Integrand size = 26, antiderivative size = 20 \[ \int \frac {x \left (b+2 c x^2\right )}{\left (-a+b x^2+c x^4\right )^8} \, dx=\frac {1}{14 \left (a-b x^2-c x^4\right )^7} \] Output:
1/14/(-c*x^4-b*x^2+a)^7
Time = 0.05 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {x \left (b+2 c x^2\right )}{\left (-a+b x^2+c x^4\right )^8} \, dx=-\frac {1}{14 \left (-a+b x^2+c x^4\right )^7} \] Input:
Integrate[(x*(b + 2*c*x^2))/(-a + b*x^2 + c*x^4)^8,x]
Output:
-1/14*1/(-a + b*x^2 + c*x^4)^7
Time = 0.17 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1576, 1104}
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.
\(\displaystyle \int \frac {x \left (b+2 c x^2\right )}{\left (-a+b x^2+c x^4\right )^8} \, dx\) |
\(\Big \downarrow \) 1576 |
\(\displaystyle \frac {1}{2} \int \frac {2 c x^2+b}{\left (-c x^4-b x^2+a\right )^8}dx^2\) |
\(\Big \downarrow \) 1104 |
\(\displaystyle \frac {1}{14 \left (a-b x^2-c x^4\right )^7}\) |
Input:
Int[(x*(b + 2*c*x^2))/(-a + b*x^2 + c*x^4)^8,x]
Output:
1/(14*(a - b*x^2 - c*x^4)^7)
Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol ] :> Simp[d*((a + b*x + c*x^2)^(p + 1)/(b*(p + 1))), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0]
Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^( p_.), x_Symbol] :> Simp[1/2 Subst[Int[(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]
Time = 0.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95
method | result | size |
gosper | \(\frac {1}{14 \left (-c \,x^{4}-b \,x^{2}+a \right )^{7}}\) | \(19\) |
default | \(\frac {1}{14 \left (-c \,x^{4}-b \,x^{2}+a \right )^{7}}\) | \(19\) |
norman | \(\frac {1}{14 \left (-c \,x^{4}-b \,x^{2}+a \right )^{7}}\) | \(19\) |
risch | \(\frac {1}{14 \left (-c \,x^{4}-b \,x^{2}+a \right )^{7}}\) | \(19\) |
parallelrisch | \(-\frac {1}{14 \left (c \,x^{4}+b \,x^{2}-a \right )^{7}}\) | \(19\) |
orering | \(\frac {-c \,x^{4}-b \,x^{2}+a}{14 \left (c \,x^{4}+b \,x^{2}-a \right )^{8}}\) | \(33\) |
Input:
int(x*(2*c*x^2+b)/(c*x^4+b*x^2-a)^8,x,method=_RETURNVERBOSE)
Output:
1/14/(-c*x^4-b*x^2+a)^7
Leaf count of result is larger than twice the leaf count of optimal. 356 vs. \(2 (18) = 36\).
