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

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

[Out]

-(e*f*p)/(24*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)

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Rubi [A]  time = 0.202369, antiderivative size = 148, normalized size of antiderivative = 1., 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 = {2475, 43, 2414, 12, 77} \[ -\frac{f \log \left (c \left (d+e x^2\right )^p\right )}{8 x^8}-\frac{g \log \left (c \left (d+e x^2\right )^p\right )}{6 x^6}-\frac{e^2 p (3 e f-4 d g)}{24 d^3 x^2}+\frac{e^3 p (3 e f-4 d g) \log \left (d+e x^2\right )}{24 d^4}-\frac{e^3 p \log (x) (3 e f-4 d g)}{12 d^4}+\frac{e p (3 e f-4 d g)}{48 d^2 x^4}-\frac{e f p}{24 d x^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(e*f*p)/(24*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 2475

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])

Rule 43

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 2414

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 12

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

Rule 77

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]))))

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} \operatorname{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) \operatorname{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) \operatorname{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) \operatorname{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]  time = 0.110856, size = 158, normalized size = 1.07 \[ -\frac{f \log \left (c \left (d+e x^2\right )^p\right )}{8 x^8}-\frac{g \log \left (c \left (d+e x^2\right )^p\right )}{6 x^6}+\frac{1}{8} e f p \left (-\frac{e^2}{d^3 x^2}+\frac{e^3 \log \left (d+e x^2\right )}{d^4}-\frac{2 e^3 \log (x)}{d^4}+\frac{e}{2 d^2 x^4}-\frac{1}{3 d x^6}\right )+\frac{1}{6} e g p \left (-\frac{e^2 \log \left (d+e x^2\right )}{d^3}+\frac{2 e^2 \log (x)}{d^3}+\frac{e}{d^2 x^2}-\frac{1}{2 d x^4}\right ) \]

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*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*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]  time = 0.388, size = 448, normalized size = 3. \begin{align*} -{\frac{ \left ( 4\,g{x}^{2}+3\,f \right ) \ln \left ( \left ( e{x}^{2}+d \right ) ^{p} \right ) }{24\,{x}^{8}}}-{\frac{-16\,\ln \left ( x \right ) d{e}^{3}gp{x}^{8}+12\,\ln \left ( x \right ){e}^{4}fp{x}^{8}+8\,\ln \left ( e{x}^{2}+d \right ) d{e}^{3}gp{x}^{8}-6\,\ln \left ( e{x}^{2}+d \right ){e}^{4}fp{x}^{8}-3\,i\pi \,{d}^{4}f \left ({\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{3}+4\,i\pi \,{d}^{4}g{x}^{2} \left ({\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +4\,i\pi \,{d}^{4}g{x}^{2}{\it csgn} \left ( i \left ( e{x}^{2}+d \right ) ^{p} \right ) \left ({\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{2}+3\,i\pi \,{d}^{4}f{\it csgn} \left ( i \left ( e{x}^{2}+d \right ) ^{p} \right ) \left ({\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{2}-8\,{d}^{2}{e}^{2}gp{x}^{6}+6\,d{e}^{3}fp{x}^{6}-4\,i\pi \,{d}^{4}g{x}^{2}{\it csgn} \left ( i \left ( e{x}^{2}+d \right ) ^{p} \right ){\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ){\it csgn} \left ( ic \right ) +3\,i\pi \,{d}^{4}f \left ({\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -4\,i\pi \,{d}^{4}g{x}^{2} \left ({\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{3}-3\,i\pi \,{d}^{4}f{\it csgn} \left ( i \left ( e{x}^{2}+d \right ) ^{p} \right ){\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ){\it csgn} \left ( ic \right ) +4\,{d}^{3}egp{x}^{4}-3\,{d}^{2}{e}^{2}fp{x}^{4}+8\,\ln \left ( c \right ){d}^{4}g{x}^{2}+2\,{d}^{3}efp{x}^{2}+6\,\ln \left ( c \right ){d}^{4}f}{48\,{d}^{4}{x}^{8}}} \end{align*}

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)

[Out]

-1/24*(4*g*x^2+3*f)/x^8*ln((e*x^2+d)^p)-1/48*(-16*ln(x)*d*e^3*g*p*x^8+12*ln(x)*e^4*f*p*x^8+8*ln(e*x^2+d)*d*e^3
*g*p*x^8-6*ln(e*x^2+d)*e^4*f*p*x^8-3*I*Pi*d^4*f*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)+4*I*Pi*d^4*g*x^2*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2+3*I*Pi*d^4*f*csgn(I*(e*x^2+d)^p)*csg
n(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)*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*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 = 1.03066, size = 178, normalized size = 1.2 \begin{align*} \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}} \end{align*}

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*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

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Fricas [A]  time = 2.23226, size = 346, normalized size = 2.34 \begin{align*} -\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}} \end{align*}

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*(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)

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

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]  time = 1.30204, size = 910, normalized size = 6.15 \begin{align*} \text{result too large to display} \end{align*}

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)