Integrand size = 94, antiderivative size = 40 \[ \int \left (a+b x^2+c x^4\right )^p \left (3 a^2 d+3 a^2 e x^2+\left (a b e (5+2 p)+3 a c d (5+4 p)-b^2 d \left (15+16 p+4 p^2\right )\right ) x^4+c (7+4 p) (a e-b d (3+2 p)) x^6\right ) \, dx=x \left (3 a d+(a e-b d (3+2 p)) x^2\right ) \left (a+b x^2+c x^4\right )^{1+p} \] Output:
x*(3*a*d+(a*e-b*d*(3+2*p))*x^2)*(c*x^4+b*x^2+a)^(p+1)
Time = 1.05 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.05 \[ \int \left (a+b x^2+c x^4\right )^p \left (3 a^2 d+3 a^2 e x^2+\left (a b e (5+2 p)+3 a c d (5+4 p)-b^2 d \left (15+16 p+4 p^2\right )\right ) x^4+c (7+4 p) (a e-b d (3+2 p)) x^6\right ) \, dx=x \left (a+b x^2+c x^4\right )^{1+p} \left (-b d (3+2 p) x^2+a \left (3 d+e x^2\right )\right ) \] Input:
Integrate[(a + b*x^2 + c*x^4)^p*(3*a^2*d + 3*a^2*e*x^2 + (a*b*e*(5 + 2*p) + 3*a*c*d*(5 + 4*p) - b^2*d*(15 + 16*p + 4*p^2))*x^4 + c*(7 + 4*p)*(a*e - b*d*(3 + 2*p))*x^6),x]
Output:
x*(a + b*x^2 + c*x^4)^(1 + p)*(-(b*d*(3 + 2*p)*x^2) + a*(3*d + e*x^2))
Time = 0.30 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.011, Rules used = {2204}
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.
\(\displaystyle \int \left (a+b x^2+c x^4\right )^p \left (3 a^2 d+3 a^2 e x^2+x^4 \left (a b e (2 p+5)+3 a c d (4 p+5)+b^2 (-d) \left (4 p^2+16 p+15\right )\right )+c (4 p+7) x^6 (a e-b d (2 p+3))\right ) \, dx\) |
\(\Big \downarrow \) 2204 |
\(\displaystyle x \left (a+b x^2+c x^4\right )^{p+1} \left (x^2 (a e-b d (2 p+3))+3 a d\right )\) |
Input:
Int[(a + b*x^2 + c*x^4)^p*(3*a^2*d + 3*a^2*e*x^2 + (a*b*e*(5 + 2*p) + 3*a* c*d*(5 + 4*p) - b^2*d*(15 + 16*p + 4*p^2))*x^4 + c*(7 + 4*p)*(a*e - b*d*(3 + 2*p))*x^6),x]
Output:
x*(3*a*d + (a*e - b*d*(3 + 2*p))*x^2)*(a + b*x^2 + c*x^4)^(1 + p)
Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{d = Coeff[Px, x, 0], e = Coeff[Px, x, 2], f = Coeff[Px, x, 4], g = Coeff[Px, x, 6]}, Simp[x*(3*a*d + (a*e - b*d*(2*p + 3))*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(3*a^2)), x] /; EqQ[3*a^2*g - c*(4*p + 7)*(a*e - b*d*(2*p + 3)), 0] && EqQ[3*a^2*f - 3*a*c*d*(4*p + 5) - b*(2*p + 5)*(a*e - b*d*(2*p + 3)), 0]] / ; FreeQ[{a, b, c, p}, x] && PolyQ[Px, x^2] && EqQ[Expon[Px, x], 6]
Time = 0.34 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.12
method | result | size |
gosper | \(x \left (c \,x^{4}+b \,x^{2}+a \right )^{p +1} \left (-2 b d p \,x^{2}+a e \,x^{2}-3 b d \,x^{2}+3 a d \right )\) | \(45\) |
