Integrand size = 23, antiderivative size = 108 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^4} \, dx=-\frac {2 e f p}{3 d x}-\frac {2 e^{3/2} f p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 d^{3/2}}+\frac {2 \sqrt {e} g p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d}}-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{3 x^3}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{x} \] Output:
-2/3*e*f*p/d/x-2/3*e^(3/2)*f*p*arctan(e^(1/2)*x/d^(1/2))/d^(3/2)+2*e^(1/2) *g*p*arctan(e^(1/2)*x/d^(1/2))/d^(1/2)-1/3*f*ln(c*(e*x^2+d)^p)/x^3-g*ln(c* (e*x^2+d)^p)/x
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.04 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.89 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^4} \, dx=\frac {2 \sqrt {e} g p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d}}-\frac {2 e f p \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-\frac {e x^2}{d}\right )}{3 d x}-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{3 x^3}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{x} \] Input:
Integrate[((f + g*x^2)*Log[c*(d + e*x^2)^p])/x^4,x]
Output:
(2*Sqrt[e]*g*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[d] - (2*e*f*p*Hypergeomet ric2F1[-1/2, 1, 1/2, -((e*x^2)/d)])/(3*d*x) - (f*Log[c*(d + e*x^2)^p])/(3* x^3) - (g*Log[c*(d + e*x^2)^p])/x
Time = 0.47 (sec) , antiderivative size = 108, 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.
\(\displaystyle \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^4} \, dx\) |
\(\Big \downarrow \) 2926 |
\(\displaystyle \int \left (\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{x^4}+\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 e^{3/2} f p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 d^{3/2}}+\frac {2 \sqrt {e} g p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d}}-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{3 x^3}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{x}-\frac {2 e f p}{3 d x}\) |
Input:
Int[((f + g*x^2)*Log[c*(d + e*x^2)^p])/x^4,x]
Output:
(-2*e*f*p)/(3*d*x) - (2*e^(3/2)*f*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(3*d^(3/2 )) + (2*Sqrt[e]*g*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[d] - (f*Log[c*(d + e *x^2)^p])/(3*x^3) - (g*Log[c*(d + e*x^2)^p])/x
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 = 1.21 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.71
method | result | size |
parts | \(-\frac {g \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{x}-\frac {f \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{3 x^{3}}-\frac {2 e p \left (\frac {\left (-3 d g +e f \right ) \arctan \left (\frac {x e}{\sqrt {d e}}\right )}{d \sqrt {d e}}+\frac {f}{d x}\right )}{3}\) | \(77\) |
