3.4.29 \(\int \frac {(f+g x^2)^2 \log (c (d+e x^2)^p)}{x^7} \, dx\) [329]

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

[Out]

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

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Rubi [A]
time = 0.14, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2525, 37, 2461, 12, 90} \begin {gather*} -\frac {\left (f+g x^2\right )^3 \log \left (c \left (d+e x^2\right )^p\right )}{6 f x^6}-\frac {p (e f-d g)^3 \log \left (d+e x^2\right )}{6 d^3 f}+\frac {e f p (e f-3 d g)}{6 d^2 x^2}+\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^2 p}{12 d x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/12*(e*f^2*p)/(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 12

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

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 90

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]))

Rule 2461

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 2525

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 )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^7} \, dx &=\frac {1}{2} \text {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) \text {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) \text {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) \text {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*}

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Mathematica [A]
time = 0.09, size = 141, normalized size = 1.08 \begin {gather*} -\frac {d e f p x^2 \left (-2 e f x^2+d \left (f+6 g x^2\right )\right )-4 e \left (e^2 f^2-3 d e f g+3 d^2 g^2\right ) p x^6 \log (x)+2 e \left (e^2 f^2-3 d e f g+3 d^2 g^2\right ) p x^6 \log \left (d+e x^2\right )+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 )}{12 d^3 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/12*(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])/(d^3*x^6)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.45, size = 656, normalized size = 5.05

method	result	size
risch	\(-\frac {\left (3 g^{2} x^{4}+3 f g \,x^{2}+f^{2}\right ) \ln \left (\left (e \,x^{2}+d \right )^{p}\right )}{6 x^{6}}+\frac {3 i \pi \,d^{3} f g \,x^{2} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}+3 i \pi \,d^{3} g^{2} x^{4} \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 ) \mathrm {csgn}\left (i c \right )-3 i \pi \,d^{3} g^{2} x^{4} \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^{2} x^{4} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )-6 \ln \left (e \,x^{2}+d \right ) d^{2} e \,g^{2} p \,x^{6}+6 \ln \left (e \,x^{2}+d \right ) d \,e^{2} f g p \,x^{6}-2 \ln \left (e \,x^{2}+d \right ) e^{3} f^{2} p \,x^{6}+12 \ln \left (x \right ) d^{2} e \,g^{2} p \,x^{6}-12 \ln \left (x \right ) d \,e^{2} f g p \,x^{6}+4 \ln \left (x \right ) e^{3} f^{2} p \,x^{6}+3 i \pi \,d^{3} g^{2} x^{4} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}+i \pi \,d^{3} f^{2} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}-3 i \pi \,d^{3} f g \,x^{2} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )-3 i \pi \,d^{3} f 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} f 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 ) \mathrm {csgn}\left (i c \right )+i \pi \,d^{3} f^{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 ) \mathrm {csgn}\left (i c \right )-i \pi \,d^{3} f^{2} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )-i \pi \,d^{3} f^{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}-6 \ln \left (c \right ) d^{3} g^{2} x^{4}-6 d^{2} e f g p \,x^{4}+2 d \,e^{2} f^{2} p \,x^{4}-6 \ln \left (c \right ) d^{3} f g \,x^{2}-d^{2} e \,f^{2} p \,x^{2}-2 \ln \left (c \right ) d^{3} f^{2}}{12 d^{3} x^{6}}\)	\(656\)

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,method=_RETURNVERBOSE)

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

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Maxima [A]
time = 0.27, size = 142, normalized size = 1.09 \begin {gather*} -\frac {1}{12} \, p {\left (\frac {2 \, {\left (3 \, d^{2} g^{2} - 3 \, d f g e + f^{2} e^{2}\right )} \log \left (x^{2} e + d\right )}{d^{3}} - \frac {2 \, {\left (3 \, d^{2} g^{2} - 3 \, d f g e + f^{2} e^{2}\right )} \log \left (x^{2}\right )}{d^{3}} + \frac {d f^{2} + 2 \, {\left (3 \, d f g - f^{2} e\right )} x^{2}}{d^{2} x^{4}}\right )} e - \frac {{\left (3 \, g^{2} x^{4} + 3 \, f g x^{2} + f^{2}\right )} \log \left ({\left (x^{2} e + d\right )}^{p} c\right )}{6 \, x^{6}} \end {gather*}

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

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Fricas [A]
time = 0.37, size = 199, normalized size = 1.53 \begin {gather*} \frac {2 \, d f^{2} p x^{4} e^{2} - {\left (6 \, d^{2} f g p x^{4} + d^{2} f^{2} p x^{2}\right )} e - 2 \, {\left (3 \, d^{2} g^{2} p x^{6} e - 3 \, d f g p x^{6} e^{2} + 3 \, d^{3} g^{2} p x^{4} + f^{2} p x^{6} e^{3} + 3 \, d^{3} f g p x^{2} + d^{3} f^{2} p\right )} \log \left (x^{2} e + 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 ) + 4 \, {\left (3 \, d^{2} g^{2} p x^{6} e - 3 \, d f g p x^{6} e^{2} + f^{2} p x^{6} e^{3}\right )} \log \left (x\right )}{12 \, d^{3} x^{6}} \end {gather*}

