Integrand size = 28, antiderivative size = 20 \[ \int \frac {x^2 \left (b+2 c x^3\right )}{\left (-a+b x^3+c x^6\right )^8} \, dx=\frac {1}{21 \left (a-b x^3-c x^6\right )^7} \] Output:
1/21/(-c*x^6-b*x^3+a)^7
Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {x^2 \left (b+2 c x^3\right )}{\left (-a+b x^3+c x^6\right )^8} \, dx=-\frac {1}{21 \left (-a+b x^3+c x^6\right )^7} \] Input:
Integrate[(x^2*(b + 2*c*x^3))/(-a + b*x^3 + c*x^6)^8,x]
Output:
-1/21*1/(-a + b*x^3 + c*x^6)^7
Time = 0.18 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1798, 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.
\(\displaystyle \int \frac {x^2 \left (b+2 c x^3\right )}{\left (-a+b x^3+c x^6\right )^8} \, dx\) |
\(\Big \downarrow \) 1798 |
\(\displaystyle \frac {1}{3} \int \frac {2 c x^3+b}{\left (-c x^6-b x^3+a\right )^8}dx^3\) |
\(\Big \downarrow \) 1104 |
\(\displaystyle \frac {1}{21 \left (a-b x^3-c x^6\right )^7}\) |
Input:
Int[(x^2*(b + 2*c*x^3))/(-a + b*x^3 + c*x^6)^8,x]
Output:
1/(21*(a - b*x^3 - c*x^6)^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_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + ( e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Simp[1/n Subst[Int[(d + e*x)^q*(a + b *x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[Simplify[m - n + 1], 0]
Time = 0.17 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95
method | result | size |
gosper | \(\frac {1}{21 \left (-c \,x^{6}-b \,x^{3}+a \right )^{7}}\) | \(19\) |
default | \(\frac {1}{21 \left (-c \,x^{6}-b \,x^{3}+a \right )^{7}}\) | \(19\) |
risch | \(\frac {1}{21 \left (-c \,x^{6}-b \,x^{3}+a \right )^{7}}\) | \(19\) |
parallelrisch | \(-\frac {1}{21 \left (c \,x^{6}+b \,x^{3}-a \right )^{7}}\) | \(19\) |
orering | \(\frac {-c \,x^{6}-b \,x^{3}+a}{21 \left (c \,x^{6}+b \,x^{3}-a \right )^{8}}\) | \(33\) |
Input:
int(x^2*(2*c*x^3+b)/(c*x^6+b*x^3-a)^8,x,method=_RETURNVERBOSE)
Output:
1/21/(-c*x^6-b*x^3+a)^7
Leaf count of result is larger than twice the leaf count of optimal. 356 vs. \(2 (18) = 36\).
