Integrand size = 23, antiderivative size = 140 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^6} \, dx=-\frac {2 e f p}{15 d x^3}+\frac {2 e^2 f p}{5 d^2 x}-\frac {2 e g p}{3 d x}+\frac {2 e^{5/2} f p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{5 d^{5/2}}-\frac {2 e^{3/2} g p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 d^{3/2}}-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{5 x^5}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{3 x^3} \] Output:
-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(e ^(1/2)*x/d^(1/2))/d^(5/2)-2/3*e^(3/2)*g*p*arctan(e^(1/2)*x/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
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 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^6} \, dx=-\frac {2 e f p \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},-\frac {e x^2}{d}\right )}{15 d x^3}-\frac {2 e g p \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-\frac {e x^2}{d}\right )}{3 d x}-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{5 x^5}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{3 x^3} \] Input:
Integrate[((f + g*x^2)*Log[c*(d + e*x^2)^p])/x^6,x]
Output:
(-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)
Time = 0.50 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2926, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^6} \, dx\) |
\(\Big \downarrow \) 2926 |
\(\displaystyle \int \left (\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{x^6}+\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{x^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 e^{5/2} f p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{5 d^{5/2}}-\frac {2 e^{3/2} g p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 d^{3/2}}-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{5 x^5}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{3 x^3}+\frac {2 e^2 f p}{5 d^2 x}-\frac {2 e f p}{15 d x^3}-\frac {2 e g p}{3 d x}\) |
Input:
Int[((f + g*x^2)*Log[c*(d + e*x^2)^p])/x^6,x]
Output:
(-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)
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]
Time = 2.40 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.69
method | result | size |
parts | \(-\frac {g \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{3 x^{3}}-\frac {f \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{5 x^{5}}-\frac {2 e p \left (\frac {e \left (5 d g -3 e f \right ) \arctan \left (\frac {x e}{\sqrt {d e}}\right )}{d^{2} \sqrt {d e}}-\frac {-5 d g +3 e f}{d^{2} x}+\frac {f}{d \,x^{3}}\right )}{15}\) | \(96\) |
risch | \(-\frac {\left (5 g \,x^{2}+3 f \right ) \ln \left (\left (e \,x^{2}+d \right )^{p}\right )}{15 x^{5}}+\frac {-5 i \pi \,d^{2} g \,x^{2} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2}+5 i \pi \,d^{2} g \,x^{2} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )+5 i \pi \,d^{2} g \,x^{2} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}-5 i \pi \,d^{2} g \,x^{2} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )-3 i \pi \,d^{2} f \,\operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2}+3 i \pi \,d^{2} f \,\operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )+3 i \pi \,d^{2} f {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}-3 i \pi \,d^{2} f {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (25 d^{2} e^{3} g^{2} p^{2}-30 d \,e^{4} f g \,p^{2}+9 e^{5} f^{2} p^{2}+d^{5} \textit {\_Z}^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (50 d^{2} e^{3} g^{2} p^{2}-60 d \,e^{4} f g \,p^{2}+18 e^{5} f^{2} p^{2}+3 \textit {\_R}^{2} d^{5}\right ) x +\left (5 d^{4} e g p -3 d^{3} e^{2} f p \right ) \textit {\_R} \right )\right ) d^{2} x^{5}-20 d e g p \,x^{4}+12 e^{2} f p \,x^{4}-10 \ln \left (c \right ) d^{2} g \,x^{2}-4 d e f p \,x^{2}-6 \ln \left (c \right ) d^{2} f}{30 d^{2} x^{5}}\) | \(483\) |
Input:
int((g*x^2+f)*ln(c*(e*x^2+d)^p)/x^6,x,method=_RETURNVERBOSE)
Output:
-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*e*p*(e*(5*d* g-3*e*f)/d^2/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))-(-5*d*g+3*e*f)/d^2/x+1/d* f/x^3)
Time = 0.09 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.85 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^6} \, dx=\left [-\frac {{\left (3 \, e^{2} f - 5 \, d e g\right )} p x^{5} \sqrt {-\frac {e}{d}} \log \left (\frac {e x^{2} - 2 \, d x \sqrt {-\frac {e}{d}} - d}{e x^{2} + d}\right ) + 2 \, d e f p x^{2} - 2 \, {\left (3 \, e^{2} f - 5 \, d e g\right )} p x^{4} + {\left (5 \, d^{2} g p x^{2} + 3 \, d^{2} f p\right )} \log \left (e x^{2} + d\right ) + {\left (5 \, d^{2} g x^{2} + 3 \, d^{2} f\right )} \log \left (c\right )}{15 \, d^{2} x^{5}}, \frac {2 \, {\left (3 \, e^{2} f - 5 \, d e g\right )} p x^{5} \sqrt {\frac {e}{d}} \arctan \left (x \sqrt {\frac {e}{d}}\right ) - 2 \, d e f p x^{2} + 2 \, {\left (3 \, e^{2} f - 5 \, d e g\right )} p x^{4} - {\left (5 \, d^{2} g p x^{2} + 3 \, d^{2} f p\right )} \log \left (e x^{2} + d\right ) - {\left (5 \, d^{2} g x^{2} + 3 \, d^{2} f\right )} \log \left (c\right )}{15 \, d^{2} x^{5}}\right ] \] Input:
integrate((g*x^2+f)*log(c*(e*x^2+d)^p)/x^6,x, algorithm="fricas")
Output:
[-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)*l og(e*x^2 + d) - (5*d^2*g*x^2 + 3*d^2*f)*log(c))/(d^2*x^5)]
Leaf count of result is larger than twice the leaf count of optimal. 1134 vs. \(2 (138) = 276\).
