\(\int \frac {(f+g x^2) \log (c (d+e x^2)^p)}{x^6} \, dx\) [322]

Optimal result

Integrand size = 23, antiderivative size = 140 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^6} \, dx=-\frac {2 e f p}{15 d x^3}+\frac {2 e^2 f p}{5 d^2 x}-\frac {2 e g p}{3 d x}+\frac {2 e^{5/2} f p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{5 d^{5/2}}-\frac {2 e^{3/2} g p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 d^{3/2}}-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{5 x^5}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{3 x^3} \]

[Out]

-2/15*e*f*p/d/x^3+2/5*e^2*f*p/d^2/x-2/3*e*g*p/d/x+2/5*e^(5/2)*f*p*arctan(x*e^(1/2)/d^(1/2))/d^(5/2)-2/3*e^(3/2
)*g*p*arctan(x*e^(1/2)/d^(1/2))/d^(3/2)-1/5*f*ln(c*(e*x^2+d)^p)/x^5-1/3*g*ln(c*(e*x^2+d)^p)/x^3

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2526, 2505, 331, 211} \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^6} \, dx=\frac {2 e^{5/2} f p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{5 d^{5/2}}-\frac {2 e^{3/2} g p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 d^{3/2}}-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{5 x^5}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{3 x^3}+\frac {2 e^2 f p}{5 d^2 x}-\frac {2 e f p}{15 d x^3}-\frac {2 e g p}{3 d x} \]

[In]

Int[((f + g*x^2)*Log[c*(d + e*x^2)^p])/x^6,x]

[Out]

(-2*e*f*p)/(15*d*x^3) + (2*e^2*f*p)/(5*d^2*x) - (2*e*g*p)/(3*d*x) + (2*e^(5/2)*f*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]]
)/(5*d^(5/2)) - (2*e^(3/2)*g*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(3*d^(3/2)) - (f*Log[c*(d + e*x^2)^p])/(5*x^5) - (
g*Log[c*(d + e*x^2)^p])/(3*x^3)

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]

Rule 2526

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, x^m*(f + g*x^s)^r, x], x] /; FreeQ[{a, b, c,
d, e, f, g, m, n, p, q, r, s}, x] && IGtQ[q, 0] && IntegerQ[m] && IntegerQ[r] && IntegerQ[s]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{x^6}+\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{x^4}\right ) \, dx \\ & = f \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{x^6} \, dx+g \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{x^4} \, dx \\ & = -\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{5 x^5}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{3 x^3}+\frac {1}{5} (2 e f p) \int \frac {1}{x^4 \left (d+e x^2\right )} \, dx+\frac {1}{3} (2 e g p) \int \frac {1}{x^2 \left (d+e x^2\right )} \, dx \\ & = -\frac {2 e f p}{15 d x^3}-\frac {2 e g p}{3 d x}-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{5 x^5}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{3 x^3}-\frac {\left (2 e^2 f p\right ) \int \frac {1}{x^2 \left (d+e x^2\right )} \, dx}{5 d}-\frac {\left (2 e^2 g p\right ) \int \frac {1}{d+e x^2} \, dx}{3 d} \\ & = -\frac {2 e f p}{15 d x^3}+\frac {2 e^2 f p}{5 d^2 x}-\frac {2 e g p}{3 d x}-\frac {2 e^{3/2} g p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 d^{3/2}}-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{5 x^5}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{3 x^3}+\frac {\left (2 e^3 f p\right ) \int \frac {1}{d+e x^2} \, dx}{5 d^2} \\ & = -\frac {2 e f p}{15 d x^3}+\frac {2 e^2 f p}{5 d^2 x}-\frac {2 e g p}{3 d x}+\frac {2 e^{5/2} f p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{5 d^{5/2}}-\frac {2 e^{3/2} g p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 d^{3/2}}-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{5 x^5}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{3 x^3} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.01 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.72 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^6} \, dx=-\frac {2 e f p \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},-\frac {e x^2}{d}\right )}{15 d x^3}-\frac {2 e g p \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-\frac {e x^2}{d}\right )}{3 d x}-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{5 x^5}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{3 x^3} \]

