3.1.11 \(\int \frac {\log (c (a+b x^2)^p)}{x^6} \, dx\) [11]

Optimal. Leaf size=74 \[ -\frac {2 b p}{15 a x^3}+\frac {2 b^2 p}{5 a^2 x}+\frac {2 b^{5/2} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{5 a^{5/2}}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{5 x^5} \]

[Out]

-2/15*b*p/a/x^3+2/5*b^2*p/a^2/x+2/5*b^(5/2)*p*arctan(x*b^(1/2)/a^(1/2))/a^(5/2)-1/5*ln(c*(b*x^2+a)^p)/x^5

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Rubi [A]
time = 0.02, antiderivative size = 74, 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 = {2505, 331, 211} \begin {gather*} \frac {2 b^{5/2} p \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{5 a^{5/2}}+\frac {2 b^2 p}{5 a^2 x}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{5 x^5}-\frac {2 b p}{15 a x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[c*(a + b*x^2)^p]/x^6,x]

[Out]

(-2*b*p)/(15*a*x^3) + (2*b^2*p)/(5*a^2*x) + (2*b^(5/2)*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(5*a^(5/2)) - Log[c*(a +
b*x^2)^p]/(5*x^5)

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 2505

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

Rubi steps

\begin {align*} \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^6} \, dx &=-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{5 x^5}+\frac {1}{5} (2 b p) \int \frac {1}{x^4 \left (a+b x^2\right )} \, dx\\ &=-\frac {2 b p}{15 a x^3}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{5 x^5}-\frac {\left (2 b^2 p\right ) \int \frac {1}{x^2 \left (a+b x^2\right )} \, dx}{5 a}\\ &=-\frac {2 b p}{15 a x^3}+\frac {2 b^2 p}{5 a^2 x}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{5 x^5}+\frac {\left (2 b^3 p\right ) \int \frac {1}{a+b x^2} \, dx}{5 a^2}\\ &=-\frac {2 b p}{15 a x^3}+\frac {2 b^2 p}{5 a^2 x}+\frac {2 b^{5/2} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{5 a^{5/2}}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{5 x^5}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.00, size = 49, normalized size = 0.66 \begin {gather*} -\frac {2 b p \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-\frac {b x^2}{a}\right )}{15 a x^3}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{5 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(a + b*x^2)^p]/x^6,x]

[Out]

(-2*b*p*Hypergeometric2F1[-3/2, 1, -1/2, -((b*x^2)/a)])/(15*a*x^3) - Log[c*(a + b*x^2)^p]/(5*x^5)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.36, size = 236, normalized size = 3.19

method	result	size
risch	\(-\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right )}{5 x^{5}}-\frac {-6 \left (\munderset {\textit {\_R} =\RootOf \left (a^{5} \textit {\_Z}^{2}+b^{5} p^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} a^{5}+2 b^{5} p^{2}\right ) x -a^{3} b^{2} p \textit {\_R} \right )\right ) a^{2} x^{5}+3 i \pi \,a^{2} \mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2}-3 i \pi \,a^{2} \mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \right )-3 i \pi \,a^{2} \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{3}+3 i \pi \,a^{2} \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )-12 b^{2} p \,x^{4}+4 a b p \,x^{2}+6 \ln \left (c \right ) a^{2}}{30 a^{2} x^{5}}\)	\(236\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*(b*x^2+a)^p)/x^6,x,method=_RETURNVERBOSE)

[Out]

-1/5/x^5*ln((b*x^2+a)^p)-1/30*(-6*sum(_R*ln((3*_R^2*a^5+2*b^5*p^2)*x-a^3*b^2*p*_R),_R=RootOf(_Z^2*a^5+b^5*p^2)
)*a^2*x^5+3*I*Pi*a^2*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2-3*I*Pi*a^2*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^
2+a)^p)*csgn(I*c)-3*I*Pi*a^2*csgn(I*c*(b*x^2+a)^p)^3+3*I*Pi*a^2*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)-12*b^2*p*x^4
+4*a*b*p*x^2+6*ln(c)*a^2)/a^2/x^5

