\(\int x^7 (a+b x^2+c x^4)^p \, dx\) [1092]

Optimal result

Integrand size = 18, antiderivative size = 257 \[ \int x^7 \left (a+b x^2+c x^4\right )^p \, dx=\frac {x^4 \left (a+b x^2+c x^4\right )^{1+p}}{4 c (2+p)}+\frac {\left (b^2 (2+p) (3+p)-2 a c (3+2 p)-2 b c (1+p) (3+p) x^2\right ) \left (a+b x^2+c x^4\right )^{1+p}}{8 c^3 (1+p) (2+p) (3+2 p)}-\frac {2^{-2+p} b \left (6 a c-b^2 (3+p)\right ) \left (-\frac {b-\sqrt {b^2-4 a c}+2 c x^2}{\sqrt {b^2-4 a c}}\right )^{-1-p} \left (a+b x^2+c x^4\right )^{1+p} \operatorname {Hypergeometric2F1}\left (-p,1+p,2+p,\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{2 \sqrt {b^2-4 a c}}\right )}{c^3 \sqrt {b^2-4 a c} (1+p) (3+2 p)} \] Output:

1/4*x^4*(c*x^4+b*x^2+a)^(p+1)/c/(2+p)+1/8*(b^2*(2+p)*(3+p)-2*a*c*(3+2*p)-2 
*b*c*(p+1)*(3+p)*x^2)*(c*x^4+b*x^2+a)^(p+1)/c^3/(p+1)/(2+p)/(3+2*p)-2^(-2+ 
p)*b*(6*a*c-b^2*(3+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^3/(-4*a*c+b^2)^(1/2)/(p+1)/(3+2*p)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.

Time = 0.47 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.63 \[ \int x^7 \left (a+b x^2+c x^4\right )^p \, dx=\frac {1}{8} x^8 \left (\frac {b-\sqrt {b^2-4 a c}+2 c x^2}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^2+c x^4\right )^p \operatorname {AppellF1}\left (4,-p,-p,5,-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},\frac {2 c x^2}{-b+\sqrt {b^2-4 a c}}\right ) \] Input:

Integrate[x^7*(a + b*x^2 + c*x^4)^p,x]
 

Output:

(x^8*(a + b*x^2 + c*x^4)^p*AppellF1[4, -p, -p, 5, (-2*c*x^2)/(b + Sqrt[b^2 
 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])/(8*((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)
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.05, 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, 1225, 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.

\(\displaystyle \int x^7 \left (a+b x^2+c x^4\right )^p \, dx\)

\(\Big \downarrow \) 1434

\(\displaystyle \frac {1}{2} \int x^6 \left (c x^4+b x^2+a\right )^pdx^2\)

\(\Big \downarrow \) 1166

\(\displaystyle \frac {1}{2} \left (\frac {\int -x^2 \left (b (p+3) x^2+2 a\right ) \left (c x^4+b x^2+a\right )^pdx^2}{2 c (p+2)}+\frac {x^4 \left (a+b x^2+c x^4\right )^{p+1}}{2 c (p+2)}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{2} \left (\frac {x^4 \left (a+b x^2+c x^4\right )^{p+1}}{2 c (p+2)}-\frac {\int x^2 \left (b (p+3) x^2+2 a\right ) \left (c x^4+b x^2+a\right )^pdx^2}{2 c (p+2)}\right )\)

\(\Big \downarrow \) 1225

\(\displaystyle \frac {1}{2} \left (\frac {x^4 \left (a+b x^2+c x^4\right )^{p+1}}{2 c (p+2)}-\frac {-\frac {b (p+2) \left (6 a c-b^2 (p+3)\right ) \int \left (c x^4+b x^2+a\right )^pdx^2}{2 c^2 (2 p+3)}-\frac {\left (-2 a c (2 p+3)+b^2 (p+2) (p+3)-2 b c (p+1) (p+3) x^2\right ) \left (a+b x^2+c x^4\right )^{p+1}}{2 c^2 (p+1) (2 p+3)}}{2 c (p+2)}\right )\)

\(\Big \downarrow \) 1096

\(\displaystyle \frac {1}{2} \left (\frac {x^4 \left (a+b x^2+c x^4\right )^{p+1}}{2 c (p+2)}-\frac {\frac {b 2^p (p+2) \left (6 a c-b^2 (p+3)\right ) \left (-\frac {-\sqrt {b^2-4 a c}+b+2 c x^2}{\sqrt {b^2-4 a c}}\right )^{-p-1} \left (a+b x^2+c x^4\right )^{p+1} \operatorname {Hypergeometric2F1}\left (-p,p+1,p+2,\frac {2 c x^2+b+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right )}{c^2 (p+1) (2 p+3) \sqrt {b^2-4 a c}}-\frac {\left (-2 a c (2 p+3)+b^2 (p+2) (p+3)-2 b c (p+1) (p+3) x^2\right ) \left (a+b x^2+c x^4\right )^{p+1}}{2 c^2 (p+1) (2 p+3)}}{2 c (p+2)}\right )\)

Input:

Int[x^7*(a + b*x^2 + c*x^4)^p,x]
 

Output:

