Integrand size = 30, antiderivative size = 203 \[ \int \frac {x^5 \left (d+e x^2+f x^4\right )}{a+b x^2+c x^4} \, dx=\frac {\left (c^2 d+b^2 f-c (b e+a f)\right ) x^2}{2 c^3}+\frac {(c e-b f) x^4}{4 c^2}+\frac {f x^6}{6 c}+\frac {\left (b^3 c e-3 a b c^2 e-b^4 f-b^2 c (c d-4 a f)+2 a c^2 (c d-a f)\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^4 \sqrt {b^2-4 a c}}+\frac {\left (b^2 c e-a c^2 e-b^3 f-b c (c d-2 a f)\right ) \log \left (a+b x^2+c x^4\right )}{4 c^4} \]
1/2*(c^2*d+b^2*f-c*(a*f+b*e))*x^2/c^3+1/4*(-b*f+c*e)*x^4/c^2+1/6*f*x^6/c+1 /4*(b^2*c*e-a*c^2*e-b^3*f-b*c*(-2*a*f+c*d))*ln(c*x^4+b*x^2+a)/c^4+1/2*(b^3 *c*e-3*a*b*c^2*e-b^4*f-b^2*c*(-4*a*f+c*d)+2*a*c^2*(-a*f+c*d))*arctanh((2*c *x^2+b)/(-4*a*c+b^2)^(1/2))/c^4/(-4*a*c+b^2)^(1/2)
Time = 0.09 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.95 \[ \int \frac {x^5 \left (d+e x^2+f x^4\right )}{a+b x^2+c x^4} \, dx=\frac {6 c \left (c^2 d+b^2 f-c (b e+a f)\right ) x^2+3 c^2 (c e-b f) x^4+2 c^3 f x^6+\frac {6 \left (-b^3 c e+3 a b c^2 e+b^4 f+b^2 c (c d-4 a f)+2 a c^2 (-c d+a f)\right ) \arctan \left (\frac {b+2 c x^2}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}-3 \left (-b^2 c e+a c^2 e+b^3 f+b c (c d-2 a f)\right ) \log \left (a+b x^2+c x^4\right )}{12 c^4} \]
(6*c*(c^2*d + b^2*f - c*(b*e + a*f))*x^2 + 3*c^2*(c*e - b*f)*x^4 + 2*c^3*f *x^6 + (6*(-(b^3*c*e) + 3*a*b*c^2*e + b^4*f + b^2*c*(c*d - 4*a*f) + 2*a*c^ 2*(-(c*d) + a*f))*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4* a*c] - 3*(-(b^2*c*e) + a*c^2*e + b^3*f + b*c*(c*d - 2*a*f))*Log[a + b*x^2 + c*x^4])/(12*c^4)
Time = 0.54 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2194, 2159, 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.