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Maxima [A]
time = 0.48, size = 62, normalized size = 0.84 \begin {gather*} \frac {2}{15} \, b p {\left (\frac {3 \, b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} + \frac {3 \, b x^{2} - a}{a^{2} x^{3}}\right )} - \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{5 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(b*x^2+a)^p)/x^6,x, algorithm="maxima")

[Out]

2/15*b*p*(3*b^2*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^2) + (3*b*x^2 - a)/(a^2*x^3)) - 1/5*log((b*x^2 + a)^p*c)/x^
5

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Fricas [A]
time = 0.39, size = 170, normalized size = 2.30 \begin {gather*} \left [\frac {3 \, b^{2} p x^{5} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) + 6 \, b^{2} p x^{4} - 2 \, a b p x^{2} - 3 \, a^{2} p \log \left (b x^{2} + a\right ) - 3 \, a^{2} \log \left (c\right )}{15 \, a^{2} x^{5}}, \frac {6 \, b^{2} p x^{5} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + 6 \, b^{2} p x^{4} - 2 \, a b p x^{2} - 3 \, a^{2} p \log \left (b x^{2} + a\right ) - 3 \, a^{2} \log \left (c\right )}{15 \, a^{2} x^{5}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(b*x^2+a)^p)/x^6,x, algorithm="fricas")

[Out]

[1/15*(3*b^2*p*x^5*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) + 6*b^2*p*x^4 - 2*a*b*p*x^2 - 3*
a^2*p*log(b*x^2 + a) - 3*a^2*log(c))/(a^2*x^5), 1/15*(6*b^2*p*x^5*sqrt(b/a)*arctan(x*sqrt(b/a)) + 6*b^2*p*x^4
- 2*a*b*p*x^2 - 3*a^2*p*log(b*x^2 + a) - 3*a^2*log(c))/(a^2*x^5)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 566 vs. \(2 (70) = 140\).
time = 169.06, size = 566, normalized size = 7.65 \begin {gather*} \begin {cases} - \frac {\log {\left (0^{p} c \right )}}{5 x^{5}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {\log {\left (a^{p} c \right )}}{5 x^{5}} & \text {for}\: b = 0 \\- \frac {2 p}{25 x^{5}} - \frac {\log {\left (c \left (b x^{2}\right )^{p} \right )}}{5 x^{5}} & \text {for}\: a = 0 \\- \frac {3 a^{3} \sqrt {- \frac {a}{b}} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{15 a^{3} x^{5} \sqrt {- \frac {a}{b}} + 15 a^{2} b x^{7} \sqrt {- \frac {a}{b}}} - \frac {2 a^{2} p x^{2} \sqrt {- \frac {a}{b}}}{\frac {15 a^{3} x^{5} \sqrt {- \frac {a}{b}}}{b} + 15 a^{2} x^{7} \sqrt {- \frac {a}{b}}} - \frac {3 a^{2} x^{2} \sqrt {- \frac {a}{b}} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{\frac {15 a^{3} x^{5} \sqrt {- \frac {a}{b}}}{b} + 15 a^{2} x^{7} \sqrt {- \frac {a}{b}}} + \frac {6 a b p x^{5} \log {\left (x - \sqrt {- \frac {a}{b}} \right )}}{\frac {15 a^{3} x^{5} \sqrt {- \frac {a}{b}}}{b} + 15 a^{2} x^{7} \sqrt {- \frac {a}{b}}} + \frac {4 a b p x^{4} \sqrt {- \frac {a}{b}}}{\frac {15 a^{3} x^{5} \sqrt {- \frac {a}{b}}}{b} + 15 a^{2} x^{7} \sqrt {- \frac {a}{b}}} - \frac {3 a b x^{5} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{\frac {15 a^{3} x^{5} \sqrt {- \frac {a}{b}}}{b} + 15 a^{2} x^{7} \sqrt {- \frac {a}{b}}} + \frac {6 b^{2} p x^{7} \log {\left (x - \sqrt {- \frac {a}{b}} \right )}}{\frac {15 a^{3} x^{5} \sqrt {- \frac {a}{b}}}{b} + 15 a^{2} x^{7} \sqrt {- \frac {a}{b}}} + \frac {6 b^{2} p x^{6} \sqrt {- \frac {a}{b}}}{\frac {15 a^{3} x^{5} \sqrt {- \frac {a}{b}}}{b} + 15 a^{2} x^{7} \sqrt {- \frac {a}{b}}} - \frac {3 b^{2} x^{7} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{\frac {15 a^{3} x^{5} \sqrt {- \frac {a}{b}}}{b} + 15 a^{2} x^{7} \sqrt {- \frac {a}{b}}} & \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**6,x)

