\(\int \frac {(f+g x^2) \log (c (d+e x^2)^p)}{x^7} \, dx\) [316]

Optimal result

Integrand size = 23, antiderivative size = 125 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^7} \, dx=-\frac {e f p}{12 d x^4}+\frac {e (2 e f-3 d g) p}{12 d^2 x^2}+\frac {e^2 (2 e f-3 d g) p \log (x)}{6 d^3}-\frac {e^2 (2 e f-3 d g) p \log \left (d+e x^2\right )}{12 d^3}-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{6 x^6}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{4 x^4} \] Output:

-1/12*e*f*p/d/x^4+1/12*e*(-3*d*g+2*e*f)*p/d^2/x^2+1/6*e^2*(-3*d*g+2*e*f)*p 
*ln(x)/d^3-1/12*e^2*(-3*d*g+2*e*f)*p*ln(e*x^2+d)/d^3-1/6*f*ln(c*(e*x^2+d)^ 
p)/x^6-1/4*g*ln(c*(e*x^2+d)^p)/x^4
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.07 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^7} \, dx=\frac {1}{4} e g p \left (-\frac {1}{d x^2}-\frac {2 e \log (x)}{d^2}+\frac {e \log \left (d+e x^2\right )}{d^2}\right )+\frac {1}{3} e f p \left (-\frac {1}{4 d x^4}+\frac {e}{2 d^2 x^2}+\frac {e^2 \log (x)}{d^3}-\frac {e^2 \log \left (d+e x^2\right )}{2 d^3}\right )-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{6 x^6}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{4 x^4} \] Input:

Integrate[((f + g*x^2)*Log[c*(d + e*x^2)^p])/x^7,x]
 

Output:

(e*g*p*(-(1/(d*x^2)) - (2*e*Log[x])/d^2 + (e*Log[d + e*x^2])/d^2))/4 + (e* 
f*p*(-1/4*1/(d*x^4) + e/(2*d^2*x^2) + (e^2*Log[x])/d^3 - (e^2*Log[d + e*x^ 
2])/(2*d^3)))/3 - (f*Log[c*(d + e*x^2)^p])/(6*x^6) - (g*Log[c*(d + e*x^2)^ 
p])/(4*x^4)
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.94, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2925, 2861, 27, 86, 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.

\(\displaystyle \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^7} \, dx\)

\(\Big \downarrow \) 2925

\(\displaystyle \frac {1}{2} \int \frac {\left (g x^2+f\right ) \log \left (c \left (e x^2+d\right )^p\right )}{x^8}dx^2\)

\(\Big \downarrow \) 2861

\(\displaystyle \frac {1}{2} \left (-e p \int -\frac {3 g x^2+2 f}{6 x^6 \left (e x^2+d\right )}dx^2-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{3 x^6}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{2 x^4}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (\frac {1}{6} e p \int \frac {3 g x^2+2 f}{x^6 \left (e x^2+d\right )}dx^2-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{3 x^6}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{2 x^4}\right )\)

\(\Big \downarrow \) 86

\(\displaystyle \frac {1}{2} \left (\frac {1}{6} e p \int \left (\frac {(3 d g-2 e f) e^2}{d^3 \left (e x^2+d\right )}-\frac {(3 d g-2 e f) e}{d^3 x^2}+\frac {3 d g-2 e f}{d^2 x^4}+\frac {2 f}{d x^6}\right )dx^2-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{3 x^6}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{2 x^4}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{3 x^6}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{2 x^4}+\frac {1}{6} e p \left (\frac {e \log \left (x^2\right ) (2 e f-3 d g)}{d^3}-\frac {e (2 e f-3 d g) \log \left (d+e x^2\right )}{d^3}+\frac {2 e f-3 d g}{d^2 x^2}-\frac {f}{d x^4}\right )\right )\)

Input:

Int[((f + g*x^2)*Log[c*(d + e*x^2)^p])/x^7,x]
 

Output:

((e*p*(-(f/(d*x^4)) + (2*e*f - 3*d*g)/(d^2*x^2) + (e*(2*e*f - 3*d*g)*Log[x 
^2])/d^3 - (e*(2*e*f - 3*d*g)*Log[d + e*x^2])/d^3))/6 - (f*Log[c*(d + e*x^ 
2)^p])/(3*x^6) - (g*Log[c*(d + e*x^2)^p])/(2*x^4))/2
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 86