risch | \(x \left (-2 b c d p \,x^{6}+a c e \,x^{6}-3 b c d \,x^{6}-2 b^{2} d p \,x^{4}+a b e \,x^{4}+3 a c d \,x^{4}-3 b^{2} d \,x^{4}-2 a b d p \,x^{2}+a^{2} e \,x^{2}+3 a^{2} d \right ) \left (c \,x^{4}+b \,x^{2}+a \right )^{p}\) | \(99\) |
norman | \(\left (-2 b^{2} d p +a b e +3 d a c -3 b^{2} d \right ) x^{5} {\mathrm e}^{p \ln \left (c \,x^{4}+b \,x^{2}+a \right )}+a \left (-2 b d p +a e \right ) x^{3} {\mathrm e}^{p \ln \left (c \,x^{4}+b \,x^{2}+a \right )}+c \left (-2 b d p +a e -3 b d \right ) x^{7} {\mathrm e}^{p \ln \left (c \,x^{4}+b \,x^{2}+a \right )}+3 a^{2} d x \,{\mathrm e}^{p \ln \left (c \,x^{4}+b \,x^{2}+a \right )}\) | \(132\) |
parallelrisch | \(\frac {-2 x^{7} \left (c \,x^{4}+b \,x^{2}+a \right )^{p} b \,c^{2} d p +x^{7} \left (c \,x^{4}+b \,x^{2}+a \right )^{p} a \,c^{2} e -3 x^{7} \left (c \,x^{4}+b \,x^{2}+a \right )^{p} b \,c^{2} d -2 x^{5} \left (c \,x^{4}+b \,x^{2}+a \right )^{p} b^{2} c d p +x^{5} \left (c \,x^{4}+b \,x^{2}+a \right )^{p} a b c e +3 x^{5} \left (c \,x^{4}+b \,x^{2}+a \right )^{p} a \,c^{2} d -3 x^{5} \left (c \,x^{4}+b \,x^{2}+a \right )^{p} b^{2} c d -2 x^{3} \left (c \,x^{4}+b \,x^{2}+a \right )^{p} a b c d p +x^{3} \left (c \,x^{4}+b \,x^{2}+a \right )^{p} a^{2} c e +3 x \left (c \,x^{4}+b \,x^{2}+a \right )^{p} a^{2} c d}{c}\) | \(242\) |
orering | \(\frac {\left (-2 b d p \,x^{2}+a e \,x^{2}-3 b d \,x^{2}+3 a d \right ) x \left (c \,x^{4}+b \,x^{2}+a \right ) \left (c \,x^{4}+b \,x^{2}+a \right )^{p} \left (3 a^{2} d +3 a^{2} e \,x^{2}+\left (a b e \left (5+2 p \right )+3 a c d \left (5+4 p \right )-b^{2} d \left (4 p^{2}+16 p +15\right )\right ) x^{4}+c \left (7+4 p \right ) \left (a e -b d \left (3+2 p \right )\right ) x^{6}\right )}{-8 b c d \,p^{2} x^{6}+4 a c e p \,x^{6}-26 b c d p \,x^{6}+7 a c e \,x^{6}-4 b^{2} d \,p^{2} x^{4}-21 b c d \,x^{6}+2 a b e p \,x^{4}+12 a c d p \,x^{4}-16 b^{2} d p \,x^{4}+5 a b e \,x^{4}+15 a c d \,x^{4}-15 b^{2} d \,x^{4}+3 a^{2} e \,x^{2}+3 a^{2} d}\) | \(262\) |
Input:
int((c*x^4+b*x^2+a)^p*(3*a^2*d+3*a^2*e*x^2+(a*b*e*(5+2*p)+3*a*c*d*(5+4*p)- b^2*d*(4*p^2+16*p+15))*x^4+c*(7+4*p)*(a*e-b*d*(3+2*p))*x^6),x,method=_RETU RNVERBOSE)
Output:
x*(c*x^4+b*x^2+a)^(p+1)*(-2*b*d*p*x^2+a*e*x^2-3*b*d*x^2+3*a*d)
Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (41) = 82\).
