\(\int \frac {(f+g x^2) \log (c (d+e x^2)^p)}{x^9} \, dx\) [317]

Optimal result

Integrand size = 23, antiderivative size = 148 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^9} \, dx=-\frac {e f p}{24 d x^6}+\frac {e (3 e f-4 d g) p}{48 d^2 x^4}-\frac {e^2 (3 e f-4 d g) p}{24 d^3 x^2}-\frac {e^3 (3 e f-4 d g) p \log (x)}{12 d^4}+\frac {e^3 (3 e f-4 d g) p \log \left (d+e x^2\right )}{24 d^4}-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{8 x^8}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{6 x^6} \]

[Out]

-1/24*e*f*p/d/x^6+1/48*e*(-4*d*g+3*e*f)*p/d^2/x^4-1/24*e^2*(-4*d*g+3*e*f)*p/d^3/x^2-1/12*e^3*(-4*d*g+3*e*f)*p*
ln(x)/d^4+1/24*e^3*(-4*d*g+3*e*f)*p*ln(e*x^2+d)/d^4-1/8*f*ln(c*(e*x^2+d)^p)/x^8-1/6*g*ln(c*(e*x^2+d)^p)/x^6

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2525, 45, 2461, 12, 78} \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^9} \, dx=-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{8 x^8}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{6 x^6}+\frac {e^3 p (3 e f-4 d g) \log \left (d+e x^2\right )}{24 d^4}-\frac {e^3 p \log (x) (3 e f-4 d g)}{12 d^4}-\frac {e^2 p (3 e f-4 d g)}{24 d^3 x^2}+\frac {e p (3 e f-4 d g)}{48 d^2 x^4}-\frac {e f p}{24 d x^6} \]

[In]

Int[((f + g*x^2)*Log[c*(d + e*x^2)^p])/x^9,x]

[Out]

-1/24*(e*f*p)/(d*x^6) + (e*(3*e*f - 4*d*g)*p)/(48*d^2*x^4) - (e^2*(3*e*f - 4*d*g)*p)/(24*d^3*x^2) - (e^3*(3*e*
f - 4*d*g)*p*Log[x])/(12*d^4) + (e^3*(3*e*f - 4*d*g)*p*Log[d + e*x^2])/(24*d^4) - (f*Log[c*(d + e*x^2)^p])/(8*
x^8) - (g*Log[c*(d + e*x^2)^p])/(6*x^6)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((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 2461

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]}, Dist[a + b*Log[c*(d + e*x)^n], u, x] - Dist[b*e*n, Int[SimplifyI
ntegrand[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] && IntegerQ[r]

Rule 2525

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(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] && IntegerQ[r] && IntegerQ[s/n] && Intege
rQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0])

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {(f+g x) \log \left (c (d+e x)^p\right )}{x^5} \, dx,x,x^2\right ) \\ & = -\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{8 x^8}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{6 x^6}-\frac {1}{2} (e p) \text {Subst}\left (\int \frac {-3 f-4 g x}{12 x^4 (d+e x)} \, dx,x,x^2\right ) \\ & = -\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{8 x^8}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{6 x^6}-\frac {1}{24} (e p) \text {Subst}\left (\int \frac {-3 f-4 g x}{x^4 (d+e x)} \, dx,x,x^2\right ) \\ & = -\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{8 x^8}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{6 x^6}-\frac {1}{24} (e p) \text {Subst}\left (\int \left (-\frac {3 f}{d x^4}+\frac {3 e f-4 d g}{d^2 x^3}+\frac {e (-3 e f+4 d g)}{d^3 x^2}-\frac {e^2 (-3 e f+4 d g)}{d^4 x}+\frac {e^3 (-3 e f+4 d g)}{d^4 (d+e x)}\right ) \, dx,x,x^2\right ) \\ & = -\frac {e f p}{24 d x^6}+\frac {e (3 e f-4 d g) p}{48 d^2 x^4}-\frac {e^2 (3 e f-4 d g) p}{24 d^3 x^2}-\frac {e^3 (3 e f-4 d g) p \log (x)}{12 d^4}+\frac {e^3 (3 e f-4 d g) p \log \left (d+e x^2\right )}{24 d^4}-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{8 x^8}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{6 x^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.09 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^9} \, dx=\frac {1}{3} e g p \left (-\frac {1}{4 d x^4}+\frac {e}{2 d^2 x^2}+\frac {e^2 \log (x)}{d^3}-\frac {e^2 \log \left (d+e x^2\right )}{2 d^3}\right )+\frac {1}{8} e f p \left (-\frac {1}{3 d x^6}+\frac {e}{2 d^2 x^4}-\frac {e^2}{d^3 x^2}-\frac {2 e^3 \log (x)}{d^4}+\frac {e^3 \log \left (d+e x^2\right )}{d^4}\right )-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{8 x^8}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{6 x^6} \]