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ 
.), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; 
 FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 
] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p 
+ 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
 

rule 2009

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

rule 2861

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*(x_)^(m_.)*((f_) + 
 (g_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = IntHide[x^m*(f + g*x^r)^q, 
 x]}, Simp[(a + b*Log[c*(d + e*x)^n])   u, x] - Simp[b*e*n   Int[SimplifyIn 
tegrand[u/(d + e*x), x], x], x] /; InverseFunctionFreeQ[u, x]] /; FreeQ[{a, 
 b, c, d, e, f, g, m, n, q, r}, x] && IntegerQ[m] && IntegerQ[q] && Integer 
Q[r]
 

rule 2925

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])
 

Maple [A] (verified)

Time = 1.81 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.86

method	result	size
parts	\(-\frac {g \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{4 x^{4}}-\frac {f \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{6 x^{6}}-\frac {e p \left (-\frac {e \left (3 d g -2 e f \right ) \ln \left (e \,x^{2}+d \right )}{2 d^{3}}-\frac {-3 d g +2 e f}{2 d^{2} x^{2}}+\frac {e \left (3 d g -2 e f \right ) \ln \left (x \right )}{d^{3}}+\frac {f}{2 d \,x^{4}}\right )}{6}\)	\(108\)
parallelrisch	\(-\frac {6 \ln \left (x \right ) x^{6} d \,e^{2} g \,p^{2}-4 \ln \left (x \right ) x^{6} e^{3} f \,p^{2}-3 x^{6} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) d \,e^{2} g p +2 x^{6} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) e^{3} f p -3 x^{6} d \,e^{2} g \,p^{2}+2 x^{6} e^{3} f \,p^{2}+3 x^{4} d^{2} e g \,p^{2}-2 x^{4} d \,e^{2} f \,p^{2}+3 x^{2} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) d^{3} g p +x^{2} d^{2} e f \,p^{2}+2 \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) d^{3} f p}{12 x^{6} p \,d^{3}}\)	\(191\)
risch	\(-\frac {\left (3 g \,x^{2}+2 f \right ) \ln \left (\left (e \,x^{2}+d \right )^{p}\right )}{12 x^{6}}-\frac {12 \ln \left (x \right ) d \,e^{2} g p \,x^{6}-8 \ln \left (x \right ) e^{3} f p \,x^{6}-6 \ln \left (-e \,x^{2}-d \right ) d \,e^{2} g p \,x^{6}+4 \ln \left (-e \,x^{2}-d \right ) e^{3} f p \,x^{6}-3 i \pi \,d^{3} g \,x^{2} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}-2 i \pi \,d^{3} f {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}-2 i \pi \,d^{3} f \,\operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )+2 i \pi \,d^{3} f {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )-3 i \pi \,d^{3} g \,x^{2} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )+2 i \pi \,d^{3} f \,\operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2}+3 i \pi \,d^{3} g \,x^{2} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2}+3 i \pi \,d^{3} g \,x^{2} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )+6 d^{2} e g p \,x^{4}-4 d \,e^{2} f p \,x^{4}+6 \ln \left (c \right ) d^{3} g \,x^{2}+2 d^{2} e f p \,x^{2}+4 \ln \left (c \right ) d^{3} f}{24 d^{3} x^{6}}\)	\(428\)

Input:

int((g*x^2+f)*ln(c*(e*x^2+d)^p)/x^7,x,method=_RETURNVERBOSE)
 

Output:

-1/4*g*ln(c*(e*x^2+d)^p)/x^4-1/6*f*ln(c*(e*x^2+d)^p)/x^6-1/6*e*p*(-1/2*e*( 
3*d*g-2*e*f)/d^3*ln(e*x^2+d)-1/2*(-3*d*g+2*e*f)/d^2/x^2+e*(3*d*g-2*e*f)/d^ 
3*ln(x)+1/2/d*f/x^4)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.03 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^7} \, dx=\frac {2 \, {\left (2 \, e^{3} f - 3 \, d e^{2} g\right )} p x^{6} \log \left (x\right ) - d^{2} e f p x^{2} + {\left (2 \, d e^{2} f - 3 \, d^{2} e g\right )} p x^{4} - {\left ({\left (2 \, e^{3} f - 3 \, d e^{2} g\right )} p x^{6} + 3 \, d^{3} g p x^{2} + 2 \, d^{3} f p\right )} \log \left (e x^{2} + d\right ) - {\left (3 \, d^{3} g x^{2} + 2 \, d^{3} f\right )} \log \left (c\right )}{12 \, d^{3} x^{6}} \] Input:

integrate((g*x^2+f)*log(c*(e*x^2+d)^p)/x^7,x, algorithm="fricas")
 

Output:

1/12*(2*(2*e^3*f - 3*d*e^2*g)*p*x^6*log(x) - d^2*e*f*p*x^2 + (2*d*e^2*f - 
3*d^2*e*g)*p*x^4 - ((2*e^3*f - 3*d*e^2*g)*p*x^6 + 3*d^3*g*p*x^2 + 2*d^3*f* 
p)*log(e*x^2 + d) - (3*d^3*g*x^2 + 2*d^3*f)*log(c))/(d^3*x^6)
                                                                                    
                                                                                    
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^7} \, dx=\text {Timed out} \] Input:

integrate((g*x**2+f)*ln(c*(e*x**2+d)**p)/x**7,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.83 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^7} \, dx=-\frac {1}{12} \, e p {\left (\frac {{\left (2 \, e^{2} f - 3 \, d e g\right )} \log \left (e x^{2} + d\right )}{d^{3}} - \frac {{\left (2 \, e^{2} f - 3 \, d e g\right )} \log \left (x^{2}\right )}{d^{3}} - \frac {{\left (2 \, e f - 3 \, d g\right )} x^{2} - d f}{d^{2} x^{4}}\right )} - \frac {{\left (3 \, g x^{2} + 2 \, f\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{12 \, x^{6}} \] Input:

integrate((g*x^2+f)*log(c*(e*x^2+d)^p)/x^7,x, algorithm="maxima")
 

Output:

-1/12*e*p*((2*e^2*f - 3*d*e*g)*log(e*x^2 + d)/d^3 - (2*e^2*f - 3*d*e*g)*lo 
g(x^2)/d^3 - ((2*e*f - 3*d*g)*x^2 - d*f)/(d^2*x^4)) - 1/12*(3*g*x^2 + 2*f) 
*log((e*x^2 + d)^p*c)/x^6
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 316 vs. \(2 (113) = 226\).

Time = 0.12 (sec) , antiderivative size = 316, normalized size of antiderivative = 2.53 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^7} \, dx=-\frac {\frac {{\left (2 \, e^{4} f p + 3 \, {\left (e x^{2} + d\right )} e^{3} g p - 3 \, d e^{3} g p\right )} \log \left (e x^{2} + d\right )}{{\left (e x^{2} + d\right )}^{3} - 3 \, {\left (e x^{2} + d\right )}^{2} d + 3 \, {\left (e x^{2} + d\right )} d^{2} - d^{3}} - \frac {2 \, {\left (e x^{2} + d\right )}^{2} e^{4} f p - 5 \, {\left (e x^{2} + d\right )} d e^{4} f p + 3 \, d^{2} e^{4} f p - 3 \, {\left (e x^{2} + d\right )}^{2} d e^{3} g p + 6 \, {\left (e x^{2} + d\right )} d^{2} e^{3} g p - 3 \, d^{3} e^{3} g p - 2 \, d^{2} e^{4} f \log \left (c\right ) - 3 \, {\left (e x^{2} + d\right )} d^{2} e^{3} g \log \left (c\right ) + 3 \, d^{3} e^{3} g \log \left (c\right )}{{\left (e x^{2} + d\right )}^{3} d^{2} - 3 \, {\left (e x^{2} + d\right )}^{2} d^{3} + 3 \, {\left (e x^{2} + d\right )} d^{4} - d^{5}} + \frac {{\left (2 \, e^{4} f p - 3 \, d e^{3} g p\right )} \log \left (e x^{2} + d\right )}{d^{3}} - \frac {{\left (2 \, e^{4} f p - 3 \, d e^{3} g p\right )} \log \left (e x^{2}\right )}{d^{3}}}{12 \, e} \] Input:

integrate((g*x^2+f)*log(c*(e*x^2+d)^p)/x^7,x, algorithm="giac")
 

