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

Optimal. Leaf size=130 \[ -\frac{\left (f+g x^2\right )^3 \log \left (c \left (d+e x^2\right )^p\right )}{6 f x^6}+\frac{e p \log (x) \left (3 d^2 g^2-3 d e f g+e^2 f^2\right )}{3 d^3}+\frac{e f p (e f-3 d g)}{6 d^2 x^2}-\frac{p (e f-d g)^3 \log \left (d+e x^2\right )}{6 d^3 f}-\frac{e f^2 p}{12 d x^4} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.208371, antiderivative size = 130, 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, 37, 2414, 12, 88} \[ -\frac{\left (f+g x^2\right )^3 \log \left (c \left (d+e x^2\right )^p\right )}{6 f x^6}+\frac{e p \log (x) \left (3 d^2 g^2-3 d e f g+e^2 f^2\right )}{3 d^3}+\frac{e f p (e f-3 d g)}{6 d^2 x^2}-\frac{p (e f-d g)^3 \log \left (d+e x^2\right )}{6 d^3 f}-\frac{e f^2 p}{12 d x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(e*f^2*p)/(12*d*x^4) + (e*f*(e*f - 3*d*g)*p)/(6*d^2*x^2) + (e*(e^2*f^2 - 3*d*e*f*g + 3*d^2*g^2)*p*Log[x])/(3*
d^3) - ((e*f - d*g)^3*p*Log[d + e*x^2])/(6*d^3*f) - ((f + g*x^2)^3*Log[c*(d + e*x^2)^p])/(6*f*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 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

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 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

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^7} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(f+g x)^2 \log \left (c (d+e x)^p\right )}{x^4} \, dx,x,x^2\right )\\ &=-\frac{\left (f+g x^2\right )^3 \log \left (c \left (d+e x^2\right )^p\right )}{6 f x^6}-\frac{1}{2} (e p) \operatorname{Subst}\left (\int -\frac{(f+g x)^3}{3 f x^3 (d+e x)} \, dx,x,x^2\right )\\ &=-\frac{\left (f+g x^2\right )^3 \log \left (c \left (d+e x^2\right )^p\right )}{6 f x^6}+\frac{(e p) \operatorname{Subst}\left (\int \frac{(f+g x)^3}{x^3 (d+e x)} \, dx,x,x^2\right )}{6 f}\\ &=-\frac{\left (f+g x^2\right )^3 \log \left (c \left (d+e x^2\right )^p\right )}{6 f x^6}+\frac{(e p) \operatorname{Subst}\left (\int \left (\frac{f^3}{d x^3}+\frac{f^2 (-e f+3 d g)}{d^2 x^2}+\frac{f \left (e^2 f^2-3 d e f g+3 d^2 g^2\right )}{d^3 x}+\frac{(-e f+d g)^3}{d^3 (d+e x)}\right ) \, dx,x,x^2\right )}{6 f}\\ &=-\frac{e f^2 p}{12 d x^4}+\frac{e f (e f-3 d g) p}{6 d^2 x^2}+\frac{e \left (e^2 f^2-3 d e f g+3 d^2 g^2\right ) p \log (x)}{3 d^3}-\frac{(e f-d g)^3 p \log \left (d+e x^2\right )}{6 d^3 f}-\frac{\left (f+g x^2\right )^3 \log \left (c \left (d+e x^2\right )^p\right )}{6 f x^6}\\ \end{align*}

