\(\int x^m (A+B x^2) (b x^2+c x^4)^p \, dx\) [278]

Optimal result

Integrand size = 24, antiderivative size = 140 \[ \int x^m \left (A+B x^2\right ) \left (b x^2+c x^4\right )^p \, dx=\frac {B x^{-1+m} \left (b x^2+c x^4\right )^{1+p}}{c (3+m+4 p)}-\frac {(b B (1+m+2 p)-A c (3+m+4 p)) x^{1+m} \left (1+\frac {c x^2}{b}\right )^{-p} \left (b x^2+c x^4\right )^p \operatorname {Hypergeometric2F1}\left (-p,\frac {1}{2} (1+m+2 p),\frac {1}{2} (3+m+2 p),-\frac {c x^2}{b}\right )}{c (1+m+2 p) (3+m+4 p)} \] Output:

B*x^(-1+m)*(c*x^4+b*x^2)^(p+1)/c/(3+m+4*p)-(b*B*(1+m+2*p)-A*c*(3+m+4*p))*x 
^(1+m)*(c*x^4+b*x^2)^p*hypergeom([-p, 1/2+1/2*m+p],[3/2+1/2*m+p],-c*x^2/b) 
/c/(1+m+2*p)/(3+m+4*p)/((1+c*x^2/b)^p)
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.96 \[ \int x^m \left (A+B x^2\right ) \left (b x^2+c x^4\right )^p \, dx=\frac {x^{1+m} \left (x^2 \left (b+c x^2\right )\right )^p \left (1+\frac {c x^2}{b}\right )^{-p} \left (A (3+m+2 p) \operatorname {Hypergeometric2F1}\left (-p,\frac {1}{2} (1+m+2 p),\frac {1}{2} (3+m+2 p),-\frac {c x^2}{b}\right )+B (1+m+2 p) x^2 \operatorname {Hypergeometric2F1}\left (-p,\frac {1}{2} (3+m+2 p),\frac {1}{2} (5+m+2 p),-\frac {c x^2}{b}\right )\right )}{(1+m+2 p) (3+m+2 p)} \] Input:

Integrate[x^m*(A + B*x^2)*(b*x^2 + c*x^4)^p,x]
 

Output:

(x^(1 + m)*(x^2*(b + c*x^2))^p*(A*(3 + m + 2*p)*Hypergeometric2F1[-p, (1 + 
 m + 2*p)/2, (3 + m + 2*p)/2, -((c*x^2)/b)] + B*(1 + m + 2*p)*x^2*Hypergeo 
metric2F1[-p, (3 + m + 2*p)/2, (5 + m + 2*p)/2, -((c*x^2)/b)]))/((1 + m + 
2*p)*(3 + m + 2*p)*(1 + (c*x^2)/b)^p)
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.94, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1945, 1431, 279, 278}

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.

\(\displaystyle \int x^m \left (A+B x^2\right ) \left (b x^2+c x^4\right )^p \, dx\)

\(\Big \downarrow \) 1945

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

\(\Big \downarrow \) 1431

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

\(\Big \downarrow \) 279

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

\(\Big \downarrow \) 278

\(\displaystyle \frac {x^{m+1} \left (\frac {c x^2}{b}+1\right )^{-p} \left (b x^2+c x^4\right )^p \left (A-\frac {b B (m+2 p+1)}{c (m+4 p+3)}\right ) \operatorname {Hypergeometric2F1}\left (-p,\frac {1}{2} (m+2 p+1),\frac {1}{2} (m+2 p+3),-\frac {c x^2}{b}\right )}{m+2 p+1}+\frac {B x^{m-1} \left (b x^2+c x^4\right )^{p+1}}{c (m+4 p+3)}\)

Input:

Int[x^m*(A + B*x^2)*(b*x^2 + c*x^4)^p,x]
 

Output:

(B*x^(-1 + m)*(b*x^2 + c*x^4)^(1 + p))/(c*(3 + m + 4*p)) + ((A - (b*B*(1 + 
 m + 2*p))/(c*(3 + m + 4*p)))*x^(1 + m)*(b*x^2 + c*x^4)^p*Hypergeometric2F 
1[-p, (1 + m + 2*p)/2, (3 + m + 2*p)/2, -((c*x^2)/b)])/((1 + m + 2*p)*(1 + 
 (c*x^2)/b)^p)
 

Defintions of rubi rules used

rule 278

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( 
c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( 
-b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && (ILtQ[p, 0 
] || GtQ[a, 0])
 

rule 279

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntP 
art[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p])   Int[(c*x)^m* 
(1 + b*(x^2/a))^p, x], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && 
!(ILtQ[p, 0] || GtQ[a, 0])
 

rule 1431

Int[((d_.)*(x_))^(m_)*((b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp 
[(b*x^2 + c*x^4)^p/((d*x)^(2*p)*(b + c*x^2)^p)   Int[(d*x)^(m + 2*p)*(b + c 
*x^2)^p, x], x] /; FreeQ[{b, c, d, m, p}, x] &&  !IntegerQ[p]
 

rule 1945

Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + 
(d_.)*(x_)^(n_.)), x_Symbol] :> Simp[d*e^(j - 1)*(e*x)^(m - j + 1)*((a*x^j 
+ b*x^(j + n))^(p + 1)/(b*(m + n + p*(j + n) + 1))), x] - Simp[(a*d*(m + j* 
p + 1) - b*c*(m + n + p*(j + n) + 1))/(b*(m + n + p*(j + n) + 1))   Int[(e* 
x)^m*(a*x^j + b*x^(j + n))^p, x], x] /; FreeQ[{a, b, c, d, e, j, m, n, p}, 
x] && EqQ[jn, j + n] &&  !IntegerQ[p] && NeQ[b*c - a*d, 0] && NeQ[m + n + p 
*(j + n) + 1, 0] && (GtQ[e, 0] || IntegerQ[j])
 

