Integrand size = 25, antiderivative size = 155 \[ \int \frac {x^3 \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=-\frac {e f p \log \left (d+e x^2\right )}{2 g^2 (e f-d g)}+\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{2 g^2 \left (f+g x^2\right )}+\frac {e f p \log \left (f+g x^2\right )}{2 g^2 (e f-d g)}+\frac {\log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )}{2 g^2}+\frac {p \operatorname {PolyLog}\left (2,-\frac {g \left (d+e x^2\right )}{e f-d g}\right )}{2 g^2} \] Output:
-1/2*e*f*p*ln(e*x^2+d)/g^2/(-d*g+e*f)+1/2*f*ln(c*(e*x^2+d)^p)/g^2/(g*x^2+f )+1/2*e*f*p*ln(g*x^2+f)/g^2/(-d*g+e*f)+1/2*ln(c*(e*x^2+d)^p)*ln(e*(g*x^2+f )/(-d*g+e*f))/g^2+1/2*p*polylog(2,-g*(e*x^2+d)/(-d*g+e*f))/g^2
Time = 0.09 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.85 \[ \int \frac {x^3 \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=\frac {\frac {e f p \log \left (d+e x^2\right )}{-e f+d g}+\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2}+\frac {e f p \log \left (f+g x^2\right )}{e f-d g}+\log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )+p \operatorname {PolyLog}\left (2,\frac {g \left (d+e x^2\right )}{-e f+d g}\right )}{2 g^2} \] Input:
Integrate[(x^3*Log[c*(d + e*x^2)^p])/(f + g*x^2)^2,x]
Output:
((e*f*p*Log[d + e*x^2])/(-(e*f) + d*g) + (f*Log[c*(d + e*x^2)^p])/(f + g*x ^2) + (e*f*p*Log[f + g*x^2])/(e*f - d*g) + Log[c*(d + e*x^2)^p]*Log[(e*(f + g*x^2))/(e*f - d*g)] + p*PolyLog[2, (g*(d + e*x^2))/(-(e*f) + d*g)])/(2* g^2)
Time = 0.68 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.94, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2925, 2863, 2009}
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.
| \(\displaystyle \int \frac {x^3 \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 2925 |
| \(\displaystyle \frac {1}{2} \int \frac {x^2 \log \left (c \left (e x^2+d\right )^p\right )}{\left (g x^2+f\right )^2}dx^2\) |
| \(\Big \downarrow \) 2863 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {\log \left (c \left (e x^2+d\right )^p\right )}{g \left (g x^2+f\right )}-\frac {f \log \left (c \left (e x^2+d\right )^p\right )}{g \left (g x^2+f\right )^2}\right )dx^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{g^2 \left (f+g x^2\right )}+\frac {\log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )}{g^2}+\frac {p \operatorname {PolyLog}\left (2,-\frac {g \left (e x^2+d\right )}{e f-d g}\right )}{g^2}-\frac {e f p \log \left (d+e x^2\right )}{g^2 (e f-d g)}+\frac {e f p \log \left (f+g x^2\right )}{g^2 (e f-d g)}\right )\) |
Input:
Int[(x^3*Log[c*(d + e*x^2)^p])/(f + g*x^2)^2,x]
Output:
(-((e*f*p*Log[d + e*x^2])/(g^2*(e*f - d*g))) + (f*Log[c*(d + e*x^2)^p])/(g ^2*(f + g*x^2)) + (e*f*p*Log[f + g*x^2])/(g^2*(e*f - d*g)) + (Log[c*(d + e *x^2)^p]*Log[(e*(f + g*x^2))/(e*f - d*g)])/g^2 + (p*PolyLog[2, -((g*(d + e *x^2))/(e*f - d*g))])/g^2)/2
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_)) ^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a, b, c , d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m _.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol] :> Simp[1/n Subst[Int[x^(Si mplify[(m + 1)/n] - 1)*(f + g*x^(s/n))^r*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r, s}, x] && Integer Q[r] && IntegerQ[s/n] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0 ] || IGtQ[q, 0])
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 4.01 (sec) , antiderivative size = 381, normalized size of antiderivative = 2.46
| method | result | size |
