Integrand size = 43, antiderivative size = 39 \[ \int \frac {a^2 e x+4 a b e x^3+3 b^2 e x^5}{c+d x^4 \left (a+b x^2\right )^4} \, dx=\frac {e \arctan \left (\frac {\sqrt {d} x^2 \left (a+b x^2\right )^2}{\sqrt {c}}\right )}{2 \sqrt {c} \sqrt {d}} \] Output:
1/2*e*arctan(d^(1/2)*x^2*(b*x^2+a)^2/c^(1/2))/c^(1/2)/d^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.11 (sec) , antiderivative size = 101, normalized size of antiderivative = 2.59 \[ \int \frac {a^2 e x+4 a b e x^3+3 b^2 e x^5}{c+d x^4 \left (a+b x^2\right )^4} \, dx=\frac {e \text {RootSum}\left [c+a^4 d \text {$\#$1}^2+4 a^3 b d \text {$\#$1}^3+6 a^2 b^2 d \text {$\#$1}^4+4 a b^3 d \text {$\#$1}^5+b^4 d \text {$\#$1}^6\&,\frac {\log \left (x^2-\text {$\#$1}\right )}{a^2 \text {$\#$1}+2 a b \text {$\#$1}^2+b^2 \text {$\#$1}^3}\&\right ]}{4 d} \] Input:
Integrate[(a^2*e*x + 4*a*b*e*x^3 + 3*b^2*e*x^5)/(c + d*x^4*(a + b*x^2)^4), x]
Output:
(e*RootSum[c + a^4*d*#1^2 + 4*a^3*b*d*#1^3 + 6*a^2*b^2*d*#1^4 + 4*a*b^3*d* #1^5 + b^4*d*#1^6 & , Log[x^2 - #1]/(a^2*#1 + 2*a*b*#1^2 + b^2*#1^3) & ])/ (4*d)
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.
\(\displaystyle \int \frac {a^2 e x+4 a b e x^3+3 b^2 e x^5}{d x^4 \left (a+b x^2\right )^4+c} \, dx\) |
\(\Big \downarrow \) 2028 |
\(\displaystyle \int \frac {x \left (a^2 e+4 a b e x^2+3 b^2 e x^4\right )}{d x^4 \left (a+b x^2\right )^4+c}dx\) |
\(\Big \downarrow \) 7266 |
\(\displaystyle \frac {1}{2} \int \frac {3 b^2 e x^4+4 a b e x^2+a^2 e}{d x^4 \left (b x^2+a\right )^4+c}dx^2\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{2} \int \left (\frac {3 b^2 e x^4}{b^4 d x^{12}+4 a b^3 d x^{10}+6 a^2 b^2 d x^8+4 a^3 b d x^6+a^4 d x^4+c}+\frac {4 a b e x^2}{b^4 d x^{12}+4 a b^3 d x^{10}+6 a^2 b^2 d x^8+4 a^3 b d x^6+a^4 d x^4+c}+\frac {a^2 e}{b^4 d x^{12}+4 a b^3 d x^{10}+6 a^2 b^2 d x^8+4 a^3 b d x^6+a^4 d x^4+c}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (a^2 e \int \frac {1}{d x^4 \left (b x^2+a\right )^4+c}dx^2+3 b^2 e \int \frac {x^4}{d x^4 \left (b x^2+a\right )^4+c}dx^2+4 a b e \int \frac {x^2}{d x^4 \left (b x^2+a\right )^4+c}dx^2\right )\) |
Input:
Int[(a^2*e*x + 4*a*b*e*x^3 + 3*b^2*e*x^5)/(c + d*x^4*(a + b*x^2)^4),x]
Output:
$Aborted
Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.) + (c_.)*(x_)^(t_.))^(p_.), x_Symbol] :> Int[x^(p*r)*(a + b*x^(s - r) + c*x^(t - r))^p*Fx, x] /; FreeQ[ {a, b, c, r, s, t}, x] && IntegerQ[p] && PosQ[s - r] && PosQ[t - r] && !(E qQ[p, 1] && EqQ[u, 1])
Int[(u_)*(x_)^(m_.), x_Symbol] :> Simp[1/(m + 1) Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /; FreeQ[m, x] && NeQ[m, -1] && Function OfQ[x^(m + 1), u, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.08 (sec) , antiderivative size = 132, normalized size of antiderivative = 3.38