Time = 0.10 (sec) , antiderivative size = 356, normalized size of antiderivative = 17.80 \[ \int \frac {x^2 \left (b+2 c x^3\right )}{\left (-a+b x^3+c x^6\right )^8} \, dx=-\frac {1}{21 \, {\left (c^{7} x^{42} + 7 \, b c^{6} x^{39} + 7 \, {\left (3 \, b^{2} c^{5} - a c^{6}\right )} x^{36} + 7 \, {\left (5 \, b^{3} c^{4} - 6 \, a b c^{5}\right )} x^{33} + 7 \, {\left (5 \, b^{4} c^{3} - 15 \, a b^{2} c^{4} + 3 \, a^{2} c^{5}\right )} x^{30} + 7 \, {\left (3 \, b^{5} c^{2} - 20 \, a b^{3} c^{3} + 15 \, a^{2} b c^{4}\right )} x^{27} + 7 \, {\left (b^{6} c - 15 \, a b^{4} c^{2} + 30 \, a^{2} b^{2} c^{3} - 5 \, a^{3} c^{4}\right )} x^{24} + {\left (b^{7} - 42 \, a b^{5} c + 210 \, a^{2} b^{3} c^{2} - 140 \, a^{3} b c^{3}\right )} x^{21} - 7 \, {\left (a b^{6} - 15 \, a^{2} b^{4} c + 30 \, a^{3} b^{2} c^{2} - 5 \, a^{4} c^{3}\right )} x^{18} + 7 \, {\left (3 \, a^{2} b^{5} - 20 \, a^{3} b^{3} c + 15 \, a^{4} b c^{2}\right )} x^{15} - 7 \, {\left (5 \, a^{3} b^{4} - 15 \, a^{4} b^{2} c + 3 \, a^{5} c^{2}\right )} x^{12} + 7 \, a^{6} b x^{3} + 7 \, {\left (5 \, a^{4} b^{3} - 6 \, a^{5} b c\right )} x^{9} - a^{7} - 7 \, {\left (3 \, a^{5} b^{2} - a^{6} c\right )} x^{6}\right )}} \] Input:
integrate(x^2*(2*c*x^3+b)/(c*x^6+b*x^3-a)^8,x, algorithm="fricas")
Output:
-1/21/(c^7*x^42 + 7*b*c^6*x^39 + 7*(3*b^2*c^5 - a*c^6)*x^36 + 7*(5*b^3*c^4 - 6*a*b*c^5)*x^33 + 7*(5*b^4*c^3 - 15*a*b^2*c^4 + 3*a^2*c^5)*x^30 + 7*(3* b^5*c^2 - 20*a*b^3*c^3 + 15*a^2*b*c^4)*x^27 + 7*(b^6*c - 15*a*b^4*c^2 + 30 *a^2*b^2*c^3 - 5*a^3*c^4)*x^24 + (b^7 - 42*a*b^5*c + 210*a^2*b^3*c^2 - 140 *a^3*b*c^3)*x^21 - 7*(a*b^6 - 15*a^2*b^4*c + 30*a^3*b^2*c^2 - 5*a^4*c^3)*x ^18 + 7*(3*a^2*b^5 - 20*a^3*b^3*c + 15*a^4*b*c^2)*x^15 - 7*(5*a^3*b^4 - 15 *a^4*b^2*c + 3*a^5*c^2)*x^12 + 7*a^6*b*x^3 + 7*(5*a^4*b^3 - 6*a^5*b*c)*x^9 - a^7 - 7*(3*a^5*b^2 - a^6*c)*x^6)
Leaf count of result is larger than twice the leaf count of optimal. 360 vs. \(2 (15) = 30\).
Time = 6.55 (sec) , antiderivative size = 360, normalized size of antiderivative = 18.00 \[ \int \frac {x^2 \left (b+2 c x^3\right )}{\left (-a+b x^3+c x^6\right )^8} \, dx=- \frac {1}{- 21 a^{7} + 147 a^{6} b x^{3} + 147 b c^{6} x^{39} + 21 c^{7} x^{42} + x^{36} \left (- 147 a c^{6} + 441 b^{2} c^{5}\right ) + x^{33} \left (- 882 a b c^{5} + 735 b^{3} c^{4}\right ) + x^{30} \cdot \left (441 a^{2} c^{5} - 2205 a b^{2} c^{4} + 735 b^{4} c^{3}\right ) + x^{27} \cdot \left (2205 a^{2} b c^{4} - 2940 a b^{3} c^{3} + 441 b^{5} c^{2}\right ) + x^{24} \left (- 735 a^{3} c^{4} + 4410 a^{2} b^{2} c^{3} - 