Integrand size = 36, antiderivative size = 119 \[ \int \frac {x^3 \left (a+b \log \left (c \left (d f-d g x^2\right )^p\right )\right )}{\sqrt {f-g x^2}} \, dx=\frac {2 b f p \sqrt {f-g x^2}}{g^2}-\frac {2 b p \left (f-g x^2\right )^{3/2}}{9 g^2}-\frac {f \sqrt {f-g x^2} \left (a+b \log \left (c \left (d f-d g x^2\right )^p\right )\right )}{g^2}+\frac {\left (f-g x^2\right )^{3/2} \left (a+b \log \left (c \left (d f-d g x^2\right )^p\right )\right )}{3 g^2} \] Output:
2*b*f*p*(-g*x^2+f)^(1/2)/g^2-2/9*b*p*(-g*x^2+f)^(3/2)/g^2-f*(-g*x^2+f)^(1/ 2)*(a+b*ln(c*(-d*g*x^2+d*f)^p))/g^2+1/3*(-g*x^2+f)^(3/2)*(a+b*ln(c*(-d*g*x ^2+d*f)^p))/g^2
Time = 0.09 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.61 \[ \int \frac {x^3 \left (a+b \log \left (c \left (d f-d g x^2\right )^p\right )\right )}{\sqrt {f-g x^2}} \, dx=\frac {\sqrt {f-g x^2} \left (-3 a \left (2 f+g x^2\right )+2 b p \left (8 f+g x^2\right )-3 b \left (2 f+g x^2\right ) \log \left (c \left (d \left (f-g x^2\right )\right )^p\right )\right )}{9 g^2} \] Input:
Integrate[(x^3*(a + b*Log[c*(d*f - d*g*x^2)^p]))/Sqrt[f - g*x^2],x]
Output:
(Sqrt[f - g*x^2]*(-3*a*(2*f + g*x^2) + 2*b*p*(8*f + g*x^2) - 3*b*(2*f + g* x^2)*Log[c*(d*(f - 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.
\(\displaystyle \int \frac {x^3 \left (a+b \log \left (c \left (d f-d g x^2\right )^p\right )\right )}{\sqrt {f-g x^2}} \, dx\) |
\(\Big \downarrow \) 2929 |
\(\displaystyle \int \frac {x^3 \left (a+b \log \left (c \left (d f-d g x^2\right )^p\right )\right )}{\sqrt {f-g x^2}}dx\) |
Input:
Int[(x^3*(a + b*Log[c*(d*f - d*g*x^2)^p]))/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]
\[\int \frac {x^{3} \left (a +b \ln \left (c \left (-d g \,x^{2}+d f \right )^{p}\right )\right )}{\sqrt {-g \,x^{2}+f}}d x\]
Input:
int(x^3*(a+b*ln(c*(-d*g*x^2+d*f)^p))/(-g*x^2+f)^(1/2),x)
Output:
int(x^3*(a+b*ln(c*(-d*g*x^2+d*f)^p))/(-g*x^2+f)^(1/2),x)
Time = 0.09 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.68 \[ \int \frac {x^3 \left (a+b \log \left (c \left (d f-d g x^2\right )^p\right )\right )}{\sqrt {f-g x^2}} \, dx=\frac {{\left (16 \, b f p + {\left (2 \, b g p - 3 \, a g\right )} x^{2} - 6 \, a f - 3 \, {\left (b g p x^{2} + 2 \, b f p\right )} \log \left (-d g x^{2} + d f\right ) - 3 \, {\left (b g x^{2} + 2 \, b f\right )} \log \left (c\right )\right )} \sqrt {-g x^{2} + f}}{9 \, g^{2}} \] Input:
integrate(x^3*(a+b*log(c*(-d*g*x^2+d*f)^p))/(-g*x^2+f)^(1/2),x, algorithm= "fricas")
Output:
1/9*(16*b*f*p + (2*b*g*p - 3*a*g)*x^2 - 6*a*f - 3*(b*g*p*x^2 + 2*b*f*p)*lo g(-d*g*x^2 + d*f) - 3*(b*g*x^2 + 2*b*f)*log(c))*sqrt(-g*x^2 + f)/g^2
\[ \int \frac {x^3 \left (a+b \log \left (c \left (d f-d g x^2\right )^p\right )\right )}{\sqrt {f-g x^2}} \, dx=\int \frac {x^{3} \left (a + b \log {\left (c \left (d f - d g x^{2}\right )^{p} \right )}\right )}{\sqrt {f - g x^{2}}}\, dx \] Input:
integrate(x**3*(a+b*ln(c*(-d*g*x**2+d*f)**p))/(-g*x**2+f)**(1/2),x)
Output:
Integral(x**3*(a + b*log(c*(d*f - d*g*x**2)**p))/sqrt(f - g*x**2), x)