[In]

Integrate[((f + g*x^2)*Log[c*(d + e*x^2)^p])/x^6,x]

[Out]

(-2*e*f*p*Hypergeometric2F1[-3/2, 1, -1/2, -((e*x^2)/d)])/(15*d*x^3) - (2*e*g*p*Hypergeometric2F1[-1/2, 1, 1/2
, -((e*x^2)/d)])/(3*d*x) - (f*Log[c*(d + e*x^2)^p])/(5*x^5) - (g*Log[c*(d + e*x^2)^p])/(3*x^3)

Maple [A] (verified)

Time = 1.14 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.69

method	result	size
parts	\(-\frac {g \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{3 x^{3}}-\frac {f \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{5 x^{5}}-\frac {2 p e \left (-\frac {-5 d g +3 e f}{d^{2} x}+\frac {f}{d \,x^{3}}+\frac {e \left (5 d g -3 e f \right ) \arctan \left (\frac {x e}{\sqrt {d e}}\right )}{d^{2} \sqrt {d e}}\right )}{15}\)	\(96\)
risch	\(-\frac {\left (5 g \,x^{2}+3 f \right ) \ln \left (\left (e \,x^{2}+d \right )^{p}\right )}{15 x^{5}}+\frac {-5 i \pi \,d^{2} g \,x^{2} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2}+5 i \pi \,d^{2} g \,x^{2} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )+5 i \pi \,d^{2} g \,x^{2} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}-5 i \pi \,d^{2} g \,x^{2} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )-3 i \pi \,d^{2} f \,\operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2}+3 i \pi \,d^{2} f \,\operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )+3 i \pi \,d^{2} f {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}-3 i \pi \,d^{2} f {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (25 d^{2} e^{3} g^{2} p^{2}-30 d \,e^{4} f g \,p^{2}+9 e^{5} f^{2} p^{2}+d^{5} \textit {\_Z}^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (50 d^{2} e^{3} g^{2} p^{2}-60 d \,e^{4} f g \,p^{2}+18 e^{5} f^{2} p^{2}+3 \textit {\_R}^{2} d^{5}\right ) x +\left (5 d^{4} e g p -3 d^{3} e^{2} f p \right ) \textit {\_R} \right )\right ) d^{2} x^{5}-20 d e g p \,x^{4}+12 e^{2} f p \,x^{4}-10 \ln \left (c \right ) d^{2} g \,x^{2}-4 d e f p \,x^{2}-6 \ln \left (c \right ) d^{2} f}{30 d^{2} x^{5}}\)	\(483\)

[In]

int((g*x^2+f)*ln(c*(e*x^2+d)^p)/x^6,x,method=_RETURNVERBOSE)

[Out]

-1/3*g*ln(c*(e*x^2+d)^p)/x^3-1/5*f*ln(c*(e*x^2+d)^p)/x^5-2/15*p*e*(-1/d^2*(-5*d*g+3*e*f)/x+f/d/x^3+e*(5*d*g-3*
e*f)/d^2/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.85 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^6} \, dx=\left [-\frac {{\left (3 \, e^{2} f - 5 \, d e g\right )} p x^{5} \sqrt {-\frac {e}{d}} \log \left (\frac {e x^{2} - 2 \, d x \sqrt {-\frac {e}{d}} - d}{e x^{2} + d}\right ) + 2 \, d e f p x^{2} - 2 \, {\left (3 \, e^{2} f - 5 \, d e g\right )} p x^{4} + {\left (5 \, d^{2} g p x^{2} + 3 \, d^{2} f p\right )} \log \left (e x^{2} + d\right ) + {\left (5 \, d^{2} g x^{2} + 3 \, d^{2} f\right )} \log \left (c\right )}{15 \, d^{2} x^{5}}, \frac {2 \, {\left (3 \, e^{2} f - 5 \, d e g\right )} p x^{5} \sqrt {\frac {e}{d}} \arctan \left (x \sqrt {\frac {e}{d}}\right ) - 2 \, d e f p x^{2} + 2 \, {\left (3 \, e^{2} f - 5 \, d e g\right )} p x^{4} - {\left (5 \, d^{2} g p x^{2} + 3 \, d^{2} f p\right )} \log \left (e x^{2} + d\right ) - {\left (5 \, d^{2} g x^{2} + 3 \, d^{2} f\right )} \log \left (c\right )}{15 \, d^{2} x^{5}}\right ] \]