((x^4*(a + b*x^2 + c*x^4)^(1 + p))/(2*c*(2 + p)) - (-1/2*((b^2*(2 + p)*(3 
+ p) - 2*a*c*(3 + 2*p) - 2*b*c*(1 + p)*(3 + p)*x^2)*(a + b*x^2 + c*x^4)^(1 
 + p))/(c^2*(1 + p)*(3 + 2*p)) + (2^p*b*(2 + p)*(6*a*c - b^2*(3 + 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^2*Sqrt[b^2 - 4*a*c]*(1 + p)*(3 + 2* 
p)))/(2*c*(2 + p)))/2
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 1096

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]
 

rule 1166

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]
 

rule 1225

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( 
x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 
 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), 
 x] + Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p 
+ 3))/(2*c^2*(2*p + 3))   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c 
, d, e, f, g, p}, x] &&  !LeQ[p, -1]
 

rule 1434

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]
 

Maple [F]

\[\int x^{7} \left (c \,x^{4}+b \,x^{2}+a \right )^{p}d x\]

Input:

int(x^7*(c*x^4+b*x^2+a)^p,x)
 

Output:

int(x^7*(c*x^4+b*x^2+a)^p,x)
 

Fricas [F]

\[ \int x^7 \left (a+b x^2+c x^4\right )^p \, dx=\int { {\left (c x^{4} + b x^{2} + a\right )}^{p} x^{7} \,d x } \] Input:

integrate(x^7*(c*x^4+b*x^2+a)^p,x, algorithm="fricas")
 

Output:

integral((c*x^4 + b*x^2 + a)^p*x^7, x)
 

Sympy [F]

\[ \int x^7 \left (a+b x^2+c x^4\right )^p \, dx=\int x^{7} \left (a + b x^{2} + c x^{4}\right )^{p}\, dx \] Input:

integrate(x**7*(c*x**4+b*x**2+a)**p,x)
 

Output:

Integral(x**7*(a + b*x**2 + c*x**4)**p, x)
 

Maxima [F]

\[ \int x^7 \left (a+b x^2+c x^4\right )^p \, dx=\int { {\left (c x^{4} + b x^{2} + a\right )}^{p} x^{7} \,d x } \] Input:

integrate(x^7*(c*x^4+b*x^2+a)^p,x, algorithm="maxima")
 

Output:

integrate((c*x^4 + b*x^2 + a)^p*x^7, x)
 

Giac [F]

\[ \int x^7 \left (a+b x^2+c x^4\right )^p \, dx=\int { {\left (c x^{4} + b x^{2} + a\right )}^{p} x^{7} \,d x } \] Input:

integrate(x^7*(c*x^4+b*x^2+a)^p,x, algorithm="giac")
 

Output:

integrate((c*x^4 + b*x^2 + a)^p*x^7, x)
 

Mupad [F(-1)]

Timed out. \[ \int x^7 \left (a+b x^2+c x^4\right )^p \, dx=\int x^7\,{\left (c\,x^4+b\,x^2+a\right )}^p \,d x \] Input:

int(x^7*(a + b*x^2 + c*x^4)^p,x)
 

Output:

int(x^7*(a + b*x^2 + c*x^4)^p, x)
 

Reduce [F]

\[ \int x^7 \left (a+b x^2+c x^4\right )^p \, dx=\text {too large to display} \] Input:

int(x^7*(c*x^4+b*x^2+a)^p,x)
 

Output:

(4*(a + b*x**2 + c*x**4)**p*a**2*c*p**2 + 20*(a + b*x**2 + c*x**4)**p*a**2 
*c*p + 18*(a + b*x**2 + c*x**4)**p*a**2*c - (a + b*x**2 + c*x**4)**p*a*b** 
2*p**2 - 5*(a + b*x**2 + c*x**4)**p*a*b**2*p - 6*(a + b*x**2 + c*x**4)**p* 
a*b**2 - 4*(a + b*x**2 + c*x**4)**p*a*b*c*p**3*x**2 - 20*(a + b*x**2 + c*x 
**4)**p*a*b*c*p**2*x**2 - 18*(a + b*x**2 + c*x**4)**p*a*b*c*p*x**2 + 8*(a 
+ b*x**2 + c*x**4)**p*a*c**2*p**3*x**4 + 16*(a + b*x**2 + c*x**4)**p*a*c** 
2*p**2*x**4 + 6*(a + b*x**2 + c*x**4)**p*a*c**2*p*x**4 + (a + b*x**2 + c*x 
**4)**p*b**3*p**3*x**2 + 5*(a + b*x**2 + c*x**4)**p*b**3*p**2*x**2 + 6*(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**3 
*x**4 - 7*(a + b*x**2 + c*x**4)**p*b**2*c*p**2*x**4 - 3*(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**3*x**6 + 6*(a 
+ b*x**2 + c*x**4)**p*b*c**2*p**2*x**6 + 2*(a + b*x**2 + c*x**4)**p*b*c**2 
*p*x**6 + 8*(a + b*x**2 + c*x**4)**p*c**3*p**3*x**8 + 24*(a + b*x**2 + c*x 
**4)**p*c**3*p**2*x**8 + 22*(a + b*x**2 + c*x**4)**p*c**3*p*x**8 + 6*(a + 
b*x**2 + c*x**4)**p*c**3*x**8 - 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**5 - 960*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 - 1680*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**...