Integrand size = 27, antiderivative size = 230 \[ \int \frac {x^9}{\left (d+e x^2\right ) \left (a+b x^2+c x^4\right )} \, dx=-\frac {(c d+b e) x^2}{2 c^2 e^2}+\frac {x^4}{4 c e}-\frac {\left (b^4 d-4 a b^2 c d+2 a^2 c^2 d-a b^3 e+3 a^2 b c e\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^3 \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )}+\frac {d^4 \log \left (d+e x^2\right )}{2 e^3 \left (c d^2-b d e+a e^2\right )}-\frac {\left (b^3 d-2 a b c d-a b^2 e+a^2 c e\right ) \log \left (a+b x^2+c x^4\right )}{4 c^3 \left (c d^2-b d e+a e^2\right )} \]
-1/2*(b*e+c*d)*x^2/c^2/e^2+1/4*x^4/c/e+1/2*d^4*ln(e*x^2+d)/e^3/(a*e^2-b*d* e+c*d^2)-1/4*(a^2*c*e-a*b^2*e-2*a*b*c*d+b^3*d)*ln(c*x^4+b*x^2+a)/c^3/(a*e^ 2-b*d*e+c*d^2)-1/2*(3*a^2*b*c*e+2*a^2*c^2*d-a*b^3*e-4*a*b^2*c*d+b^4*d)*arc tanh((2*c*x^2+b)/(-4*a*c+b^2)^(1/2))/c^3/(a*e^2-b*d*e+c*d^2)/(-4*a*c+b^2)^ (1/2)
Time = 0.18 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.99 \[ \int \frac {x^9}{\left (d+e x^2\right ) \left (a+b x^2+c x^4\right )} \, dx=\frac {1}{4} \left (-\frac {2 (c d+b e) x^2}{c^2 e^2}+\frac {x^4}{c e}-\frac {2 \left (b^4 d-4 a b^2 c d+2 a^2 c^2 d-a b^3 e+3 a^2 b c e\right ) \arctan \left (\frac {b+2 c x^2}{\sqrt {-b^2+4 a c}}\right )}{c^3 \sqrt {-b^2+4 a c} \left (-c d^2+e (b d-a e)\right )}+\frac {2 d^4 \log \left (d+e x^2\right )}{e^3 \left (c d^2+e (-b d+a e)\right )}+\frac {\left (-b^3 d+2 a b c d+a b^2 e-a^2 c e\right ) \log \left (a+b x^2+c x^4\right )}{c^3 \left (c d^2+e (-b d+a e)\right )}\right ) \]
((-2*(c*d + b*e)*x^2)/(c^2*e^2) + x^4/(c*e) - (2*(b^4*d - 4*a*b^2*c*d + 2* a^2*c^2*d - a*b^3*e + 3*a^2*b*c*e)*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c] ])/(c^3*Sqrt[-b^2 + 4*a*c]*(-(c*d^2) + e*(b*d - a*e))) + (2*d^4*Log[d + e* x^2])/(e^3*(c*d^2 + e*(-(b*d) + a*e))) + ((-(b^3*d) + 2*a*b*c*d + a*b^2*e - a^2*c*e)*Log[a + b*x^2 + c*x^4])/(c^3*(c*d^2 + e*(-(b*d) + a*e))))/4
Time = 0.56 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1578, 1200, 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.