Time = 0.08 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.25 \[ \int \left (a+b x^2+c x^4\right )^p \left (3 a^2 d+3 a^2 e x^2+\left (a b e (5+2 p)+3 a c d (5+4 p)-b^2 d \left (15+16 p+4 p^2\right )\right ) x^4+c (7+4 p) (a e-b d (3+2 p)) x^6\right ) \, dx=-{\left ({\left (2 \, b c d p + 3 \, b c d - a c e\right )} x^{7} + {\left (2 \, b^{2} d p - a b e + 3 \, {\left (b^{2} - a c\right )} d\right )} x^{5} - 3 \, a^{2} d x + {\left (2 \, a b d p - a^{2} e\right )} x^{3}\right )} {\left (c x^{4} + b x^{2} + a\right )}^{p} \] Input:
integrate((c*x^4+b*x^2+a)^p*(3*a^2*d+3*a^2*e*x^2+(a*b*e*(5+2*p)+3*a*c*d*(5 +4*p)-b^2*d*(4*p^2+16*p+15))*x^4+c*(7+4*p)*(a*e-b*d*(3+2*p))*x^6),x, algor ithm="fricas")
Output:
-((2*b*c*d*p + 3*b*c*d - a*c*e)*x^7 + (2*b^2*d*p - a*b*e + 3*(b^2 - a*c)*d )*x^5 - 3*a^2*d*x + (2*a*b*d*p - a^2*e)*x^3)*(c*x^4 + b*x^2 + a)^p
Leaf count of result is larger than twice the leaf count of optimal. 235 vs. \(2 (36) = 72\).
Time = 169.22 (sec) , antiderivative size = 235, normalized size of antiderivative = 5.88 \[ \int \left (a+b x^2+c x^4\right )^p \left (3 a^2 d+3 a^2 e x^2+\left (a b e (5+2 p)+3 a c d (5+4 p)-b^2 d \left (15+16 p+4 p^2\right )\right ) x^4+c (7+4 p) (a e-b d (3+2 p)) x^6\right ) \, dx=3 a^{2} d x \left (a + b x^{2} + c x^{4}\right )^{p} + a^{2} e x^{3} \left (a + b x^{2} + c x^{4}\right )^{p} - 2 a b d p x^{3} \left (a + b x^{2} + c x^{4}\right )^{p} + a b e x^{5} \left (a + b x^{2} + c x^{4}\right )^{p} + 3 a c d x^{5} \left (a + b x^{2} + c x^{4}\right )^{p} + a c e x^{7} \left (a + b x^{2} + c x^{4}\right )^{p} - 2 b^{2} d p x^{5} \left (a + b x^{2} + c x^{4}\right )^{p} - 3 b^{2} d x^{5} \left (a + b x^{2} + c x^{4}\right )^{p} - 2 b c d p x^{7} \left (a + b x^{2} + c x^{4}\right )^{p} - 3 b c d x^{7} \left (a + b x^{2} + c x^{4}\right )^{p} \] Input:
integrate((c*x**4+b*x**2+a)**p*(3*a**2*d+3*a**2*e*x**2+(a*b*e*(5+2*p)+3*a* c*d*(5+4*p)-b**2*d*(4*p**2+16*p+15))*x**4+c*(7+4*p)*(a*e-b*d*(3+2*p))*x**6 ),x)
Output:
3*a**2*d*x*(a + b*x**2 + c*x**4)**p + a**2*e*x**3*(a + b*x**2 + c*x**4)**p - 2*a*b*d*p*x**3*(a + b*x**2 + c*x**4)**p + a*b*e*x**5*(a + b*x**2 + c*x* *4)**p + 3*a*c*d*x**5*(a + b*x**2 + c*x**4)**p + a*c*e*x**7*(a + b*x**2 + c*x**4)**p - 2*b**2*d*p*x**5*(a + b*x**2 + c*x**4)**p - 3*b**2*d*x**5*(a + b*x**2 + c*x**4)**p - 2*b*c*d*p*x**7*(a + b*x**2 + c*x**4)**p - 3*b*c*d*x **7*(a + b*x**2 + c*x**4)**p
Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (41) = 82\).