risch | \(-\frac {\left (3 g \,x^{2}+f \right ) \ln \left (\left (e \,x^{2}+d \right )^{p}\right )}{3 x^{3}}-\frac {3 i \pi \,d^{2} g \,x^{2} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2}-3 i \pi \,d^{2} g \,x^{2} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )-3 i \pi \,d^{2} g \,x^{2} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}+3 i \pi \,d^{2} g \,x^{2} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )+i \pi \,d^{2} f \,\operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2}-i \pi \,d^{2} f \,\operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )-i \pi \,d^{2} f {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}+i \pi \,d^{2} f {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )-6 \sqrt {-d e}\, p \ln \left (-e x -\sqrt {-d e}\right ) g d \,x^{3}+2 \sqrt {-d e}\, p \ln \left (-e x -\sqrt {-d e}\right ) e f \,x^{3}+6 \sqrt {-d e}\, p \ln \left (-e x +\sqrt {-d e}\right ) g d \,x^{3}-2 \sqrt {-d e}\, p \ln \left (-e x +\sqrt {-d e}\right ) e f \,x^{3}+6 \ln \left (c \right ) d^{2} g \,x^{2}+4 d e f p \,x^{2}+2 \ln \left (c \right ) d^{2} f}{6 d^{2} x^{3}}\) | \(442\) |
Input:
int((g*x^2+f)*ln(c*(e*x^2+d)^p)/x^4,x,method=_RETURNVERBOSE)
Output:
-g*ln(c*(e*x^2+d)^p)/x-1/3*f*ln(c*(e*x^2+d)^p)/x^3-2/3*e*p*((-3*d*g+e*f)/d /(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))+1/d*f/x)
Time = 0.12 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.77 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^4} \, dx=\left [-\frac {{\left (e f - 3 \, d g\right )} p x^{3} \sqrt {-\frac {e}{d}} \log \left (\frac {e x^{2} + 2 \, d x \sqrt {-\frac {e}{d}} - d}{e x^{2} + d}\right ) + 2 \, e f p x^{2} + {\left (3 \, d g p x^{2} + d f p\right )} \log \left (e x^{2} + d\right ) + {\left (3 \, d g x^{2} + d f\right )} \log \left (c\right )}{3 \, d x^{3}}, -\frac {2 \, {\left (e f - 3 \, d g\right )} p x^{3} \sqrt {\frac {e}{d}} \arctan \left (x \sqrt {\frac {e}{d}}\right ) + 2 \, e f p x^{2} + {\left (3 \, d g p x^{2} + d f p\right )} \log \left (e x^{2} + d\right ) + {\left (3 \, d g x^{2} + d f\right )} \log \left (c\right )}{3 \, d x^{3}}\right ] \] Input:
integrate((g*x^2+f)*log(c*(e*x^2+d)^p)/x^4,x, algorithm="fricas")
Output:
[-1/3*((e*f - 3*d*g)*p*x^3*sqrt(-e/d)*log((e*x^2 + 2*d*x*sqrt(-e/d) - d)/( e*x^2 + d)) + 2*e*f*p*x^2 + (3*d*g*p*x^2 + d*f*p)*log(e*x^2 + d) + (3*d*g* x^2 + d*f)*log(c))/(d*x^3), -1/3*(2*(e*f - 3*d*g)*p*x^3*sqrt(e/d)*arctan(x *sqrt(e/d)) + 2*e*f*p*x^2 + (3*d*g*p*x^2 + d*f*p)*log(e*x^2 + d) + (3*d*g* x^2 + d*f)*log(c))/(d*x^3)]
Leaf count of result is larger than twice the leaf count of optimal. 901 vs. \(2 (104) = 208\).
Time = 39.14 (sec) , antiderivative size = 901, normalized size of antiderivative = 8.34 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^4} \, dx =\text {Too large to display} \] Input:
integrate((g*x**2+f)*ln(c*(e*x**2+d)**p)/x**4,x)
Output:
Piecewise(((-f/(3*x**3) - g/x)*log(0**p*c), Eq(d, 0) & Eq(e, 0)), ((-f/(3* x**3) - g/x)*log(c*d**p), Eq(e, 0)), (-2*f*p/(9*x**3) - f*log(c*(e*x**2)** p)/(3*x**3) - 2*g*p/x - g*log(c*(e*x**2)**p)/x, Eq(d, 0)), ((-f/(3*x**3) - g/x)*log(0**p*c), Eq(d, -e*x**2)), (-d**2*f*sqrt(-d/e)*log(c*(d + e*x**2) **p)/(3*d**2*x**3*sqrt(-d/e) + 3*d*e*x**5*sqrt(-d/e)) + 6*d**2*g*p*x**3*lo g(x - sqrt(-d/e))/(3*d**2*x**3*sqrt(-d/e) + 3*d*e*x**5*sqrt(-d/e)) - 3*d** 2*g*x**3*log(c*(d + e*x**2)**p)/(3*d**2*x**3*sqrt(-d/e) + 3*d*e*x**5*sqrt( -d/e)) - 3*d**2*g*x**2*sqrt(-d/e)*log(c*(d + e*x**2)**p)/(3*d**2*x**3*sqrt (-d/e) + 3*d*e*x**5*sqrt(-d/e)) - 