3.2.16 \(\int \frac {x^3}{\log ^3(c (a+b x^2)^p)} \, dx\) [116]

3.2.16.1 Optimal result

Integrand size = 18, antiderivative size = 204 \[ \int \frac {x^3}{\log ^3\left (c \left (a+b x^2\right )^p\right )} \, dx=-\frac {a \left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-1/p} \operatorname {ExpIntegralEi}\left (\frac {\log \left (c \left (a+b x^2\right )^p\right )}{p}\right )}{4 b^2 p^3}+\frac {\left (a+b x^2\right )^2 \left (c \left (a+b x^2\right )^p\right )^{-2/p} \operatorname {ExpIntegralEi}\left (\frac {2 \log \left (c \left (a+b x^2\right )^p\right )}{p}\right )}{b^2 p^3}-\frac {x^2 \left (a+b x^2\right )}{4 b p \log ^2\left (c \left (a+b x^2\right )^p\right )}-\frac {a \left (a+b x^2\right )}{4 b^2 p^2 \log \left (c \left (a+b x^2\right )^p\right )}-\frac {x^2 \left (a+b x^2\right )}{2 b p^2 \log \left (c \left (a+b x^2\right )^p\right )} \]

output

-1/4*a*(b*x^2+a)*Ei(ln(c*(b*x^2+a)^p)/p)/b^2/p^3/((c*(b*x^2+a)^p)^(1/p))+( 
b*x^2+a)^2*Ei(2*ln(c*(b*x^2+a)^p)/p)/b^2/p^3/((c*(b*x^2+a)^p)^(2/p))-1/4*x 
^2*(b*x^2+a)/b/p/ln(c*(b*x^2+a)^p)^2-1/4*a*(b*x^2+a)/b^2/p^2/ln(c*(b*x^2+a 
)^p)-1/2*x^2*(b*x^2+a)/b/p^2/ln(c*(b*x^2+a)^p)
 

3.2.16.2 Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.91 \[ \int \frac {x^3}{\log ^3\left (c \left (a+b x^2\right )^p\right )} \, dx=-\frac {\left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-2/p} \left (a \left (c \left (a+b x^2\right )^p\right )^{\frac {1}{p}} \operatorname {ExpIntegralEi}\left (\frac {\log \left (c \left (a+b x^2\right )^p\right )}{p}\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )-4 \left (a+b x^2\right ) \operatorname {ExpIntegralEi}\left (\frac {2 \log \left (c \left (a+b x^2\right )^p\right )}{p}\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )+p \left (c \left (a+b x^2\right )^p\right )^{2/p} \left (b p x^2+\left (a+2 b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )\right )\right )}{4 b^2 p^3 \log ^2\left (c \left (a+b x^2\right )^p\right )} \]

input

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

output

-1/4*((a + b*x^2)*(a*(c*(a + b*x^2)^p)^p^(-1)*ExpIntegralEi[Log[c*(a + b*x 
^2)^p]/p]*Log[c*(a + b*x^2)^p]^2 - 4*(a + b*x^2)*ExpIntegralEi[(2*Log[c*(a 
 + b*x^2)^p])/p]*Log[c*(a + b*x^2)^p]^2 + p*(c*(a + b*x^2)^p)^(2/p)*(b*p*x 
^2 + (a + 2*b*x^2)*Log[c*(a + b*x^2)^p])))/(b^2*p^3*(c*(a + b*x^2)^p)^(2/p 
)*Log[c*(a + b*x^2)^p]^2)
 

3.2.16.3 Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.54, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {2904, 2847, 2836, 2734, 2737, 2609, 2847, 2836, 2737, 2609, 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.

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

\(\Big \downarrow \) 2904

\(\displaystyle \frac {1}{2} \int \frac {x^2}{\log ^3\left (c \left (b x^2+a\right )^p\right )}dx^2\)

\(\Big \downarrow \) 2847

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

\(\Big \downarrow \) 2836

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

\(\Big \downarrow \) 2734

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

\(\Big \downarrow \) 2737

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

\(\Big \downarrow \) 2609

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

\(\Big \downarrow \) 2847

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

\(\Big \downarrow \) 2836

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

\(\Big \downarrow \) 2737

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

\(\Big \downarrow \) 2609

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

\(\Big \downarrow \) 2846

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

\(\Big \downarrow \) 2009

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

input

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

output

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

3.2.16.3.1 Defintions of rubi rules used

rule 2009

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

rule 2609

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Si 
mp[(F^(g*(e - c*(f/d)))/d)*ExpIntegralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; F 
reeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]
 

rule 2734

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[x*((a + b 
*Log[c*x^n])^(p + 1)/(b*n*(p + 1))), x] - Simp[1/(b*n*(p + 1))   Int[(a + b 
*Log[c*x^n])^(p + 1), x], x] /; FreeQ[{a, b, c, n}, x] && LtQ[p, -1] && Int 
egerQ[2*p]
 

rule 2737

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[x/(n*(c*x 
^n)^(1/n))   Subst[Int[E^(x/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ 
[{a, b, c, n, p}, x]
 

rule 2836

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] : 
> Simp[1/e   Subst[Int[(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{ 
a, b, c, d, e, n, p}, x]
 