| parts | \(\frac {\ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) \ln \left (g \,x^{2}+f \right )}{2 g^{2}}+\frac {f \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{2 g^{2} \left (g \,x^{2}+f \right )}-e p \left (\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+d \right )}{\sum }\left (\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (g \,x^{2}+f \right )-\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \left (\ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2} e g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =1\right )-x +\underline {\hspace {1.25 ex}}\alpha }{\operatorname {RootOf}\left (\textit {\_Z}^{2} e g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =1\right )}\right )+\ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2} e g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =2\right )-x +\underline {\hspace {1.25 ex}}\alpha }{\operatorname {RootOf}\left (\textit {\_Z}^{2} e g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =2\right )}\right )\right )-\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2} e g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =1\right )-x +\underline {\hspace {1.25 ex}}\alpha }{\operatorname {RootOf}\left (\textit {\_Z}^{2} e g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =1\right )}\right )-\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2} e g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =2\right )-x +\underline {\hspace {1.25 ex}}\alpha }{\operatorname {RootOf}\left (\textit {\_Z}^{2} e g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =2\right )}\right )\right )}{2 g^{2} e}+\frac {f \left (-\frac {\ln \left (e \,x^{2}+d \right )}{2 \left (d g -e f \right )}+\frac {\ln \left (g \,x^{2}+f \right )}{2 d g -2 e f}\right )}{g^{2}}\right )\) | \(381\) |
| risch | \(\frac {\ln \left (\left (e \,x^{2}+d \right )^{p}\right ) \ln \left (g \,x^{2}+f \right )}{2 g^{2}}+\frac {\ln \left (\left (e \,x^{2}+d \right )^{p}\right ) f}{2 g^{2} \left (g \,x^{2}+f \right )}-\frac {p \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+d \right )}{\sum }\left (\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (g \,x^{2}+f \right )-\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \left (\ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2} e g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =1\right )-x +\underline {\hspace {1.25 ex}}\alpha }{\operatorname {RootOf}\left (\textit {\_Z}^{2} e g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =1\right )}\right )+\ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2} e g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =2\right )-x +\underline {\hspace {1.25 ex}}\alpha }{\operatorname {RootOf}\left (\textit {\_Z}^{2} e g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =2\right )}\right )\right )-\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2} e g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =1\right )-x +\underline {\hspace {1.25 ex}}\alpha }{\operatorname {RootOf}\left (\textit {\_Z}^{2} e g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =1\right )}\right )-\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2} e g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =2\right )-x +\underline {\hspace {1.25 ex}}\alpha }{\operatorname {RootOf}\left (\textit {\_Z}^{2} e g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =2\right )}\right )\right )\right )}{2 g^{2}}+\frac {e p f \ln \left (e \,x^{2}+d \right )}{2 g^{2} \left (d g -e f \right )}-\frac {e p f \ln \left (g \,x^{2}+f \right )}{2 g^{2} \left (d g -e f \right )}+\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (\frac {\ln \left (g \,x^{2}+f \right )}{2 g^{2}}+\frac {f}{2 g^{2} \left (g \,x^{2}+f \right )}\right )\) | \(523\) |
Input:
int(x^3*ln(c*(e*x^2+d)^p)/(g*x^2+f)^2,x,method=_RETURNVERBOSE)
Output:
1/2*ln(c*(e*x^2+d)^p)/g^2*ln(g*x^2+f)+1/2*f*ln(c*(e*x^2+d)^p)/g^2/(g*x^2+f )-e*p*(1/2/g^2/e*sum(ln(x-_alpha)*ln(g*x^2+f)-ln(x-_alpha)*(ln((RootOf(_Z^ 2*e*g+2*_Z*_alpha*e*g-d*g+e*f,index=1)-x+_alpha)/RootOf(_Z^2*e*g+2*_Z*_alp ha*e*g-d*g+e*f,index=1))+ln((RootOf(_Z^2*e*g+2*_Z*_alpha*e*g-d*g+e*f,index =2)-x+_alpha)/RootOf(_Z^2*e*g+2*_Z*_alpha*e*g-d*g+e*f,index=2)))-dilog((Ro otOf(_Z^2*e*g+2*_Z*_alpha*e*g-d*g+e*f,index=1)-x+_alpha)/RootOf(_Z^2*e*g+2 *_Z*_alpha*e*g-d*g+e*f,index=1))-dilog((RootOf(_Z^2*e*g+2*_Z*_alpha*e*g-d* g+e*f,index=2)-x+_alpha)/RootOf(_Z^2*e*g+2*_Z*_alpha*e*g-d*g+e*f,index=2)) ,_alpha=RootOf(_Z^2*e+d))+f/g^2*(-1/2/(d*g-e*f)*ln(e*x^2+d)+1/2/(d*g-e*f)* ln(g*x^2+f)))
\[ \int \frac {x^3 \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=\int { \frac {x^{3} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{{\left (g x^{2} + f\right )}^{2}} \,d x } \] Input:
integrate(x^3*log(c*(e*x^2+d)^p)/(g*x^2+f)^2,x, algorithm="fricas")
Output:
integral(x^3*log((e*x^2 + d)^p*c)/(g^2*x^4 + 2*f*g*x^2 + f^2), x)
Timed out. \[ \int \frac {x^3 \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate(x**3*ln(c*(e*x**2+d)**p)/(g*x**2+f)**2,x)
Output:
Timed out
\[ \int \frac {x^3 \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=\int { \frac {x^{3} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{{\left (g x^{2} + f\right )}^{2}} \,d x } \] Input:
integrate(x^3*log(c*(e*x^2+d)^p)/(g*x^2+f)^2,x, algorithm="maxima")
Output:
integrate(x^3*log((e*x^2 + d)^p*c)/(g*x^2 + f)^2, x)
\[ \int \frac {x^3 \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=\int { \frac {x^{3} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{{\left (g x^{2} + f\right )}^{2}} \,d x } \] Input:
integrate(x^3*log(c*(e*x^2+d)^p)/(g*x^2+f)^2,x, algorithm="giac")
Output:
integrate(x^3*log((e*x^2 + d)^p*c)/(g*x^2 + f)^2, x)
Timed out. \[ \int \frac {x^3 \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=\int \frac {x^3\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}{{\left (g\,x^2+f\right )}^2} \,d x \] Input:
int((x^3*log(c*(d + e*x^2)^p))/(f + g*x^2)^2,x)
Output:
int((x^3*log(c*(d + e*x^2)^p))/(f + g*x^2)^2, x)
\[ \int \frac {x^3 \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx =\text {Too large to display} \] Input:
int(x^3*log(c*(e*x^2+d)^p)/(g*x^2+f)^2,x)
Output:
(4*int((log((d + e*x**2)**p*c)*x**3)/(d*f**2 + 2*d*f*g*x**2 + d*g**2*x**4 + e*f**2*x**2 + 2*e*f*g*x**4 + e*g**2*x**6),x)*d**3*f*g**4*p + 4*int((log( (d + e*x**2)**p*c)*x**3)/(d*f**2 + 2*d*f*g*x**2 + d*g**2*x**4 + e*f**2*x** 2 + 2*e*f*g*x**4 + e*g**2*x**6),x)*d**3*g**5*p*x**2 - 12*int((log((d + e*x **2)**p*c)*x**3)/(d*f**2 + 2*d*f*g*x**2 + d*g**2*x**4 + e*f**2*x**2 + 2*e* f*g*x**4 + e*g**2*x**6),x)*d**2*e*f**2*g**3*p - 12*int((log((d + e*x**2)** p*c)*x**3)/(d*f**2 + 2*d*f*g*x**2 + d*g**2*x**4 + e*f**2*x**2 + 2*e*f*g*x* *4 + e*g**2*x**6),x)*d**2*e*f*g**4*p*x**2 + 12*int((log((d + e*x**2)**p*c) *x**3)/(d*f**2 + 2*d*f*g*x**2 + d*g**2*x**4 + e*f**2*x**2 + 2*e*f*g*x**4 + e*g**2*x**6),x)*d*e**2*f**3*g**2*p + 12*int((log((d + e*x**2)**p*c)*x**3) /(d*f**2 + 2*d*f*g*x**2 + d*g**2*x**4 + e*f**2*x**2 + 2*e*f*g*x**4 + e*g** 2*x**6),x)*d*e**2*f**2*g**3*p*x**2 - 4*int((log((d + e*x**2)**p*c)*x**3)/( d*f**2 + 2*d*f*g*x**2 + d*g**2*x**4 + e*f**2*x**2 + 2*e*f*g*x**4 + e*g**2* x**6),x)*e**3*f**4*g*p - 4*int((log((d + e*x**2)**p*c)*x**3)/(d*f**2 + 2*d *f*g*x**2 + d*g**2*x**4 + e*f**2*x**2 + 2*e*f*g*x**4 + e*g**2*x**6),x)*e** 3*f**3*g**2*p*x**2 + 2*log(d + e*x**2)*d*e*f**2*g*p**2 + 2*log(d + e*x**2) *d*e*f*g**2*p**2*x**2 - 2*log(f + g*x**2)*e**2*f**3*p**2 - 2*log(f + g*x** 2)*e**2*f**2*g*p**2*x**2 + log((d + e*x**2)**p*c)**2*d**2*f*g**2 + log((d + e*x**2)**p*c)**2*d**2*g**3*x**2 - log((d + e*x**2)**p*c)**2*d*e*f**2*g - log((d + e*x**2)**p*c)**2*d*e*f*g**2*x**2 - 2*log((d + e*x**2)**p*c)*d...