Maple [F]

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

Input:

int(x^m*(B*x^2+A)*(c*x^4+b*x^2)^p,x)
 

Output:

int(x^m*(B*x^2+A)*(c*x^4+b*x^2)^p,x)
 

Fricas [F]

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

integrate(x^m*(B*x^2+A)*(c*x^4+b*x^2)^p,x, algorithm="fricas")
 

Output:

integral((B*x^2 + A)*(c*x^4 + b*x^2)^p*x^m, x)
 

Sympy [F]

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

integrate(x**m*(B*x**2+A)*(c*x**4+b*x**2)**p,x)
 

Output:

Integral(x**m*(x**2*(b + c*x**2))**p*(A + B*x**2), x)
 

Maxima [F]

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

integrate(x^m*(B*x^2+A)*(c*x^4+b*x^2)^p,x, algorithm="maxima")
 

Output:

integrate((B*x^2 + A)*(c*x^4 + b*x^2)^p*x^m, x)
 

Giac [F]

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

integrate(x^m*(B*x^2+A)*(c*x^4+b*x^2)^p,x, algorithm="giac")
 

Output:

integrate((B*x^2 + A)*(c*x^4 + b*x^2)^p*x^m, x)
 

Mupad [F(-1)]

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

int(x^m*(A + B*x^2)*(b*x^2 + c*x^4)^p,x)
 

Output:

int(x^m*(A + B*x^2)*(b*x^2 + c*x^4)^p, x)
 

Reduce [F]

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

int(x^m*(B*x^2+A)*(c*x^4+b*x^2)^p,x)
 

Output:

(x**m*(b*x**2 + c*x**4)**p*a*c*m*x + 4*x**m*(b*x**2 + c*x**4)**p*a*c*p*x + 
 3*x**m*(b*x**2 + c*x**4)**p*a*c*x + 2*x**m*(b*x**2 + c*x**4)**p*b**2*p*x 
+ x**m*(b*x**2 + c*x**4)**p*b*c*m*x**3 + 4*x**m*(b*x**2 + c*x**4)**p*b*c*p 
*x**3 + x**m*(b*x**2 + c*x**4)**p*b*c*x**3 + 2*int((x**m*(b*x**2 + c*x**4) 
**p)/(b*m**2 + 8*b*m*p + 4*b*m + 16*b*p**2 + 16*b*p + 3*b + c*m**2*x**2 + 
8*c*m*p*x**2 + 4*c*m*x**2 + 16*c*p**2*x**2 + 16*c*p*x**2 + 3*c*x**2),x)*a* 
b*c*m**3*p + 24*int((x**m*(b*x**2 + c*x**4)**p)/(b*m**2 + 8*b*m*p + 4*b*m 
+ 16*b*p**2 + 16*b*p + 3*b + c*m**2*x**2 + 8*c*m*p*x**2 + 4*c*m*x**2 + 16* 
c*p**2*x**2 + 16*c*p*x**2 + 3*c*x**2),x)*a*b*c*m**2*p**2 + 14*int((x**m*(b 
*x**2 + c*x**4)**p)/(b*m**2 + 8*b*m*p + 4*b*m + 16*b*p**2 + 16*b*p + 3*b + 
 c*m**2*x**2 + 8*c*m*p*x**2 + 4*c*m*x**2 + 16*c*p**2*x**2 + 16*c*p*x**2 + 
3*c*x**2),x)*a*b*c*m**2*p + 96*int((x**m*(b*x**2 + c*x**4)**p)/(b*m**2 + 8 
*b*m*p + 4*b*m + 16*b*p**2 + 16*b*p + 3*b + c*m**2*x**2 + 8*c*m*p*x**2 + 4 
*c*m*x**2 + 16*c*p**2*x**2 + 16*c*p*x**2 + 3*c*x**2),x)*a*b*c*m*p**3 + 112 
*int((x**m*(b*x**2 + c*x**4)**p)/(b*m**2 + 8*b*m*p + 4*b*m + 16*b*p**2 + 1 
6*b*p + 3*b + c*m**2*x**2 + 8*c*m*p*x**2 + 4*c*m*x**2 + 16*c*p**2*x**2 + 1 
6*c*p*x**2 + 3*c*x**2),x)*a*b*c*m*p**2 + 30*int((x**m*(b*x**2 + c*x**4)**p 
)/(b*m**2 + 8*b*m*p + 4*b*m + 16*b*p**2 + 16*b*p + 3*b + c*m**2*x**2 + 8*c 
*m*p*x**2 + 4*c*m*x**2 + 16*c*p**2*x**2 + 16*c*p*x**2 + 3*c*x**2),x)*a*b*c 
*m*p + 128*int((x**m*(b*x**2 + c*x**4)**p)/(b*m**2 + 8*b*m*p + 4*b*m + ...