Integrand size = 27, antiderivative size = 189 \[ \int \frac {x^7}{\left (d+e x^2\right ) \left (a+b x^2+c x^4\right )} \, dx=\frac {x^2}{2 c e}+\frac {\left (b^3 d-3 a b c d-a b^2 e+2 a^2 c e\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^2 \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )}-\frac {d^3 \log \left (d+e x^2\right )}{2 e^2 \left (c d^2-b d e+a e^2\right )}+\frac {\left (b^2 d-a c d-a b e\right ) \log \left (a+b x^2+c x^4\right )}{4 c^2 \left (c d^2-b d e+a e^2\right )} \]
1/2*x^2/c/e-1/2*d^3*ln(e*x^2+d)/e^2/(a*e^2-b*d*e+c*d^2)+1/4*(-a*b*e-a*c*d+ b^2*d)*ln(c*x^4+b*x^2+a)/c^2/(a*e^2-b*d*e+c*d^2)+1/2*(2*a^2*c*e-a*b^2*e-3* a*b*c*d+b^3*d)*arctanh((2*c*x^2+b)/(-4*a*c+b^2)^(1/2))/c^2/(a*e^2-b*d*e+c* d^2)/(-4*a*c+b^2)^(1/2)
Time = 0.12 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.98 \[ \int \frac {x^7}{\left (d+e x^2\right ) \left (a+b x^2+c x^4\right )} \, dx=\frac {2 e^2 \left (b^3 d-3 a b c d-a b^2 e+2 a^2 c e\right ) \arctan \left (\frac {b+2 c x^2}{\sqrt {-b^2+4 a c}}\right )+\sqrt {-b^2+4 a c} \left (2 c^2 d^3 \log \left (d+e x^2\right )+e \left (-2 c \left (c d^2-b d e+a e^2\right ) x^2+e \left (-b^2 d+a c d+a b e\right ) \log \left (a+b x^2+c x^4\right )\right )\right )}{4 c^2 \sqrt {-b^2+4 a c} e^2 \left (-c d^2+e (b d-a e)\right )} \]
(2*e^2*(b^3*d - 3*a*b*c*d - a*b^2*e + 2*a^2*c*e)*ArcTan[(b + 2*c*x^2)/Sqrt [-b^2 + 4*a*c]] + Sqrt[-b^2 + 4*a*c]*(2*c^2*d^3*Log[d + e*x^2] + e*(-2*c*( c*d^2 - b*d*e + a*e^2)*x^2 + e*(-(b^2*d) + a*c*d + a*b*e)*Log[a + b*x^2 + c*x^4])))/(4*c^2*Sqrt[-b^2 + 4*a*c]*e^2*(-(c*d^2) + e*(b*d - a*e)))
Time = 0.43 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.98, 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.
\(\displaystyle \int \frac {x^7}{\left (d+e x^2\right ) \left (a+b x^2+c x^4\right )} \, dx\) |
\(\Big \downarrow \) 1578 |
\(\displaystyle \frac {1}{2} \int \frac {x^6}{\left (e x^2+d\right ) \left (c x^4+b x^2+a\right )}dx^2\) |
\(\Big \downarrow \) 1200 |
\(\displaystyle \frac {1}{2} \int \left (-\frac {d^3}{e \left (c d^2-b e d+a e^2\right ) \left (e x^2+d\right )}+\frac {1}{c e}+\frac {\left (d b^2-a e b-a c d\right ) x^2+a (b d-a e)}{c \left (c d^2-b e d+a e^2\right ) \left (c x^4+b x^2+a\right )}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (\frac {\text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (2 a^2 c e-a b^2 e-3 a b c d+b^3 d\right )}{c^2 \sqrt {b^2-4 a c} \left (a e^2-b d e+c d^2\right )}+\frac {\left (-a b e-a c d+b^2 d\right ) \log \left (a+b x^2+c x^4\right )}{2 c^2 \left (a e^2-b d e+c d^2\right )}-\frac {d^3 \log \left (d+e x^2\right )}{e^2 \left (a e^2-b d e+c d^2\right )}+\frac {x^2}{c e}\right )\) |
(x^2/(c*e) + ((b^3*d - 3*a*b*c*d - a*b^2*e + 2*a^2*c*e)*ArcTanh[(b + 2*c*x ^2)/Sqrt[b^2 - 4*a*c]])/(c^2*Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)) - (d^3*Log[d + e*x^2])/(e^2*(c*d^2 - b*d*e + a*e^2)) + ((b^2*d - a*c*d - a*b *e)*Log[a + b*x^2 + c*x^4])/(2*c^2*(c*d^2 - b*d*e + a*e^2)))/2
3.3.96.3.1 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 = 0.56 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.91
method | result | size |