Time = 0.12 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.03 \[ \int \frac {x^3 \left (a+b \log \left (c \left (d f-d g x^2\right )^p\right )\right )}{\sqrt {f-g x^2}} \, dx=-\frac {1}{3} \, {\left (\frac {\sqrt {-g x^{2} + f} x^{2}}{g} + \frac {2 \, \sqrt {-g x^{2} + f} f}{g^{2}}\right )} b \log \left ({\left (-d g x^{2} + d f\right )}^{p} c\right ) - \frac {1}{3} \, {\left (\frac {\sqrt {-g x^{2} + f} x^{2}}{g} + \frac {2 \, \sqrt {-g x^{2} + f} f}{g^{2}}\right )} a - \frac {2 \, {\left ({\left (-g x^{2} + f\right )}^{\frac {3}{2}} - 9 \, \sqrt {-g x^{2} + f} f\right )} b p}{9 \, g^{2}} \] Input:
integrate(x^3*(a+b*log(c*(-d*g*x^2+d*f)^p))/(-g*x^2+f)^(1/2),x, algorithm= "maxima")
Output:
-1/3*(sqrt(-g*x^2 + f)*x^2/g + 2*sqrt(-g*x^2 + f)*f/g^2)*b*log((-d*g*x^2 + d*f)^p*c) - 1/3*(sqrt(-g*x^2 + f)*x^2/g + 2*sqrt(-g*x^2 + f)*f/g^2)*a - 2 /9*((-g*x^2 + f)^(3/2) - 9*sqrt(-g*x^2 + f)*f)*b*p/g^2
Time = 0.13 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.16 \[ \int \frac {x^3 \left (a+b \log \left (c \left (d f-d g x^2\right )^p\right )\right )}{\sqrt {f-g x^2}} \, dx=\frac {\frac {{\left (3 \, {\left ({\left (-g x^{2} + f\right )}^{\frac {3}{2}} - 3 \, \sqrt {-g x^{2} + f} f\right )} \log \left (-d g x^{2} + d f\right ) - 2 \, {\left (-g x^{2} + f\right )}^{\frac {3}{2}} + 18 \, \sqrt {-g x^{2} + f} f\right )} b p}{g} + \frac {3 \, {\left ({\left (-g x^{2} + f\right )}^{\frac {3}{2}} - 3 \, \sqrt {-g x^{2} + f} f\right )} b \log \left (c\right )}{g} + \frac {3 \, {\left ({\left (-g x^{2} + f\right )}^{\frac {3}{2}} - 3 \, \sqrt {-g x^{2} + f} f\right )} a}{g}}{9 \, g} \] Input:
integrate(x^3*(a+b*log(c*(-d*g*x^2+d*f)^p))/(-g*x^2+f)^(1/2),x, algorithm= "giac")
Output:
1/9*((3*((-g*x^2 + f)^(3/2) - 3*sqrt(-g*x^2 + f)*f)*log(-d*g*x^2 + d*f) - 2*(-g*x^2 + f)^(3/2) + 18*sqrt(-g*x^2 + f)*f)*b*p/g + 3*((-g*x^2 + f)^(3/2 ) - 3*sqrt(-g*x^2 + f)*f)*b*log(c)/g + 3*((-g*x^2 + f)^(3/2) - 3*sqrt(-g*x ^2 + f)*f)*a/g)/g
Timed out. \[ \int \frac {x^3 \left (a+b \log \left (c \left (d f-d g x^2\right )^p\right )\right )}{\sqrt {f-g x^2}} \, dx=\int \frac {x^3\,\left (a+b\,\ln \left (c\,{\left (d\,f-d\,g\,x^2\right )}^p\right )\right )}{\sqrt {f-g\,x^2}} \,d x \] Input:
int((x^3*(a + b*log(c*(d*f - d*g*x^2)^p)))/(f - g*x^2)^(1/2),x)
Output:
int((x^3*(a + b*log(c*(d*f - d*g*x^2)^p)))/(f - g*x^2)^(1/2), x)
Time = 0.17 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.69 \[ \int \frac {x^3 \left (a+b \log \left (c \left (d f-d g x^2\right )^p\right )\right )}{\sqrt {f-g x^2}} \, dx=\frac {\sqrt {-g \,x^{2}+f}\, \left (-6 \,\mathrm {log}\left (d^{p} \left (-g \,x^{2}+f \right )^{p} c \right ) b f -3 \,\mathrm {log}\left (d^{p} \left (-g \,x^{2}+f \right )^{p} c \right ) b g \,x^{2}-6 a f -3 a g \,x^{2}+16 b f p +2 b g p \,x^{2}\right )}{9 g^{2}} \] Input:
int(x^3*(a+b*log(c*(-d*g*x^2+d*f)^p))/(-g*x^2+f)^(1/2),x)
Output:
(sqrt(f - g*x**2)*( - 6*log(d**p*(f - g*x**2)**p*c)*b*f - 3*log(d**p*(f - g*x**2)**p*c)*b*g*x**2 - 6*a*f - 3*a*g*x**2 + 16*b*f*p + 2*b*g*p*x**2))/(9 *g**2)