Integrand size = 18, antiderivative size = 63 \[ \int \frac {x^7}{a+b x^4+c x^8} \, dx=\frac {b \text {arctanh}\left (\frac {b+2 c x^4}{\sqrt {b^2-4 a c}}\right )}{4 c \sqrt {b^2-4 a c}}+\frac {\log \left (a+b x^4+c x^8\right )}{8 c} \] Output:
1/4*b*arctanh((2*c*x^4+b)/(-4*a*c+b^2)^(1/2))/c/(-4*a*c+b^2)^(1/2)+1/8*ln( c*x^8+b*x^4+a)/c
Time = 0.02 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.98 \[ \int \frac {x^7}{a+b x^4+c x^8} \, dx=\frac {-\frac {2 b \arctan \left (\frac {b+2 c x^4}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}+\log \left (a+b x^4+c x^8\right )}{8 c} \] Input:
Integrate[x^7/(a + b*x^4 + c*x^8),x]
Output:
((-2*b*ArcTan[(b + 2*c*x^4)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + Log[ a + b*x^4 + c*x^8])/(8*c)
Time = 0.23 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {1693, 1142, 1083, 219, 1103}
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.
\(\displaystyle \int \frac {x^7}{a+b x^4+c x^8} \, dx\) |
\(\Big \downarrow \) 1693 |
\(\displaystyle \frac {1}{4} \int \frac {x^4}{c x^8+b x^4+a}dx^4\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle \frac {1}{4} \left (\frac {\int \frac {2 c x^4+b}{c x^8+b x^4+a}dx^4}{2 c}-\frac {b \int \frac {1}{c x^8+b x^4+a}dx^4}{2 c}\right )\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {1}{4} \left (\frac {b \int \frac {1}{-x^8+b^2-4 a c}d\left (2 c x^4+b\right )}{c}+\frac {\int \frac {2 c x^4+b}{c x^8+b x^4+a}dx^4}{2 c}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{4} \left (\frac {\int \frac {2 c x^4+b}{c x^8+b x^4+a}dx^4}{2 c}+\frac {b \text {arctanh}\left (\frac {b+2 c x^4}{\sqrt {b^2-4 a c}}\right )}{c \sqrt {b^2-4 a c}}\right )\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {1}{4} \left (\frac {b \text {arctanh}\left (\frac {b+2 c x^4}{\sqrt {b^2-4 a c}}\right )}{c \sqrt {b^2-4 a c}}+\frac {\log \left (a+b x^4+c x^8\right )}{2 c}\right )\) |
Input:
Int[x^7/(a + b*x^4 + c*x^8),x]
Output:
((b*ArcTanh[(b + 2*c*x^4)/Sqrt[b^2 - 4*a*c]])/(c*Sqrt[b^2 - 4*a*c]) + Log[ a + b*x^4 + c*x^8]/(2*c))/4
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol ] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && IntegerQ [Simplify[(m + 1)/n]]
Time = 0.05 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.95
method | result | size |
default | \(\frac {\ln \left (c \,x^{8}+b \,x^{4}+a \right )}{8 c}-\frac {b \arctan \left (\frac {2 c \,x^{4}+b}{\sqrt {4 a c -b^{2}}}\right )}{4 c \sqrt {4 a c -b^{2}}}\) | \(60\) |
risch | \(\frac {\ln \left (\left (-4 a b c +b^{3}+\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b \right ) x^{4}+2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a \right ) a}{8 a c -2 b^{2}}-\frac {\ln \left (\left (-4 a b c +b^{3}+\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b \right ) x^{4}+2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a \right ) b^{2}}{8 c \left (4 a c -b^{2}\right )}+\frac {\ln \left (\left (-4 a b c +b^{3}+\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b \right ) x^{4}+2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a \right ) \sqrt {-b^{2} \left (4 a c -b^{2}\right )}}{8 c \left (4 a c -b^{2}\right )}+\frac {\ln \left (\left (-4 a b c +b^{3}-\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b \right ) x^{4}-2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a \right ) a}{8 a c -2 b^{2}}-\frac {\ln \left (\left (-4 a b c +b^{3}-\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b \right ) x^{4}-2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a \right ) b^{2}}{8 c \left (4 a c -b^{2}\right )}-\frac {\ln \left (\left (-4 a b c +b^{3}-\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b \right ) x^{4}-2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a \right ) \sqrt {-b^{2} \left (4 a c -b^{2}\right )}}{8 c \left (4 a c -b^{2}\right )}\) | \(467\) |
Input:
int(x^7/(c*x^8+b*x^4+a),x,method=_RETURNVERBOSE)
Output:
1/8*ln(c*x^8+b*x^4+a)/c-1/4*b/c/(4*a*c-b^2)^(1/2)*arctan((2*c*x^4+b)/(4*a* c-b^2)^(1/2))
Time = 0.08 (sec) , antiderivative size = 197, normalized size of antiderivative = 3.13 \[ \int \frac {x^7}{a+b x^4+c x^8} \, dx=\left [\frac {\sqrt {b^{2} - 4 \, a c} b \log \left (\frac {2 \, c^{2} x^{8} + 2 \, b c x^{4} + b^{2} - 2 \, a c + {\left (2 \, c x^{4} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{8} + b x^{4} + a}\right ) + {\left (b^{2} - 4 \, a c\right )} \log \left (c x^{8} + b x^{4} + a\right )}{8 \, {\left (b^{2} c - 4 \, a c^{2}\right )}}, \frac {2 \, \sqrt {-b^{2} + 4 \, a c} b \arctan \left (-\frac {{\left (2 \, c x^{4} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) + {\left (b^{2} - 4 \, a c\right )} \log \left (c x^{8} + b x^{4} + a\right )}{8 \, {\left (b^{2} c - 4 \, a c^{2}\right )}}\right ] \] Input:
integrate(x^7/(c*x^8+b*x^4+a),x, algorithm="fricas")
Output:
[1/8*(sqrt(b^2 - 4*a*c)*b*log((2*c^2*x^8 + 2*b*c*x^4 + b^2 - 2*a*c + (2*c* x^4 + b)*sqrt(b^2 - 4*a*c))/(c*x^8 + b*x^4 + a)) + (b^2 - 4*a*c)*log(c*x^8 + b*x^4 + a))/(b^2*c - 4*a*c^2), 1/8*(2*sqrt(-b^2 + 4*a*c)*b*arctan(-(2*c *x^4 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) + (b^2 - 4*a*c)*log(c*x^8 + b* x^4 + a))/(b^2*c - 4*a*c^2)]
Leaf count of result is larger than twice the leaf count of optimal. 223 vs. \(2 (54) = 108\).
Time = 1.58 (sec) , antiderivative size = 223, normalized size of antiderivative = 3.54 \[ \int \frac {x^7}{a+b x^4+c x^8} \, dx=\left (- \frac {b \sqrt {- 4 a c + b^{2}}}{8 c \left (4 a c - b^{2}\right )} + \frac {1}{8 c}\right ) \log {\left (x^{4} + \frac {- 16 a c \left (- \frac {b \sqrt {- 4 a c + b^{2}}}{8 c \left (4 a c - b^{2}\right )} + \frac {1}{8 c}\right ) + 2 a + 4 b^{2} \left (- \frac {b \sqrt {- 4 a c + b^{2}}}{8 c \left (4 a c - b^{2}\right )} + \frac {1}{8 c}\right )}{b} \right )} + \left (\frac {b \sqrt {- 4 a c + b^{2}}}{8 c \left (4 a c - b^{2}\right )} + \frac {1}{8 c}\right ) \log {\left (x^{4} + \frac {- 16 a c \left (\frac {b \sqrt {- 4 a c + b^{2}}}{8 c \left (4 a c - b^{2}\right )} + \frac {1}{8 c}\right ) + 2 a + 4 b^{2} \left (\frac {b \sqrt {- 4 a c + b^{2}}}{8 c \left (4 a c - b^{2}\right )} + \frac {1}{8 c}\right )}{b} \right )} \] Input:
integrate(x**7/(c*x**8+b*x**4+a),x)
Output:
(-b*sqrt(-4*a*c + b**2)/(8*c*(4*a*c - b**2)) + 1/(8*c))*log(x**4 + (-16*a* c*(-b*sqrt(-4*a*c + b**2)/(8*c*(4*a*c - b**2)) + 1/(8*c)) + 2*a + 4*b**2*( -b*sqrt(-4*a*c + b**2)/(8*c*(4*a*c - b**2)) + 1/(8*c)))/b) + (b*sqrt(-4*a* c + b**2)/(8*c*(4*a*c - b**2)) + 1/(8*c))*log(x**4 + (-16*a*c*(b*sqrt(-4*a *c + b**2)/(8*c*(4*a*c - b**2)) + 1/(8*c)) + 2*a + 4*b**2*(b*sqrt(-4*a*c + b**2)/(8*c*(4*a*c - b**2)) + 1/(8*c)))/b)
Exception generated. \[ \int \frac {x^7}{a+b x^4+c x^8} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^7/(c*x^8+b*x^4+a),x, algorithm="maxima")
Output:
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 = 1.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.94 \[ \int \frac {x^7}{a+b x^4+c x^8} \, dx=-\frac {b \arctan \left (\frac {2 \, c x^{4} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{4 \, \sqrt {-b^{2} + 4 \, a c} c} + \frac {\log \left (c x^{8} + b x^{4} + a\right )}{8 \, c} \] Input:
integrate(x^7/(c*x^8+b*x^4+a),x, algorithm="giac")
Output:
-1/4*b*arctan((2*c*x^4 + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*c) + 1 /8*log(c*x^8 + b*x^4 + a)/c
Time = 21.22 (sec) , antiderivative size = 2654, normalized size of antiderivative = 42.13 \[ \int \frac {x^7}{a+b x^4+c x^8} \, dx=\text {Too large to display} \] Input:
int(x^7/(a + b*x^4 + c*x^8),x)
Output:
(log(a + b*x^4 + c*x^8)*(16*a*c - 4*b^2))/(2*(64*a*c^2 - 16*b^2*c)) - (b*a tan((8*x^4*(((a*c - b^2)*(((((16*a*c - 4*b^2)*((b*(448*b^3*c^3 - (256*b^3* c^4*(16*a*c - 4*b^2))/(64*a*c^2 - 16*b^2*c)))/(8*c*(4*a*c - b^2)^(1/2)) - (32*b^4*c^3*(16*a*c - 4*b^2))/((64*a*c^2 - 16*b^2*c)*(4*a*c - b^2)^(1/2))) )/(2*(64*a*c^2 - 16*b^2*c)) - (b*(144*b^3*c^2 - ((448*b^3*c^3 - (256*b^3*c ^4*(16*a*c - 4*b^2))/(64*a*c^2 - 16*b^2*c))*(16*a*c - 4*b^2))/(2*(64*a*c^2 - 16*b^2*c))))/(8*c*(4*a*c - b^2)^(1/2)))*(16*a*c - 4*b^2))/(2*(64*a*c^2 - 16*b^2*c)) - (b*((b*((b*(448*b^3*c^3 - (256*b^3*c^4*(16*a*c - 4*b^2))/(6 4*a*c^2 - 16*b^2*c)))/(8*c*(4*a*c - b^2)^(1/2)) - (32*b^4*c^3*(16*a*c - 4* b^2))/((64*a*c^2 - 16*b^2*c)*(4*a*c - b^2)^(1/2))))/(8*c*(4*a*c - b^2)^(1/ 2)) - (4*b^5*c^2*(16*a*c - 4*b^2))/((64*a*c^2 - 16*b^2*c)*(4*a*c - b^2)))) /(8*c*(4*a*c - b^2)^(1/2)) + (b*(20*b^3*c - ((144*b^3*c^2 - ((448*b^3*c^3 - (256*b^3*c^4*(16*a*c - 4*b^2))/(64*a*c^2 - 16*b^2*c))*(16*a*c - 4*b^2))/ (2*(64*a*c^2 - 16*b^2*c)))*(16*a*c - 4*b^2))/(2*(64*a*c^2 - 16*b^2*c))))/( 8*c*(4*a*c - b^2)^(1/2)) + (b^6*c*(16*a*c - 4*b^2))/(2*(64*a*c^2 - 16*b^2* c)*(4*a*c - b^2)^(3/2))))/(8*a^3*c^2) + ((b^3 - 3*a*b*c)*(b^7/(8*(4*a*c - b^2)^2) + b^3 - ((20*b^3*c - ((144*b^3*c^2 - ((448*b^3*c^3 - (256*b^3*c^4* (16*a*c - 4*b^2))/(64*a*c^2 - 16*b^2*c))*(16*a*c - 4*b^2))/(2*(64*a*c^2 - 16*b^2*c)))*(16*a*c - 4*b^2))/(2*(64*a*c^2 - 16*b^2*c)))*(16*a*c - 4*b^2)) /(2*(64*a*c^2 - 16*b^2*c)) + ((16*a*c - 4*b^2)*((b*((b*(448*b^3*c^3 - (...
\[ \int \frac {x^7}{a+b x^4+c x^8} \, dx=\int \frac {x^{7}}{c \,x^{8}+b \,x^{4}+a}d x \] Input:
int(x^7/(c*x^8+b*x^4+a),x)
Output:
int(x^7/(c*x^8+b*x^4+a),x)