Integrand size = 20, antiderivative size = 308 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{d+e x} \, dx=-\frac {p \log \left (-\frac {e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right ) \log (d+e x)}{e}-\frac {p \log \left (-\frac {e \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right ) \log (d+e x)}{e}-\frac {p \log \left (\frac {\sqrt [3]{-1} e \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right ) \log (d+e x)}{e}+\frac {\log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{e}-\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{e}-\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right )}{e}-\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{e} \] Output:
-p*ln(-e*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*d-a^(1/3)*e))*ln(e*x+d)/e-p*ln(-e*(( -1)^(2/3)*a^(1/3)+b^(1/3)*x)/(b^(1/3)*d-(-1)^(2/3)*a^(1/3)*e))*ln(e*x+d)/e -p*ln((-1)^(1/3)*e*(a^(1/3)+(-1)^(2/3)*b^(1/3)*x)/(b^(1/3)*d+(-1)^(1/3)*a^ (1/3)*e))*ln(e*x+d)/e+ln(e*x+d)*ln(c*(b*x^3+a)^p)/e-p*polylog(2,b^(1/3)*(e *x+d)/(b^(1/3)*d-a^(1/3)*e))/e-p*polylog(2,b^(1/3)*(e*x+d)/(b^(1/3)*d+(-1) ^(1/3)*a^(1/3)*e))/e-p*polylog(2,b^(1/3)*(e*x+d)/(b^(1/3)*d-(-1)^(2/3)*a^( 1/3)*e))/e
Time = 0.18 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.02 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{d+e x} \, dx=-\frac {p \log \left (-\frac {e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right ) \log (d+e x)}{e}-\frac {p \log \left (-\frac {(-1)^{2/3} e \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right ) \log (d+e x)}{e}-\frac {p \log \left (\frac {\sqrt [3]{-1} e \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right ) \log (d+e x)}{e}+\frac {\log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{e}-\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{e}-\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right )}{e}-\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{e} \] Input:
Integrate[Log[c*(a + b*x^3)^p]/(d + e*x),x]
Output:
-((p*Log[-((e*(a^(1/3) + b^(1/3)*x))/(b^(1/3)*d - a^(1/3)*e))]*Log[d + e*x ])/e) - (p*Log[-(((-1)^(2/3)*e*(a^(1/3) - (-1)^(1/3)*b^(1/3)*x))/(b^(1/3)* d - (-1)^(2/3)*a^(1/3)*e))]*Log[d + e*x])/e - (p*Log[((-1)^(1/3)*e*(a^(1/3 ) + (-1)^(2/3)*b^(1/3)*x))/(b^(1/3)*d + (-1)^(1/3)*a^(1/3)*e)]*Log[d + e*x ])/e + (Log[d + e*x]*Log[c*(a + b*x^3)^p])/e - (p*PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1/3)*d - a^(1/3)*e)])/e - (p*PolyLog[2, (b^(1/3)*(d + e*x))/(b^ (1/3)*d + (-1)^(1/3)*a^(1/3)*e)])/e - (p*PolyLog[2, (b^(1/3)*(d + e*x))/(b ^(1/3)*d - (-1)^(2/3)*a^(1/3)*e)])/e
Time = 1.19 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.05, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2912, 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.
| \(\displaystyle \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{d+e x} \, dx\) |
| \(\Big \downarrow \) 2912 |
| \(\displaystyle \frac {\log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{e}-\frac {3 b p \int \frac {x^2 \log (d+e x)}{b x^3+a}dx}{e}\) |
| \(\Big \downarrow \) 2863 |
| \(\displaystyle \frac {\log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{e}-\frac {3 b p \int \left (\frac {\log (d+e x)}{3 b^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}+\frac {\log (d+e x)}{3 b^{2/3} \left (\sqrt [3]{b} x-\sqrt [3]{-1} \sqrt [3]{a}\right )}+\frac {\log (d+e x)}{3 b^{2/3} \left (\sqrt [3]{b} x+(-1)^{2/3} \sqrt [3]{a}\right )}\right )dx}{e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{e}-\frac {3 b p \left (\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{3 b}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right )}{3 b}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{3 b}+\frac {\log (d+e x) \log \left (-\frac {e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{3 b}+\frac {\log (d+e x) \log \left (-\frac {e \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{3 b}+\frac {\log (d+e x) \log \left (\frac {\sqrt [3]{-1} e \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{a} e+\sqrt [3]{b} d}\right )}{3 b}\right )}{e}\) |
Input:
Int[Log[c*(a + b*x^3)^p]/(d + e*x),x]
Output:
