3.1.12 \(\int \frac {\log (c (a+b x^2)^p)}{x^7} \, dx\) [12]

Optimal. Leaf size=78 \[ -\frac {b p}{12 a x^4}+\frac {b^2 p}{6 a^2 x^2}+\frac {b^3 p \log (x)}{3 a^3}-\frac {b^3 p \log \left (a+b x^2\right )}{6 a^3}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{6 x^6} \]

[Out]

-1/12*b*p/a/x^4+1/6*b^2*p/a^2/x^2+1/3*b^3*p*ln(x)/a^3-1/6*b^3*p*ln(b*x^2+a)/a^3-1/6*ln(c*(b*x^2+a)^p)/x^6

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Rubi [A]
time = 0.04, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2504, 2442, 46} \begin {gather*} -\frac {b^3 p \log \left (a+b x^2\right )}{6 a^3}+\frac {b^3 p \log (x)}{3 a^3}+\frac {b^2 p}{6 a^2 x^2}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{6 x^6}-\frac {b p}{12 a x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[c*(a + b*x^2)^p]/x^7,x]

[Out]

-1/12*(b*p)/(a*x^4) + (b^2*p)/(6*a^2*x^2) + (b^3*p*Log[x])/(3*a^3) - (b^3*p*Log[a + b*x^2])/(6*a^3) - Log[c*(a
+ b*x^2)^p]/(6*x^6)

Rule 46

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 2442

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[(f + g*
x)^(q + 1)*((a + b*Log[c*(d + e*x)^n])/(g*(q + 1))), x] - Dist[b*e*(n/(g*(q + 1))), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

Rule 2504

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rubi steps

\begin {align*} \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^7} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {\log \left (c (a+b x)^p\right )}{x^4} \, dx,x,x^2\right )\\ &=-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{6 x^6}+\frac {1}{6} (b p) \text {Subst}\left (\int \frac {1}{x^3 (a+b x)} \, dx,x,x^2\right )\\ &=-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{6 x^6}+\frac {1}{6} (b p) \text {Subst}\left (\int \left (\frac {1}{a x^3}-\frac {b}{a^2 x^2}+\frac {b^2}{a^3 x}-\frac {b^3}{a^3 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {b p}{12 a x^4}+\frac {b^2 p}{6 a^2 x^2}+\frac {b^3 p \log (x)}{3 a^3}-\frac {b^3 p \log \left (a+b x^2\right )}{6 a^3}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{6 x^6}\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 68, normalized size = 0.87 \begin {gather*} -\frac {\frac {b p x^2 \left (a \left (a-2 b x^2\right )-4 b^2 x^4 \log (x)+2 b^2 x^4 \log \left (a+b x^2\right )\right )}{a^3}+2 \log \left (c \left (a+b x^2\right )^p\right )}{12 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(a + b*x^2)^p]/x^7,x]

[Out]

-1/12*((b*p*x^2*(a*(a - 2*b*x^2) - 4*b^2*x^4*Log[x] + 2*b^2*x^4*Log[a + b*x^2]))/a^3 + 2*Log[c*(a + b*x^2)^p])
/x^6

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.26, size = 206, normalized size = 2.64

method	result	size
risch	\(-\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right )}{6 x^{6}}-\frac {-4 b^{3} p \ln \left (x \right ) x^{6}+2 b^{3} p \ln \left (b \,x^{2}+a \right ) x^{6}+i \pi \,a^{3} \mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2}-i \pi \,a^{3} \mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \right )-i \pi \,a^{3} \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{3}+i \pi \,a^{3} \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )-2 a \,b^{2} p \,x^{4}+a^{2} b p \,x^{2}+2 \ln \left (c \right ) a^{3}}{12 a^{3} x^{6}}\)	\(206\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*(b*x^2+a)^p)/x^7,x,method=_RETURNVERBOSE)

[Out]

-1/6/x^6*ln((b*x^2+a)^p)-1/12*(-4*b^3*p*ln(x)*x^6+2*b^3*p*ln(b*x^2+a)*x^6+I*Pi*a^3*csgn(I*(b*x^2+a)^p)*csgn(I*
c*(b*x^2+a)^p)^2-I*Pi*a^3*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)*csgn(I*c)-I*Pi*a^3*csgn(I*c*(b*x^2+a)^p)^3
+I*Pi*a^3*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)-2*a*b^2*p*x^4+a^2*b*p*x^2+2*ln(c)*a^3)/a^3/x^6

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Maxima [A]
time = 0.27, size = 69, normalized size = 0.88 \begin {gather*} -\frac {1}{12} \, b p {\left (\frac {2 \, b^{2} \log \left (b x^{2} + a\right )}{a^{3}} - \frac {2 \, b^{2} \log \left (x^{2}\right )}{a^{3}} - \frac {2 \, b x^{2} - a}{a^{2} x^{4}}\right )} - \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{6 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(b*x^2+a)^p)/x^7,x, algorithm="maxima")