Time = 156.92 (sec) , antiderivative size = 1134, normalized size of antiderivative = 8.10 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^6} \, dx=\text {Too large to display} \] Input:
integrate((g*x**2+f)*ln(c*(e*x**2+d)**p)/x**6,x)
Output:
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*l og(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*sqr t(-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 + 15*d**2 *x**7*sqrt(-d/e)) - 10*d**2*g*p*x**5*log(x - sqrt(-d/e))/(15*d**3*x**5*sqr t(-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*s qrt(-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*sqr t(-d/e)) - 10*d*e*g*p*x**6*sqrt(-d/e)/(15*d**3*x**5*sqrt(-d/e)/e + 15*d...
Exception generated. \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^6} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((g*x^2+f)*log(c*(e*x^2+d)^p)/x^6,x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Time = 0.12 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.84 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^6} \, dx=\frac {2 \, {\left (3 \, e^{3} f p - 5 \, d e^{2} g p\right )} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{15 \, \sqrt {d e} d^{2}} - \frac {{\left (5 \, g p x^{2} + 3 \, f p\right )} \log \left (e x^{2} + d\right )}{15 \, x^{5}} + \frac {6 \, e^{2} f p x^{4} - 10 \, d e g p x^{4} - 2 \, d e f p x^{2} - 5 \, d^{2} g x^{2} \log \left (c\right ) - 3 \, d^{2} f \log \left (c\right )}{15 \, d^{2} x^{5}} \] Input:
integrate((g*x^2+f)*log(c*(e*x^2+d)^p)/x^6,x, algorithm="giac")
Output:
2/15*(3*e^3*f*p - 5*d*e^2*g*p)*arctan(e*x/sqrt(d*e))/(sqrt(d*e)*d^2) - 1/1 5*(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)
Time = 26.36 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.63 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^6} \, dx=-\frac {\frac {2\,e\,f\,p}{d}+\frac {2\,e\,p\,x^2\,\left (5\,d\,g-3\,e\,f\right )}{d^2}}{15\,x^3}-\frac {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\,\left (\frac {g\,x^2}{3}+\frac {f}{5}\right )}{x^5}-\frac {2\,e^{3/2}\,p\,\mathrm {atan}\left (\frac {\sqrt {e}\,x}{\sqrt {d}}\right )\,\left (5\,d\,g-3\,e\,f\right )}{15\,d^{5/2}} \] Input:
int((log(c*(d + e*x^2)^p)*(f + g*x^2))/x^6,x)
Output:
- ((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))
Time = 0.16 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.94 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^6} \, dx=\frac {-10 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) d e g p \,x^{5}+6 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) e^{2} f p \,x^{5}-3 \,\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) d^{3} f -5 \,\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) d^{3} g \,x^{2}-2 d^{2} e f p \,x^{2}-10 d^{2} e g p \,x^{4}+6 d \,e^{2} f p \,x^{4}}{15 d^{3} x^{5}} \] Input:
int((g*x^2+f)*log(c*(e*x^2+d)^p)/x^6,x)
Output:
( - 10*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*d*e*g*p*x**5 + 6*sqrt (e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*e**2*f*p*x**5 - 3*log((d + e*x** 2)**p*c)*d**3*f - 5*log((d + e*x**2)**p*c)*d**3*g*x**2 - 2*d**2*e*f*p*x**2 - 10*d**2*e*g*p*x**4 + 6*d*e**2*f*p*x**4)/(15*d**3*x**5)