)^(1/2),x, algorithm= "maxima")
Output:
-1/3*(sqrt(-g*x^2 + 4)*x^2/g + 8*sqrt(-g*x^2 + 4)/g^2)*b*log((-d*g*x^2 + 4 *d)^p*c) - 1/3*(sqrt(-g*x^2 + 4)*x^2/g + 8*sqrt(-g*x^2 + 4)/g^2)*a - 2/9*( (-g*x^2 + 4)^(3/2) - 36*sqrt(-g*x^2 + 4))*b*p/g^2
Time = 0.12 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.15 \[ \int \frac {x^3 \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{\sqrt {4-g x^2}} \, dx=\frac {\frac {{\left (3 \, {\left ({\left (-g x^{2} + 4\right )}^{\frac {3}{2}} - 12 \, \sqrt {-g x^{2} + 4}\right )} \log \left (-d g x^{2} + 4 \, d\right ) - 2 \, {\left (-g x^{2} + 4\right )}^{\frac {3}{2}} + 72 \, \sqrt {-g x^{2} + 4}\right )} b p}{g} + \frac {3 \, {\left ({\left (-g x^{2} + 4\right )}^{\frac {3}{2}} - 12 \, \sqrt {-g x^{2} + 4}\right )} b \log \left (c\right )}{g} + \frac {3 \, {\left ({\left (-g x^{2} + 4\right )}^{\frac {3}{2}} - 12 \, \sqrt {-g x^{2} + 4}\right )} a}{g}}{9 \, g} \] Input:
integrate(x^3*(a+b*log(c*(-d*g*x^2+4*d)^p))/(-g*x^2+4)^(1/2),x, algorithm= "giac")
Output:
1/9*((3*((-g*x^2 + 4)^(3/2) - 12*sqrt(-g*x^2 + 4))*log(-d*g*x^2 + 4*d) - 2 *(-g*x^2 + 4)^(3/2) + 72*sqrt(-g*x^2 + 4))*b*p/g + 3*((-g*x^2 + 4)^(3/2) - 12*sqrt(-g*x^2 + 4))*b*log(c)/g + 3*((-g*x^2 + 4)^(3/2) - 12*sqrt(-g*x^2 + 4))*a/g)/g
Timed out. \[ \int \frac {x^3 \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{\sqrt {4-g x^2}} \, dx=\int \frac {x^3\,\left (a+b\,\ln \left (c\,{\left (4\,d-d\,g\,x^2\right )}^p\right )\right )}{\sqrt {4-g\,x^2}} \,d x \] Input:
int((x^3*(a + b*log(c*(4*d - d*g*x^2)^p)))/(4 - g*x^2)^(1/2),x)
Output:
int((x^3*(a + b*log(c*(4*d - d*g*x^2)^p)))/(4 - g*x^2)^(1/2), x)
Time = 0.16 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.85 \[ \int \frac {x^3 \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{\sqrt {4-g x^2}} \, dx=\frac {\sqrt {-g \,x^{2}+4}\, \left (-3 \,\mathrm {log}\left (\frac {d^{p} \left (-g \,x^{2}+4\right )^{p} 4^{p} c}{2^{2 p}}\right ) b g \,x^{2}-24 \,\mathrm {log}\left (\frac {d^{p} \left (-g \,x^{2}+4\right )^{p} 4^{p} c}{2^{2 p}}\right ) b -3 a g \,x^{2}-24 a +2 b g p \,x^{2}+64 b p \right )}{9 g^{2}} \] Input:
int(x^3*(a+b*log(c*(-d*g*x^2+4*d)^p))/(-g*x^2+4)^(1/2),x)
Output:
(sqrt( - g*x**2 + 4)*( - 3*log((d**p*( - g*x**2 + 4)**p*4**p*c)/2**(2*p))* b*g*x**2 - 24*log((d**p*( - g*x**2 + 4)**p*4**p*c)/2**(2*p))*b - 3*a*g*x** 2 - 24*a + 2*b*g*p*x**2 + 64*b*p))/(9*g**2)