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]

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

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Rubi [A]  time = 0.16, antiderivative size = 125, 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 = {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 12

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

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 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]))))

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 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])

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*}

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Mathematica [A]  time = 0.07, size = 130, normalized size = 1.04 \[ -\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{6 x^6}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{4 x^4}+\frac {1}{4} e g p \left (\frac {e \log \left (d+e x^2\right )}{d^2}-\frac {2 e \log (x)}{d^2}-\frac {1}{d x^2}\right )+\frac {1}{6} e f p \left (-\frac {e^2 \log \left (d+e x^2\right )}{d^3}+\frac {2 e^2 \log (x)}{d^3}+\frac {e}{d^2 x^2}-\frac {1}{2 d x^4}\right ) \]

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*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 - (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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fricas [A]  time = 0.69, size = 129, normalized size = 1.03 \[ \frac {2 \, {\left (2 \, e^{3} f - 3 \, d e^{2} g\right )} p x^{6} \log \relax (x) - d^{2} e f p x^{2} + {\left (2 \, d e^{2} f - 3 \, d^{2} e g\right )} p x^{4} - {\left ({\left (2 \, e^{3} f - 3 \, d e^{2} g\right )} p x^{6} + 3 \, d^{3} g p x^{2} + 2 \, d^{3} f p\right )} \log \left (e x^{2} + d\right ) - {\left (3 \, d^{3} g x^{2} + 2 \, d^{3} f\right )} \log \relax (c)}{12 \, d^{3} x^{6}} \]

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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giac [B]  time = 0.24, size = 515, normalized size = 4.12 \[ \frac {{\left (3 \, {\left (x^{2} e + d\right )}^{3} d g p e^{3} \log \left (x^{2} e + d\right ) - 9 \, {\left (x^{2} e + d\right )}^{2} d^{2} g p e^{3} \log \left (x^{2} e + d\right ) + 6 \, {\left (x^{2} e + d\right )} d^{3} g p e^{3} \log \left (x^{2} e + d\right ) - 3 \, {\left (x^{2} e + d\right )}^{3} d g p e^{3} \log \left (x^{2} e\right ) + 9 \, {\left (x^{2} e + d\right )}^{2} d^{2} g p e^{3} \log \left (x^{2} e\right ) - 9 \, {\left (x^{2} e + d\right )} d^{3} g p e^{3} \log \left (x^{2} e\right ) + 3 \, d^{4} g p e^{3} \log \left (x^{2} e\right ) - 3 \, {\left (x^{2} e + d\right )}^{2} d^{2} g p e^{3} + 6 \, {\left (x^{2} e + d\right )} d^{3} g p e^{3} - 3 \, d^{4} g p e^{3} - 2 \, {\left (x^{2} e + d\right )}^{3} f p e^{4} \log \left (x^{2} e + d\right ) + 6 \, {\left (x^{2} e + d\right )}^{2} d f p e^{4} \log \left (x^{2} e + d\right ) - 6 \, {\left (x^{2} e + d\right )} d^{2} f p e^{4} \log \left (x^{2} e + d\right ) + 2 \, {\left (x^{2} e + d\right )}^{3} f p e^{4} \log \left (x^{2} e\right ) - 6 \, {\left (x^{2} e + d\right )}^{2} d f p e^{4} \log \left (x^{2} e\right ) + 6 \, {\left (x^{2} e + d\right )} d^{2} f p e^{4} \log \left (x^{2} e\right ) - 2 \, d^{3} f p e^{4} \log \left (x^{2} e\right ) - 3 \, {\left (x^{2} e + d\right )} d^{3} g e^{3} \log \relax (c) + 3 \, d^{4} g e^{3} \log \relax (c) + 2 \, {\left (x^{2} e + d\right )}^{2} d f p e^{4} - 5 \, {\left (x^{2} e + d\right )} d^{2} f p e^{4} + 3 \, d^{3} f p e^{4} - 2 \, d^{3} f e^{4} \log \relax (c)\right )} e^{\left (-1\right )}}{12 \, {\left ({\left (x^{2} e + d\right )}^{3} d^{3} - 3 \, {\left (x^{2} e + d\right )}^{2} d^{4} + 3 \, {\left (x^{2} e + d\right )} d^{5} - d^{6}\right )}} \]

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)

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maple [C]  time = 0.41, size = 428, normalized size = 3.42 \[ -\frac {\left (3 g \,x^{2}+2 f \right ) \ln \left (\left (e \,x^{2}+d \right )^{p}\right )}{12 x^{6}}-\frac {12 d \,e^{2} g p \,x^{6} \ln \relax (x )-6 d \,e^{2} g p \,x^{6} \ln \left (-e \,x^{2}-d \right )-8 e^{3} f p \,x^{6} \ln \relax (x )+4 e^{3} f p \,x^{6} \ln \left (-e \,x^{2}-d \right )-3 i \pi \,d^{3} g \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )+3 i \pi \,d^{3} g \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2}+3 i \pi \,d^{3} g \,x^{2} \mathrm {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2}-3 i \pi \,d^{3} g \,x^{2} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}+6 d^{2} e g p \,x^{4}-4 d \,e^{2} f p \,x^{4}-2 i \pi \,d^{3} f \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )+2 i \pi \,d^{3} f \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2}+2 i \pi \,d^{3} f \,\mathrm {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2}-2 i \pi \,d^{3} f \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}+6 d^{3} g \,x^{2} \ln \relax (c )+2 d^{2} e f p \,x^{2}+4 d^{3} f \ln \relax (c )}{24 d^{3} x^{6}} \]

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^6-2*I*Pi*d^3*f*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*c*(e*x^2+d)^p)^2*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)*csgn(I*c)+3*I*Pi*d^3*g*x^2*csgn(I*c*(e*x^2+d)^p)^2*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)^3+2*I*Pi*d^3*f*csgn(I*(e*x^2+d)^p)*c
sgn(I*c*(e*x^2+d)^p)^2+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 = 0.45, size = 104, normalized size = 0.83 \[ -\frac {1}{12} \, e p {\left (\frac {{\left (2 \, e^{2} f - 3 \, d e g\right )} \log \left (e x^{2} + d\right )}{d^{3}} - \frac {{\left (2 \, e^{2} f - 3 \, d e g\right )} \log \left (x^{2}\right )}{d^{3}} - \frac {{\left (2 \, e f - 3 \, d g\right )} x^{2} - d f}{d^{2} x^{4}}\right )} - \frac {{\left (3 \, g x^{2} + 2 \, f\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{12 \, x^{6}} \]

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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mupad [B]  time = 0.37, size = 113, normalized size = 0.90 \[ \frac {\ln \relax (x)\,\left (2\,e^3\,f\,p-3\,d\,e^2\,g\,p\right )}{6\,d^3}-\frac {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\,\left (\frac {g\,x^2}{4}+\frac {f}{6}\right )}{x^6}-\frac {\ln \left (e\,x^2+d\right )\,\left (2\,e^3\,f\,p-3\,d\,e^2\,g\,p\right )}{12\,d^3}-\frac {\frac {e\,f\,p}{2\,d}+\frac {e\,p\,x^2\,\left (3\,d\,g-2\,e\,f\right )}{2\,d^2}}{6\,x^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(c*(d + e*x^2)^p)*(f + g*x^2))/x^7,x)

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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