method | result | size |
default | \(\frac {e \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b^{4} d \,\textit {\_Z}^{6}+4 d a \,b^{3} \textit {\_Z}^{5}+6 a^{2} d \,b^{2} \textit {\_Z}^{4}+4 a^{3} d b \,\textit {\_Z}^{3}+a^{4} d \,\textit {\_Z}^{2}+c \right )}{\sum }\frac {\left (3 \textit {\_R}^{2} b^{2}+4 \textit {\_R} a b +a^{2}\right ) \ln \left (x^{2}-\textit {\_R} \right )}{3 b^{4} \textit {\_R}^{5}+10 a \,b^{3} \textit {\_R}^{4}+12 a^{2} b^{2} \textit {\_R}^{3}+6 a^{3} b \,\textit {\_R}^{2}+a^{4} \textit {\_R}}\right )}{4 d}\) | \(132\) |
risch | \(-\frac {e \ln \left (\left (-4 a^{3} b^{2} d -27 \sqrt {-c d}\, b^{3}\right ) x^{6}+\left (-8 a^{4} b d -54 \sqrt {-c d}\, a \,b^{2}\right ) x^{4}+\left (-4 d \,a^{5}-27 \sqrt {-c d}\, a^{2} b \right ) x^{2}-4 \sqrt {-c d}\, a^{3}+27 b c \right )}{4 \sqrt {-c d}}+\frac {e \ln \left (\left (4 a^{3} b^{2} d -27 \sqrt {-c d}\, b^{3}\right ) x^{6}+\left (8 a^{4} b d -54 \sqrt {-c d}\, a \,b^{2}\right ) x^{4}+\left (4 d \,a^{5}-27 \sqrt {-c d}\, a^{2} b \right ) x^{2}-4 \sqrt {-c d}\, a^{3}-27 b c \right )}{4 \sqrt {-c d}}\) | \(198\) |
Input:
int((3*b^2*e*x^5+4*a*b*e*x^3+a^2*e*x)/(c+d*x^4*(b*x^2+a)^4),x,method=_RETU RNVERBOSE)
Output:
1/4*e/d*sum((3*_R^2*b^2+4*_R*a*b+a^2)/(3*_R^5*b^4+10*_R^4*a*b^3+12*_R^3*a^ 2*b^2+6*_R^2*a^3*b+_R*a^4)*ln(x^2-_R),_R=RootOf(_Z^6*b^4*d+4*_Z^5*a*b^3*d+ 6*_Z^4*a^2*b^2*d+4*_Z^3*a^3*b*d+_Z^2*a^4*d+c))
Time = 0.12 (sec) , antiderivative size = 200, normalized size of antiderivative = 5.13 \[ \int \frac {a^2 e x+4 a b e x^3+3 b^2 e x^5}{c+d x^4 \left (a+b x^2\right )^4} \, dx=\left [-\frac {\sqrt {-c d} e \log \left (\frac {b^{4} d x^{12} + 4 \, a b^{3} d x^{10} + 6 \, a^{2} b^{2} d x^{8} + 4 \, a^{3} b d x^{6} + a^{4} d x^{4} - 2 \, {\left (b^{2} x^{6} + 2 \, a b x^{4} + a^{2} x^{2}\right )} \sqrt {-c d} - c}{b^{4} d x^{12} + 4 \, a b^{3} d x^{10} + 6 \, a^{2} b^{2} d x^{8} + 4 \, a^{3} b d x^{6} + a^{4} d x^{4} + c}\right )}{4 \, c d}, -\frac {\sqrt {c d} e \arctan \left (\frac {\sqrt {c d}}{b^{2} d x^{6} + 2 \, a b d x^{4} + a^{2} d x^{2}}\right )}{2 \, c d}\right ] \] Input:
integrate((3*b^2*e*x^5+4*a*b*e*x^3+a^2*e*x)/(c+d*x^4*(b*x^2+a)^4),x, algor ithm="fricas")
Output:
[-1/4*sqrt(-c*d)*e*log((b^4*d*x^12 + 4*a*b^3*d*x^10 + 6*a^2*b^2*d*x^8 + 4* a^3*b*d*x^6 + a^4*d*x^4 - 2*(b^2*x^6 + 2*a*b*x^4 + a^2*x^2)*sqrt(-c*d) - c )/(b^4*d*x^12 + 4*a*b^3*d*x^10 + 6*a^2*b^2*d*x^8 + 4*a^3*b*d*x^6 + a^4*d*x ^4 + c))/(c*d), -1/2*sqrt(c*d)*e*arctan(sqrt(c*d)/(b^2*d*x^6 + 2*a*b*d*x^4 + a^2*d*x^2))/(c*d)]
Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (36) = 72\).