\(\displaystyle \int \frac {x^5 \left (d+e x^2+f x^4\right )}{a+b x^2+c x^4} \, dx\) |
\(\Big \downarrow \) 2194 |
\(\displaystyle \frac {1}{2} \int \frac {x^4 \left (f x^4+e x^2+d\right )}{c x^4+b x^2+a}dx^2\) |
\(\Big \downarrow \) 2159 |
\(\displaystyle \frac {1}{2} \int \left (\frac {f x^4}{c}+\frac {(c e-b f) x^2}{c^2}+\frac {f b^2+c^2 d-c (b e+a f)}{c^3}-\frac {a \left (f b^2+c^2 d-c (b e+a f)\right )-\left (-f b^3+c e b^2-c (c d-2 a f) b-a c^2 e\right ) x^2}{c^3 \left (c x^4+b x^2+a\right )}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (\frac {\text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (-b^2 c (c d-4 a f)-3 a b c^2 e+2 a c^2 (c d-a f)+b^4 (-f)+b^3 c e\right )}{c^4 \sqrt {b^2-4 a c}}+\frac {x^2 \left (-c (a f+b e)+b^2 f+c^2 d\right )}{c^3}+\frac {\log \left (a+b x^2+c x^4\right ) \left (-b c (c d-2 a f)-a c^2 e+b^3 (-f)+b^2 c e\right )}{2 c^4}+\frac {x^4 (c e-b f)}{2 c^2}+\frac {f x^6}{3 c}\right )\) |
(((c^2*d + b^2*f - c*(b*e + a*f))*x^2)/c^3 + ((c*e - b*f)*x^4)/(2*c^2) + ( f*x^6)/(3*c) + ((b^3*c*e - 3*a*b*c^2*e - b^4*f - b^2*c*(c*d - 4*a*f) + 2*a *c^2*(c*d - a*f))*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(c^4*Sqrt[b^2 - 4*a*c]) + ((b^2*c*e - a*c^2*e - b^3*f - b*c*(c*d - 2*a*f))*Log[a + b*x^2 + c*x^4])/(2*c^4))/2
3.1.48.3.1 Defintions of rubi rules used
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x ], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] : > Simp[1/2 Subst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x + c*x^2) ^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x^2] && IntegerQ [(m - 1)/2]
Time = 0.22 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.10
method | result | size |
default | \(-\frac {-\frac {1}{3} f \,x^{6} c^{2}+\frac {1}{2} b c f \,x^{4}-\frac {1}{2} c^{2} e \,x^{4}+a c f \,x^{2}-b^{2} f \,x^{2}+b c e \,x^{2}-c^{2} d \,x^{2}}{2 c^{3}}+\frac {\frac {\left (2 a b c f -a \,c^{2} e -b^{3} f +b^{2} c e -b \,c^{2} d \right ) \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{2 c}+\frac {2 \left (a^{2} c f -a \,b^{2} f +a b c e -a \,c^{2} d -\frac {\left (2 a b c f -a \,c^{2} e -b^{3} f +b^{2} c e -b \,c^{2} d \right ) b}{2 c}\right ) \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{2 c^{3}}\) | \(224\) |
risch | \(\text {Expression too large to display}\) | \(4876\) |
-1/2/c^3*(-1/3*f*x^6*c^2+1/2*b*c*f*x^4-1/2*c^2*e*x^4+a*c*f*x^2-b^2*f*x^2+b *c*e*x^2-c^2*d*x^2)+1/2/c^3*(1/2*(2*a*b*c*f-a*c^2*e-b^3*f+b^2*c*e-b*c^2*d) /c*ln(c*x^4+b*x^2+a)+2*(a^2*c*f-a*b^2*f+a*b*c*e-a*c^2*d-1/2*(2*a*b*c*f-a*c ^2*e-b^3*f+b^2*c*e-b*c^2*d)*b/c)/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a *c-b^2)^(1/2)))
