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

Optimal. Leaf size=125 \[ -\frac{f \log \left (c \left (d+e x^2\right )^p\right )}{6 x^6}-\frac{g \log \left (c \left (d+e x^2\right )^p\right )}{4 x^4}-\frac{e^2 p (2 e f-3 d g) \log \left (d+e x^2\right )}{12 d^3}+\frac{e^2 p \log (x) (2 e f-3 d g)}{6 d^3}+\frac{e p (2 e f-3 d g)}{12 d^2 x^2}-\frac{e f p}{12 d x^4} \]

[Out]

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

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Rubi [A]  time = 0.162969, antiderivative size = 125, 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} \[ -\frac{f \log \left (c \left (d+e x^2\right )^p\right )}{6 x^6}-\frac{g \log \left (c \left (d+e x^2\right )^p\right )}{4 x^4}-\frac{e^2 p (2 e f-3 d g) \log \left (d+e x^2\right )}{12 d^3}+\frac{e^2 p \log (x) (2 e f-3 d g)}{6 d^3}+\frac{e p (2 e f-3 d g)}{12 d^2 x^2}-\frac{e f p}{12 d x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0628308, size = 130, normalized size = 1.04 \[ -\frac{f \log \left (c \left (d+e x^2\right )^p\right )}{6 x^6}-\frac{g \log \left (c \left (d+e x^2\right )^p\right )}{4 x^4}+\frac{1}{6} e f p \left (-\frac{e^2 \log \left (d+e x^2\right )}{d^3}+\frac{2 e^2 \log (x)}{d^3}+\frac{e}{d^2 x^2}-\frac{1}{2 d x^4}\right )+\frac{1}{4} e g p \left (\frac{e \log \left (d+e x^2\right )}{d^2}-\frac{2 e \log (x)}{d^2}-\frac{1}{d x^2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.395, size = 428, normalized size = 3.4 \begin{align*} -{\frac{ \left ( 3\,g{x}^{2}+2\,f \right ) \ln \left ( \left ( e{x}^{2}+d \right ) ^{p} \right ) }{12\,{x}^{6}}}-{\frac{12\,\ln \left ( x \right ) d{e}^{2}gp{x}^{6}-8\,\ln \left ( x \right ){e}^{3}fp{x}^{6}-6\,\ln \left ( -e{x}^{2}-d \right ) d{e}^{2}gp{x}^{6}+4\,\ln \left ( -e{x}^{2}-d \right ){e}^{3}fp{x}^{6}-3\,i\pi \,{d}^{3}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}^{3}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}+2\,i\pi \,{d}^{3}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}-3\,i\pi \,{d}^{3}g{x}^{2} \left ({\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{3}+2\,i\pi \,{d}^{3}f \left ({\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -2\,i\pi \,{d}^{3}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 ) -2\,i\pi \,{d}^{3}f \left ({\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{3}+3\,i\pi \,{d}^{3}g{x}^{2} \left ({\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +6\,{d}^{2}egp{x}^{4}-4\,d{e}^{2}fp{x}^{4}+6\,\ln \left ( c \right ){d}^{3}g{x}^{2}+2\,{d}^{2}efp{x}^{2}+4\,\ln \left ( c \right ){d}^{3}f}{24\,{d}^{3}{x}^{6}}} \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^7,x)

[Out]

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

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Maxima [A]  time = 1.02451, size = 140, normalized size = 1.12 \begin{align*} -\frac{1}{12} \, e p{\left (\frac{{\left (2 \, e^{2} f - 3 \, d e g\right )} \log \left (e x^{2} + d\right )}{d^{3}} - \frac{{\left (2 \, e^{2} f - 3 \, d e g\right )} \log \left (x^{2}\right )}{d^{3}} - \frac{{\left (2 \, e f - 3 \, d g\right )} x^{2} - d f}{d^{2} x^{4}}\right )} - \frac{{\left (3 \, g x^{2} + 2 \, f\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{12 \, x^{6}} \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^7,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.04722, size = 285, normalized size = 2.28 \begin{align*} \frac{2 \,{\left (2 \, e^{3} f - 3 \, d e^{2} g\right )} p x^{6} \log \left (x\right ) - d^{2} e f p x^{2} +{\left (2 \, d e^{2} f - 3 \, d^{2} e g\right )} p x^{4} -{\left ({\left (2 \, e^{3} f - 3 \, d e^{2} g\right )} p x^{6} + 3 \, d^{3} g p x^{2} + 2 \, d^{3} f p\right )} \log \left (e x^{2} + d\right ) -{\left (3 \, d^{3} g x^{2} + 2 \, d^{3} f\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)*log(c*(e*x^2+d)^p)/x^7,x, algorithm="fricas")