Output:

-1/12*((2*e^4*f*p + 3*(e*x^2 + d)*e^3*g*p - 3*d*e^3*g*p)*log(e*x^2 + d)/(( 
e*x^2 + d)^3 - 3*(e*x^2 + d)^2*d + 3*(e*x^2 + d)*d^2 - d^3) - (2*(e*x^2 + 
d)^2*e^4*f*p - 5*(e*x^2 + d)*d*e^4*f*p + 3*d^2*e^4*f*p - 3*(e*x^2 + d)^2*d 
*e^3*g*p + 6*(e*x^2 + d)*d^2*e^3*g*p - 3*d^3*e^3*g*p - 2*d^2*e^4*f*log(c) 
- 3*(e*x^2 + d)*d^2*e^3*g*log(c) + 3*d^3*e^3*g*log(c))/((e*x^2 + d)^3*d^2 
- 3*(e*x^2 + d)^2*d^3 + 3*(e*x^2 + d)*d^4 - d^5) + (2*e^4*f*p - 3*d*e^3*g* 
p)*log(e*x^2 + d)/d^3 - (2*e^4*f*p - 3*d*e^3*g*p)*log(e*x^2)/d^3)/e
 

Mupad [B] (verification not implemented)

Time = 26.32 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.90 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^7} \, dx=\frac {\ln \left (x\right )\,\left (2\,e^3\,f\,p-3\,d\,e^2\,g\,p\right )}{6\,d^3}-\frac {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\,\left (\frac {g\,x^2}{4}+\frac {f}{6}\right )}{x^6}-\frac {\ln \left (e\,x^2+d\right )\,\left (2\,e^3\,f\,p-3\,d\,e^2\,g\,p\right )}{12\,d^3}-\frac {\frac {e\,f\,p}{2\,d}+\frac {e\,p\,x^2\,\left (3\,d\,g-2\,e\,f\right )}{2\,d^2}}{6\,x^4} \] Input:

int((log(c*(d + e*x^2)^p)*(f + g*x^2))/x^7,x)
 

Output:

(log(x)*(2*e^3*f*p - 3*d*e^2*g*p))/(6*d^3) - (log(c*(d + e*x^2)^p)*(f/6 + 
(g*x^2)/4))/x^6 - (log(d + e*x^2)*(2*e^3*f*p - 3*d*e^2*g*p))/(12*d^3) - (( 
e*f*p)/(2*d) + (e*p*x^2*(3*d*g - 2*e*f))/(2*d^2))/(6*x^4)
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.19 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^7} \, dx=\frac {-2 \,\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) d^{3} f -3 \,\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) d^{3} g \,x^{2}+3 \,\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) d \,e^{2} g \,x^{6}-2 \,\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) e^{3} f \,x^{6}-6 \,\mathrm {log}\left (x \right ) d \,e^{2} g p \,x^{6}+4 \,\mathrm {log}\left (x \right ) e^{3} f p \,x^{6}-d^{2} e f p \,x^{2}-3 d^{2} e g p \,x^{4}+2 d \,e^{2} f p \,x^{4}}{12 d^{3} x^{6}} \] Input:

int((g*x^2+f)*log(c*(e*x^2+d)^p)/x^7,x)
 

Output:

( - 2*log((d + e*x**2)**p*c)*d**3*f - 3*log((d + e*x**2)**p*c)*d**3*g*x**2 
 + 3*log((d + e*x**2)**p*c)*d*e**2*g*x**6 - 2*log((d + e*x**2)**p*c)*e**3* 
f*x**6 - 6*log(x)*d*e**2*g*p*x**6 + 4*log(x)*e**3*f*p*x**6 - d**2*e*f*p*x* 
*2 - 3*d**2*e*g*p*x**4 + 2*d*e**2*f*p*x**4)/(12*d**3*x**6)