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

Optimal. Leaf size=148 \[ -\frac {e f p}{24 d x^6}+\frac {e (3 e f-4 d g) p}{48 d^2 x^4}-\frac {e^2 (3 e f-4 d g) p}{24 d^3 x^2}-\frac {e^3 (3 e f-4 d g) p \log (x)}{12 d^4}+\frac {e^3 (3 e f-4 d g) p \log \left (d+e x^2\right )}{24 d^4}-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{8 x^8}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{6 x^6} \]

[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

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Rubi [A]
time = 0.13, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2525, 45, 2461, 12, 78} \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^9} \, dx &=\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*}

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Mathematica [A]
time = 0.08, size = 158, normalized size = 1.07 \begin {gather*} \frac {1}{6} e g p \left (-\frac {1}{2 d x^4}+\frac {e}{d^2 x^2}+\frac {2 e^2 \log (x)}{d^3}-\frac {e^2 \log \left (d+e x^2\right )}{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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.19, size = 448, normalized size = 3.03

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(4*g*x^2+3*f)/x^8*ln((e*x^2+d)^p)-1/48*(8*ln(e*x^2+d)*d*e^3*g*p*x^8-6*ln(e*x^2+d)*e^4*f*p*x^8-16*ln(x)*d
*e^3*g*p*x^8+12*ln(x)*e^4*f*p*x^8-4*I*Pi*d^4*g*x^2*csgn(I*c*(e*x^2+d)^p)^3+4*I*Pi*d^4*g*x^2*csgn(I*c*(e*x^2+d)
^p)^2*csgn(I*c)+3*I*Pi*d^4*f*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)+4*I*Pi*d^4*g*x^2*csgn(I*(e*x^2+d)^p)*csgn(I*c*(
e*x^2+d)^p)^2-8*d^2*e^2*g*p*x^6+6*d*e^3*f*p*x^6-4*I*Pi*d^4*g*x^2*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csg
n(I*c)+3*I*Pi*d^4*f*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2-3*I*Pi*d^4*f*csgn(I*c*(e*x^2+d)^p)^3-3*I*Pi*d^
4*f*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)+4*d^3*e*g*p*x^4-3*d^2*e^2*f*p*x^4+8*ln(c)*d^4*g*x^2+2*
d^3*e*f*p*x^2+6*ln(c)*d^4*f)/d^4/x^8