[In]

Integrate[((f + g*x^2)*Log[c*(d + e*x^2)^p])/x^9,x]

[Out]

(e*g*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 + (e*f*p*(-1/3*1/
(d*x^6) + e/(2*d^2*x^4) - e^2/(d^3*x^2) - (2*e^3*Log[x])/d^4 + (e^3*Log[d + e*x^2])/d^4))/8 - (f*Log[c*(d + e*
x^2)^p])/(8*x^8) - (g*Log[c*(d + e*x^2)^p])/(6*x^6)

Maple [A] (verified)

Time = 1.52 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.89

method	result	size
parts	\(-\frac {g \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{6 x^{6}}-\frac {f \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{8 x^{8}}-\frac {p e \left (-\frac {-4 d g +3 e f}{4 d^{2} x^{4}}-\frac {\left (4 d g -3 e f \right ) e}{2 d^{3} x^{2}}+\frac {f}{2 d \,x^{6}}-\frac {\left (4 d g -3 e f \right ) e^{2} \ln \left (x \right )}{d^{4}}+\frac {e^{2} \left (4 d g -3 e f \right ) \ln \left (e \,x^{2}+d \right )}{2 d^{4}}\right )}{12}\)	\(131\)
parallelrisch	\(\frac {16 \ln \left (x \right ) x^{8} d \,e^{3} g \,p^{2}-12 \ln \left (x \right ) x^{8} e^{4} f \,p^{2}-8 x^{8} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) d \,e^{3} g p +6 x^{8} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) e^{4} f p -8 x^{8} d \,e^{3} g \,p^{2}+6 x^{8} e^{4} f \,p^{2}+8 x^{6} d^{2} e^{2} g \,p^{2}-6 x^{6} d \,e^{3} f \,p^{2}-4 x^{4} d^{3} e g \,p^{2}+3 x^{4} d^{2} e^{2} f \,p^{2}-8 x^{2} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) d^{4} g p -2 x^{2} d^{3} e f \,p^{2}-6 \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) d^{4} f p}{48 x^{8} d^{4} p}\)	\(222\)
risch	\(-\frac {\left (4 g \,x^{2}+3 f \right ) \ln \left (\left (e \,x^{2}+d \right )^{p}\right )}{24 x^{8}}-\frac {8 \ln \left (e \,x^{2}+d \right ) d \,e^{3} g p \,x^{8}-6 \ln \left (e \,x^{2}+d \right ) e^{4} f p \,x^{8}-16 \ln \left (x \right ) d \,e^{3} g p \,x^{8}+12 \ln \left (x \right ) e^{4} f p \,x^{8}+4 i \pi \,d^{4} g \,x^{2} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )-3 i \pi \,d^{4} f \,\operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )-3 i \pi \,d^{4} f {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}+3 i \pi \,d^{4} f {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )-8 d^{2} e^{2} g p \,x^{6}+6 d \,e^{3} f p \,x^{6}+4 i \pi \,d^{4} g \,x^{2} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2}-4 i \pi \,d^{4} g \,x^{2} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )+3 i \pi \,d^{4} f \,\operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2}-4 i \pi \,d^{4} g \,x^{2} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}+4 d^{3} e g p \,x^{4}-3 d^{2} e^{2} f p \,x^{4}+8 \ln \left (c \right ) d^{4} g \,x^{2}+2 d^{3} e f p \,x^{2}+6 \ln \left (c \right ) d^{4} f}{48 d^{4} x^{8}}\)	\(448\)

[In]

int((g*x^2+f)*ln(c*(e*x^2+d)^p)/x^9,x,method=_RETURNVERBOSE)

[Out]

