Integrand size = 18, antiderivative size = 325 \[ \int \frac {x^6}{a+b x^4+c x^8} \, dx=-\frac {\left (-b-\sqrt {b^2-4 a c}\right )^{3/4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt {b^2-4 a c}}+\frac {\left (-b+\sqrt {b^2-4 a c}\right )^{3/4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt {b^2-4 a c}}+\frac {\left (-b-\sqrt {b^2-4 a c}\right )^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt {b^2-4 a c}}-\frac {\left (-b+\sqrt {b^2-4 a c}\right )^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt {b^2-4 a c}} \] Output:
-1/4*(-b-(-4*a*c+b^2)^(1/2))^(3/4)*arctan(2^(1/4)*c^(1/4)*x/(-b-(-4*a*c+b^ 2)^(1/2))^(1/4))*2^(1/4)/c^(3/4)/(-4*a*c+b^2)^(1/2)+1/4*(-b+(-4*a*c+b^2)^( 1/2))^(3/4)*arctan(2^(1/4)*c^(1/4)*x/(-b+(-4*a*c+b^2)^(1/2))^(1/4))*2^(1/4 )/c^(3/4)/(-4*a*c+b^2)^(1/2)+1/4*(-b-(-4*a*c+b^2)^(1/2))^(3/4)*arctanh(2^( 1/4)*c^(1/4)*x/(-b-(-4*a*c+b^2)^(1/2))^(1/4))*2^(1/4)/c^(3/4)/(-4*a*c+b^2) ^(1/2)-1/4*(-b+(-4*a*c+b^2)^(1/2))^(3/4)*arctanh(2^(1/4)*c^(1/4)*x/(-b+(-4 *a*c+b^2)^(1/2))^(1/4))*2^(1/4)/c^(3/4)/(-4*a*c+b^2)^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.14 \[ \int \frac {x^6}{a+b x^4+c x^8} \, dx=\frac {1}{4} \text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {\log (x-\text {$\#$1}) \text {$\#$1}^3}{b+2 c \text {$\#$1}^4}\&\right ] \] Input:
Integrate[x^6/(a + b*x^4 + c*x^8),x]
Output:
RootSum[a + b*#1^4 + c*#1^8 & , (Log[x - #1]*#1^3)/(b + 2*c*#1^4) & ]/4
Time = 0.45 (sec) , antiderivative size = 314, normalized size of antiderivative = 0.97, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {1710, 27, 827, 218, 221}
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.
\(\displaystyle \int \frac {x^6}{a+b x^4+c x^8} \, dx\) |
\(\Big \downarrow \) 1710 |
\(\displaystyle \frac {1}{2} \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \int \frac {2 x^2}{2 c x^4+b-\sqrt {b^2-4 a c}}dx+\frac {1}{2} \left (\frac {b}{\sqrt {b^2-4 a c}}+1\right ) \int \frac {2 x^2}{2 c x^4+b+\sqrt {b^2-4 a c}}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \int \frac {x^2}{2 c x^4+b-\sqrt {b^2-4 a c}}dx+\left (\frac {b}{\sqrt {b^2-4 a c}}+1\right ) \int \frac {x^2}{2 c x^4+b+\sqrt {b^2-4 a c}}dx\) |
\(\Big \downarrow \) 827 |
\(\displaystyle \left (\frac {b}{\sqrt {b^2-4 a c}}+1\right ) \left (\frac {\int \frac {1}{\sqrt {2} \sqrt {c} x^2+\sqrt {-b-\sqrt {b^2-4 a c}}}dx}{2 \sqrt {2} \sqrt {c}}-\frac {\int \frac {1}{\sqrt {-b-\sqrt {b^2-4 a c}}-\sqrt {2} \sqrt {c} x^2}dx}{2 \sqrt {2} \sqrt {c}}\right )+\left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \left (\frac {\int \frac {1}{\sqrt {2} \sqrt {c} x^2+\sqrt {\sqrt {b^2-4 a c}-b}}dx}{2 \sqrt {2} \sqrt {c}}-\frac {\int \frac {1}{\sqrt {\sqrt {b^2-4 a c}-b}-\sqrt {2} \sqrt {c} x^2}dx}{2 \sqrt {2} \sqrt {c}}\right )\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \left (\frac {b}{\sqrt {b^2-4 a c}}+1\right ) \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}-\frac {\int \frac {1}{\sqrt {-b-\sqrt {b^2-4 a c}}-\sqrt {2} \sqrt {c} x^2}dx}{2 \sqrt {2} \sqrt {c}}\right )+\left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{\sqrt {b^2-4 a c}-b}}-\frac {\int \frac {1}{\sqrt {\sqrt {b^2-4 a c}-b}-\sqrt {2} \sqrt {c} x^2}dx}{2 \sqrt {2} \sqrt {c}}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \left (\frac {b}{\sqrt {b^2-4 a c}}+1\right ) \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )+\left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{\sqrt {b^2-4 a c}-b}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )\) |