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Maxima [A]
time = 0.33, size = 132, normalized size = 0.89 \begin {gather*} -\frac {1}{48} \, p {\left (\frac {2 \, {\left (4 \, d g e^{2} - 3 \, f e^{3}\right )} \log \left (x^{2} e + d\right )}{d^{4}} - \frac {2 \, {\left (4 \, d g e^{2} - 3 \, f e^{3}\right )} \log \left (x^{2}\right )}{d^{4}} - \frac {2 \, {\left (4 \, d g e - 3 \, f e^{2}\right )} x^{4} - 2 \, d^{2} f - {\left (4 \, d^{2} g - 3 \, d f e\right )} x^{2}}{d^{3} x^{6}}\right )} e - \frac {{\left (4 \, g x^{2} + 3 \, f\right )} \log \left ({\left (x^{2} e + d\right )}^{p} c\right )}{24 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.34, size = 160, normalized size = 1.08 \begin {gather*} -\frac {6 \, d f p x^{6} e^{3} - {\left (8 \, d^{2} g p x^{6} + 3 \, d^{2} f p x^{4}\right )} e^{2} + 2 \, {\left (2 \, d^{3} g p x^{4} + d^{3} f p x^{2}\right )} e + 2 \, {\left (4 \, d g p x^{8} e^{3} - 3 \, f p x^{8} e^{4} + 4 \, d^{4} g p x^{2} + 3 \, d^{4} f p\right )} \log \left (x^{2} e + d\right ) + 2 \, {\left (4 \, d^{4} g x^{2} + 3 \, d^{4} f\right )} \log \left (c\right ) - 4 \, {\left (4 \, d g p x^{8} e^{3} - 3 \, f p x^{8} e^{4}\right )} \log \left (x\right )}{48 \, d^{4} x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 674 vs. \(2 (140) = 280\).
time = 3.57, size = 674, normalized size = 4.55 \begin {gather*} -\frac {{\left (8 \, {\left (x^{2} e + d\right )}^{4} d g p e^{4} \log \left (x^{2} e + d\right ) - 32 \, {\left (x^{2} e + d\right )}^{3} d^{2} g p e^{4} \log \left (x^{2} e + d\right ) + 48 \, {\left (x^{2} e + d\right )}^{2} d^{3} g p e^{4} \log \left (x^{2} e + d\right ) - 24 \, {\left (x^{2} e + d\right )} d^{4} g p e^{4} \log \left (x^{2} e + d\right ) - 8 \, {\left (x^{2} e + d\right )}^{4} d g p e^{4} \log \left (x^{2} e\right ) + 32 \, {\left (x^{2} e + d\right )}^{3} d^{2} g p e^{4} \log \left (x^{2} e\right ) - 48 \, {\left (x^{2} e + d\right )}^{2} d^{3} g p e^{4} \log \left (x^{2} e\right ) + 32 \, {\left (x^{2} e + d\right )} d^{4} g p e^{4} \log \left (x^{2} e\right ) - 8 \, d^{5} g p e^{4} \log \left (x^{2} e\right ) - 8 \, {\left (x^{2} e + d\right )}^{3} d^{2} g p e^{4} + 28 \, {\left (x^{2} e + d\right )}^{2} d^{3} g p e^{4} - 32 \, {\left (x^{2} e + d\right )} d^{4} g p e^{4} + 12 \, d^{5} g p e^{4} - 6 \, {\left (x^{2} e + d\right )}^{4} f p e^{5} \log \left (x^{2} e + d\right ) + 24 \, {\left (x^{2} e + d\right )}^{3} d f p e^{5} \log \left (x^{2} e + d\right ) - 36 \, {\left (x^{2} e + d\right )}^{2} d^{2} f p e^{5} \log \left (x^{2} e + d\right ) + 24 \, {\left (x^{2} e + d\right )} d^{3} f p e^{5} \log \left (x^{2} e + d\right ) + 6 \, {\left (x^{2} e + d\right )}^{4} f p e^{5} \log \left (x^{2} e\right ) - 24 \, {\left (x^{2} e + d\right )}^{3} d f p e^{5} \log \left (x^{2} e\right ) + 36 \, {\left (x^{2} e + d\right )}^{2} d^{2} f p e^{5} \log \left (x^{2} e\right ) - 24 \, {\left (x^{2} e + d\right )} d^{3} f p e^{5} \log \left (x^{2} e\right ) + 6 \, d^{4} f p e^{5} \log \left (x^{2} e\right ) + 8 \, {\left (x^{2} e + d\right )} d^{4} g e^{4} \log \left (c\right ) - 8 \, d^{5} g e^{4} \log \left (c\right ) + 6 \, {\left (x^{2} e + d\right )}^{3} d f p e^{5} - 21 \, {\left (x^{2} e + d\right )}^{2} d^{2} f p e^{5} + 26 \, {\left (x^{2} e + d\right )} d^{3} f p e^{5} - 11 \, d^{4} f p e^{5} + 6 \, d^{4} f e^{5} \log \left (c\right )\right )} e^{\left (-1\right )}}{48 \, {\left ({\left (x^{2} e + d\right )}^{4} d^{4} - 4 \, {\left (x^{2} e + d\right )}^{3} d^{5} + 6 \, {\left (x^{2} e + d\right )}^{2} d^{6} - 4 \, {\left (x^{2} e + d\right )} d^{7} + d^{8}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/48*(8*(x^2*e + d)^4*d*g*p*e^4*log(x^2*e + d) - 32*(x^2*e + d)^3*d^2*g*p*e^4*log(x^2*e + d) + 48*(x^2*e + d)
^2*d^3*g*p*e^4*log(x^2*e + d) - 24*(x^2*e + d)*d^4*g*p*e^4*log(x^2*e + d) - 8*(x^2*e + d)^4*d*g*p*e^4*log(x^2*
e) + 32*(x^2*e + d)^3*d^2*g*p*e^4*log(x^2*e) - 48*(x^2*e + d)^2*d^3*g*p*e^4*log(x^2*e) + 32*(x^2*e + d)*d^4*g*
p*e^4*log(x^2*e) - 8*d^5*g*p*e^4*log(x^2*e) - 8*(x^2*e + d)^3*d^2*g*p*e^4 + 28*(x^2*e + d)^2*d^3*g*p*e^4 - 32*
(x^2*e + d)*d^4*g*p*e^4 + 12*d^5*g*p*e^4 - 6*(x^2*e + d)^4*f*p*e^5*log(x^2*e + d) + 24*(x^2*e + d)^3*d*f*p*e^5
*log(x^2*e + d) - 36*(x^2*e + d)^2*d^2*f*p*e^5*log(x^2*e + d) + 24*(x^2*e + d)*d^3*f*p*e^5*log(x^2*e + d) + 6*
(x^2*e + d)^4*f*p*e^5*log(x^2*e) - 24*(x^2*e + d)^3*d*f*p*e^5*log(x^2*e) + 36*(x^2*e + d)^2*d^2*f*p*e^5*log(x^
2*e) - 24*(x^2*e + d)*d^3*f*p*e^5*log(x^2*e) + 6*d^4*f*p*e^5*log(x^2*e) + 8*(x^2*e + d)*d^4*g*e^4*log(c) - 8*d
^5*g*e^4*log(c) + 6*(x^2*e + d)^3*d*f*p*e^5 - 21*(x^2*e + d)^2*d^2*f*p*e^5 + 26*(x^2*e + d)*d^3*f*p*e^5 - 11*d
^4*f*p*e^5 + 6*d^4*f*e^5*log(c))*e^(-1)/((x^2*e + d)^4*d^4 - 4*(x^2*e + d)^3*d^5 + 6*(x^2*e + d)^2*d^6 - 4*(x^
2*e + d)*d^7 + d^8)

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Mupad [B]
time = 0.40, size = 134, normalized size = 0.91 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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