-1/6*g*ln(c*(e*x^2+d)^p)/x^6-1/8*f*ln(c*(e*x^2+d)^p)/x^8-1/12*p*e*(-1/4*(-4*d*g+3*e*f)/d^2/x^4-1/2*(4*d*g-3*e*
f)/d^3*e/x^2+1/2*f/d/x^6-(4*d*g-3*e*f)/d^4*e^2*ln(x)+1/2*e^2*(4*d*g-3*e*f)/d^4*ln(e*x^2+d))

Fricas [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.05 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^9} \, dx=-\frac {4 \, {\left (3 \, e^{4} f - 4 \, d e^{3} g\right )} p x^{8} \log \left (x\right ) + 2 \, d^{3} e f p x^{2} + 2 \, {\left (3 \, d e^{3} f - 4 \, d^{2} e^{2} g\right )} p x^{6} - {\left (3 \, d^{2} e^{2} f - 4 \, d^{3} e g\right )} p x^{4} - 2 \, {\left ({\left (3 \, e^{4} f - 4 \, d e^{3} g\right )} p x^{8} - 4 \, d^{4} g p x^{2} - 3 \, d^{4} f p\right )} \log \left (e x^{2} + d\right ) + 2 \, {\left (4 \, d^{4} g x^{2} + 3 \, d^{4} f\right )} \log \left (c\right )}{48 \, d^{4} x^{8}} \]

[In]

integrate((g*x^2+f)*log(c*(e*x^2+d)^p)/x^9,x, algorithm="fricas")

[Out]

-1/48*(4*(3*e^4*f - 4*d*e^3*g)*p*x^8*log(x) + 2*d^3*e*f*p*x^2 + 2*(3*d*e^3*f - 4*d^2*e^2*g)*p*x^6 - (3*d^2*e^2
*f - 4*d^3*e*g)*p*x^4 - 2*((3*e^4*f - 4*d*e^3*g)*p*x^8 - 4*d^4*g*p*x^2 - 3*d^4*f*p)*log(e*x^2 + d) + 2*(4*d^4*
g*x^2 + 3*d^4*f)*log(c))/(d^4*x^8)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^9} \, dx=\text {Timed out} \]

[In]

integrate((g*x**2+f)*ln(c*(e*x**2+d)**p)/x**9,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.89 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^9} \, dx=\frac {1}{48} \, e p {\left (\frac {2 \, {\left (3 \, e^{3} f - 4 \, d e^{2} g\right )} \log \left (e x^{2} + d\right )}{d^{4}} - \frac {2 \, {\left (3 \, e^{3} f - 4 \, d e^{2} g\right )} \log \left (x^{2}\right )}{d^{4}} - \frac {2 \, {\left (3 \, e^{2} f - 4 \, d e g\right )} x^{4} + 2 \, d^{2} f - {\left (3 \, d e f - 4 \, d^{2} g\right )} x^{2}}{d^{3} x^{6}}\right )} - \frac {{\left (4 \, g x^{2} + 3 \, f\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{24 \, x^{8}} \]

[In]

integrate((g*x^2+f)*log(c*(e*x^2+d)^p)/x^9,x, algorithm="maxima")

[Out]

1/48*e*p*(2*(3*e^3*f - 4*d*e^2*g)*log(e*x^2 + d)/d^4 - 2*(3*e^3*f - 4*d*e^2*g)*log(x^2)/d^4 - (2*(3*e^2*f - 4*
d*e*g)*x^4 + 2*d^2*f - (3*d*e*f - 4*d^2*g)*x^2)/(d^3*x^6)) - 1/24*(4*g*x^2 + 3*f)*log((e*x^2 + d)^p*c)/x^8

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 379 vs. \(2 (134) = 268\).