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 2847

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_. 
)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)*(f + g*x)^q*((a + b*Log[c*(d + e 
*x)^n])^(p + 1)/(b*e*n*(p + 1))), x] + (-Simp[(q + 1)/(b*n*(p + 1))   Int[( 
f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^(p + 1), x], x] + Simp[q*((e*f - d*g) 
/(b*e*n*(p + 1)))   Int[(f + g*x)^(q - 1)*(a + b*Log[c*(d + e*x)^n])^(p + 1 
), x], x]) /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] && Lt 
Q[p, -1] && GtQ[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])
 

3.2.16.4 Maple [C] (warning: unable to verify)

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

Time = 0.90 (sec) , antiderivative size = 1969, normalized size of antiderivative = 9.65

method	result	size
risch	\(\text {Expression too large to display}\)	\(1969\)

input

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

output

-1/2*(2*b^2*p*x^4+2*a*b*p*x^2+I*Pi*a^2*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)+3 
*I*Pi*a*b*x^2*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)-2*I*Pi*b^2*x^4*csgn(I*c*(b 
*x^2+a)^p)^3+2*I*Pi*b^2*x^4*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)-I*Pi*a^2*csg 
n(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)*csgn(I*c)+I*Pi*a^2*csgn(I*(b*x^2+a) 
^p)*csgn(I*c*(b*x^2+a)^p)^2-3*I*Pi*a*b*x^2*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b 
*x^2+a)^p)*csgn(I*c)-I*Pi*a^2*csgn(I*c*(b*x^2+a)^p)^3-2*I*Pi*b^2*x^4*csgn( 
I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)*csgn(I*c)-3*I*Pi*a*b*x^2*csgn(I*c*(b* 
x^2+a)^p)^3+3*I*Pi*a*b*x^2*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2+2*I 
*Pi*b^2*x^4*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2+4*ln(c)*b^2*x^4+4* 
b^2*x^4*ln((b*x^2+a)^p)+6*ln(c)*a*b*x^2+6*a*b*x^2*ln((b*x^2+a)^p)+2*ln(c)* 
a^2+2*a^2*ln((b*x^2+a)^p))/(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/b^2/p^2-1/p^3*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)*x^4-2/b/p^3*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))...
 

3.2.16.5 Fricas [A] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.32 \[ \int \frac {x^3}{\log ^3\left (c \left (a+b x^2\right )^p\right )} \, dx=-\frac {{\left (a p^{2} \log \left (b x^{2} + a\right )^{2} + 2 \, a p \log \left (b x^{2} + a\right ) \log \left (c\right ) + a \log \left (c\right )^{2}\right )} c^{\left (\frac {1}{p}\right )} \operatorname {log\_integral}\left ({\left (b x^{2} + a\right )} c^{\left (\frac {1}{p}\right )}\right ) + {\left (b^{2} p^{2} x^{4} + a b p^{2} x^{2} + {\left (2 \, b^{2} p^{2} x^{4} + 3 \, a b p^{2} x^{2} + a^{2} p^{2}\right )} \log \left (b x^{2} + a\right ) + {\left (2 \, b^{2} p x^{4} + 3 \, a b p x^{2} + a^{2} p\right )} \log \left (c\right )\right )} c^{\frac {2}{p}} - 4 \, {\left (p^{2} \log \left (b x^{2} + a\right )^{2} + 2 \, p \log \left (b x^{2} + a\right ) \log \left (c\right ) + \log \left (c\right )^{2}\right )} \operatorname {log\_integral}\left ({\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} c^{\frac {2}{p}}\right )}{4 \, {\left (b^{2} p^{5} \log \left (b x^{2} + a\right )^{2} + 2 \, b^{2} p^{4} \log \left (b x^{2} + a\right ) \log \left (c\right ) + b^{2} p^{3} \log \left (c\right )^{2}\right )} c^{\frac {2}{p}}} \]

input

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

output

-1/4*((a*p^2*log(b*x^2 + a)^2 + 2*a*p*log(b*x^2 + a)*log(c) + a*log(c)^2)* 
c^(1/p)*log_integral((b*x^2 + a)*c^(1/p)) + (b^2*p^2*x^4 + a*b*p^2*x^2 + ( 
2*b^2*p^2*x^4 + 3*a*b*p^2*x^2 + a^2*p^2)*log(b*x^2 + a) + (2*b^2*p*x^4 + 3 
*a*b*p*x^2 + a^2*p)*log(c))*c^(2/p) - 4*(p^2*log(b*x^2 + a)^2 + 2*p*log(b* 
x^2 + a)*log(c) + log(c)^2)*log_integral((b^2*x^4 + 2*a*b*x^2 + a^2)*c^(2/ 
p)))/((b^2*p^5*log(b*x^2 + a)^2 + 2*b^2*p^4*log(b*x^2 + a)*log(c) + b^2*p^ 
3*log(c)^2)*c^(2/p))
 