Time = 0.07 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.15 \[ \int \left (a+b x^2+c x^4\right )^p \left (3 a^2 d+3 a^2 e x^2+\left (a b e (5+2 p)+3 a c d (5+4 p)-b^2 d \left (15+16 p+4 p^2\right )\right ) x^4+c (7+4 p) (a e-b d (3+2 p)) x^6\right ) \, dx=-{\left ({\left (b c d {\left (2 \, p + 3\right )} - a c e\right )} x^{7} + {\left (b^{2} d {\left (2 \, p + 3\right )} - {\left (3 \, c d + b e\right )} a\right )} x^{5} - 3 \, a^{2} d x + {\left (2 \, a b d p - a^{2} e\right )} x^{3}\right )} {\left (c x^{4} + b x^{2} + a\right )}^{p} \] Input:
integrate((c*x^4+b*x^2+a)^p*(3*a^2*d+3*a^2*e*x^2+(a*b*e*(5+2*p)+3*a*c*d*(5 +4*p)-b^2*d*(4*p^2+16*p+15))*x^4+c*(7+4*p)*(a*e-b*d*(3+2*p))*x^6),x, algor ithm="maxima")
Output:
-((b*c*d*(2*p + 3) - a*c*e)*x^7 + (b^2*d*(2*p + 3) - (3*c*d + b*e)*a)*x^5 - 3*a^2*d*x + (2*a*b*d*p - a^2*e)*x^3)*(c*x^4 + b*x^2 + a)^p
Leaf count of result is larger than twice the leaf count of optimal. 223 vs. \(2 (41) = 82\).
Time = 0.21 (sec) , antiderivative size = 223, normalized size of antiderivative = 5.58 \[ \int \left (a+b x^2+c x^4\right )^p \left (3 a^2 d+3 a^2 e x^2+\left (a b e (5+2 p)+3 a c d (5+4 p)-b^2 d \left (15+16 p+4 p^2\right )\right ) x^4+c (7+4 p) (a e-b d (3+2 p)) x^6\right ) \, dx=-2 \, {\left (c x^{4} + b x^{2} + a\right )}^{p} b c d p x^{7} - 3 \, {\left (c x^{4} + b x^{2} + a\right )}^{p} b c d x^{7} + {\left (c x^{4} + b x^{2} + a\right )}^{p} a c e x^{7} - 2 \, {\left (c x^{4} + b x^{2} + a\right )}^{p} b^{2} d p x^{5} - 3 \, {\left (c x^{4} + b x^{2} + a\right )}^{p} b^{2} d x^{5} + 3 \, {\left (c x^{4} + b x^{2} + a\right )}^{p} a c d x^{5} + {\left (c x^{4} + b x^{2} + a\right )}^{p} a b e x^{5} - 2 \, {\left (c x^{4} + b x^{2} + a\right )}^{p} a b d p x^{3} + {\left (c x^{4} + b x^{2} + a\right )}^{p} a^{2} e x^{3} + 3 \, {\left (c x^{4} + b x^{2} + a\right )}^{p} a^{2} d x \] Input:
integrate((c*x^4+b*x^2+a)^p*(3*a^2*d+3*a^2*e*x^2+(a*b*e*(5+2*p)+3*a*c*d*(5 +4*p)-b^2*d*(4*p^2+16*p+15))*x^4+c*(7+4*p)*(a*e-b*d*(3+2*p))*x^6),x, algor ithm="giac")