Time = 0.50 (sec) , antiderivative size = 677, normalized size of antiderivative = 3.33 \[ \int \frac {x^5 \left (d+e x^2+f x^4\right )}{a+b x^2+c x^4} \, dx=\left [\frac {2 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} f x^{6} + 3 \, {\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} e - {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} f\right )} x^{4} + 6 \, {\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d - {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} e + {\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} f\right )} x^{2} + 3 \, \sqrt {b^{2} - 4 \, a c} {\left ({\left (b^{2} c^{2} - 2 \, a c^{3}\right )} d - {\left (b^{3} c - 3 \, a b c^{2}\right )} e + {\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} f\right )} \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 ) - 3 \, {\left ({\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d - {\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} e + {\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} f\right )} \log \left (c x^{4} + b x^{2} + a\right )}{12 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )}}, \frac {2 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} f x^{6} + 3 \, {\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} e - {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} f\right )} x^{4} + 6 \, {\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d - {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} e + {\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} f\right )} x^{2} - 6 \, \sqrt {-b^{2} + 4 \, a c} {\left ({\left (b^{2} c^{2} - 2 \, a c^{3}\right )} d - {\left (b^{3} c - 3 \, a b c^{2}\right )} e + {\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} f\right )} \arctan \left (-\frac {{\left (2 \, c x^{2} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) - 3 \, {\left ({\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d - {\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} e + {\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} f\right )} \log \left (c x^{4} + b x^{2} + a\right )}{12 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )}}\right ] \]
[1/12*(2*(b^2*c^3 - 4*a*c^4)*f*x^6 + 3*((b^2*c^3 - 4*a*c^4)*e - (b^3*c^2 - 4*a*b*c^3)*f)*x^4 + 6*((b^2*c^3 - 4*a*c^4)*d - (b^3*c^2 - 4*a*b*c^3)*e + (b^4*c - 5*a*b^2*c^2 + 4*a^2*c^3)*f)*x^2 + 3*sqrt(b^2 - 4*a*c)*((b^2*c^2 - 2*a*c^3)*d - (b^3*c - 3*a*b*c^2)*e + (b^4 - 4*a*b^2*c + 2*a^2*c^2)*f)*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)) - 3*((b^3*c^2 - 4*a*b*c^3)*d - (b^4*c - 5*a*b^2*c^2 + 4*a^2*c^3)*e + (b^5 - 6*a*b^3*c + 8*a^2*b*c^2)*f)*log(c*x^4 + b*x^2 + a))/ (b^2*c^4 - 4*a*c^5), 1/12*(2*(b^2*c^3 - 4*a*c^4)*f*x^6 + 3*((b^2*c^3 - 4*a *c^4)*e - (b^3*c^2 - 4*a*b*c^3)*f)*x^4 + 6*((b^2*c^3 - 4*a*c^4)*d - (b^3*c ^2 - 4*a*b*c^3)*e + (b^4*c - 5*a*b^2*c^2 + 4*a^2*c^3)*f)*x^2 - 6*sqrt(-b^2 + 4*a*c)*((b^2*c^2 - 2*a*c^3)*d - (b^3*c - 3*a*b*c^2)*e + (b^4 - 4*a*b^2* c + 2*a^2*c^2)*f)*arctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) - 3*((b^3*c^2 - 4*a*b*c^3)*d - (b^4*c - 5*a*b^2*c^2 + 4*a^2*c^3)*e + (b^5 - 6*a*b^3*c + 8*a^2*b*c^2)*f)*log(c*x^4 + b*x^2 + a))/(b^2*c^4 - 4*a*c^5)]
