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

Optimal. Leaf size=216 \[ -\frac{f^2 \log \left (c \left (d+e x^2\right )^p\right )}{8 x^8}-\frac{f g \log \left (c \left (d+e x^2\right )^p\right )}{3 x^6}-\frac{g^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 x^4}-\frac{e p \left (6 d^2 g^2-8 d e f g+3 e^2 f^2\right )}{24 d^3 x^2}+\frac{e^2 p \left (6 d^2 g^2-8 d e f g+3 e^2 f^2\right ) \log \left (d+e x^2\right )}{24 d^4}-\frac{e^2 p \log (x) \left (6 d^2 g^2-8 d e f g+3 e^2 f^2\right )}{12 d^4}+\frac{e f p (3 e f-8 d g)}{48 d^2 x^4}-\frac{e f^2 p}{24 d x^6} \]

[Out]

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

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Rubi [A]  time = 0.289703, antiderivative size = 216, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2475, 43, 2414, 12, 893} \[ -\frac{f^2 \log \left (c \left (d+e x^2\right )^p\right )}{8 x^8}-\frac{f g \log \left (c \left (d+e x^2\right )^p\right )}{3 x^6}-\frac{g^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 x^4}-\frac{e p \left (6 d^2 g^2-8 d e f g+3 e^2 f^2\right )}{24 d^3 x^2}+\frac{e^2 p \left (6 d^2 g^2-8 d e f g+3 e^2 f^2\right ) \log \left (d+e x^2\right )}{24 d^4}-\frac{e^2 p \log (x) \left (6 d^2 g^2-8 d e f g+3 e^2 f^2\right )}{12 d^4}+\frac{e f p (3 e f-8 d g)}{48 d^2 x^4}-\frac{e f^2 p}{24 d x^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 893

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rubi steps

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

Mathematica [A]  time = 0.160982, size = 184, normalized size = 0.85 \[ -\frac{2 d^4 \left (3 f^2+8 f g x^2+6 g^2 x^4\right ) \log \left (c \left (d+e x^2\right )^p\right )+d e p x^2 \left (2 d^2 \left (f^2+4 f g x^2+6 g^2 x^4\right )-d e f x^2 \left (3 f+16 g x^2\right )+6 e^2 f^2 x^4\right )+4 e^2 p x^8 \log (x) \left (6 d^2 g^2-8 d e f g+3 e^2 f^2\right )-2 e^2 p x^8 \left (6 d^2 g^2-8 d e f g+3 e^2 f^2\right ) \log \left (d+e x^2\right )}{48 d^4 x^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d*e*p*x^2*(6*e^2*f^2*x^4 - d*e*f*x^2*(3*f + 16*g*x^2) + 2*d^2*(f^2 + 4*f*g*x^2 + 6*g^2*x^4)) + 4*e^2*(3*e^2*
f^2 - 8*d*e*f*g + 6*d^2*g^2)*p*x^8*Log[x] - 2*e^2*(3*e^2*f^2 - 8*d*e*f*g + 6*d^2*g^2)*p*x^8*Log[d + e*x^2] + 2
*d^4*(3*f^2 + 8*f*g*x^2 + 6*g^2*x^4)*Log[c*(d + e*x^2)^p])/(48*d^4*x^8)