default | \(\frac {x^{2}}{2 c e}-\frac {\frac {\left (a b e +a c d -b^{2} d \right ) \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{2 c}+\frac {2 \left (e \,a^{2}-d a b -\frac {\left (a b e +a c d -b^{2} d \right ) b}{2 c}\right ) \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c}-\frac {d^{3} \ln \left (e \,x^{2}+d \right )}{2 e^{2} \left (a \,e^{2}-b d e +c \,d^{2}\right )}\) | \(172\) |
risch | \(\text {Expression too large to display}\) | \(15081\) |
1/2*x^2/c/e-1/2/(a*e^2-b*d*e+c*d^2)/c*(1/2*(a*b*e+a*c*d-b^2*d)/c*ln(c*x^4+ b*x^2+a)+2*(e*a^2-d*a*b-1/2*(a*b*e+a*c*d-b^2*d)*b/c)/(4*a*c-b^2)^(1/2)*arc tan((2*c*x^2+b)/(4*a*c-b^2)^(1/2)))-1/2*d^3*ln(e*x^2+d)/e^2/(a*e^2-b*d*e+c *d^2)
Time = 113.05 (sec) , antiderivative size = 616, normalized size of antiderivative = 3.26 \[ \int \frac {x^7}{\left (d+e x^2\right ) \left (a+b x^2+c x^4\right )} \, dx=\left [-\frac {2 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{3} \log \left (e x^{2} + d\right ) - 2 \, {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{2} e - {\left (b^{3} c - 4 \, a b c^{2}\right )} d e^{2} + {\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} e^{3}\right )} x^{2} - {\left ({\left (b^{3} - 3 \, a b c\right )} d e^{2} - {\left (a b^{2} - 2 \, a^{2} c\right )} e^{3}\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c + {\left (2 \, c x^{2} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) - {\left ({\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d e^{2} - {\left (a b^{3} - 4 \, a^{2} b c\right )} e^{3}\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, {\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d^{2} e^{2} - {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d e^{3} + {\left (a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )} e^{4}\right )}}, -\frac {2 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{3} \log \left (e x^{2} + d\right ) - 2 \, {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{2} e - {\left (b^{3} c - 4 \, a b c^{2}\right )} d e^{2} + {\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} e^{3}\right )} x^{2} - 2 \, {\left ({\left (b^{3} - 3 \, a b c\right )} d e^{2} - {\left (a b^{2} - 2 \, a^{2} c\right )} e^{3}\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {{\left (2 \, c x^{2} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) - {\left ({\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d e^{2} - {\left (a b^{3} - 4 \, a^{2} b c\right )} e^{3}\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, {\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d^{2} e^{2} - {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d e^{3} + {\left (a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )} e^{4}\right )}}\right ] \]
[-1/4*(2*(b^2*c^2 - 4*a*c^3)*d^3*log(e*x^2 + d) - 2*((b^2*c^2 - 4*a*c^3)*d ^2*e - (b^3*c - 4*a*b*c^2)*d*e^2 + (a*b^2*c - 4*a^2*c^2)*e^3)*x^2 - ((b^3 - 3*a*b*c)*d*e^2 - (a*b^2 - 2*a^2*c)*e^3)*sqrt(b^2 - 4*a*c)*log((2*c^2*x^4 + 2*b*c*x^2 + b^2 - 2*a*c + (2*c*x^2 + b)*sqrt(b^2 - 4*a*c))/(c*x^4 + b*x ^2 + a)) - ((b^4 - 5*a*b^2*c + 4*a^2*c^2)*d*e^2 - (a*b^3 - 4*a^2*b*c)*e^3) *log(c*x^4 + b*x^2 + a))/((b^2*c^3 - 4*a*c^4)*d^2*e^2 - (b^3*c^2 - 4*a*b*c ^3)*d*e^3 + (a*b^2*c^2 - 4*a^2*c^3)*e^4), -1/4*(2*(b^2*c^2 - 4*a*c^3)*d^3* log(e*x^2 + d) - 2*((b^2*c^2 - 4*a*c^3)*d^2*e - (b^3*c - 4*a*b*c^2)*d*e^2 + (a*b^2*c - 4*a^2*c^2)*e^3)*x^2 - 2*((b^3 - 3*a*b*c)*d*e^2 - (a*b^2 - 2*a ^2*c)*e^3)*sqrt(-b^2 + 4*a*c)*arctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^ 2 - 4*a*c)) - ((b^4 - 5*a*b^2*c + 4*a^2*c^2)*d*e^2 - (a*b^3 - 4*a^2*b*c)*e ^3)*log(c*x^4 + b*x^2 + a))/((b^2*c^3 - 4*a*c^4)*d^2*e^2 - (b^3*c^2 - 4*a* b*c^3)*d*e^3 + (a*b^2*c^2 - 4*a^2*c^3)*e^4)]