Mathematica [A]  time = 0.121187, size = 141, normalized size = 1.08 \[ -\frac{2 d^3 \left (f^2+3 f g x^2+3 g^2 x^4\right ) \log \left (c \left (d+e x^2\right )^p\right )-4 e p x^6 \log (x) \left (3 d^2 g^2-3 d e f g+e^2 f^2\right )+2 e p x^6 \left (3 d^2 g^2-3 d e f g+e^2 f^2\right ) \log \left (d+e x^2\right )+d e f p x^2 \left (d \left (f+6 g x^2\right )-2 e f x^2\right )}{12 d^3 x^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [C]  time = 0.403, size = 656, normalized size = 5.1 \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^7,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.02354, size = 185, normalized size = 1.42 \begin{align*} -\frac{1}{12} \, e p{\left (\frac{2 \,{\left (e^{2} f^{2} - 3 \, d e f g + 3 \, d^{2} g^{2}\right )} \log \left (e x^{2} + d\right )}{d^{3}} - \frac{2 \,{\left (e^{2} f^{2} - 3 \, d e f g + 3 \, d^{2} g^{2}\right )} \log \left (x^{2}\right )}{d^{3}} + \frac{d f^{2} - 2 \,{\left (e f^{2} - 3 \, d f g\right )} x^{2}}{d^{2} x^{4}}\right )} - \frac{{\left (3 \, g^{2} x^{4} + 3 \, f g x^{2} + f^{2}\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{6 \, x^{6}} \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^7,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.80621, size = 393, normalized size = 3.02 \begin{align*} \frac{4 \,{\left (e^{3} f^{2} - 3 \, d e^{2} f g + 3 \, d^{2} e g^{2}\right )} p x^{6} \log \left (x\right ) - d^{2} e f^{2} p x^{2} + 2 \,{\left (d e^{2} f^{2} - 3 \, d^{2} e f g\right )} p x^{4} - 2 \,{\left (3 \, d^{3} g^{2} p x^{4} + 3 \, d^{3} f g p x^{2} +{\left (e^{3} f^{2} - 3 \, d e^{2} f g + 3 \, d^{2} e g^{2}\right )} p x^{6} + d^{3} f^{2} p\right )} \log \left (e x^{2} + d\right ) - 2 \,{\left (3 \, d^{3} g^{2} x^{4} + 3 \, d^{3} f g x^{2} + d^{3} f^{2}\right )} \log \left (c\right )}{12 \, d^{3} x^{6}} \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^7,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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**7,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.33923, size = 1068, normalized size = 8.22 \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^7,x, algorithm="giac")

[Out]

-1/12*(6*(x^2*e + d)^3*d^2*g^2*p*e^2*log(x^2*e + d) - 12*(x^2*e + d)^2*d^3*g^2*p*e^2*log(x^2*e + d) + 6*(x^2*e
+ d)*d^4*g^2*p*e^2*log(x^2*e + d) - 6*(x^2*e + d)^3*d^2*g^2*p*e^2*log(x^2*e) + 18*(x^2*e + d)^2*d^3*g^2*p*e^2
*log(x^2*e) - 18*(x^2*e + d)*d^4*g^2*p*e^2*log(x^2*e) + 6*d^5*g^2*p*e^2*log(x^2*e) - 6*(x^2*e + d)^3*d*f*g*p*e
^3*log(x^2*e + d) + 18*(x^2*e + d)^2*d^2*f*g*p*e^3*log(x^2*e + d) - 12*(x^2*e + d)*d^3*f*g*p*e^3*log(x^2*e + d
) + 6*(x^2*e + d)^3*d*f*g*p*e^3*log(x^2*e) - 18*(x^2*e + d)^2*d^2*f*g*p*e^3*log(x^2*e) + 18*(x^2*e + d)*d^3*f*
g*p*e^3*log(x^2*e) - 6*d^4*f*g*p*e^3*log(x^2*e) + 6*(x^2*e + d)^2*d^3*g^2*e^2*log(c) - 12*(x^2*e + d)*d^4*g^2*
e^2*log(c) + 6*d^5*g^2*e^2*log(c) + 6*(x^2*e + d)^2*d^2*f*g*p*e^3 - 12*(x^2*e + d)*d^3*f*g*p*e^3 + 6*d^4*f*g*p
*e^3 + 2*(x^2*e + d)^3*f^2*p*e^4*log(x^2*e + d) - 6*(x^2*e + d)^2*d*f^2*p*e^4*log(x^2*e + d) + 6*(x^2*e + d)*d
^2*f^2*p*e^4*log(x^2*e + d) - 2*(x^2*e + d)^3*f^2*p*e^4*log(x^2*e) + 6*(x^2*e + d)^2*d*f^2*p*e^4*log(x^2*e) -
6*(x^2*e + d)*d^2*f^2*p*e^4*log(x^2*e) + 2*d^3*f^2*p*e^4*log(x^2*e) + 6*(x^2*e + d)*d^3*f*g*e^3*log(c) - 6*d^4
*f*g*e^3*log(c) - 2*(x^2*e + d)^2*d*f^2*p*e^4 + 5*(x^2*e + d)*d^2*f^2*p*e^4 - 3*d^3*f^2*p*e^4 + 2*d^3*f^2*e^4*
log(c))*e^(-1)/((x^2*e + d)^3*d^3 - 3*(x^2*e + d)^2*d^4 + 3*(x^2*e + d)*d^5 - d^6)