Integrand size = 18, antiderivative size = 223 \[ \int x^5 \left (a+b x^2+c x^4\right )^p \, dx=-\frac {b (2+p) \left (a+b x^2+c x^4\right )^{1+p}}{4 c^2 (1+p) (3+2 p)}+\frac {x^2 \left (a+b x^2+c x^4\right )^{1+p}}{2 c (3+2 p)}+\frac {2^{-1+p} \left (2 a c-b^2 (2+p)\right ) \left (-\frac {b-\sqrt {b^2-4 a c}+2 c x^2}{\sqrt {b^2-4 a c}}\right )^{-1-p} \left (a+b x^2+c x^4\right )^{1+p} \operatorname {Hypergeometric2F1}\left (-p,1+p,2+p,\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{2 \sqrt {b^2-4 a c}}\right )}{c^2 \sqrt {b^2-4 a c} (1+p) (3+2 p)} \] Output:
-1/4*b*(2+p)*(c*x^4+b*x^2+a)^(p+1)/c^2/(p+1)/(3+2*p)+1/2*x^2*(c*x^4+b*x^2+ a)^(p+1)/c/(3+2*p)+2^(-1+p)*(2*a*c-b^2*(2+p))*(-(b-(-4*a*c+b^2)^(1/2)+2*c* x^2)/(-4*a*c+b^2)^(1/2))^(-1-p)*(c*x^4+b*x^2+a)^(p+1)*hypergeom([-p, p+1], [2+p],1/2*(b+(-4*a*c+b^2)^(1/2)+2*c*x^2)/(-4*a*c+b^2)^(1/2))/c^2/(-4*a*c+b ^2)^(1/2)/(p+1)/(3+2*p)
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 0.43 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.73 \[ \int x^5 \left (a+b x^2+c x^4\right )^p \, dx=\frac {1}{6} x^6 \left (\frac {b-\sqrt {b^2-4 a c}+2 c x^2}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^2+c x^4\right )^p \operatorname {AppellF1}\left (3,-p,-p,4,-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},\frac {2 c x^2}{-b+\sqrt {b^2-4 a c}}\right ) \] Input:
Integrate[x^5*(a + b*x^2 + c*x^4)^p,x]
Output:
(x^6*(a + b*x^2 + c*x^4)^p*AppellF1[3, -p, -p, 4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])/(6*((b - Sqrt[b^2 - 4*a*c ] + 2*c*x^2)/(b - Sqrt[b^2 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^2) /(b + Sqrt[b^2 - 4*a*c]))^p)
Time = 0.56 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {1434, 1166, 25, 1160, 1096}
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.
\(\displaystyle \int x^5 \left (a+b x^2+c x^4\right )^p \, dx\) |
\(\Big \downarrow \) 1434 |
\(\displaystyle \frac {1}{2} \int x^4 \left (c x^4+b x^2+a\right )^pdx^2\) |
\(\Big \downarrow \) 1166 |
\(\displaystyle \frac {1}{2} \left (\frac {\int -\left (\left (b (p+2) x^2+a\right ) \left (c x^4+b x^2+a\right )^p\right )dx^2}{c (2 p+3)}+\frac {x^2 \left (a+b x^2+c x^4\right )^{p+1}}{c (2 p+3)}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{2} \left (\frac {x^2 \left (a+b x^2+c x^4\right )^{p+1}}{c (2 p+3)}-\frac {\int \left (b (p+2) x^2+a\right ) \left (c x^4+b x^2+a\right )^pdx^2}{c (2 p+3)}\right )\) |
\(\Big \downarrow \) 1160 |