\(\displaystyle \int \frac {x^9}{\left (d+e x^2\right ) \left (a+b x^2+c x^4\right )} \, dx\) |
\(\Big \downarrow \) 1578 |
\(\displaystyle \frac {1}{2} \int \frac {x^8}{\left (e x^2+d\right ) \left (c x^4+b x^2+a\right )}dx^2\) |
\(\Big \downarrow \) 1200 |
\(\displaystyle \frac {1}{2} \int \left (\frac {d^4}{e^2 \left (c d^2-b e d+a e^2\right ) \left (e x^2+d\right )}+\frac {x^2}{c e}+\frac {-c d-b e}{c^2 e^2}+\frac {-\left (\left (d b^3-a e b^2-2 a c d b+a^2 c e\right ) x^2\right )-a \left (d b^2-a e b-a c d\right )}{c^2 \left (c d^2-b e d+a e^2\right ) \left (c x^4+b x^2+a\right )}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (-\frac {\text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (3 a^2 b c e+2 a^2 c^2 d-a b^3 e-4 a b^2 c d+b^4 d\right )}{c^3 \sqrt {b^2-4 a c} \left (a e^2-b d e+c d^2\right )}-\frac {\left (a^2 c e-a b^2 e-2 a b c d+b^3 d\right ) \log \left (a+b x^2+c x^4\right )}{2 c^3 \left (a e^2-b d e+c d^2\right )}+\frac {d^4 \log \left (d+e x^2\right )}{e^3 \left (a e^2-b d e+c d^2\right )}-\frac {x^2 (b e+c d)}{c^2 e^2}+\frac {x^4}{2 c e}\right )\) |
(-(((c*d + b*e)*x^2)/(c^2*e^2)) + x^4/(2*c*e) - ((b^4*d - 4*a*b^2*c*d + 2* a^2*c^2*d - a*b^3*e + 3*a^2*b*c*e)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c] ])/(c^3*Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)) + (d^4*Log[d + e*x^2])/ (e^3*(c*d^2 - b*d*e + a*e^2)) - ((b^3*d - 2*a*b*c*d - a*b^2*e + a^2*c*e)*L og[a + b*x^2 + c*x^4])/(2*c^3*(c*d^2 - b*d*e + a*e^2)))/2
3.3.95.3.1 Defintions of rubi rules used
Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* (x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g* x)^n/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && In tegersQ[n]
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ )^4)^(p_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && Int egerQ[(m - 1)/2]
Time = 0.51 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.94
method | result | size |
default | \(\frac {\left (-c \,x^{2} e +b e +c d \right )^{2}}{4 e^{3} c^{3}}+\frac {\frac {\left (-a^{2} c e +a \,b^{2} e +2 a b c d -b^{3} d \right ) \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{2 c}+\frac {2 \left (a^{2} b e +a^{2} c d -a \,b^{2} d -\frac {\left (-a^{2} c e +a \,b^{2} e +2 a b c d -b^{3} d \right ) b}{2 c}\right ) \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c^{2}}+\frac {d^{4} \ln \left (e \,x^{2}+d \right )}{2 e^{3} \left (a \,e^{2}-b d e +c \,d^{2}\right )}\) | \(216\) |