[Out]

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

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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**7,x)

[Out]

Timed out

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Giac [B]  time = 1.28447, size = 695, normalized size = 5.56 \begin{align*} \frac{{\left (3 \,{\left (x^{2} e + d\right )}^{3} d g p e^{3} \log \left (x^{2} e + d\right ) - 9 \,{\left (x^{2} e + d\right )}^{2} d^{2} g p e^{3} \log \left (x^{2} e + d\right ) + 6 \,{\left (x^{2} e + d\right )} d^{3} g p e^{3} \log \left (x^{2} e + d\right ) - 3 \,{\left (x^{2} e + d\right )}^{3} d g p e^{3} \log \left (x^{2} e\right ) + 9 \,{\left (x^{2} e + d\right )}^{2} d^{2} g p e^{3} \log \left (x^{2} e\right ) - 9 \,{\left (x^{2} e + d\right )} d^{3} g p e^{3} \log \left (x^{2} e\right ) + 3 \, d^{4} g p e^{3} \log \left (x^{2} e\right ) - 3 \,{\left (x^{2} e + d\right )}^{2} d^{2} g p e^{3} + 6 \,{\left (x^{2} e + d\right )} d^{3} g p e^{3} - 3 \, d^{4} g p e^{3} - 2 \,{\left (x^{2} e + d\right )}^{3} f p e^{4} \log \left (x^{2} e + d\right ) + 6 \,{\left (x^{2} e + d\right )}^{2} d f p e^{4} \log \left (x^{2} e + d\right ) - 6 \,{\left (x^{2} e + d\right )} d^{2} f p e^{4} \log \left (x^{2} e + d\right ) + 2 \,{\left (x^{2} e + d\right )}^{3} f p e^{4} \log \left (x^{2} e\right ) - 6 \,{\left (x^{2} e + d\right )}^{2} d f p e^{4} \log \left (x^{2} e\right ) + 6 \,{\left (x^{2} e + d\right )} d^{2} f p e^{4} \log \left (x^{2} e\right ) - 2 \, d^{3} f p e^{4} \log \left (x^{2} e\right ) - 3 \,{\left (x^{2} e + d\right )} d^{3} g e^{3} \log \left (c\right ) + 3 \, d^{4} g e^{3} \log \left (c\right ) + 2 \,{\left (x^{2} e + d\right )}^{2} d f p e^{4} - 5 \,{\left (x^{2} e + d\right )} d^{2} f p e^{4} + 3 \, d^{3} f p e^{4} - 2 \, d^{3} f e^{4} \log \left (c\right )\right )} e^{\left (-1\right )}}{12 \,{\left ({\left (x^{2} e + d\right )}^{3} d^{3} - 3 \,{\left (x^{2} e + d\right )}^{2} d^{4} + 3 \,{\left (x^{2} e + d\right )} d^{5} - d^{6}\right )}} \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^7,x, algorithm="giac")

[Out]

1/12*(3*(x^2*e + d)^3*d*g*p*e^3*log(x^2*e + d) - 9*(x^2*e + d)^2*d^2*g*p*e^3*log(x^2*e + d) + 6*(x^2*e + d)*d^
3*g*p*e^3*log(x^2*e + d) - 3*(x^2*e + d)^3*d*g*p*e^3*log(x^2*e) + 9*(x^2*e + d)^2*d^2*g*p*e^3*log(x^2*e) - 9*(
x^2*e + d)*d^3*g*p*e^3*log(x^2*e) + 3*d^4*g*p*e^3*log(x^2*e) - 3*(x^2*e + d)^2*d^2*g*p*e^3 + 6*(x^2*e + d)*d^3
*g*p*e^3 - 3*d^4*g*p*e^3 - 2*(x^2*e + d)^3*f*p*e^4*log(x^2*e + d) + 6*(x^2*e + d)^2*d*f*p*e^4*log(x^2*e + d) -
6*(x^2*e + d)*d^2*f*p*e^4*log(x^2*e + d) + 2*(x^2*e + d)^3*f*p*e^4*log(x^2*e) - 6*(x^2*e + d)^2*d*f*p*e^4*log
(x^2*e) + 6*(x^2*e + d)*d^2*f*p*e^4*log(x^2*e) - 2*d^3*f*p*e^4*log(x^2*e) - 3*(x^2*e + d)*d^3*g*e^3*log(c) + 3
*d^4*g*e^3*log(c) + 2*(x^2*e + d)^2*d*f*p*e^4 - 5*(x^2*e + d)*d^2*f*p*e^4 + 3*d^3*f*p*e^4 - 2*d^3*f*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)