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 791 vs. \(2 (128) = 256\).
time = 5.80, size = 791, normalized size = 6.08 \begin {gather*} -\frac {{\left (6 \, {\left (x^{2} e + d\right )}^{3} d^{2} g^{2} p e^{2} \log \left (x^{2} e + d\right ) - 12 \, {\left (x^{2} e + d\right )}^{2} d^{3} g^{2} p e^{2} \log \left (x^{2} e + d\right ) + 6 \, {\left (x^{2} e + d\right )} d^{4} g^{2} p e^{2} \log \left (x^{2} e + d\right ) - 6 \, {\left (x^{2} e + d\right )}^{3} d^{2} g^{2} p e^{2} \log \left (x^{2} e\right ) + 18 \, {\left (x^{2} e + d\right )}^{2} d^{3} g^{2} p e^{2} \log \left (x^{2} e\right ) - 18 \, {\left (x^{2} e + d\right )} d^{4} g^{2} p e^{2} \log \left (x^{2} e\right ) + 6 \, d^{5} g^{2} p e^{2} \log \left (x^{2} e\right ) - 6 \, {\left (x^{2} e + d\right )}^{3} d f g p e^{3} \log \left (x^{2} e + d\right ) + 18 \, {\left (x^{2} e + d\right )}^{2} d^{2} f g p e^{3} \log \left (x^{2} e + d\right ) - 12 \, {\left (x^{2} e + d\right )} d^{3} f g p e^{3} \log \left (x^{2} e + d\right ) + 6 \, {\left (x^{2} e + d\right )}^{3} d f g p e^{3} \log \left (x^{2} e\right ) - 18 \, {\left (x^{2} e + d\right )}^{2} d^{2} f g p e^{3} \log \left (x^{2} e\right ) + 18 \, {\left (x^{2} e + d\right )} d^{3} f g p e^{3} \log \left (x^{2} e\right ) - 6 \, d^{4} f g p e^{3} \log \left (x^{2} e\right ) + 6 \, {\left (x^{2} e + d\right )}^{2} d^{3} g^{2} e^{2} \log \left (c\right ) - 12 \, {\left (x^{2} e + d\right )} d^{4} g^{2} e^{2} \log \left (c\right ) + 6 \, d^{5} g^{2} e^{2} \log \left (c\right ) + 6 \, {\left (x^{2} e + d\right )}^{2} d^{2} f g p e^{3} - 12 \, {\left (x^{2} e + d\right )} d^{3} f g p e^{3} + 6 \, d^{4} f g p e^{3} + 2 \, {\left (x^{2} e + d\right )}^{3} f^{2} p e^{4} \log \left (x^{2} e + d\right ) - 6 \, {\left (x^{2} e + d\right )}^{2} d f^{2} p e^{4} \log \left (x^{2} e + d\right ) + 6 \, {\left (x^{2} e + d\right )} d^{2} f^{2} p e^{4} \log \left (x^{2} e + d\right ) - 2 \, {\left (x^{2} e + d\right )}^{3} f^{2} p e^{4} \log \left (x^{2} e\right ) + 6 \, {\left (x^{2} e + d\right )}^{2} d f^{2} p e^{4} \log \left (x^{2} e\right ) - 6 \, {\left (x^{2} e + d\right )} d^{2} f^{2} p e^{4} \log \left (x^{2} e\right ) + 2 \, d^{3} f^{2} p e^{4} \log \left (x^{2} e\right ) + 6 \, {\left (x^{2} e + d\right )} d^{3} f g e^{3} \log \left (c\right ) - 6 \, d^{4} f g e^{3} \log \left (c\right ) - 2 \, {\left (x^{2} e + d\right )}^{2} d f^{2} p e^{4} + 5 \, {\left (x^{2} e + d\right )} d^{2} f^{2} p e^{4} - 3 \, d^{3} f^{2} p e^{4} + 2 \, d^{3} f^{2} 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 {gather*}

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)

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Mupad [B]
time = 0.41, size = 151, normalized size = 1.16 \begin {gather*} \frac {\ln \left (x\right )\,\left (3\,p\,d^2\,e\,g^2-3\,p\,d\,e^2\,f\,g+p\,e^3\,f^2\right )}{3\,d^3}-\frac {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\,\left (\frac {f^2}{6}+\frac {f\,g\,x^2}{2}+\frac {g^2\,x^4}{2}\right )}{x^6}-\frac {\ln \left (e\,x^2+d\right )\,\left (3\,p\,d^2\,e\,g^2-3\,p\,d\,e^2\,f\,g+p\,e^3\,f^2\right )}{6\,d^3}-\frac {\frac {e\,f^2\,p}{4\,d}+\frac {e\,f\,p\,x^2\,\left (3\,d\,g-e\,f\right )}{2\,d^2}}{3\,x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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