\(\int \frac {x^3}{\log (c (a+b x^2)^p)} \, dx\) [102]

Optimal result

Integrand size = 18, antiderivative size = 107 \[ \int \frac {x^3}{\log \left (c \left (a+b x^2\right )^p\right )} \, dx=-\frac {a \left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-1/p} \operatorname {ExpIntegralEi}\left (\frac {\log \left (c \left (a+b x^2\right )^p\right )}{p}\right )}{2 b^2 p}+\frac {\left (a+b x^2\right )^2 \left (c \left (a+b x^2\right )^p\right )^{-2/p} \operatorname {ExpIntegralEi}\left (\frac {2 \log \left (c \left (a+b x^2\right )^p\right )}{p}\right )}{2 b^2 p} \] Output:

-1/2*a*(b*x^2+a)*Ei(ln(c*(b*x^2+a)^p)/p)/b^2/p/((c*(b*x^2+a)^p)^(1/p))+1/2 
*(b*x^2+a)^2*Ei(2*ln(c*(b*x^2+a)^p)/p)/b^2/p/((c*(b*x^2+a)^p)^(2/p))
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.90 \[ \int \frac {x^3}{\log \left (c \left (a+b x^2\right )^p\right )} \, dx=-\frac {\left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-2/p} \left (a \left (c \left (a+b x^2\right )^p\right )^{\frac {1}{p}} \operatorname {ExpIntegralEi}\left (\frac {\log \left (c \left (a+b x^2\right )^p\right )}{p}\right )-\left (a+b x^2\right ) \operatorname {ExpIntegralEi}\left (\frac {2 \log \left (c \left (a+b x^2\right )^p\right )}{p}\right )\right )}{2 b^2 p} \] Input:

Integrate[x^3/Log[c*(a + b*x^2)^p],x]
 

Output:

-1/2*((a + b*x^2)*(a*(c*(a + b*x^2)^p)^p^(-1)*ExpIntegralEi[Log[c*(a + b*x 
^2)^p]/p] - (a + b*x^2)*ExpIntegralEi[(2*Log[c*(a + b*x^2)^p])/p]))/(b^2*p 
*(c*(a + b*x^2)^p)^(2/p))
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2904, 2846, 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.

\(\displaystyle \int \frac {x^3}{\log \left (c \left (a+b x^2\right )^p\right )} \, dx\)

\(\Big \downarrow \) 2904

\(\displaystyle \frac {1}{2} \int \frac {x^2}{\log \left (c \left (b x^2+a\right )^p\right )}dx^2\)

\(\Big \downarrow \) 2846

\(\displaystyle \frac {1}{2} \int \left (\frac {b x^2+a}{b \log \left (c \left (b x^2+a\right )^p\right )}-\frac {a}{b \log \left (c \left (b x^2+a\right )^p\right )}\right )dx^2\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (\frac {\left (a+b x^2\right )^2 \left (c \left (a+b x^2\right )^p\right )^{-2/p} \operatorname {ExpIntegralEi}\left (\frac {2 \log \left (c \left (b x^2+a\right )^p\right )}{p}\right )}{b^2 p}-\frac {a \left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-1/p} \operatorname {ExpIntegralEi}\left (\frac {\log \left (c \left (b x^2+a\right )^p\right )}{p}\right )}{b^2 p}\right )\)

Input:

Int[x^3/Log[c*(a + b*x^2)^p],x]
 

Output:

(-((a*(a + b*x^2)*ExpIntegralEi[Log[c*(a + b*x^2)^p]/p])/(b^2*p*(c*(a + b* 
x^2)^p)^p^(-1))) + ((a + b*x^2)^2*ExpIntegralEi[(2*Log[c*(a + b*x^2)^p])/p 
])/(b^2*p*(c*(a + b*x^2)^p)^(2/p)))/2
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2846

Int[((f_.) + (g_.)*(x_))^(q_.)/((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.) 
]*(b_.)), x_Symbol] :> Int[ExpandIntegrand[(f + g*x)^q/(a + b*Log[c*(d + e* 
x)^n]), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] & 
& IGtQ[q, 0]
 

rule 2904

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m 
_.), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*L 
og[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, 
 x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) & 
&  !(EqQ[q, 1] && ILtQ[n, 0] && IGtQ[m, 0])
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 1.00 (sec) , antiderivative size = 547, normalized size of antiderivative = 5.11

method	result	size
risch	\(-\frac {\left (b \,x^{2}+a \right )^{2} c^{-\frac {2}{p}} {\left (\left (b \,x^{2}+a \right )^{p}\right )}^{-\frac {2}{p}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \left (-\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )+\operatorname {csgn}\left (i c \right )\right ) \left (-\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )+\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right )\right )}{p}} \operatorname {expIntegral}_{1}\left (-2 \ln \left (b \,x^{2}+a \right )-\frac {i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2}-i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )-i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{3}+i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )+2 \ln \left (c \right )+2 \ln \left (\left (b \,x^{2}+a \right )^{p}\right )-2 p \ln \left (b \,x^{2}+a \right )}{p}\right )}{2 b^{2} p}+\frac {a \left (b \,x^{2}+a \right ) c^{-\frac {1}{p}} {\left (\left (b \,x^{2}+a \right )^{p}\right )}^{-\frac {1}{p}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \left (-\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )+\operatorname {csgn}\left (i c \right )\right ) \left (-\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )+\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right )\right )}{2 p}} \operatorname {expIntegral}_{1}\left (-\ln \left (b \,x^{2}+a \right )-\frac {i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2}-i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )-i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{3}+i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )+2 \ln \left (c \right )+2 \ln \left (\left (b \,x^{2}+a \right )^{p}\right )-2 p \ln \left (b \,x^{2}+a \right )}{2 p}\right )}{2 b^{2} p}\)	\(547\)