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Maple [C]  time = 0.408, size = 713, normalized size = 3.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(6*g^2*x^4+8*f*g*x^2+3*f^2)/x^8*ln((e*x^2+d)^p)-1/48*(6*ln(c)*d^4*f^2+12*ln(c)*d^4*g^2*x^4+6*I*Pi*d^4*g^
2*x^4*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2+6*I*Pi*d^4*g^2*x^4*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)-6*I*Pi*
d^4*g^2*x^4*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)+8*I*Pi*d^4*f*g*x^2*csgn(I*(e*x^2+d)^p)*csgn(I*
c*(e*x^2+d)^p)^2+8*I*Pi*d^4*f*g*x^2*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)-8*I*Pi*d^4*f*g*x^2*csgn(I*(e*x^2+d)^p)*c
sgn(I*c*(e*x^2+d)^p)*csgn(I*c)+12*d^3*e*g^2*p*x^6+6*d*e^3*f^2*p*x^6-3*d^2*e^2*f^2*p*x^4+2*d^3*e*f^2*p*x^2-6*ln
(-e*x^2-d)*e^4*f^2*p*x^8+12*ln(x)*e^4*f^2*p*x^8+16*ln(c)*d^4*f*g*x^2-3*I*Pi*d^4*f^2*csgn(I*c*(e*x^2+d)^p)^3+3*
I*Pi*d^4*f^2*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)+16*ln(-e*x^2-d)*d*e^3*f*g*p*x^8-16*d^2*e^2*f*g*p*x^6+8*d^3*e*f*
g*p*x^4-8*I*Pi*d^4*f*g*x^2*csgn(I*c*(e*x^2+d)^p)^3-12*ln(-e*x^2-d)*d^2*e^2*g^2*p*x^8+24*ln(x)*d^2*e^2*g^2*p*x^
8-6*I*Pi*d^4*g^2*x^4*csgn(I*c*(e*x^2+d)^p)^3+3*I*Pi*d^4*f^2*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2-32*ln(
x)*d*e^3*f*g*p*x^8-3*I*Pi*d^4*f^2*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c))/d^4/x^8

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Maxima [A]  time = 1.02201, size = 247, normalized size = 1.14 \begin{align*} \frac{1}{48} \, e p{\left (\frac{2 \,{\left (3 \, e^{3} f^{2} - 8 \, d e^{2} f g + 6 \, d^{2} e g^{2}\right )} \log \left (e x^{2} + d\right )}{d^{4}} - \frac{2 \,{\left (3 \, e^{3} f^{2} - 8 \, d e^{2} f g + 6 \, d^{2} e g^{2}\right )} \log \left (x^{2}\right )}{d^{4}} - \frac{2 \,{\left (3 \, e^{2} f^{2} - 8 \, d e f g + 6 \, d^{2} g^{2}\right )} x^{4} + 2 \, d^{2} f^{2} -{\left (3 \, d e f^{2} - 8 \, d^{2} f g\right )} x^{2}}{d^{3} x^{6}}\right )} - \frac{{\left (6 \, g^{2} x^{4} + 8 \, f g x^{2} + 3 \, f^{2}\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)^2*log(c*(e*x^2+d)^p)/x^9,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.85303, size = 489, normalized size = 2.26 \begin{align*} -\frac{4 \,{\left (3 \, e^{4} f^{2} - 8 \, d e^{3} f g + 6 \, d^{2} e^{2} g^{2}\right )} p x^{8} \log \left (x\right ) + 2 \, d^{3} e f^{2} p x^{2} + 2 \,{\left (3 \, d e^{3} f^{2} - 8 \, d^{2} e^{2} f g + 6 \, d^{3} e g^{2}\right )} p x^{6} -{\left (3 \, d^{2} e^{2} f^{2} - 8 \, d^{3} e f g\right )} p x^{4} + 2 \,{\left (6 \, d^{4} g^{2} p x^{4} -{\left (3 \, e^{4} f^{2} - 8 \, d e^{3} f g + 6 \, d^{2} e^{2} g^{2}\right )} p x^{8} + 8 \, d^{4} f g p x^{2} + 3 \, d^{4} f^{2} p\right )} \log \left (e x^{2} + d\right ) + 2 \,{\left (6 \, d^{4} g^{2} x^{4} + 8 \, d^{4} f g x^{2} + 3 \, d^{4} f^{2}\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)^2*log(c*(e*x^2+d)^p)/x^9,x, algorithm="fricas")

[Out]

-1/48*(4*(3*e^4*f^2 - 8*d*e^3*f*g + 6*d^2*e^2*g^2)*p*x^8*log(x) + 2*d^3*e*f^2*p*x^2 + 2*(3*d*e^3*f^2 - 8*d^2*e
^2*f*g + 6*d^3*e*g^2)*p*x^6 - (3*d^2*e^2*f^2 - 8*d^3*e*f*g)*p*x^4 + 2*(6*d^4*g^2*p*x^4 - (3*e^4*f^2 - 8*d*e^3*
f*g + 6*d^2*e^2*g^2)*p*x^8 + 8*d^4*f*g*p*x^2 + 3*d^4*f^2*p)*log(e*x^2 + d) + 2*(6*d^4*g^2*x^4 + 8*d^4*f*g*x^2
+ 3*d^4*f^2)*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)**2*ln(c*(e*x**2+d)**p)/x**9,x)