Time = 0.09 (sec) , antiderivative size = 356, normalized size of antiderivative = 17.80 \[ \int \frac {x \left (b+2 c x^2\right )}{\left (-a+b x^2+c x^4\right )^8} \, dx=-\frac {1}{14 \, {\left (c^{7} x^{28} + 7 \, b c^{6} x^{26} + 7 \, {\left (3 \, b^{2} c^{5} - a c^{6}\right )} x^{24} + 7 \, {\left (5 \, b^{3} c^{4} - 6 \, a b c^{5}\right )} x^{22} + 7 \, {\left (5 \, b^{4} c^{3} - 15 \, a b^{2} c^{4} + 3 \, a^{2} c^{5}\right )} x^{20} + 7 \, {\left (3 \, b^{5} c^{2} - 20 \, a b^{3} c^{3} + 15 \, a^{2} b c^{4}\right )} x^{18} + 7 \, {\left (b^{6} c - 15 \, a b^{4} c^{2} + 30 \, a^{2} b^{2} c^{3} - 5 \, a^{3} c^{4}\right )} x^{16} + {\left (b^{7} - 42 \, a b^{5} c + 210 \, a^{2} b^{3} c^{2} - 140 \, a^{3} b c^{3}\right )} x^{14} - 7 \, {\left (a b^{6} - 15 \, a^{2} b^{4} c + 30 \, a^{3} b^{2} c^{2} - 5 \, a^{4} c^{3}\right )} x^{12} + 7 \, {\left (3 \, a^{2} b^{5} - 20 \, a^{3} b^{3} c + 15 \, a^{4} b c^{2}\right )} x^{10} + 7 \, a^{6} b x^{2} - 7 \, {\left (5 \, a^{3} b^{4} - 15 \, a^{4} b^{2} c + 3 \, a^{5} c^{2}\right )} x^{8} - a^{7} + 7 \, {\left (5 \, a^{4} b^{3} - 6 \, a^{5} b c\right )} x^{6} - 7 \, {\left (3 \, a^{5} b^{2} - a^{6} c\right )} x^{4}\right )}} \] Input:
integrate(x*(2*c*x^2+b)/(c*x^4+b*x^2-a)^8,x, algorithm="fricas")
Output:
-1/14/(c^7*x^28 + 7*b*c^6*x^26 + 7*(3*b^2*c^5 - a*c^6)*x^24 + 7*(5*b^3*c^4 - 6*a*b*c^5)*x^22 + 7*(5*b^4*c^3 - 15*a*b^2*c^4 + 3*a^2*c^5)*x^20 + 7*(3* b^5*c^2 - 20*a*b^3*c^3 + 15*a^2*b*c^4)*x^18 + 7*(b^6*c - 15*a*b^4*c^2 + 30 *a^2*b^2*c^3 - 5*a^3*c^4)*x^16 + (b^7 - 42*a*b^5*c + 210*a^2*b^3*c^2 - 140 *a^3*b*c^3)*x^14 - 7*(a*b^6 - 15*a^2*b^4*c + 30*a^3*b^2*c^2 - 5*a^4*c^3)*x ^12 + 7*(3*a^2*b^5 - 20*a^3*b^3*c + 15*a^4*b*c^2)*x^10 + 7*a^6*b*x^2 - 7*( 5*a^3*b^4 - 15*a^4*b^2*c + 3*a^5*c^2)*x^8 - a^7 + 7*(5*a^4*b^3 - 6*a^5*b*c )*x^6 - 7*(3*a^5*b^2 - a^6*c)*x^4)
Leaf count of result is larger than twice the leaf count of optimal. 360 vs. \(2 (15) = 30\).
Time = 4.87 (sec) , antiderivative size = 360, normalized size of antiderivative = 18.00 \[ \int \frac {x \left (b+2 c x^2\right )}{\left (-a+b x^2+c x^4\right )^8} \, dx=- \frac {1}{- 14 a^{7} + 98 a^{6} b x^{2} + 98 b c^{6} x^{26} + 14 c^{7} x^{28} + x^{24} \left (- 98 a c^{6} + 294 b^{2} c^{5}\right ) + x^{22} \left (- 588 a b c^{5} + 490 b^{3} c^{4}\right ) + x^{20} \cdot \left (294 a^{2} c^{5} - 1470 a b^{2} c^{4} + 490 b^{4} c^{3}\right ) + x^{18} \cdot \left (1470 a^{2} b c^{4} - 1960 a b^{3} c^{3} + 294 b^{5} c^{2}\right ) + x^{16} \left (- 490 a^{3} c^{4} + 2940 a^{2} b^{2} c^{3} - 1470 a b^{4} c^{2} + 98 b^{6} c\right ) + x^{14} \left (- 1960 a^{3} b c^{3} + 2940 a^{2} b^{3} c^{2} - 588 a b^{5} c + 14 b^{7}\right ) + x^{12} \cdot \left (490 a^{4} c^{3} - 2940 a^{3} b^{2} c^{2} + 1470 a^{2} b^{4} c - 98 a b^{6}\right ) + x^{10} \cdot \left (1470 a^{4} b c^{2} - 1960 a^{3} b^{3} c + 294 a^{2} b^{5}\right ) + x^{8} \left (- 294 a^{5} c^{2} + 1470 a^{4} b^{2} c - 490 a^{3} b^{4}\right ) + x^{6} \left (- 588 a^{5} b c + 490 a^{4} b^{3}\right ) + x^{4} \cdot \left (98 a^{6} c - 294 a^{5} b^{2}\right )} \] Input:
integrate(x*(2*c*x**2+b)/(c*x**4+b*x**2-a)**8,x)
Output:
-1/(-14*a**7 + 98*a**6*b*x**2 + 98*b*c**6*x**26 + 14*c**7*x**28 + x**24*(- 98*a*c**6 + 294*b**2*c**5) + x**22*(-588*a*b*c**5 + 490*b**3*c**4) + x**20 *(294*a**2*c**5 - 1470*a*b**2*c**4 + 490*b**4*c**3) + x**18*(1470*a**2*b*c **4 - 1960*a*b**3*c**3 + 294*b**5*c**2) + x**16*(-490*a**3*c**4 + 2940*a** 2*b**2*c**3 - 1470*a*b**4*c**2 + 98*b**6*c) + x**14*(-1960*a**3*b*c**3 + 2 940*a**2*b**3*c**2 - 588*a*b**5*c + 14*b**7) + x**12*(490*a**4*c**3 - 2940 *a**3*b**2*c**2 + 1470*a**2*b**4*c - 98*a*b**6) + x**10*(1470*a**4*b*c**2 - 1960*a**3*b**3*c + 294*a**2*b**5) + x**8*(-294*a**5*c**2 + 1470*a**4*b** 2*c - 490*a**3*b**4) + x**6*(-588*a**5*b*c + 490*a**4*b**3) + x**4*(98*a** 6*c - 294*a**5*b**2))
Leaf count of result is larger than twice the leaf count of optimal. 356 vs. \(2 (18) = 36\).
Time = 0.12 (sec) , antiderivative size = 356, normalized size of antiderivative = 17.80 \[ \int \frac {x \left (b+2 c x^2\right )}{\left (-a+b x^2+c x^4\right )^8} \, dx=-\frac {1}{14 \, {\left (c^{7} x^{28} + 7 \, b c^{6} x^{26} + 7 \, {\left (3 \, b^{2} c^{5} - a c^{6}\right )} x^{24} + 7 \, {\left (5 \, b^{3} c^{4} - 6 \, a b c^{5}\right )} x^{22} + 7 \, {\left (5 \, b^{4} c^{3} - 15 \, a b^{2} c^{4} + 3 \, a^{2} c^{5}\right )} x^{20} + 7 \, {\left (3 \, b^{5} c^{2} - 20 \, a b^{3} c^{3} + 15 \, a^{2} b c^{4}\right )} x^{18} + 7 \, {\left (b^{6} c - 15 \, a b^{4} c^{2} + 30 \, a^{2} b^{2} c^{3} - 5 \, a^{3} c^{4}\right )} x^{16} + {\left (b^{7} - 42 \, a b^{5} c + 210 \, a^{2} b^{3} c^{2} - 140 \, a^{3} b c^{3}\right )} x^{14} - 7 \, {\left (a b^{6} - 15 \, a^{2} b^{4} c + 30 \, a^{3} b^{2} c^{2} - 5 \, a^{4} c^{3}\right )} x^{12} + 7 \, {\left (3 \, a^{2} b^{5} - 20 \, a^{3} b^{3} c + 15 \, a^{4} b c^{2}\right )} x^{10} + 7 \, a^{6} b x^{2} - 7 \, {\left (5 \, a^{3} b^{4} - 15 \, a^{4} b^{2} c + 3 \, a^{5} c^{2}\right )} x^{8} - a^{7} + 7 \, {\left (5 \, a^{4} b^{3} - 6 \, a^{5} b c\right )} x^{6} - 7 \, {\left (3 \, a^{5} b^{2} - a^{6} c\right )} x^{4}\right )}} \] Input:
integrate(x*(2*c*x^2+b)/(c*x^4+b*x^2-a)^8,x, algorithm="maxima")
Output:
-1/14/(c^7*x^28 + 7*b*c^6*x^26 + 7*(3*b^2*c^5 - a*c^6)*x^24 + 7*(5*b^3*c^4 - 6*a*b*c^5)*x^22 + 7*(5*b^4*c^3 - 15*a*b^2*c^4 + 3*a^2*c^5)*x^20 + 7*(3* b^5*c^2 - 20*a*b^3*c^3 + 15*a^2*b*c^4)*x^18 + 7*(b^6*c - 15*a*b^4*c^2 + 30 *a^2*b^2*c^3 - 5*a^3*c^4)*x^16 + (b^7 - 42*a*b^5*c + 210*a^2*b^3*c^2 - 140 *a^3*b*c^3)*x^14 - 7*(a*b^6 - 15*a^2*b^4*c + 30*a^3*b^2*c^2 - 5*a^4*c^3)*x ^12 + 7*(3*a^2*b^5 - 20*a^3*b^3*c + 15*a^4*b*c^2)*x^10 + 7*a^6*b*x^2 - 7*( 5*a^3*b^4 - 15*a^4*b^2*c + 3*a^5*c^2)*x^8 - a^7 + 7*(5*a^4*b^3 - 6*a^5*b*c )*x^6 - 7*(3*a^5*b^2 - a^6*c)*x^4)
Time = 0.97 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {x \left (b+2 c x^2\right )}{\left (-a+b x^2+c x^4\right )^8} \, dx=-\frac {1}{14 \, {\left (c x^{4} + b x^{2} - a\right )}^{7}} \] Input:
integrate(x*(2*c*x^2+b)/(c*x^4+b*x^2-a)^8,x, algorithm="giac")
Output:
-1/14/(c*x^4 + b*x^2 - a)^7
Time = 15.24 (sec) , antiderivative size = 360, normalized size of antiderivative = 18.00 \[ \int \frac {x \left (b+2 c x^2\right )}{\left (-a+b x^2+c x^4\right )^8} \, dx=-\frac {1}{14\,\left (x^{10}\,\left (105\,a^4\,b\,c^2-140\,a^3\,b^3\,c+21\,a^2\,b^5\right )+x^{18}\,\left (105\,a^2\,b\,c^4-140\,a\,b^3\,c^3+21\,b^5\,c^2\right )+x^{14}\,\left (-140\,a^3\,b\,c^3+210\,a^2\,b^3\,c^2-42\,a\,b^5\,c+b^7\right )+x^6\,\left (35\,a^4\,b^3-42\,a^5\,b\,c\right )+x^{22}\,\left (35\,b^3\,c^4-42\,a\,b\,c^5\right )-x^8\,\left (21\,a^5\,c^2-105\,a^4\,b^2\,c+35\,a^3\,b^4\right )+x^{20}\,\left (21\,a^2\,c^5-105\,a\,b^2\,c^4+35\,b^4\,c^3\right )-a^7-x^{12}\,\left (-35\,a^4\,c^3+210\,a^3\,b^2\,c^2-105\,a^2\,b^4\,c+7\,a\,b^6\right )+x^{16}\,\left (-35\,a^3\,c^4+210\,a^2\,b^2\,c^3-105\,a\,b^4\,c^2+7\,b^6\,c\right )+c^7\,x^{28}+x^4\,\left (7\,a^6\,c-21\,a^5\,b^2\right )-x^{24}\,\left (7\,a\,c^6-21\,b^2\,c^5\right )+7\,a^6\,b\,x^2+7\,b\,c^6\,x^{26}\right )} \] Input:
int((x*(b + 2*c*x^2))/(b*x^2 - a + c*x^4)^8,x)
Output:
-1/(14*(x^10*(21*a^2*b^5 - 140*a^3*b^3*c + 105*a^4*b*c^2) + x^18*(21*b^5*c ^2 - 140*a*b^3*c^3 + 105*a^2*b*c^4) + x^14*(b^7 - 140*a^3*b*c^3 + 210*a^2* b^3*c^2 - 42*a*b^5*c) + x^6*(35*a^4*b^3 - 42*a^5*b*c) + x^22*(35*b^3*c^4 - 42*a*b*c^5) - x^8*(35*a^3*b^4 + 21*a^5*c^2 - 105*a^4*b^2*c) + x^20*(21*a^ 2*c^5 + 35*b^4*c^3 - 105*a*b^2*c^4) - a^7 - x^12*(7*a*b^6 - 35*a^4*c^3 - 1 05*a^2*b^4*c + 210*a^3*b^2*c^2) + x^16*(7*b^6*c - 35*a^3*c^4 - 105*a*b^4*c ^2 + 210*a^2*b^2*c^3) + c^7*x^28 + x^4*(7*a^6*c - 21*a^5*b^2) - x^24*(7*a* c^6 - 21*b^2*c^5) + 7*a^6*b*x^2 + 7*b*c^6*x^26))
Time = 0.20 (sec) , antiderivative size = 390, normalized size of antiderivative = 19.50 \[ \int \frac {x \left (b+2 c x^2\right )}{\left (-a+b x^2+c x^4\right )^8} \, dx=\frac {1}{-14 c^{7} x^{28}-98 b \,c^{6} x^{26}+98 a \,c^{6} x^{24}-294 b^{2} c^{5} x^{24}+588 a b \,c^{5} x^{22}-490 b^{3} c^{4} x^{22}-294 a^{2} c^{5} x^{20}+1470 a \,b^{2} c^{4} x^{20}-490 b^{4} c^{3} x^{20}-1470 a^{2} b \,c^{4} x^{18}+1960 a \,b^{3} c^{3} x^{18}-294 b^{5} c^{2} x^{18}+490 a^{3} c^{4} x^{16}-2940 a^{2} b^{2} c^{3} x^{16}+1470 a \,b^{4} c^{2} x^{16}-98 b^{6} c \,x^{16}+1960 a^{3} b \,c^{3} x^{14}-2940 a^{2} b^{3} c^{2} x^{14}+588 a \,b^{5} c \,x^{14}-14 b^{7} x^{14}-490 a^{4} c^{3} x^{12}+2940 a^{3} b^{2} c^{2} x^{12}-1470 a^{2} b^{4} c \,x^{12}+98 a \,b^{6} x^{12}-1470 a^{4} b \,c^{2} x^{10}+1960 a^{3} b^{3} c \,x^{10}-294 a^{2} b^{5} x^{10}+294 a^{5} c^{2} x^{8}-1470 a^{4} b^{2} c \,x^{8}+490 a^{3} b^{4} x^{8}+588 a^{5} b c \,x^{6}-490 a^{4} b^{3} x^{6}-98 a^{6} c \,x^{4}+294 a^{5} b^{2} x^{4}-98 a^{6} b \,x^{2}+14 a^{7}} \] Input:
int(x*(2*c*x^2+b)/(c*x^4+b*x^2-a)^8,x)
Output:
1/(14*(a**7 - 7*a**6*b*x**2 - 7*a**6*c*x**4 + 21*a**5*b**2*x**4 + 42*a**5* b*c*x**6 + 21*a**5*c**2*x**8 - 35*a**4*b**3*x**6 - 105*a**4*b**2*c*x**8 - 105*a**4*b*c**2*x**10 - 35*a**4*c**3*x**12 + 35*a**3*b**4*x**8 + 140*a**3* b**3*c*x**10 + 210*a**3*b**2*c**2*x**12 + 140*a**3*b*c**3*x**14 + 35*a**3* c**4*x**16 - 21*a**2*b**5*x**10 - 105*a**2*b**4*c*x**12 - 210*a**2*b**3*c* *2*x**14 - 210*a**2*b**2*c**3*x**16 - 105*a**2*b*c**4*x**18 - 21*a**2*c**5 *x**20 + 7*a*b**6*x**12 + 42*a*b**5*c*x**14 + 105*a*b**4*c**2*x**16 + 140* a*b**3*c**3*x**18 + 105*a*b**2*c**4*x**20 + 42*a*b*c**5*x**22 + 7*a*c**6*x **24 - b**7*x**14 - 7*b**6*c*x**16 - 21*b**5*c**2*x**18 - 35*b**4*c**3*x** 20 - 35*b**3*c**4*x**22 - 21*b**2*c**5*x**24 - 7*b*c**6*x**26 - c**7*x**28 ))