Timed out. \[ \int \frac {x^5 \left (d+e x^2+f x^4\right )}{a+b x^2+c x^4} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {x^5 \left (d+e x^2+f x^4\right )}{a+b x^2+c x^4} \, dx=\text {Exception raised: ValueError} \]
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 = 0.61 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.02 \[ \int \frac {x^5 \left (d+e x^2+f x^4\right )}{a+b x^2+c x^4} \, dx=\frac {2 \, c^{2} f x^{6} + 3 \, c^{2} e x^{4} - 3 \, b c f x^{4} + 6 \, c^{2} d x^{2} - 6 \, b c e x^{2} + 6 \, b^{2} f x^{2} - 6 \, a c f x^{2}}{12 \, c^{3}} - \frac {{\left (b c^{2} d - b^{2} c e + a c^{2} e + b^{3} f - 2 \, a b c f\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, c^{4}} + \frac {{\left (b^{2} c^{2} d - 2 \, a c^{3} d - b^{3} c e + 3 \, a b c^{2} e + b^{4} f - 4 \, a b^{2} c f + 2 \, a^{2} c^{2} f\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt {-b^{2} + 4 \, a c} c^{4}} \]
1/12*(2*c^2*f*x^6 + 3*c^2*e*x^4 - 3*b*c*f*x^4 + 6*c^2*d*x^2 - 6*b*c*e*x^2 + 6*b^2*f*x^2 - 6*a*c*f*x^2)/c^3 - 1/4*(b*c^2*d - b^2*c*e + a*c^2*e + b^3* f - 2*a*b*c*f)*log(c*x^4 + b*x^2 + a)/c^4 + 1/2*(b^2*c^2*d - 2*a*c^3*d - b ^3*c*e + 3*a*b*c^2*e + b^4*f - 4*a*b^2*c*f + 2*a^2*c^2*f)*arctan((2*c*x^2 + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*c^4)
Time = 8.37 (sec) , antiderivative size = 2295, normalized size of antiderivative = 11.31 \[ \int \frac {x^5 \left (d+e x^2+f x^4\right )}{a+b x^2+c x^4} \, dx=\text {Too large to display} \]
x^4*(e/(4*c) - (b*f)/(4*c^2)) - x^2*((b*(e/c - (b*f)/c^2))/(2*c) - d/(2*c) + (a*f)/(2*c^2)) + (log(a + b*x^2 + c*x^4)*(2*b^5*f - 8*a^2*c^3*e + 2*b^3 *c^2*d - 2*b^4*c*e - 8*a*b*c^3*d - 12*a*b^3*c*f + 10*a*b^2*c^2*e + 16*a^2* b*c^2*f))/(2*(16*a*c^5 - 4*b^2*c^4)) + (f*x^6)/(6*c) + (atan((2*c^6*(4*a*c - b^2)*(x^2*(((((6*b^2*c^6*d + 4*a^2*c^6*f - 6*b^3*c^5*e + 6*b^4*c^4*f - 4*a*c^7*d + 10*a*b*c^6*e - 16*a*b^2*c^5*f)/c^6 + (4*b*c^2*(2*b^5*f - 8*a^2 *c^3*e + 2*b^3*c^2*d - 2*b^4*c*e - 8*a*b*c^3*d - 12*a*b^3*c*f + 10*a*b^2*c ^2*e + 16*a^2*b*c^2*f))/(16*a*c^5 - 4*b^2*c^4))*(b^4*f + b^2*c^2*d + 2*a^2 *c^2*f - 2*a*c^3*d - b^3*c*e + 3*a*b*c^2*e - 4*a*b^2*c*f))/(8*c^4*(4*a*c - b^2)^(1/2)) + (b*(b^4*f + b^2*c^2*d + 2*a^2*c^2*f - 2*a*c^3*d - b^3*c*e + 3*a*b*c^2*e - 4*a*b^2*c*f)*(2*b^5*f - 8*a^2*c^3*e + 2*b^3*c^2*d - 2*b^4*c *e - 8*a*b*c^3*d - 12*a*b^3*c*f + 10*a*b^2*c^2*e + 16*a^2*b*c^2*f))/(2*c^2 *(4*a*c - b^2)^(1/2)*(16*a*c^5 - 4*b^2*c^4)))/a - (b*((b^7*f^2 + b^3*c^4*d ^2 + b^5*c^2*e^2 - 3*a*b^3*c^3*e^2 + 2*a^2*b*c^4*e^2 - 2*a^3*b*c^3*f^2 - 2 *b^6*c*e*f + 7*a^2*b^3*c^2*f^2 - a*b*c^5*d^2 - 5*a*b^5*c*f^2 - a^2*c^5*d*e - 2*b^4*c^3*d*e + a^3*c^4*e*f + 2*b^5*c^2*d*f + 4*a*b^2*c^4*d*e - 6*a*b^3 *c^3*d*f + 3*a^2*b*c^4*d*f + 8*a*b^4*c^2*e*f - 8*a^2*b^2*c^3*e*f)/c^6 + (( (6*b^2*c^6*d + 4*a^2*c^6*f - 6*b^3*c^5*e + 6*b^4*c^4*f - 4*a*c^7*d + 10*a* b*c^6*e - 16*a*b^2*c^5*f)/c^6 + (4*b*c^2*(2*b^5*f - 8*a^2*c^3*e + 2*b^3*c^ 2*d - 2*b^4*c*e - 8*a*b*c^3*d - 12*a*b^3*c*f + 10*a*b^2*c^2*e + 16*a^2*...