[Out]

Timed out

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Giac [B]  time = 1.35522, size = 1470, normalized size = 6.81 \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)^2*log(c*(e*x^2+d)^p)/x^9,x, algorithm="giac")

[Out]

1/48*(12*(x^2*e + d)^4*d^2*g^2*p*e^3*log(x^2*e + d) - 48*(x^2*e + d)^3*d^3*g^2*p*e^3*log(x^2*e + d) + 60*(x^2*
e + d)^2*d^4*g^2*p*e^3*log(x^2*e + d) - 24*(x^2*e + d)*d^5*g^2*p*e^3*log(x^2*e + d) - 12*(x^2*e + d)^4*d^2*g^2
*p*e^3*log(x^2*e) + 48*(x^2*e + d)^3*d^3*g^2*p*e^3*log(x^2*e) - 72*(x^2*e + d)^2*d^4*g^2*p*e^3*log(x^2*e) + 48
*(x^2*e + d)*d^5*g^2*p*e^3*log(x^2*e) - 12*d^6*g^2*p*e^3*log(x^2*e) - 12*(x^2*e + d)^3*d^3*g^2*p*e^3 + 36*(x^2
*e + d)^2*d^4*g^2*p*e^3 - 36*(x^2*e + d)*d^5*g^2*p*e^3 + 12*d^6*g^2*p*e^3 - 16*(x^2*e + d)^4*d*f*g*p*e^4*log(x
^2*e + d) + 64*(x^2*e + d)^3*d^2*f*g*p*e^4*log(x^2*e + d) - 96*(x^2*e + d)^2*d^3*f*g*p*e^4*log(x^2*e + d) + 48
*(x^2*e + d)*d^4*f*g*p*e^4*log(x^2*e + d) + 16*(x^2*e + d)^4*d*f*g*p*e^4*log(x^2*e) - 64*(x^2*e + d)^3*d^2*f*g
*p*e^4*log(x^2*e) + 96*(x^2*e + d)^2*d^3*f*g*p*e^4*log(x^2*e) - 64*(x^2*e + d)*d^4*f*g*p*e^4*log(x^2*e) + 16*d
^5*f*g*p*e^4*log(x^2*e) - 12*(x^2*e + d)^2*d^4*g^2*e^3*log(c) + 24*(x^2*e + d)*d^5*g^2*e^3*log(c) - 12*d^6*g^2
*e^3*log(c) + 16*(x^2*e + d)^3*d^2*f*g*p*e^4 - 56*(x^2*e + d)^2*d^3*f*g*p*e^4 + 64*(x^2*e + d)*d^4*f*g*p*e^4 -
 24*d^5*f*g*p*e^4 + 6*(x^2*e + d)^4*f^2*p*e^5*log(x^2*e + d) - 24*(x^2*e + d)^3*d*f^2*p*e^5*log(x^2*e + d) + 3
6*(x^2*e + d)^2*d^2*f^2*p*e^5*log(x^2*e + d) - 24*(x^2*e + d)*d^3*f^2*p*e^5*log(x^2*e + d) - 6*(x^2*e + d)^4*f
^2*p*e^5*log(x^2*e) + 24*(x^2*e + d)^3*d*f^2*p*e^5*log(x^2*e) - 36*(x^2*e + d)^2*d^2*f^2*p*e^5*log(x^2*e) + 24
*(x^2*e + d)*d^3*f^2*p*e^5*log(x^2*e) - 6*d^4*f^2*p*e^5*log(x^2*e) - 16*(x^2*e + d)*d^4*f*g*e^4*log(c) + 16*d^
5*f*g*e^4*log(c) - 6*(x^2*e + d)^3*d*f^2*p*e^5 + 21*(x^2*e + d)^2*d^2*f^2*p*e^5 - 26*(x^2*e + d)*d^3*f^2*p*e^5
 + 11*d^4*f^2*p*e^5 - 6*d^4*f^2*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)