Time = 1.20 (sec) , antiderivative size = 102, normalized size of antiderivative = 2.62 \[ \int \frac {a^2 e x+4 a b e x^3+3 b^2 e x^5}{c+d x^4 \left (a+b x^2\right )^4} \, dx=e \left (- \frac {\sqrt {- \frac {1}{c d}} \log {\left (\frac {a^{2} x^{2}}{b^{2}} + \frac {2 a x^{4}}{b} + x^{6} - \frac {c \sqrt {- \frac {1}{c d}}}{b^{2}} \right )}}{4} + \frac {\sqrt {- \frac {1}{c d}} \log {\left (\frac {a^{2} x^{2}}{b^{2}} + \frac {2 a x^{4}}{b} + x^{6} + \frac {c \sqrt {- \frac {1}{c d}}}{b^{2}} \right )}}{4}\right ) \] Input:
integrate((3*b**2*e*x**5+4*a*b*e*x**3+a**2*e*x)/(c+d*x**4*(b*x**2+a)**4),x )
Output:
e*(-sqrt(-1/(c*d))*log(a**2*x**2/b**2 + 2*a*x**4/b + x**6 - c*sqrt(-1/(c*d ))/b**2)/4 + sqrt(-1/(c*d))*log(a**2*x**2/b**2 + 2*a*x**4/b + x**6 + c*sqr t(-1/(c*d))/b**2)/4)
\[ \int \frac {a^2 e x+4 a b e x^3+3 b^2 e x^5}{c+d x^4 \left (a+b x^2\right )^4} \, dx=\int { \frac {3 \, b^{2} e x^{5} + 4 \, a b e x^{3} + a^{2} e x}{{\left (b x^{2} + a\right )}^{4} d x^{4} + c} \,d x } \] Input:
integrate((3*b^2*e*x^5+4*a*b*e*x^3+a^2*e*x)/(c+d*x^4*(b*x^2+a)^4),x, algor ithm="maxima")
Output:
integrate((3*b^2*e*x^5 + 4*a*b*e*x^3 + a^2*e*x)/((b*x^2 + a)^4*d*x^4 + c), x)
Time = 7.48 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.13 \[ \int \frac {a^2 e x+4 a b e x^3+3 b^2 e x^5}{c+d x^4 \left (a+b x^2\right )^4} \, dx=\frac {e \arctan \left (\frac {{\left (b^{2} e x^{6} + 2 \, a b e x^{4} + a^{2} e x^{2}\right )} d}{\sqrt {c d} e}\right )}{2 \, \sqrt {c d}} \] Input:
integrate((3*b^2*e*x^5+4*a*b*e*x^3+a^2*e*x)/(c+d*x^4*(b*x^2+a)^4),x, algor ithm="giac")
Output:
1/2*e*arctan((b^2*e*x^6 + 2*a*b*e*x^4 + a^2*e*x^2)*d/(sqrt(c*d)*e))/sqrt(c *d)
Time = 23.48 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.15 \[ \int \frac {a^2 e x+4 a b e x^3+3 b^2 e x^5}{c+d x^4 \left (a+b x^2\right )^4} \, dx=\frac {e\,\mathrm {atan}\left (\frac {a^2\,\sqrt {d}\,x^2+b^2\,\sqrt {d}\,x^6+2\,a\,b\,\sqrt {d}\,x^4}{\sqrt {c}}\right )}{2\,\sqrt {c}\,\sqrt {d}} \] Input:
int((3*b^2*e*x^5 + a^2*e*x + 4*a*b*e*x^3)/(c + d*x^4*(a + b*x^2)^4),x)
Output:
(e*atan((a^2*d^(1/2)*x^2 + b^2*d^(1/2)*x^6 + 2*a*b*d^(1/2)*x^4)/c^(1/2)))/ (2*c^(1/2)*d^(1/2))
\[ \int \frac {a^2 e x+4 a b e x^3+3 b^2 e x^5}{c+d x^4 \left (a+b x^2\right )^4} \, dx=e \left (3 \left (\int \frac {x^{5}}{b^{4} d \,x^{12}+4 a \,b^{3} d \,x^{10}+6 a^{2} b^{2} d \,x^{8}+4 a^{3} b d \,x^{6}+a^{4} d \,x^{4}+c}d x \right ) b^{2}+4 \left (\int \frac {x^{3}}{b^{4} d \,x^{12}+4 a \,b^{3} d \,x^{10}+6 a^{2} b^{2} d \,x^{8}+4 a^{3} b d \,x^{6}+a^{4} d \,x^{4}+c}d x \right ) a b +\left (\int \frac {x}{b^{4} d \,x^{12}+4 a \,b^{3} d \,x^{10}+6 a^{2} b^{2} d \,x^{8}+4 a^{3} b d \,x^{6}+a^{4} d \,x^{4}+c}d x \right ) a^{2}\right ) \] Input:
int((3*b^2*e*x^5+4*a*b*e*x^3+a^2*e*x)/(c+d*x^4*(b*x^2+a)^4),x)
Output:
e*(3*int(x**5/(a**4*d*x**4 + 4*a**3*b*d*x**6 + 6*a**2*b**2*d*x**8 + 4*a*b* *3*d*x**10 + b**4*d*x**12 + c),x)*b**2 + 4*int(x**3/(a**4*d*x**4 + 4*a**3* b*d*x**6 + 6*a**2*b**2*d*x**8 + 4*a*b**3*d*x**10 + b**4*d*x**12 + c),x)*a* b + int(x/(a**4*d*x**4 + 4*a**3*b*d*x**6 + 6*a**2*b**2*d*x**8 + 4*a*b**3*d *x**10 + b**4*d*x**12 + c),x)*a**2)