[In]

integrate((g*x^2+f)*log(c*(e*x^2+d)^p)/x^6,x, algorithm="fricas")

[Out]

[-1/15*((3*e^2*f - 5*d*e*g)*p*x^5*sqrt(-e/d)*log((e*x^2 - 2*d*x*sqrt(-e/d) - d)/(e*x^2 + d)) + 2*d*e*f*p*x^2 -
2*(3*e^2*f - 5*d*e*g)*p*x^4 + (5*d^2*g*p*x^2 + 3*d^2*f*p)*log(e*x^2 + d) + (5*d^2*g*x^2 + 3*d^2*f)*log(c))/(d
^2*x^5), 1/15*(2*(3*e^2*f - 5*d*e*g)*p*x^5*sqrt(e/d)*arctan(x*sqrt(e/d)) - 2*d*e*f*p*x^2 + 2*(3*e^2*f - 5*d*e*
g)*p*x^4 - (5*d^2*g*p*x^2 + 3*d^2*f*p)*log(e*x^2 + d) - (5*d^2*g*x^2 + 3*d^2*f)*log(c))/(d^2*x^5)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1134 vs. \(2 (138) = 276\).

Time = 155.65 (sec) , antiderivative size = 1134, normalized size of antiderivative = 8.10 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^6} \, dx=\text {Too large to display} \]

[In]

integrate((g*x**2+f)*ln(c*(e*x**2+d)**p)/x**6,x)

[Out]

Piecewise(((-f/(5*x**5) - g/(3*x**3))*log(0**p*c), Eq(d, 0) & Eq(e, 0)), ((-f/(5*x**5) - g/(3*x**3))*log(c*d**
p), Eq(e, 0)), (-2*f*p/(25*x**5) - f*log(c*(e*x**2)**p)/(5*x**5) - 2*g*p/(9*x**3) - g*log(c*(e*x**2)**p)/(3*x*
*3), Eq(d, 0)), ((-f/(5*x**5) - g/(3*x**3))*log(0**p*c), Eq(d, -e*x**2)), (-3*d**3*f*sqrt(-d/e)*log(c*(d + e*x
**2)**p)/(15*d**3*x**5*sqrt(-d/e) + 15*d**2*e*x**7*sqrt(-d/e)) - 5*d**3*g*x**2*sqrt(-d/e)*log(c*(d + e*x**2)**
p)/(15*d**3*x**5*sqrt(-d/e) + 15*d**2*e*x**7*sqrt(-d/e)) - 2*d**2*f*p*x**2*sqrt(-d/e)/(15*d**3*x**5*sqrt(-d/e)
/e + 15*d**2*x**7*sqrt(-d/e)) - 3*d**2*f*x**2*sqrt(-d/e)*log(c*(d + e*x**2)**p)/(15*d**3*x**5*sqrt(-d/e)/e + 1
5*d**2*x**7*sqrt(-d/e)) - 10*d**2*g*p*x**5*log(x - sqrt(-d/e))/(15*d**3*x**5*sqrt(-d/e)/e + 15*d**2*x**7*sqrt(
-d/e)) - 10*d**2*g*p*x**4*sqrt(-d/e)/(15*d**3*x**5*sqrt(-d/e)/e + 15*d**2*x**7*sqrt(-d/e)) + 5*d**2*g*x**5*log
(c*(d + e*x**2)**p)/(15*d**3*x**5*sqrt(-d/e)/e + 15*d**2*x**7*sqrt(-d/e)) - 5*d**2*g*x**4*sqrt(-d/e)*log(c*(d
+ e*x**2)**p)/(15*d**3*x**5*sqrt(-d/e)/e + 15*d**2*x**7*sqrt(-d/e)) + 6*d*e*f*p*x**5*log(x - sqrt(-d/e))/(15*d
**3*x**5*sqrt(-d/e)/e + 15*d**2*x**7*sqrt(-d/e)) + 4*d*e*f*p*x**4*sqrt(-d/e)/(15*d**3*x**5*sqrt(-d/e)/e + 15*d
**2*x**7*sqrt(-d/e)) - 3*d*e*f*x**5*log(c*(d + e*x**2)**p)/(15*d**3*x**5*sqrt(-d/e)/e + 15*d**2*x**7*sqrt(-d/e
)) - 10*d*e*g*p*x**7*log(x - sqrt(-d/e))/(15*d**3*x**5*sqrt(-d/e)/e + 15*d**2*x**7*sqrt(-d/e)) - 10*d*e*g*p*x*
*6*sqrt(-d/e)/(15*d**3*x**5*sqrt(-d/e)/e + 15*d**2*x**7*sqrt(-d/e)) + 5*d*e*g*x**7*log(c*(d + e*x**2)**p)/(15*
d**3*x**5*sqrt(-d/e)/e + 15*d**2*x**7*sqrt(-d/e)) + 6*e**2*f*p*x**7*log(x - sqrt(-d/e))/(15*d**3*x**5*sqrt(-d/
e)/e + 15*d**2*x**7*sqrt(-d/e)) + 6*e**2*f*p*x**6*sqrt(-d/e)/(15*d**3*x**5*sqrt(-d/e)/e + 15*d**2*x**7*sqrt(-d
/e)) - 3*e**2*f*x**7*log(c*(d + e*x**2)**p)/(15*d**3*x**5*sqrt(-d/e)/e + 15*d**2*x**7*sqrt(-d/e)), True))