[Out]

-1/12*b*p*(2*b^2*log(b*x^2 + a)/a^3 - 2*b^2*log(x^2)/a^3 - (2*b*x^2 - a)/(a^2*x^4)) - 1/6*log((b*x^2 + a)^p*c)
/x^6

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Fricas [A]
time = 0.38, size = 71, normalized size = 0.91 \begin {gather*} \frac {4 \, b^{3} p x^{6} \log \left (x\right ) + 2 \, a b^{2} p x^{4} - a^{2} b p x^{2} - 2 \, a^{3} \log \left (c\right ) - 2 \, {\left (b^{3} p x^{6} + a^{3} p\right )} \log \left (b x^{2} + a\right )}{12 \, a^{3} x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(b*x^2+a)^p)/x^7,x, algorithm="fricas")

[Out]

1/12*(4*b^3*p*x^6*log(x) + 2*a*b^2*p*x^4 - a^2*b*p*x^2 - 2*a^3*log(c) - 2*(b^3*p*x^6 + a^3*p)*log(b*x^2 + a))/
(a^3*x^6)

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Sympy [A]
time = 8.34, size = 97, normalized size = 1.24 \begin {gather*} \begin {cases} - \frac {\log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{6 x^{6}} - \frac {b p}{12 a x^{4}} + \frac {b^{2} p}{6 a^{2} x^{2}} + \frac {b^{3} p \log {\left (x \right )}}{3 a^{3}} - \frac {b^{3} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{6 a^{3}} & \text {for}\: a \neq 0 \\- \frac {p}{18 x^{6}} - \frac {\log {\left (c \left (b x^{2}\right )^{p} \right )}}{6 x^{6}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(c*(b*x**2+a)**p)/x**7,x)

[Out]

Piecewise((-log(c*(a + b*x**2)**p)/(6*x**6) - b*p/(12*a*x**4) + b**2*p/(6*a**2*x**2) + b**3*p*log(x)/(3*a**3)
- b**3*log(c*(a + b*x**2)**p)/(6*a**3), Ne(a, 0)), (-p/(18*x**6) - log(c*(b*x**2)**p)/(6*x**6), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 191 vs. \(2 (68) = 136\).
time = 3.65, size = 191, normalized size = 2.45 \begin {gather*} -\frac {\frac {2 \, b^{4} p \log \left (b x^{2} + a\right )}{{\left (b x^{2} + a\right )}^{3} - 3 \, {\left (b x^{2} + a\right )}^{2} a + 3 \, {\left (b x^{2} + a\right )} a^{2} - a^{3}} + \frac {2 \, b^{4} p \log \left (b x^{2} + a\right )}{a^{3}} - \frac {2 \, b^{4} p \log \left (b x^{2}\right )}{a^{3}} - \frac {2 \, {\left (b x^{2} + a\right )}^{2} b^{4} p - 5 \, {\left (b x^{2} + a\right )} a b^{4} p + 3 \, a^{2} b^{4} p - 2 \, a^{2} b^{4} \log \left (c\right )}{{\left (b x^{2} + a\right )}^{3} a^{2} - 3 \, {\left (b x^{2} + a\right )}^{2} a^{3} + 3 \, {\left (b x^{2} + a\right )} a^{4} - a^{5}}}{12 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(b*x^2+a)^p)/x^7,x, algorithm="giac")

[Out]

-1/12*(2*b^4*p*log(b*x^2 + a)/((b*x^2 + a)^3 - 3*(b*x^2 + a)^2*a + 3*(b*x^2 + a)*a^2 - a^3) + 2*b^4*p*log(b*x^
2 + a)/a^3 - 2*b^4*p*log(b*x^2)/a^3 - (2*(b*x^2 + a)^2*b^4*p - 5*(b*x^2 + a)*a*b^4*p + 3*a^2*b^4*p - 2*a^2*b^4
*log(c))/((b*x^2 + a)^3*a^2 - 3*(b*x^2 + a)^2*a^3 + 3*(b*x^2 + a)*a^4 - a^5))/b

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Mupad [B]
time = 0.26, size = 68, normalized size = 0.87 \begin {gather*} \frac {b^2\,p}{6\,a^2\,x^2}-\frac {b^3\,p\,\ln \left (b\,x^2+a\right )}{6\,a^3}-\frac {\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}{6\,x^6}+\frac {b^3\,p\,\ln \left (x\right )}{3\,a^3}-\frac {b\,p}{12\,a\,x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*(a + b*x^2)^p)/x^7,x)

[Out]

(b^2*p)/(6*a^2*x^2) - (b^3*p*log(a + b*x^2))/(6*a^3) - log(c*(a + b*x^2)^p)/(6*x^6) + (b^3*p*log(x))/(3*a^3) -
(b*p)/(12*a*x^4)

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