2*d*f*p*x**3*log(x - sqrt(-d/e))/(3*d**2 *x**3*sqrt(-d/e)/e + 3*d*x**5*sqrt(-d/e)) - 2*d*f*p*x**2*sqrt(-d/e)/(3*d** 2*x**3*sqrt(-d/e)/e + 3*d*x**5*sqrt(-d/e)) + d*f*x**3*log(c*(d + e*x**2)** p)/(3*d**2*x**3*sqrt(-d/e)/e + 3*d*x**5*sqrt(-d/e)) - d*f*x**2*sqrt(-d/e)* log(c*(d + e*x**2)**p)/(3*d**2*x**3*sqrt(-d/e)/e + 3*d*x**5*sqrt(-d/e)) + 6*d*g*p*x**5*log(x - sqrt(-d/e))/(3*d**2*x**3*sqrt(-d/e)/e + 3*d*x**5*sqrt (-d/e)) - 3*d*g*x**5*log(c*(d + e*x**2)**p)/(3*d**2*x**3*sqrt(-d/e)/e + 3* d*x**5*sqrt(-d/e)) - 3*d*g*x**4*sqrt(-d/e)*log(c*(d + e*x**2)**p)/(3*d**2* x**3*sqrt(-d/e)/e + 3*d*x**5*sqrt(-d/e)) - 2*e*f*p*x**5*log(x - sqrt(-d/e) )/(3*d**2*x**3*sqrt(-d/e)/e + 3*d*x**5*sqrt(-d/e)) - 2*e*f*p*x**4*sqrt(-d/ e)/(3*d**2*x**3*sqrt(-d/e)/e + 3*d*x**5*sqrt(-d/e)) + e*f*x**5*log(c*(d + e*x**2)**p)/(3*d**2*x**3*sqrt(-d/e)/e + 3*d*x**5*sqrt(-d/e)), True))
Exception generated. \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^4} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((g*x^2+f)*log(c*(e*x^2+d)^p)/x^4,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 = 88, normalized size of antiderivative = 0.81 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^4} \, dx=-\frac {2 \, {\left (e^{2} f p - 3 \, d e g p\right )} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{3 \, \sqrt {d e} d} - \frac {{\left (3 \, g p x^{2} + f p\right )} \log \left (e x^{2} + d\right )}{3 \, x^{3}} - \frac {2 \, e f p x^{2} + 3 \, d g x^{2} \log \left (c\right ) + d f \log \left (c\right )}{3 \, d x^{3}} \] Input:
integrate((g*x^2+f)*log(c*(e*x^2+d)^p)/x^4,x, algorithm="giac")
Output:
-2/3*(e^2*f*p - 3*d*e*g*p)*arctan(e*x/sqrt(d*e))/(sqrt(d*e)*d) - 1/3*(3*g* p*x^2 + f*p)*log(e*x^2 + d)/x^3 - 1/3*(2*e*f*p*x^2 + 3*d*g*x^2*log(c) + d* f*log(c))/(d*x^3)
Time = 26.51 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.60 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^4} \, dx=\frac {2\,\sqrt {e}\,p\,\mathrm {atan}\left (\frac {\sqrt {e}\,x}{\sqrt {d}}\right )\,\left (3\,d\,g-e\,f\right )}{3\,d^{3/2}}-\frac {2\,e\,f\,p}{3\,d\,x}-\frac {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\,\left (g\,x^2+\frac {f}{3}\right )}{x^3} \] Input:
int((log(c*(d + e*x^2)^p)*(f + g*x^2))/x^4,x)
Output:
(2*e^(1/2)*p*atan((e^(1/2)*x)/d^(1/2))*(3*d*g - e*f))/(3*d^(3/2)) - (2*e*f *p)/(3*d*x) - (log(c*(d + e*x^2)^p)*(f/3 + g*x^2))/x^3
Time = 0.16 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.97 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^4} \, dx=\frac {6 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) d g p \,x^{3}-2 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) e f p \,x^{3}-\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) d^{2} f -3 \,\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) d^{2} g \,x^{2}-2 d e f p \,x^{2}}{3 d^{2} x^{3}} \] Input:
int((g*x^2+f)*log(c*(e*x^2+d)^p)/x^4,x)
Output:
(6*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*d*g*p*x**3 - 2*sqrt(e)*sq rt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*e*f*p*x**3 - log((d + e*x**2)**p*c)*d* *2*f - 3*log((d + e*x**2)**p*c)*d**2*g*x**2 - 2*d*e*f*p*x**2)/(3*d**2*x**3 )