[Out]

Piecewise((-log(0**p*c)/(5*x**5), Eq(a, 0) & Eq(b, 0)), (-log(a**p*c)/(5*x**5), Eq(b, 0)), (-2*p/(25*x**5) - l
og(c*(b*x**2)**p)/(5*x**5), Eq(a, 0)), (-3*a**3*sqrt(-a/b)*log(c*(a + b*x**2)**p)/(15*a**3*x**5*sqrt(-a/b) + 1
5*a**2*b*x**7*sqrt(-a/b)) - 2*a**2*p*x**2*sqrt(-a/b)/(15*a**3*x**5*sqrt(-a/b)/b + 15*a**2*x**7*sqrt(-a/b)) - 3
*a**2*x**2*sqrt(-a/b)*log(c*(a + b*x**2)**p)/(15*a**3*x**5*sqrt(-a/b)/b + 15*a**2*x**7*sqrt(-a/b)) + 6*a*b*p*x
**5*log(x - sqrt(-a/b))/(15*a**3*x**5*sqrt(-a/b)/b + 15*a**2*x**7*sqrt(-a/b)) + 4*a*b*p*x**4*sqrt(-a/b)/(15*a*
*3*x**5*sqrt(-a/b)/b + 15*a**2*x**7*sqrt(-a/b)) - 3*a*b*x**5*log(c*(a + b*x**2)**p)/(15*a**3*x**5*sqrt(-a/b)/b
+ 15*a**2*x**7*sqrt(-a/b)) + 6*b**2*p*x**7*log(x - sqrt(-a/b))/(15*a**3*x**5*sqrt(-a/b)/b + 15*a**2*x**7*sqrt
(-a/b)) + 6*b**2*p*x**6*sqrt(-a/b)/(15*a**3*x**5*sqrt(-a/b)/b + 15*a**2*x**7*sqrt(-a/b)) - 3*b**2*x**7*log(c*(
a + b*x**2)**p)/(15*a**3*x**5*sqrt(-a/b)/b + 15*a**2*x**7*sqrt(-a/b)), True))

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Giac [A]
time = 3.53, size = 71, normalized size = 0.96 \begin {gather*} \frac {2 \, b^{3} p \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{5 \, \sqrt {a b} a^{2}} - \frac {p \log \left (b x^{2} + a\right )}{5 \, x^{5}} + \frac {6 \, b^{2} p x^{4} - 2 \, a b p x^{2} - 3 \, a^{2} \log \left (c\right )}{15 \, a^{2} x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(b*x^2+a)^p)/x^6,x, algorithm="giac")

[Out]

2/5*b^3*p*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^2) - 1/5*p*log(b*x^2 + a)/x^5 + 1/15*(6*b^2*p*x^4 - 2*a*b*p*x^2 -
3*a^2*log(c))/(a^2*x^5)

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Mupad [B]
time = 0.26, size = 61, normalized size = 0.82 \begin {gather*} \frac {2\,b^{5/2}\,p\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{5\,a^{5/2}}-\frac {\frac {2\,b\,p}{3\,a}-\frac {2\,b^2\,p\,x^2}{a^2}}{5\,x^3}-\frac {\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}{5\,x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*(a + b*x^2)^p)/x^6,x)

[Out]

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

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