Input:

int(x^3/ln(c*(b*x^2+a)^p),x,method=_RETURNVERBOSE)
 

Output:

-1/2/b^2/p*(b*x^2+a)^2*c^(-2/p)*((b*x^2+a)^p)^(-2/p)*exp(I*Pi*csgn(I*c*(b* 
x^2+a)^p)*(-csgn(I*c*(b*x^2+a)^p)+csgn(I*c))*(-csgn(I*c*(b*x^2+a)^p)+csgn( 
I*(b*x^2+a)^p))/p)*Ei(1,-2*ln(b*x^2+a)-(I*Pi*csgn(I*(b*x^2+a)^p)*csgn(I*c* 
(b*x^2+a)^p)^2-I*Pi*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)*csgn(I*c)-I* 
Pi*csgn(I*c*(b*x^2+a)^p)^3+I*Pi*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)+2*ln(c)+ 
2*ln((b*x^2+a)^p)-2*p*ln(b*x^2+a))/p)+1/2/b^2*a/p*(b*x^2+a)*c^(-1/p)*((b*x 
^2+a)^p)^(-1/p)*exp(1/2*I*Pi*csgn(I*c*(b*x^2+a)^p)*(-csgn(I*c*(b*x^2+a)^p) 
+csgn(I*c))*(-csgn(I*c*(b*x^2+a)^p)+csgn(I*(b*x^2+a)^p))/p)*Ei(1,-ln(b*x^2 
+a)-1/2*(I*Pi*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2-I*Pi*csgn(I*(b*x 
^2+a)^p)*csgn(I*c*(b*x^2+a)^p)*csgn(I*c)-I*Pi*csgn(I*c*(b*x^2+a)^p)^3+I*Pi 
*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)+2*ln(c)+2*ln((b*x^2+a)^p)-2*p*ln(b*x^2+ 
a))/p)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.64 \[ \int \frac {x^3}{\log \left (c \left (a+b x^2\right )^p\right )} \, dx=-\frac {a c^{\left (\frac {1}{p}\right )} \operatorname {log\_integral}\left ({\left (b x^{2} + a\right )} c^{\left (\frac {1}{p}\right )}\right ) - \operatorname {log\_integral}\left ({\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} c^{\frac {2}{p}}\right )}{2 \, b^{2} c^{\frac {2}{p}} p} \] Input:

integrate(x^3/log(c*(b*x^2+a)^p),x, algorithm="fricas")
 

Output:

-1/2*(a*c^(1/p)*log_integral((b*x^2 + a)*c^(1/p)) - log_integral((b^2*x^4 
+ 2*a*b*x^2 + a^2)*c^(2/p)))/(b^2*c^(2/p)*p)
 

Sympy [F]

\[ \int \frac {x^3}{\log \left (c \left (a+b x^2\right )^p\right )} \, dx=\int \frac {x^{3}}{\log {\left (c \left (a + b x^{2}\right )^{p} \right )}}\, dx \] Input:

integrate(x**3/ln(c*(b*x**2+a)**p),x)
 

Output:

Integral(x**3/log(c*(a + b*x**2)**p), x)
 

Maxima [F]

\[ \int \frac {x^3}{\log \left (c \left (a+b x^2\right )^p\right )} \, dx=\int { \frac {x^{3}}{\log \left ({\left (b x^{2} + a\right )}^{p} c\right )} \,d x } \] Input:

integrate(x^3/log(c*(b*x^2+a)^p),x, algorithm="maxima")
 

Output:

integrate(x^3/log((b*x^2 + a)^p*c), x)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.64 \[ \int \frac {x^3}{\log \left (c \left (a+b x^2\right )^p\right )} \, dx=-\frac {a {\rm Ei}\left (\frac {\log \left (c\right )}{p} + \log \left (b x^{2} + a\right )\right )}{2 \, b^{2} c^{\left (\frac {1}{p}\right )} p} + \frac {{\rm Ei}\left (\frac {2 \, \log \left (c\right )}{p} + 2 \, \log \left (b x^{2} + a\right )\right )}{2 \, b^{2} c^{\frac {2}{p}} p} \] Input:

integrate(x^3/log(c*(b*x^2+a)^p),x, algorithm="giac")
 

Output:

-1/2*a*Ei(log(c)/p + log(b*x^2 + a))/(b^2*c^(1/p)*p) + 1/2*Ei(2*log(c)/p + 
 2*log(b*x^2 + a))/(b^2*c^(2/p)*p)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3}{\log \left (c \left (a+b x^2\right )^p\right )} \, dx=\int \frac {x^3}{\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )} \,d x \] Input:

int(x^3/log(c*(a + b*x^2)^p),x)
 

Output:

int(x^3/log(c*(a + b*x^2)^p), x)
 

Reduce [F]

\[ \int \frac {x^3}{\log \left (c \left (a+b x^2\right )^p\right )} \, dx=\int \frac {x^{3}}{\mathrm {log}\left (\left (b \,x^{2}+a \right )^{p} c \right )}d x \] Input:

int(x^3/log(c*(b*x^2+a)^p),x)
                                                                                    
                                                                                    
 

Output:

int(x**3/log((a + b*x**2)**p*c),x)