\(\displaystyle \frac {1}{2} \left (\frac {x^2 \left (a+b x^2+c x^4\right )^{p+1}}{c (2 p+3)}-\frac {\frac {\left (2 a c-b^2 (p+2)\right ) \int \left (c x^4+b x^2+a\right )^pdx^2}{2 c}+\frac {b (p+2) \left (a+b x^2+c x^4\right )^{p+1}}{2 c (p+1)}}{c (2 p+3)}\right )\) |
\(\Big \downarrow \) 1096 |
\(\displaystyle \frac {1}{2} \left (\frac {x^2 \left (a+b x^2+c x^4\right )^{p+1}}{c (2 p+3)}-\frac {\frac {b (p+2) \left (a+b x^2+c x^4\right )^{p+1}}{2 c (p+1)}-\frac {2^p \left (2 a c-b^2 (p+2)\right ) \left (-\frac {-\sqrt {b^2-4 a c}+b+2 c x^2}{\sqrt {b^2-4 a c}}\right )^{-p-1} \left (a+b x^2+c x^4\right )^{p+1} \operatorname {Hypergeometric2F1}\left (-p,p+1,p+2,\frac {2 c x^2+b+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right )}{c (p+1) \sqrt {b^2-4 a c}}}{c (2 p+3)}\right )\) |
Input:
Int[x^5*(a + b*x^2 + c*x^4)^p,x]
Output:
((x^2*(a + b*x^2 + c*x^4)^(1 + p))/(c*(3 + 2*p)) - ((b*(2 + p)*(a + b*x^2 + c*x^4)^(1 + p))/(2*c*(1 + p)) - (2^p*(2*a*c - b^2*(2 + p))*(-((b - Sqrt[ b^2 - 4*a*c] + 2*c*x^2)/Sqrt[b^2 - 4*a*c]))^(-1 - p)*(a + b*x^2 + c*x^4)^( 1 + p)*Hypergeometric2F1[-p, 1 + p, 2 + p, (b + Sqrt[b^2 - 4*a*c] + 2*c*x^ 2)/(2*Sqrt[b^2 - 4*a*c])])/(c*Sqrt[b^2 - 4*a*c]*(1 + p)))/(c*(3 + 2*p)))/2
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(-(a + b*x + c*x^2)^(p + 1)/(q*(p + 1)*((q - b - 2*c*x) /(2*q))^(p + 1)))*Hypergeometric2F1[-p, p + 1, p + 2, (b + q + 2*c*x)/(2*q) ], x]] /; FreeQ[{a, b, c, p}, x] && !IntegerQ[4*p] && !IntegerQ[3*p]
Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b *e)/(2*c) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Simp[1/(c*(m + 2*p + 1)) Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]* (a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && If[Ration alQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadrat icQ[a, b, c, d, e, m, p, x]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp [1/2 Subst[Int[x^((m - 1)/2)*(a + b*x + c*x^2)^p, x], x, x^2], x] /; Free Q[{a, b, c, p}, x] && IntegerQ[(m - 1)/2]
\[\int x^{5} \left (c \,x^{4}+b \,x^{2}+a \right )^{p}d x\]
Input:
int(x^5*(c*x^4+b*x^2+a)^p,x)
Output:
int(x^5*(c*x^4+b*x^2+a)^p,x)
\[ \int x^5 \left (a+b x^2+c x^4\right )^p \, dx=\int { {\left (c x^{4} + b x^{2} + a\right )}^{p} x^{5} \,d x } \] Input:
integrate(x^5*(c*x^4+b*x^2+a)^p,x, algorithm="fricas")
Output:
integral((c*x^4 + b*x^2 + a)^p*x^5, x)