3.2.16.6 Sympy [F]

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

input

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

output

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

3.2.16.7 Maxima [F]

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

input

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

output

-1/4*(b^2*(p + 2*log(c))*x^4 + a*b*(p + 3*log(c))*x^2 + a^2*log(c) + (2*b^ 
2*p*x^4 + 3*a*b*p*x^2 + a^2*p)*log(b*x^2 + a))/(b^2*p^4*log(b*x^2 + a)^2 + 
 2*b^2*p^3*log(b*x^2 + a)*log(c) + b^2*p^2*log(c)^2) + integrate(1/2*(4*b* 
x^3 + 3*a*x)/(b*p^3*log(b*x^2 + a) + b*p^2*log(c)), x)
 

3.2.16.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 874 vs. \(2 (198) = 396\).

Time = 0.38 (sec) , antiderivative size = 874, normalized size of antiderivative = 4.28 \[ \int \frac {x^3}{\log ^3\left (c \left (a+b x^2\right )^p\right )} \, dx=\text {Too large to display} \]

input

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

output

1/4*((b*x^2 + a)*p^2*log(b*x^2 + a)/(b^2*p^5*log(b*x^2 + a)^2 + 2*b^2*p^4* 
log(b*x^2 + a)*log(c) + b^2*p^3*log(c)^2) - p^2*Ei(log(c)/p + log(b*x^2 + 
a))*log(b*x^2 + a)^2/((b^2*p^5*log(b*x^2 + a)^2 + 2*b^2*p^4*log(b*x^2 + a) 
*log(c) + b^2*p^3*log(c)^2)*c^(1/p)) + (b*x^2 + a)*p^2/(b^2*p^5*log(b*x^2 
+ a)^2 + 2*b^2*p^4*log(b*x^2 + a)*log(c) + b^2*p^3*log(c)^2) + (b*x^2 + a) 
*p*log(c)/(b^2*p^5*log(b*x^2 + a)^2 + 2*b^2*p^4*log(b*x^2 + a)*log(c) + b^ 
2*p^3*log(c)^2) - 2*p*Ei(log(c)/p + log(b*x^2 + a))*log(b*x^2 + a)*log(c)/ 
((b^2*p^5*log(b*x^2 + a)^2 + 2*b^2*p^4*log(b*x^2 + a)*log(c) + b^2*p^3*log 
(c)^2)*c^(1/p)) - Ei(log(c)/p + log(b*x^2 + a))*log(c)^2/((b^2*p^5*log(b*x 
^2 + a)^2 + 2*b^2*p^4*log(b*x^2 + a)*log(c) + b^2*p^3*log(c)^2)*c^(1/p)))* 
a - 1/4*(2*(b*x^2 + a)^2*p^2*log(b*x^2 + a)/(b*p^5*log(b*x^2 + a)^2 + 2*b* 
p^4*log(b*x^2 + a)*log(c) + b*p^3*log(c)^2) + (b*x^2 + a)^2*p^2/(b*p^5*log 
(b*x^2 + a)^2 + 2*b*p^4*log(b*x^2 + a)*log(c) + b*p^3*log(c)^2) - 4*p^2*Ei 
(2*log(c)/p + 2*log(b*x^2 + a))*log(b*x^2 + a)^2/((b*p^5*log(b*x^2 + a)^2 
+ 2*b*p^4*log(b*x^2 + a)*log(c) + b*p^3*log(c)^2)*c^(2/p)) + 2*(b*x^2 + a) 
^2*p*log(c)/(b*p^5*log(b*x^2 + a)^2 + 2*b*p^4*log(b*x^2 + a)*log(c) + b*p^ 
3*log(c)^2) - 8*p*Ei(2*log(c)/p + 2*log(b*x^2 + a))*log(b*x^2 + a)*log(c)/ 
((b*p^5*log(b*x^2 + a)^2 + 2*b*p^4*log(b*x^2 + a)*log(c) + b*p^3*log(c)^2) 
*c^(2/p)) - 4*Ei(2*log(c)/p + 2*log(b*x^2 + a))*log(c)^2/((b*p^5*log(b*x^2 
 + a)^2 + 2*b*p^4*log(b*x^2 + a)*log(c) + b*p^3*log(c)^2)*c^(2/p)))/b
 

3.2.16.9 Mupad [F(-1)]

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

input

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

output

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