Output:
-2*(c*x^4 + b*x^2 + a)^p*b*c*d*p*x^7 - 3*(c*x^4 + b*x^2 + a)^p*b*c*d*x^7 + (c*x^4 + b*x^2 + a)^p*a*c*e*x^7 - 2*(c*x^4 + b*x^2 + a)^p*b^2*d*p*x^5 - 3 *(c*x^4 + b*x^2 + a)^p*b^2*d*x^5 + 3*(c*x^4 + b*x^2 + a)^p*a*c*d*x^5 + (c* x^4 + b*x^2 + a)^p*a*b*e*x^5 - 2*(c*x^4 + b*x^2 + a)^p*a*b*d*p*x^3 + (c*x^ 4 + b*x^2 + a)^p*a^2*e*x^3 + 3*(c*x^4 + b*x^2 + a)^p*a^2*d*x
Time = 18.40 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.20 \[ \int \left (a+b x^2+c x^4\right )^p \left (3 a^2 d+3 a^2 e x^2+\left (a b e (5+2 p)+3 a c d (5+4 p)-b^2 d \left (15+16 p+4 p^2\right )\right ) x^4+c (7+4 p) (a e-b d (3+2 p)) x^6\right ) \, dx={\left (c\,x^4+b\,x^2+a\right )}^p\,\left (x^3\,\left (a^2\,e-2\,a\,b\,d\,p\right )-x^5\,\left (3\,b^2\,d-a\,b\,e-3\,a\,c\,d+2\,b^2\,d\,p\right )-c\,x^7\,\left (3\,b\,d-a\,e+2\,b\,d\,p\right )+3\,a^2\,d\,x\right ) \] Input:
int((a + b*x^2 + c*x^4)^p*(3*a^2*d + x^4*(a*b*e*(2*p + 5) - b^2*d*(16*p + 4*p^2 + 15) + 3*a*c*d*(4*p + 5)) + 3*a^2*e*x^2 + c*x^6*(4*p + 7)*(a*e - b* d*(2*p + 3))),x)
Output:
(a + b*x^2 + c*x^4)^p*(x^3*(a^2*e - 2*a*b*d*p) - x^5*(3*b^2*d - a*b*e - 3* a*c*d + 2*b^2*d*p) - c*x^7*(3*b*d - a*e + 2*b*d*p) + 3*a^2*d*x)
Time = 0.18 (sec) , antiderivative size = 98, normalized size of antiderivative = 2.45 \[ \int \left (a+b x^2+c x^4\right )^p \left (3 a^2 d+3 a^2 e x^2+\left (a b e (5+2 p)+3 a c d (5+4 p)-b^2 d \left (15+16 p+4 p^2\right )\right ) x^4+c (7+4 p) (a e-b d (3+2 p)) x^6\right ) \, dx=\left (c \,x^{4}+b \,x^{2}+a \right )^{p} x \left (-2 b c d p \,x^{6}+a c e \,x^{6}-3 b c d \,x^{6}-2 b^{2} d p \,x^{4}+a b e \,x^{4}+3 a c d \,x^{4}-3 b^{2} d \,x^{4}-2 a b d p \,x^{2}+a^{2} e \,x^{2}+3 a^{2} d \right ) \] Input:
int((c*x^4+b*x^2+a)^p*(3*a^2*d+3*a^2*e*x^2+(a*b*e*(5+2*p)+3*a*c*d*(5+4*p)- b^2*d*(4*p^2+16*p+15))*x^4+c*(7+4*p)*(a*e-b*d*(3+2*p))*x^6),x)
Output:
(a + b*x**2 + c*x**4)**p*x*(3*a**2*d + a**2*e*x**2 - 2*a*b*d*p*x**2 + a*b* e*x**4 + 3*a*c*d*x**4 + a*c*e*x**6 - 2*b**2*d*p*x**4 - 3*b**2*d*x**4 - 2*b *c*d*p*x**6 - 3*b*c*d*x**6)