\[ \int x^5 \left (a+b x^2+c x^4\right )^p \, dx=\int x^{5} \left (a + b x^{2} + c x^{4}\right )^{p}\, dx \] Input:
integrate(x**5*(c*x**4+b*x**2+a)**p,x)
Output:
Integral(x**5*(a + b*x**2 + c*x**4)**p, x)
\[ \int x^5 \left (a+b x^2+c x^4\right )^p \, dx=\int { {\left (c x^{4} + b x^{2} + a\right )}^{p} x^{5} \,d x } \] Input:
integrate(x^5*(c*x^4+b*x^2+a)^p,x, algorithm="maxima")
Output:
integrate((c*x^4 + b*x^2 + a)^p*x^5, x)
\[ \int x^5 \left (a+b x^2+c x^4\right )^p \, dx=\int { {\left (c x^{4} + b x^{2} + a\right )}^{p} x^{5} \,d x } \] Input:
integrate(x^5*(c*x^4+b*x^2+a)^p,x, algorithm="giac")
Output:
integrate((c*x^4 + b*x^2 + a)^p*x^5, x)
Timed out. \[ \int x^5 \left (a+b x^2+c x^4\right )^p \, dx=\int x^5\,{\left (c\,x^4+b\,x^2+a\right )}^p \,d x \] Input:
int(x^5*(a + b*x^2 + c*x^4)^p,x)
Output:
int(x^5*(a + b*x^2 + c*x^4)^p, x)
\[ \int x^5 \left (a+b x^2+c x^4\right )^p \, dx =\text {Too large to display} \] Input:
int(x^5*(c*x^4+b*x^2+a)^p,x)
Output:
( - 4*(a + b*x**2 + c*x**4)**p*a**2*c*p - 4*(a + b*x**2 + c*x**4)**p*a**2* c + (a + b*x**2 + c*x**4)**p*a*b**2*p + 2*(a + b*x**2 + c*x**4)**p*a*b**2 + 4*(a + b*x**2 + c*x**4)**p*a*b*c*p**2*x**2 + 4*(a + b*x**2 + c*x**4)**p* a*b*c*p*x**2 - (a + b*x**2 + c*x**4)**p*b**3*p**2*x**2 - 2*(a + b*x**2 + c *x**4)**p*b**3*p*x**2 + 2*(a + b*x**2 + c*x**4)**p*b**2*c*p**2*x**4 + (a + b*x**2 + c*x**4)**p*b**2*c*p*x**4 + 4*(a + b*x**2 + c*x**4)**p*b*c**2*p** 2*x**6 + 6*(a + b*x**2 + c*x**4)**p*b*c**2*p*x**6 + 2*(a + b*x**2 + c*x**4 )**p*b*c**2*x**6 + 64*int(((a + b*x**2 + c*x**4)**p*x**3)/(4*a*p**2 + 8*a* p + 3*a + 4*b*p**2*x**2 + 8*b*p*x**2 + 3*b*x**2 + 4*c*p**2*x**4 + 8*c*p*x* *4 + 3*c*x**4),x)*a**2*c**2*p**4 + 192*int(((a + b*x**2 + c*x**4)**p*x**3) /(4*a*p**2 + 8*a*p + 3*a + 4*b*p**2*x**2 + 8*b*p*x**2 + 3*b*x**2 + 4*c*p** 2*x**4 + 8*c*p*x**4 + 3*c*x**4),x)*a**2*c**2*p**3 + 176*int(((a + b*x**2 + c*x**4)**p*x**3)/(4*a*p**2 + 8*a*p + 3*a + 4*b*p**2*x**2 + 8*b*p*x**2 + 3 *b*x**2 + 4*c*p**2*x**4 + 8*c*p*x**4 + 3*c*x**4),x)*a**2*c**2*p**2 + 48*in t(((a + b*x**2 + c*x**4)**p*x**3)/(4*a*p**2 + 8*a*p + 3*a + 4*b*p**2*x**2 + 8*b*p*x**2 + 3*b*x**2 + 4*c*p**2*x**4 + 8*c*p*x**4 + 3*c*x**4),x)*a**2*c **2*p - 32*int(((a + b*x**2 + c*x**4)**p*x**3)/(4*a*p**2 + 8*a*p + 3*a + 4 *b*p**2*x**2 + 8*b*p*x**2 + 3*b*x**2 + 4*c*p**2*x**4 + 8*c*p*x**4 + 3*c*x* *4),x)*a*b**2*c*p**5 - 176*int(((a + b*x**2 + c*x**4)**p*x**3)/(4*a*p**2 + 8*a*p + 3*a + 4*b*p**2*x**2 + 8*b*p*x**2 + 3*b*x**2 + 4*c*p**2*x**4 + ...