(Log[d + e*x]*Log[c*(a + b*x^3)^p])/e - (3*b*p*((Log[-((e*(a^(1/3) + b^(1/ 3)*x))/(b^(1/3)*d - a^(1/3)*e))]*Log[d + e*x])/(3*b) + (Log[-((e*((-1)^(2/ 3)*a^(1/3) + b^(1/3)*x))/(b^(1/3)*d - (-1)^(2/3)*a^(1/3)*e))]*Log[d + e*x] )/(3*b) + (Log[((-1)^(1/3)*e*(a^(1/3) + (-1)^(2/3)*b^(1/3)*x))/(b^(1/3)*d + (-1)^(1/3)*a^(1/3)*e)]*Log[d + e*x])/(3*b) + PolyLog[2, (b^(1/3)*(d + e* x))/(b^(1/3)*d - a^(1/3)*e)]/(3*b) + PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1/ 3)*d + (-1)^(1/3)*a^(1/3)*e)]/(3*b) + PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1 /3)*d - (-1)^(2/3)*a^(1/3)*e)]/(3*b)))/e
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_.))/((f_.) + (g_. )*(x_)), x_Symbol] :> Simp[Log[f + g*x]*((a + b*Log[c*(d + e*x^n)^p])/g), x ] - Simp[b*e*n*(p/g) Int[x^(n - 1)*(Log[f + g*x]/(d + e*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && RationalQ[n]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.88 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.33
| method | result | size |
| parts | \(\frac {\ln \left (e x +d \right ) \ln \left (c \left (b \,x^{3}+a \right )^{p}\right )}{e}-\frac {p \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b -3 b d \,\textit {\_Z}^{2}+3 b \,d^{2} \textit {\_Z} +a \,e^{3}-b \,d^{3}\right )}{\sum }\left (\ln \left (e x +d \right ) \ln \left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )\right )\right )}{e}\) | \(101\) |
| risch | \(\frac {\ln \left (\left (b \,x^{3}+a \right )^{p}\right ) \ln \left (e x +d \right )}{e}-\frac {p \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b -3 b d \,\textit {\_Z}^{2}+3 b \,d^{2} \textit {\_Z} +a \,e^{3}-b \,d^{3}\right )}{\sum }\left (\ln \left (e x +d \right ) \ln \left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )\right )\right )}{e}+\frac {\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (b \,x^{3}+a \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (b \,x^{3}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \ln \left (e x +d \right )}{e}\) | \(226\) |
Input:
int(ln(c*(b*x^3+a)^p)/(e*x+d),x,method=_RETURNVERBOSE)
Output:
ln(e*x+d)*ln(c*(b*x^3+a)^p)/e-p/e*sum(ln(e*x+d)*ln((-e*x+_R1-d)/_R1)+dilog ((-e*x+_R1-d)/_R1),_R1=RootOf(_Z^3*b-3*_Z^2*b*d+3*_Z*b*d^2+a*e^3-b*d^3))
\[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{d+e x} \, dx=\int { \frac {\log \left ({\left (b x^{3} + a\right )}^{p} c\right )}{e x + d} \,d x } \] Input:
integrate(log(c*(b*x^3+a)^p)/(e*x+d),x, algorithm="fricas")
Output:
integral(log((b*x^3 + a)^p*c)/(e*x + d), x)
Timed out. \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{d+e x} \, dx=\text {Timed out} \] Input:
integrate(ln(c*(b*x**3+a)**p)/(e*x+d),x)
Output:
Timed out
\[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{d+e x} \, dx=\int { \frac {\log \left ({\left (b x^{3} + a\right )}^{p} c\right )}{e x + d} \,d x } \] Input:
integrate(log(c*(b*x^3+a)^p)/(e*x+d),x, algorithm="maxima")
Output:
integrate(log((b*x^3 + a)^p*c)/(e*x + d), x)
\[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{d+e x} \, dx=\int { \frac {\log \left ({\left (b x^{3} + a\right )}^{p} c\right )}{e x + d} \,d x } \] Input:
integrate(log(c*(b*x^3+a)^p)/(e*x+d),x, algorithm="giac")
Output:
integrate(log((b*x^3 + a)^p*c)/(e*x + d), x)
Timed out. \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{d+e x} \, dx=\int \frac {\ln \left (c\,{\left (b\,x^3+a\right )}^p\right )}{d+e\,x} \,d x \] Input:
int(log(c*(a + b*x^3)^p)/(d + e*x),x)
Output:
int(log(c*(a + b*x^3)^p)/(d + e*x), x)
\[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{d+e x} \, dx=\frac {6 \left (\int \frac {\mathrm {log}\left (\left (b \,x^{3}+a \right )^{p} c \right )}{b e \,x^{4}+b d \,x^{3}+a e x +a d}d x \right ) a e p -6 \left (\int \frac {\mathrm {log}\left (\left (b \,x^{3}+a \right )^{p} c \right ) x^{2}}{b e \,x^{4}+b d \,x^{3}+a e x +a d}d x \right ) b d p +{\mathrm {log}\left (\left (b \,x^{3}+a \right )^{p} c \right )}^{2}}{6 e p} \] Input:
int(log(c*(b*x^3+a)^p)/(e*x+d),x)
Output:
(6*int(log((a + b*x**3)**p*c)/(a*d + a*e*x + b*d*x**3 + b*e*x**4),x)*a*e*p - 6*int((log((a + b*x**3)**p*c)*x**2)/(a*d + a*e*x + b*d*x**3 + b*e*x**4) ,x)*b*d*p + log((a + b*x**3)**p*c)**2)/(6*e*p)