Time = 0.32 (sec) , antiderivative size = 379, normalized size of antiderivative = 2.56 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^9} \, dx=-\frac {\frac {2 \, {\left (3 \, e^{5} f p + 4 \, {\left (e x^{2} + d\right )} e^{4} g p - 4 \, d e^{4} g p\right )} \log \left (e x^{2} + d\right )}{{\left (e x^{2} + d\right )}^{4} - 4 \, {\left (e x^{2} + d\right )}^{3} d + 6 \, {\left (e x^{2} + d\right )}^{2} d^{2} - 4 \, {\left (e x^{2} + d\right )} d^{3} + d^{4}} + \frac {6 \, {\left (e x^{2} + d\right )}^{3} e^{5} f p - 21 \, {\left (e x^{2} + d\right )}^{2} d e^{5} f p + 26 \, {\left (e x^{2} + d\right )} d^{2} e^{5} f p - 11 \, d^{3} e^{5} f p - 8 \, {\left (e x^{2} + d\right )}^{3} d e^{4} g p + 28 \, {\left (e x^{2} + d\right )}^{2} d^{2} e^{4} g p - 32 \, {\left (e x^{2} + d\right )} d^{3} e^{4} g p + 12 \, d^{4} e^{4} g p + 6 \, d^{3} e^{5} f \log \left (c\right ) + 8 \, {\left (e x^{2} + d\right )} d^{3} e^{4} g \log \left (c\right ) - 8 \, d^{4} e^{4} g \log \left (c\right )}{{\left (e x^{2} + d\right )}^{4} d^{3} - 4 \, {\left (e x^{2} + d\right )}^{3} d^{4} + 6 \, {\left (e x^{2} + d\right )}^{2} d^{5} - 4 \, {\left (e x^{2} + d\right )} d^{6} + d^{7}} - \frac {2 \, {\left (3 \, e^{5} f p - 4 \, d e^{4} g p\right )} \log \left (e x^{2} + d\right )}{d^{4}} + \frac {2 \, {\left (3 \, e^{5} f p - 4 \, d e^{4} g p\right )} \log \left (e x^{2}\right )}{d^{4}}}{48 \, e} \]

[In]

integrate((g*x^2+f)*log(c*(e*x^2+d)^p)/x^9,x, algorithm="giac")

[Out]

-1/48*(2*(3*e^5*f*p + 4*(e*x^2 + d)*e^4*g*p - 4*d*e^4*g*p)*log(e*x^2 + d)/((e*x^2 + d)^4 - 4*(e*x^2 + d)^3*d +
6*(e*x^2 + d)^2*d^2 - 4*(e*x^2 + d)*d^3 + d^4) + (6*(e*x^2 + d)^3*e^5*f*p - 21*(e*x^2 + d)^2*d*e^5*f*p + 26*(
e*x^2 + d)*d^2*e^5*f*p - 11*d^3*e^5*f*p - 8*(e*x^2 + d)^3*d*e^4*g*p + 28*(e*x^2 + d)^2*d^2*e^4*g*p - 32*(e*x^2
+ d)*d^3*e^4*g*p + 12*d^4*e^4*g*p + 6*d^3*e^5*f*log(c) + 8*(e*x^2 + d)*d^3*e^4*g*log(c) - 8*d^4*e^4*g*log(c))
/((e*x^2 + d)^4*d^3 - 4*(e*x^2 + d)^3*d^4 + 6*(e*x^2 + d)^2*d^5 - 4*(e*x^2 + d)*d^6 + d^7) - 2*(3*e^5*f*p - 4*
d*e^4*g*p)*log(e*x^2 + d)/d^4 + 2*(3*e^5*f*p - 4*d*e^4*g*p)*log(e*x^2)/d^4)/e

Mupad [B] (verification not implemented)

Time = 1.68 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.91 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^9} \, dx=\frac {\ln \left (e\,x^2+d\right )\,\left (3\,e^4\,f\,p-4\,d\,e^3\,g\,p\right )}{24\,d^4}-\frac {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\,\left (\frac {g\,x^2}{6}+\frac {f}{8}\right )}{x^8}-\frac {\frac {e\,f\,p}{2\,d}+\frac {e\,p\,x^2\,\left (4\,d\,g-3\,e\,f\right )}{4\,d^2}-\frac {e^2\,p\,x^4\,\left (4\,d\,g-3\,e\,f\right )}{2\,d^3}}{12\,x^6}-\frac {\ln \left (x\right )\,\left (3\,e^4\,f\,p-4\,d\,e^3\,g\,p\right )}{12\,d^4} \]

[In]

int((log(c*(d + e*x^2)^p)*(f + g*x^2))/x^9,x)

[Out]

(log(d + e*x^2)*(3*e^4*f*p - 4*d*e^3*g*p))/(24*d^4) - (log(c*(d + e*x^2)^p)*(f/8 + (g*x^2)/6))/x^8 - ((e*f*p)/
(2*d) + (e*p*x^2*(4*d*g - 3*e*f))/(4*d^2) - (e^2*p*x^4*(4*d*g - 3*e*f))/(2*d^3))/(12*x^6) - (log(x)*(3*e^4*f*p
- 4*d*e^3*g*p))/(12*d^4)