2205 a b^{4} c^{2} + 147 b^{6} c\right ) + x^{21} \left (- 2940 a^{3} b c^{3} + 4410 a^{2} b^{3} c^{2} - 882 a b^{5} c + 21 b^{7}\right ) + x^{18} \cdot \left (735 a^{4} c^{3} - 4410 a^{3} b^{2} c^{2} + 2205 a^{2} b^{4} c - 147 a b^{6}\right ) + x^{15} \cdot \left (2205 a^{4} b c^{2} - 2940 a^{3} b^{3} c + 441 a^{2} b^{5}\right ) + x^{12} \left (- 441 a^{5} c^{2} + 2205 a^{4} b^{2} c - 735 a^{3} b^{4}\right ) + x^{9} \left (- 882 a^{5} b c + 735 a^{4} b^{3}\right ) + x^{6} \cdot \left (147 a^{6} c - 441 a^{5} b^{2}\right )} \] Input:
integrate(x**2*(2*c*x**3+b)/(c*x**6+b*x**3-a)**8,x)
Output:
-1/(-21*a**7 + 147*a**6*b*x**3 + 147*b*c**6*x**39 + 21*c**7*x**42 + x**36* (-147*a*c**6 + 441*b**2*c**5) + x**33*(-882*a*b*c**5 + 735*b**3*c**4) + x* *30*(441*a**2*c**5 - 2205*a*b**2*c**4 + 735*b**4*c**3) + x**27*(2205*a**2* b*c**4 - 2940*a*b**3*c**3 + 441*b**5*c**2) + x**24*(-735*a**3*c**4 + 4410* a**2*b**2*c**3 - 2205*a*b**4*c**2 + 147*b**6*c) + x**21*(-2940*a**3*b*c**3 + 4410*a**2*b**3*c**2 - 882*a*b**5*c + 21*b**7) + x**18*(735*a**4*c**3 - 4410*a**3*b**2*c**2 + 2205*a**2*b**4*c - 147*a*b**6) + x**15*(2205*a**4*b* c**2 - 2940*a**3*b**3*c + 441*a**2*b**5) + x**12*(-441*a**5*c**2 + 2205*a* *4*b**2*c - 735*a**3*b**4) + x**9*(-882*a**5*b*c + 735*a**4*b**3) + x**6*( 147*a**6*c - 441*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 \[ \int \frac {x^2 \left (b+2 c x^3\right )}{\left (-a+b x^3+c x^6\right )^8} \, dx=-\frac {1}{21 \, {\left (c^{7} x^{42} + 7 \, b c^{6} x^{39} + 7 \, {\left (3 \, b^{2} c^{5} - a c^{6}\right )} x^{36} + 7 \, {\left (5 \, b^{3} c^{4} - 6 \, a b c^{5}\right )} x^{33} + 7 \, {\left (5 \, b^{4} c^{3} - 15 \, a b^{2} c^{4} + 3 \, a^{2} c^{5}\right )} x^{30} + 7 \, {\left (3 \, b^{5} c^{2} - 20 \, a b^{3} c^{3} + 15 \, a^{2} b c^{4}\right )} x^{27} + 7 \, {\left (b^{6} c - 15 \, a b^{4} c^{2} + 30 \, a^{2} b^{2} c^{3} - 5 \, a^{3} c^{4}\right )} x^{24} + {\left (b^{7} - 42 \, a b^{5} c + 210 \, a^{2} b^{3} c^{2} - 140 \, a^{3} b c^{3}\right )} x^{21} - 7 \, {\left (a b^{6} - 15 \, a^{2} b^{4} c + 30 \, a^{3} b^{2} c^{2} - 5 \, a^{4} c^{3}\right )} x^{18} + 7 \, {\left (3 \, a^{2} b^{5} - 20 \, a^{3} b^{3} c + 15 \, a^{4} b c^{2}\right )} x^{15} - 7 \, {\left (5 \, a^{3} b^{4} - 15 \, a^{4} b^{2} c + 3 \, a^{5} c^{2}\right )} x^{12} + 7 \, a^{6} b x^{3} + 7 \, {\left (5 \, a^{4} b^{3} - 6 \, a^{5} b c\right )} x^{9} - a^{7} - 7 \, {\left (3 \, a^{5} b^{2} - a^{6} c\right )} x^{6}\right )}} \] Input:
integrate(x^2*(2*c*x^3+b)/(c*x^6+b*x^3-a)^8,x, algorithm="maxima")
Output:
-1/21/(c^7*x^42 + 7*b*c^6*x^39 + 7*(3*b^2*c^5 - a*c^6)*x^36 + 7*(5*b^3*c^4 - 6*a*b*c^5)*x^33 + 7*(5*b^4*c^3 - 15*a*b^2*c^4 + 3*a^2*c^5)*x^30 + 7*(3* b^5*c^2 - 20*a*b^3*c^3 + 15*a^2*b*c^4)*x^27 + 7*(b^6*c - 15*a*b^4*c^2 + 30 *a^2*b^2*c^3 - 5*a^3*c^4)*x^24 + (b^7 - 42*a*b^5*c + 210*a^2*b^3*c^2 - 140 *a^3*b*c^3)*x^21 - 7*(a*b^6 - 15*a^2*b^4*c + 30*a^3*b^2*c^2 - 5*a^4*c^3)*x ^18 + 7*(3*a^2*b^5 - 20*a^3*b^3*c + 15*a^4*b*c^2)*x^15 - 7*(5*a^3*b^4 - 15 *a^4*b^2*c + 3*a^5*c^2)*x^12 + 7*a^6*b*x^3 + 7*(5*a^4*b^3 - 6*a^5*b*c)*x^9 - a^7 - 7*(3*a^5*b^2 - a^6*c)*x^6)
Time = 1.90 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {x^2 \left (b+2 c x^3\right )}{\left (-a+b x^3+c x^6\right )^8} \, dx=-\frac {1}{21 \, {\left (c x^{6} + b x^{3} - a\right )}^{7}} \] Input:
integrate(x^2*(2*c*x^3+b)/(c*x^6+b*x^3-a)^8,x, algorithm="giac")
Output:
-1/21/(c*x^6 + b*x^3 - a)^7
Time = 40.17 (sec) , antiderivative size = 360, normalized size of antiderivative = 18.00 \[ \int \frac {x^2 \left (b+2 c x^3\right )}{\left (-a+b x^3+c x^6\right )^8} \, dx=-\frac {1}{21\,\left (x^{15}\,\left (105\,a^4\,b\,c^2-140\,a^3\,b^3\,c+21\,a^2\,b^5\right )+x^{27}\,\left (105\,a^2\,b\,c^4-140\,a\,b^3\,c^3+21\,b^5\,c^2\right )+x^{21}\,\left (-140\,a^3\,b\,c^3+210\,a^2\,b^3\,c^2-42\,a\,b^5\,c+b^7\right )+x^9\,\left (35\,a^4\,b^3-42\,a^5\,b\,c\right )+x^{33}\,\left (35\,b^3\,c^4-42\,a\,b\,c^5\right )-x^{12}\,\left (21\,a^5\,c^2-105\,a^4\,b^2\,c+35\,a^3\,b^4\right )+x^{30}\,\left (21\,a^2\,c^5-105\,a\,b^2\,c^4+35\,b^4\,c^3\right )-a^7-x^{18}\,\left (-35\,a^4\,c^3+210\,a^3\,b^2\,c^2-105\,a^2\,b^4\,c+7\,a\,b^6\right )+x^{24}\,\left (-35\,a^3\,c^4+210\,a^2\,b^2\,c^3-105\,a\,b^4\,c^2+7\,b^6\,c\right )+c^7\,x^{42}+x^6\,\left (7\,a^6\,c-21\,a^5\,b^2\right )-x^{36}\,\left (7\,a\,c^6-21\,b^2\,c^5\right )+7\,a^6\,b\,x^3+7\,b\,c^6\,x^{39}\right )} \] Input:
int((x^2*(b + 2*c*x^3))/(b*x^3 - a + c*x^6)^8,x)
Output:
-1/(21*(x^15*(21*a^2*b^5 - 140*a^3*b^3*c + 105*a^4*b*c^2) + x^27*(21*b^5*c ^2 - 140*a*b^3*c^3 + 105*a^2*b*c^4) + x^21*(b^7 - 140*a^3*b*c^3 + 210*a^2* b^3*c^2 - 42*a*b^5*c) + x^9*(35*a^4*b^3 - 42*a^5*b*c) + x^33*(35*b^3*c^4 - 42*a*b*c^5) - x^12*(35*a^3*b^4 + 21*a^5*c^2 - 105*a^4*b^2*c) + x^30*(21*a ^2*c^5 + 35*b^4*c^3 - 105*a*b^2*c^4) - a^7 - x^18*(7*a*b^6 - 35*a^4*c^3 - 105*a^2*b^4*c + 210*a^3*b^2*c^2) + x^24*(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^42 + x^6*(7*a^6*c - 21*a^5*b^2) - x^36*(7*a *c^6 - 21*b^2*c^5) + 7*a^6*b*x^3 + 7*b*c^6*x^39))
Time = 0.18 (sec) , antiderivative size = 390, normalized size of antiderivative = 19.50 \[ \int \frac {x^2 \left (b+2 c x^3\right )}{\left (-a+b x^3+c x^6\right )^8} \, dx=\frac {1}{-21 c^{7} x^{42}-147 b \,c^{6} x^{39}+147 a \,c^{6} x^{36}-441 b^{2} c^{5} x^{36}+882 a b \,c^{5} x^{33}-735 b^{3} c^{4} x^{33}-441 a^{2} c^{5} x^{30}+2205 a \,b^{2} c^{4} x^{30}-735 b^{4} c^{3} x^{30}-2205 a^{2} b \,c^{4} x^{27}+2940 a \,b^{3} c^{3} x^{27}-441 b^{5} c^{2} x^{27}+735 a^{3} c^{4} x^{24}-4410 a^{2} b^{2} c^{3} x^{24}+2205 a \,b^{4} c^{2} x^{24}-147 b^{6} c \,x^{24}+2940 a^{3} b \,c^{3} x^{21}-4410 a^{2} b^{3} c^{2} x^{21}+882 a \,b^{5} c \,x^{21}-21 b^{7} x^{21}-735 a^{4} c^{3} x^{18}+4410 a^{3} b^{2} c^{2} x^{18}-2205 a^{2} b^{4} c \,x^{18}+147 a \,b^{6} x^{18}-2205 a^{4} b \,c^{2} x^{15}+2940 a^{3} b^{3} c \,x^{15}-441 a^{2} b^{5} x^{15}+441 a^{5} c^{2} x^{12}-2205 a^{4} b^{2} c \,x^{12}+735 a^{3} b^{4} x^{12}+882 a^{5} b c \,x^{9}-735 a^{4} b^{3} x^{9}-147 a^{6} c \,x^{6}+441 a^{5} b^{2} x^{6}-147 a^{6} b \,x^{3}+21 a^{7}} \] Input:
int(x^2*(2*c*x^3+b)/(c*x^6+b*x^3-a)^8,x)
Output:
1/(21*(a**7 - 7*a**6*b*x**3 - 7*a**6*c*x**6 + 21*a**5*b**2*x**6 + 42*a**5* b*c*x**9 + 21*a**5*c**2*x**12 - 35*a**4*b**3*x**9 - 105*a**4*b**2*c*x**12 - 105*a**4*b*c**2*x**15 - 35*a**4*c**3*x**18 + 35*a**3*b**4*x**12 + 140*a* *3*b**3*c*x**15 + 210*a**3*b**2*c**2*x**18 + 140*a**3*b*c**3*x**21 + 35*a* *3*c**4*x**24 - 21*a**2*b**5*x**15 - 105*a**2*b**4*c*x**18 - 210*a**2*b**3 *c**2*x**21 - 210*a**2*b**2*c**3*x**24 - 105*a**2*b*c**4*x**27 - 21*a**2*c **5*x**30 + 7*a*b**6*x**18 + 42*a*b**5*c*x**21 + 105*a*b**4*c**2*x**24 + 1 40*a*b**3*c**3*x**27 + 105*a*b**2*c**4*x**30 + 42*a*b*c**5*x**33 + 7*a*c** 6*x**36 - b**7*x**21 - 7*b**6*c*x**24 - 21*b**5*c**2*x**27 - 35*b**4*c**3* x**30 - 35*b**3*c**4*x**33 - 21*b**2*c**5*x**36 - 7*b*c**6*x**39 - c**7*x* *42))