Timed out. \[ \int \frac {x^7}{\left (d+e x^2\right ) \left (a+b x^2+c x^4\right )} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {x^7}{\left (d+e x^2\right ) \left (a+b x^2+c x^4\right )} \, dx=\text {Exception raised: ValueError} \]
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 = 0.59 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.03 \[ \int \frac {x^7}{\left (d+e x^2\right ) \left (a+b x^2+c x^4\right )} \, dx=-\frac {d^{3} \log \left ({\left | e x^{2} + d \right |}\right )}{2 \, {\left (c d^{2} e^{2} - b d e^{3} + a e^{4}\right )}} + \frac {{\left (b^{2} d - a c d - a b e\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, {\left (c^{3} d^{2} - b c^{2} d e + a c^{2} e^{2}\right )}} - \frac {{\left (b^{3} d - 3 \, a b c d - a b^{2} e + 2 \, a^{2} c e\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, {\left (c^{3} d^{2} - b c^{2} d e + a c^{2} e^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {x^{2}}{2 \, c e} \]
-1/2*d^3*log(abs(e*x^2 + d))/(c*d^2*e^2 - b*d*e^3 + a*e^4) + 1/4*(b^2*d - a*c*d - a*b*e)*log(c*x^4 + b*x^2 + a)/(c^3*d^2 - b*c^2*d*e + a*c^2*e^2) - 1/2*(b^3*d - 3*a*b*c*d - a*b^2*e + 2*a^2*c*e)*arctan((2*c*x^2 + b)/sqrt(-b ^2 + 4*a*c))/((c^3*d^2 - b*c^2*d*e + a*c^2*e^2)*sqrt(-b^2 + 4*a*c)) + 1/2* x^2/(c*e)
Time = 18.41 (sec) , antiderivative size = 2304, normalized size of antiderivative = 12.19 \[ \int \frac {x^7}{\left (d+e x^2\right ) \left (a+b x^2+c x^4\right )} \, dx=\text {Too large to display} \]
x^2/(2*c*e) - (d^3*log(d + e*x^2))/(2*(a*e^4 + c*d^2*e^2 - b*d*e^3)) - (lo g(a*e^5*x^2*(b^2 - 4*a*c)^(7/2) - 128*a^5*c^3*e^5 - 8*c^3*d^5*(b^2 - 4*a*c )^(5/2) - 512*a^3*c^5*d^4*e + 8*b^2*c^3*d^5*(b^2 - 4*a*c)^(3/2) + 6*b^3*d^ 2*e^3*(b^2 - 4*a*c)^(5/2) - 3*b^5*d^2*e^3*(b^2 - 4*a*c)^(3/2) + 32*a^4*b^2 *c^2*e^5 + 384*a^4*c^4*d^2*e^3 + 256*a^2*c^6*d^5*x^2 + 16*b^4*c^4*d^5*x^2 + 3*a*d*e^4*(b^2 - 4*a*c)^(7/2) - 3*b*d^2*e^3*(b^2 - 4*a*c)^(7/2) - 3*c*d^ 3*e^2*(b^2 - 4*a*c)^(7/2) - 16*a^2*b^3*c^3*d^3*e^2 + 48*a^2*b^4*c^2*d^2*e^ 3 - 288*a^3*b^2*c^3*d^2*e^3 + 16*a^3*b^3*c^2*e^5*x^2 - 384*a^3*c^5*d^3*e^2 *x^2 + 16*b^6*c^2*d^3*e^2*x^2 - 6*a*b^2*d*e^4*(b^2 - 4*a*c)^(5/2) + 3*a*b^ 4*d*e^4*(b^2 - 4*a*c)^(3/2) + 8*b*c^2*d^4*e*(b^2 - 4*a*c)^(5/2) - 32*a*b^4 *c^3*d^4*e + 192*a^4*b*c^3*d*e^4 - 2*b^2*c*d^3*e^2*(b^2 - 4*a*c)^(5/2) - 8 *b^3*c^2*d^4*e*(b^2 - 4*a*c)^(3/2) + 5*b^4*c*d^3*e^2*(b^2 - 4*a*c)^(3/2) - 2*a*b^2*e^5*x^2*(b^2 - 4*a*c)^(5/2) + a*b^4*e^5*x^2*(b^2 - 4*a*c)^(3/2) + 16*b*c^4*d^5*x^2*(b^2 - 4*a*c)^(3/2) - 3*c*d^2*e^3*x^2*(b^2 - 4*a*c)^(7/2 ) - 16*c^3*d^4*e*x^2*(b^2 - 4*a*c)^(5/2) + 256*a^2*b^2*c^4*d^4*e + 64*a^3* b*c^4*d^3*e^2 - 48*a^3*b^3*c^2*d*e^4 - 128*a*b^2*c^5*d^5*x^2 - 64*a^4*b*c^ 3*e^5*x^2 + 384*a^4*c^4*d*e^4*x^2 - 32*b^5*c^3*d^4*e*x^2 + 480*a^2*b^2*c^4 *d^3*e^2*x^2 + 48*a^2*b^3*c^3*d^2*e^3*x^2 + 256*a*b^3*c^4*d^4*e*x^2 - 512* a^2*b*c^5*d^4*e*x^2 + 8*b*c^2*d^3*e^2*x^2*(b^2 - 4*a*c)^(5/2) + 6*b^2*c*d^ 2*e^3*x^2*(b^2 - 4*a*c)^(5/2) - 16*b^2*c^3*d^4*e*x^2*(b^2 - 4*a*c)^(3/2...