Input:
Int[x^6/(a + b*x^4 + c*x^8),x]
Output:
(1 + b/Sqrt[b^2 - 4*a*c])*(ArcTan[(2^(1/4)*c^(1/4)*x)/(-b - Sqrt[b^2 - 4*a *c])^(1/4)]/(2*2^(3/4)*c^(3/4)*(-b - Sqrt[b^2 - 4*a*c])^(1/4)) - ArcTanh[( 2^(1/4)*c^(1/4)*x)/(-b - Sqrt[b^2 - 4*a*c])^(1/4)]/(2*2^(3/4)*c^(3/4)*(-b - Sqrt[b^2 - 4*a*c])^(1/4))) + (1 - b/Sqrt[b^2 - 4*a*c])*(ArcTan[(2^(1/4)* c^(1/4)*x)/(-b + Sqrt[b^2 - 4*a*c])^(1/4)]/(2*2^(3/4)*c^(3/4)*(-b + Sqrt[b ^2 - 4*a*c])^(1/4)) - ArcTanh[(2^(1/4)*c^(1/4)*x)/(-b + Sqrt[b^2 - 4*a*c]) ^(1/4)]/(2*2^(3/4)*c^(3/4)*(-b + Sqrt[b^2 - 4*a*c])^(1/4)))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Int[((d_.)*(x_))^(m_)/((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_)), x_Symbo l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(d^n/2)*(b/q + 1) Int[(d*x)^(m - n)/(b/2 + q/2 + c*x^n), x], x] - Simp[(d^n/2)*(b/q - 1) Int[(d*x)^(m - n)/(b/2 - q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d}, x] && EqQ[n2, 2*n] & & NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GeQ[m, n]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.04 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.13
method | result | size |
default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8} c +\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\textit {\_R}^{6} \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}\right )}{4}\) | \(43\) |
risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8} c +\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\textit {\_R}^{6} \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}\right )}{4}\) | \(43\) |
Input:
int(x^6/(c*x^8+b*x^4+a),x,method=_RETURNVERBOSE)
Output:
1/4*sum(_R^6/(2*_R^7*c+_R^3*b)*ln(x-_R),_R=RootOf(_Z^8*c+_Z^4*b+a))
Leaf count of result is larger than twice the leaf count of optimal. 4433 vs. \(2 (245) = 490\).
Time = 0.13 (sec) , antiderivative size = 4433, normalized size of antiderivative = 13.64 \[ \int \frac {x^6}{a+b x^4+c x^8} \, dx=\text {Too large to display} \] Input:
integrate(x^6/(c*x^8+b*x^4+a),x, algorithm="fricas")
Output:
Too large to include
Time = 48.43 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.71 \[ \int \frac {x^6}{a+b x^4+c x^8} \, dx=\operatorname {RootSum} {\left (t^{8} \cdot \left (16777216 a^{4} c^{7} - 16777216 a^{3} b^{2} c^{6} + 6291456 a^{2} b^{4} c^{5} - 1048576 a b^{6} c^{4} + 65536 b^{8} c^{3}\right ) + t^{4} \left (- 12288 a^{3} b c^{3} + 10240 a^{2} b^{3} c^{2} - 2816 a b^{5} c + 256 b^{7}\right ) + a^{3}, \left ( t \mapsto t \log {\left (x + \frac {2097152 t^{7} a^{4} c^{7} - 2621440 t^{7} a^{3} b^{2} c^{6} + 1179648 t^{7} a^{2} b^{4} c^{5} - 229376 t^{7} a b^{6} c^{4} + 16384 t^{7} b^{8} c^{3} - 1280 t^{3} a^{3} b c^{3} + 1600 t^{3} a^{2} b^{3} c^{2} - 576 t^{3} a b^{5} c + 64 t^{3} b^{7}}{a^{3} c - a^{2} b^{2}} \right )} \right )\right )} \] Input:
integrate(x**6/(c*x**8+b*x**4+a),x)
Output:
RootSum(_t**8*(16777216*a**4*c**7 - 16777216*a**3*b**2*c**6 + 6291456*a**2 *b**4*c**5 - 1048576*a*b**6*c**4 + 65536*b**8*c**3) + _t**4*(-12288*a**3*b *c**3 + 10240*a**2*b**3*c**2 - 2816*a*b**5*c + 256*b**7) + a**3, Lambda(_t , _t*log(x + (2097152*_t**7*a**4*c**7 - 2621440*_t**7*a**3*b**2*c**6 + 117 9648*_t**7*a**2*b**4*c**5 - 229376*_t**7*a*b**6*c**4 + 16384*_t**7*b**8*c* *3 - 1280*_t**3*a**3*b*c**3 + 1600*_t**3*a**2*b**3*c**2 - 576*_t**3*a*b**5 *c + 64*_t**3*b**7)/(a**3*c - a**2*b**2))))
\[ \int \frac {x^6}{a+b x^4+c x^8} \, dx=\int { \frac {x^{6}}{c x^{8} + b x^{4} + a} \,d x } \] Input:
integrate(x^6/(c*x^8+b*x^4+a),x, algorithm="maxima")
Output:
integrate(x^6/(c*x^8 + b*x^4 + a), x)
\[ \int \frac {x^6}{a+b x^4+c x^8} \, dx=\int { \frac {x^{6}}{c x^{8} + b x^{4} + a} \,d x } \] Input:
integrate(x^6/(c*x^8+b*x^4+a),x, algorithm="giac")
Output:
integrate(x^6/(c*x^8 + b*x^4 + a), x)
Time = 21.31 (sec) , antiderivative size = 8033, normalized size of antiderivative = 24.72 \[ \int \frac {x^6}{a+b x^4+c x^8} \, dx=\text {Too large to display} \] Input:
int(x^6/(a + b*x^4 + c*x^8),x)
Output:
atan((((-(b^7 + b^2*(-(4*a*c - b^2)^5)^(1/2) - 48*a^3*b*c^3 + 40*a^2*b^3*c ^2 - 11*a*b^5*c - a*c*(-(4*a*c - b^2)^5)^(1/2))/(512*(256*a^4*c^7 + b^8*c^ 3 - 16*a*b^6*c^4 + 96*a^2*b^4*c^5 - 256*a^3*b^2*c^6)))^(3/4)*(4096*a^5*c^5 + 256*a^3*b^4*c^3 - 2048*a^4*b^2*c^4 + x*(-(b^7 + b^2*(-(4*a*c - b^2)^5)^ (1/2) - 48*a^3*b*c^3 + 40*a^2*b^3*c^2 - 11*a*b^5*c - a*c*(-(4*a*c - b^2)^5 )^(1/2))/(512*(256*a^4*c^7 + b^8*c^3 - 16*a*b^6*c^4 + 96*a^2*b^4*c^5 - 256 *a^3*b^2*c^6)))^(1/4)*(32768*a^5*c^6 + 2048*a^3*b^4*c^4 - 16384*a^4*b^2*c^ 5)) + x*(4*a^3*b^3*c - 12*a^4*b*c^2))*(-(b^7 + b^2*(-(4*a*c - b^2)^5)^(1/2 ) - 48*a^3*b*c^3 + 40*a^2*b^3*c^2 - 11*a*b^5*c - a*c*(-(4*a*c - b^2)^5)^(1 /2))/(512*(256*a^4*c^7 + b^8*c^3 - 16*a*b^6*c^4 + 96*a^2*b^4*c^5 - 256*a^3 *b^2*c^6)))^(1/4)*1i - ((-(b^7 + b^2*(-(4*a*c - b^2)^5)^(1/2) - 48*a^3*b*c ^3 + 40*a^2*b^3*c^2 - 11*a*b^5*c - a*c*(-(4*a*c - b^2)^5)^(1/2))/(512*(256 *a^4*c^7 + b^8*c^3 - 16*a*b^6*c^4 + 96*a^2*b^4*c^5 - 256*a^3*b^2*c^6)))^(3 /4)*(4096*a^5*c^5 + 256*a^3*b^4*c^3 - 2048*a^4*b^2*c^4 - x*(-(b^7 + b^2*(- (4*a*c - b^2)^5)^(1/2) - 48*a^3*b*c^3 + 40*a^2*b^3*c^2 - 11*a*b^5*c - a*c* (-(4*a*c - b^2)^5)^(1/2))/(512*(256*a^4*c^7 + b^8*c^3 - 16*a*b^6*c^4 + 96* a^2*b^4*c^5 - 256*a^3*b^2*c^6)))^(1/4)*(32768*a^5*c^6 + 2048*a^3*b^4*c^4 - 16384*a^4*b^2*c^5)) - x*(4*a^3*b^3*c - 12*a^4*b*c^2))*(-(b^7 + b^2*(-(4*a *c - b^2)^5)^(1/2) - 48*a^3*b*c^3 + 40*a^2*b^3*c^2 - 11*a*b^5*c - a*c*(-(4 *a*c - b^2)^5)^(1/2))/(512*(256*a^4*c^7 + b^8*c^3 - 16*a*b^6*c^4 + 96*a...
\[ \int \frac {x^6}{a+b x^4+c x^8} \, dx=\int \frac {x^{6}}{c \,x^{8}+b \,x^{4}+a}d x \] Input:
int(x^6/(c*x^8+b*x^4+a),x)
Output:
int(x^6/(c*x^8+b*x^4+a),x)