risch | \(\frac {x^{4}}{4 c e}-\frac {b \,x^{2}}{2 e \,c^{2}}-\frac {d \,x^{2}}{2 c \,e^{2}}+\frac {b^{2}}{4 e \,c^{3}}+\frac {b d}{2 e^{2} c^{2}}+\frac {d^{2}}{4 c \,e^{3}}+\frac {d^{4} \ln \left (e \,x^{2}+d \right )}{2 e^{3} \left (a \,e^{2}-b d e +c \,d^{2}\right )}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (4 a^{2} c^{2} e^{2}-a \,b^{2} c \,e^{2}-4 a b \,c^{2} d e +4 a \,c^{3} d^{2}+b^{3} c d e -b^{2} c^{2} d^{2}\right ) \textit {\_Z}^{2}+\left (4 a^{3} c^{2} e^{3}-5 a^{2} b^{2} c \,e^{3}-8 a^{2} b \,c^{2} d \,e^{2}+a \,b^{4} e^{3}+6 a \,b^{3} c d \,e^{2}-b^{5} d \,e^{2}\right ) \textit {\_Z} +a^{4} c^{2} e^{4}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (4 e^{4} c^{2} a^{2}-b^{2} e^{4} c a +3 b \,c^{2} e^{3} d a -4 c^{3} e^{2} d^{2} a -b^{3} c d \,e^{3}+2 b^{2} c^{2} d^{2} e^{2}-b \,c^{3} e \,d^{3}\right ) \textit {\_R}^{2}+\left (4 a^{3} c^{2} e^{5}-5 a^{2} b^{2} c \,e^{5}-b d \,e^{4} a^{2} c^{2}-5 a^{2} c^{3} d^{2} e^{3}+a \,b^{4} e^{5}-3 a \,b^{3} c d \,e^{4}+4 a \,b^{2} c^{2} d^{2} e^{3}+4 e \,d^{4} a \,c^{4}+b^{5} d \,e^{4}-b^{4} c \,d^{2} e^{3}-b^{2} c^{3} d^{4} e +b \,c^{4} d^{5}\right ) \textit {\_R} +a^{4} e^{6} c^{2}+b \,e^{5} d \,a^{3} c^{2}-e^{4} a^{3} d^{2} c^{3}+b^{2} e^{4} d^{2} a^{2} c^{2}-2 b \,e^{3} d^{3} a^{2} c^{3}+e^{2} a^{2} d^{4} c^{4}+b^{3} e^{3} d^{3} a \,c^{2}-3 b^{2} e^{2} d^{4} a \,c^{3}+b^{4} c^{2} d^{4} e^{2}\right ) x^{2}+\left (6 a^{2} c^{2} d \,e^{3}-2 b^{2} c d \,e^{3} a +2 b \,c^{2} d^{2} e^{2} a -2 c^{3} d^{3} e a \right ) \textit {\_R}^{2}+\left (5 a^{3} c^{2} d \,e^{4}-8 a^{2} b^{2} c d \,e^{4}-2 a^{2} b \,c^{2} d^{2} e^{3}-4 a^{2} c^{3} d^{3} e^{2}+2 a \,b^{4} d \,e^{4}+a \,b^{2} c^{2} d^{3} e^{2}+2 a \,c^{4} d^{5}\right ) \textit {\_R} +a^{4} c^{2} d \,e^{5}+a^{3} b \,c^{2} d^{2} e^{4}-a^{3} c^{3} d^{3} e^{3}+a^{2} b^{2} c^{2} d^{3} e^{3}-2 a^{2} b \,c^{3} d^{4} e^{2}+a \,b^{3} c^{2} d^{4} e^{2}\right )}{2 e^{2} c^{2}}\) | \(831\) |
1/4*(-c*e*x^2+b*e+c*d)^2/e^3/c^3+1/2/(a*e^2-b*d*e+c*d^2)/c^2*(1/2*(-a^2*c* e+a*b^2*e+2*a*b*c*d-b^3*d)/c*ln(c*x^4+b*x^2+a)+2*(a^2*b*e+a^2*c*d-a*b^2*d- 1/2*(-a^2*c*e+a*b^2*e+2*a*b*c*d-b^3*d)*b/c)/(4*a*c-b^2)^(1/2)*arctan((2*c* x^2+b)/(4*a*c-b^2)^(1/2)))+1/2*d^4*ln(e*x^2+d)/e^3/(a*e^2-b*d*e+c*d^2)
Timed out. \[ \int \frac {x^9}{\left (d+e x^2\right ) \left (a+b x^2+c x^4\right )} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {x^9}{\left (d+e x^2\right ) \left (a+b x^2+c x^4\right )} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {x^9}{\left (d+e x^2\right ) \left (a+b x^2+c x^4\right )} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Time = 0.60 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.01 \[ \int \frac {x^9}{\left (d+e x^2\right ) \left (a+b x^2+c x^4\right )} \, dx=\frac {d^{4} \log \left ({\left | e x^{2} + d \right |}\right )}{2 \, {\left (c d^{2} e^{3} - b d e^{4} + a e^{5}\right )}} - \frac {{\left (b^{3} d - 2 \, a b c d - a b^{2} e + a^{2} c e\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, {\left (c^{4} d^{2} - b c^{3} d e + a c^{3} e^{2}\right )}} + \frac {{\left (b^{4} d - 4 \, a b^{2} c d + 2 \, a^{2} c^{2} d - a b^{3} e + 3 \, a^{2} b c e\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, {\left (c^{4} d^{2} - b c^{3} d e + a c^{3} e^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {c e x^{4} - 2 \, c d x^{2} - 2 \, b e x^{2}}{4 \, c^{2} e^{2}} \]
1/2*d^4*log(abs(e*x^2 + d))/(c*d^2*e^3 - b*d*e^4 + a*e^5) - 1/4*(b^3*d - 2 *a*b*c*d - a*b^2*e + a^2*c*e)*log(c*x^4 + b*x^2 + a)/(c^4*d^2 - b*c^3*d*e + a*c^3*e^2) + 1/2*(b^4*d - 4*a*b^2*c*d + 2*a^2*c^2*d - a*b^3*e + 3*a^2*b* c*e)*arctan((2*c*x^2 + b)/sqrt(-b^2 + 4*a*c))/((c^4*d^2 - b*c^3*d*e + a*c^ 3*e^2)*sqrt(-b^2 + 4*a*c)) + 1/4*(c*e*x^4 - 2*c*d*x^2 - 2*b*e*x^2)/(c^2*e^ 2)
Time = 73.40 (sec) , antiderivative size = 7024, normalized size of antiderivative = 30.54 \[ \int \frac {x^9}{\left (d+e x^2\right ) \left (a+b x^2+c x^4\right )} \, dx=\text {Too large to display} \]
(d^4*log(d + e*x^2))/(2*a*e^5 + 2*c*d^2*e^3 - 2*b*d*e^4) + (log((x^2*(a^7* e^7 + b^7*d^7 - 2*a^3*b*c^3*d^7 - a^4*c^3*d^6*e - 2*a^6*c*d^2*e^5 + 7*a^2* b^3*c^2*d^7 + 3*a^2*b^5*d^5*e^2 + 4*a^3*b^4*d^4*e^3 + 4*a^4*b^3*d^3*e^4 + 3*a^5*b^2*d^2*e^5 + 2*a^5*c^2*d^4*e^3 - 5*a*b^5*c*d^7 + 2*a*b^6*d^6*e + 2* a^6*b*d*e^6 - 8*a^2*b^4*c*d^6*e - 6*a^5*b*c*d^3*e^4 + 8*a^3*b^2*c^2*d^6*e - 9*a^3*b^3*c*d^5*e^2 + 5*a^4*b*c^2*d^5*e^2 - 9*a^4*b^2*c*d^4*e^3))/(c^4*e ^4) + (a*d*(a^3*e^3 + b^3*d^3 - 2*a*b*c*d^3 + a*b^2*d^2*e + a^2*b*d*e^2 - a^2*c*d^2*e)^2)/(c^4*e^4) + (((x^2*(4*a^2*c^6*d^8 + 6*a^4*b^4*e^8 + 18*a^6 *c^2*e^8 + 6*b^4*c^4*d^8 + 6*b^8*d^4*e^4 - 16*a*b^2*c^5*d^8 - 26*a^5*b^2*c *e^8 + 8*a*b^7*d^3*e^5 + 8*a^3*b^5*d*e^7 - 2*b^5*c^3*d^7*e - 2*b^7*c*d^5*e ^3 + 8*a^2*b^6*d^2*e^6 - 20*a^3*c^5*d^6*e^2 + 40*a^4*c^4*d^4*e^4 - 36*a^5* c^3*d^2*e^6 + 2*b^6*c^2*d^6*e^2 + 42*a^2*b^2*c^4*d^6*e^2 - 28*a^2*b^3*c^3* d^5*e^3 + 80*a^2*b^4*c^2*d^4*e^4 - 64*a^3*b^2*c^3*d^4*e^4 + 80*a^3*b^3*c^2 *d^3*e^5 + 48*a^4*b^2*c^2*d^2*e^6 + 18*a*b^3*c^4*d^7*e - 40*a*b^6*c*d^4*e^ 4 - 26*a^2*b*c^5*d^7*e - 32*a^4*b^3*c*d*e^7 + 12*a^5*b*c^2*d*e^7 - 16*a*b^ 4*c^3*d^6*e^2 + 10*a*b^5*c^2*d^5*e^3 - 48*a^2*b^5*c*d^3*e^5 + 46*a^3*b*c^4 *d^5*e^3 - 40*a^3*b^4*c*d^2*e^6 - 48*a^4*b*c^3*d^3*e^5))/(c^4*e^4) + (((x^ 2*(8*a*b^8*e^9 + 8*b*c^8*d^9 + 8*b^9*d*e^8 + 120*a^5*c^4*e^9 - 72*a^2*b^6* c*e^9 - 8*b^2*c^7*d^8*e - 8*b^8*c*d^2*e^7 + 212*a^3*b^4*c^2*e^9 - 240*a^4* b^2*c^3*e^9 - 112*a^2*c^7*d^6*e^3 + 240*a^3*c^6*d^4*e^5 - 228*a^4*c^5*d...