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^6} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((g*x^2+f)*log(c*(e*x^2+d)^p)/x^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.84 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^6} \, dx=\frac {2 \, {\left (3 \, e^{3} f p - 5 \, d e^{2} g p\right )} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{15 \, \sqrt {d e} d^{2}} - \frac {{\left (5 \, g p x^{2} + 3 \, f p\right )} \log \left (e x^{2} + d\right )}{15 \, x^{5}} + \frac {6 \, e^{2} f p x^{4} - 10 \, d e g p x^{4} - 2 \, d e f p x^{2} - 5 \, d^{2} g x^{2} \log \left (c\right ) - 3 \, d^{2} f \log \left (c\right )}{15 \, d^{2} x^{5}} \]

[In]

integrate((g*x^2+f)*log(c*(e*x^2+d)^p)/x^6,x, algorithm="giac")

[Out]

2/15*(3*e^3*f*p - 5*d*e^2*g*p)*arctan(e*x/sqrt(d*e))/(sqrt(d*e)*d^2) - 1/15*(5*g*p*x^2 + 3*f*p)*log(e*x^2 + d)
/x^5 + 1/15*(6*e^2*f*p*x^4 - 10*d*e*g*p*x^4 - 2*d*e*f*p*x^2 - 5*d^2*g*x^2*log(c) - 3*d^2*f*log(c))/(d^2*x^5)

Mupad [B] (verification not implemented)

Time = 1.63 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.63 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^6} \, dx=-\frac {\frac {2\,e\,f\,p}{d}+\frac {2\,e\,p\,x^2\,\left (5\,d\,g-3\,e\,f\right )}{d^2}}{15\,x^3}-\frac {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\,\left (\frac {g\,x^2}{3}+\frac {f}{5}\right )}{x^5}-\frac {2\,e^{3/2}\,p\,\mathrm {atan}\left (\frac {\sqrt {e}\,x}{\sqrt {d}}\right )\,\left (5\,d\,g-3\,e\,f\right )}{15\,d^{5/2}} \]

[In]

int((log(c*(d + e*x^2)^p)*(f + g*x^2))/x^6,x)

[Out]

- ((2*e*f*p)/d + (2*e*p*x^2*(5*d*g - 3*e*f))/d^2)/(15*x^3) - (log(c*(d + e*x^2)^p)*(f/5 + (g*x^2)/3))/x^5 - (2
*e^(3/2)*p*atan((e^(1/2)*x)/d^(1/2))*(5*d*g - 3*e*f))/(15*d^(5/2))