\(\int (A+B x^2) (a+c x^2+b x^4)^p \, dx\) [19]

Optimal result

Integrand size = 22, antiderivative size = 274 \[ \int \left (A+B x^2\right ) \left (a+c x^2+b x^4\right )^p \, dx=A x \left (1+\frac {2 b x^2}{c-\sqrt {-4 a b+c^2}}\right )^{-p} \left (1+\frac {2 b x^2}{c+\sqrt {-4 a b+c^2}}\right )^{-p} \left (a+c x^2+b x^4\right )^p \operatorname {AppellF1}\left (\frac {1}{2},-p,-p,\frac {3}{2},-\frac {2 b x^2}{c-\sqrt {-4 a b+c^2}},-\frac {2 b x^2}{c+\sqrt {-4 a b+c^2}}\right )+\frac {1}{3} B x^3 \left (1+\frac {2 b x^2}{c-\sqrt {-4 a b+c^2}}\right )^{-p} \left (1+\frac {2 b x^2}{c+\sqrt {-4 a b+c^2}}\right )^{-p} \left (a+c x^2+b x^4\right )^p \operatorname {AppellF1}\left (\frac {3}{2},-p,-p,\frac {5}{2},-\frac {2 b x^2}{c-\sqrt {-4 a b+c^2}},-\frac {2 b x^2}{c+\sqrt {-4 a b+c^2}}\right ) \] Output:

A*x*(b*x^4+c*x^2+a)^p*AppellF1(1/2,-p,-p,3/2,-2*b*x^2/(c-(-4*a*b+c^2)^(1/2 
)),-2*b*x^2/(c+(-4*a*b+c^2)^(1/2)))/((1+2*b*x^2/(c-(-4*a*b+c^2)^(1/2)))^p) 
/((1+2*b*x^2/(c+(-4*a*b+c^2)^(1/2)))^p)+1/3*B*x^3*(b*x^4+c*x^2+a)^p*Appell 
F1(3/2,-p,-p,5/2,-2*b*x^2/(c-(-4*a*b+c^2)^(1/2)),-2*b*x^2/(c+(-4*a*b+c^2)^ 
(1/2)))/((1+2*b*x^2/(c-(-4*a*b+c^2)^(1/2)))^p)/((1+2*b*x^2/(c+(-4*a*b+c^2) 
^(1/2)))^p)
 

Mathematica [A] (warning: unable to verify)

Time = 0.67 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.85 \[ \int \left (A+B x^2\right ) \left (a+c x^2+b x^4\right )^p \, dx=\frac {1}{3} x \left (\frac {c-\sqrt {-4 a b+c^2}+2 b x^2}{c-\sqrt {-4 a b+c^2}}\right )^{-p} \left (\frac {c+\sqrt {-4 a b+c^2}+2 b x^2}{c+\sqrt {-4 a b+c^2}}\right )^{-p} \left (a+c x^2+b x^4\right )^p \left (3 A \operatorname {AppellF1}\left (\frac {1}{2},-p,-p,\frac {3}{2},-\frac {2 b x^2}{c+\sqrt {-4 a b+c^2}},\frac {2 b x^2}{-c+\sqrt {-4 a b+c^2}}\right )+B x^2 \operatorname {AppellF1}\left (\frac {3}{2},-p,-p,\frac {5}{2},-\frac {2 b x^2}{c+\sqrt {-4 a b+c^2}},\frac {2 b x^2}{-c+\sqrt {-4 a b+c^2}}\right )\right ) \] Input:

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

Output:

(x*(a + c*x^2 + b*x^4)^p*(3*A*AppellF1[1/2, -p, -p, 3/2, (-2*b*x^2)/(c + S 
qrt[-4*a*b + c^2]), (2*b*x^2)/(-c + Sqrt[-4*a*b + c^2])] + B*x^2*AppellF1[ 
3/2, -p, -p, 5/2, (-2*b*x^2)/(c + Sqrt[-4*a*b + c^2]), (2*b*x^2)/(-c + Sqr 
t[-4*a*b + c^2])]))/(3*((c - Sqrt[-4*a*b + c^2] + 2*b*x^2)/(c - Sqrt[-4*a* 
b + c^2]))^p*((c + Sqrt[-4*a*b + c^2] + 2*b*x^2)/(c + Sqrt[-4*a*b + c^2])) 
^p)
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1515, 2009}

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.

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

\(\Big \downarrow \) 1515

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

\(\Big \downarrow \) 2009

\(\displaystyle A x \left (\frac {2 b x^2}{c-\sqrt {c^2-4 a b}}+1\right )^{-p} \left (a+b x^4+c x^2\right )^p \left (\frac {2 b x^2}{\sqrt {c^2-4 a b}+c}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},-p,-p,\frac {3}{2},-\frac {2 b x^2}{c-\sqrt {c^2-4 a b}},-\frac {2 b x^2}{c+\sqrt {c^2-4 a b}}\right )+\frac {1}{3} B x^3 \left (\frac {2 b x^2}{c-\sqrt {c^2-4 a b}}+1\right )^{-p} \left (a+b x^4+c x^2\right )^p \left (\frac {2 b x^2}{\sqrt {c^2-4 a b}+c}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {3}{2},-p,-p,\frac {5}{2},-\frac {2 b x^2}{c-\sqrt {c^2-4 a b}},-\frac {2 b x^2}{c+\sqrt {c^2-4 a b}}\right )\)

Input:

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

Output:

(A*x*(a + c*x^2 + b*x^4)^p*AppellF1[1/2, -p, -p, 3/2, (-2*b*x^2)/(c - Sqrt 
[-4*a*b + c^2]), (-2*b*x^2)/(c + Sqrt[-4*a*b + c^2])])/((1 + (2*b*x^2)/(c 
- Sqrt[-4*a*b + c^2]))^p*(1 + (2*b*x^2)/(c + Sqrt[-4*a*b + c^2]))^p) + (B* 
x^3*(a + c*x^2 + b*x^4)^p*AppellF1[3/2, -p, -p, 5/2, (-2*b*x^2)/(c - Sqrt[ 
-4*a*b + c^2]), (-2*b*x^2)/(c + Sqrt[-4*a*b + c^2])])/(3*(1 + (2*b*x^2)/(c 
 - Sqrt[-4*a*b + c^2]))^p*(1 + (2*b*x^2)/(c + Sqrt[-4*a*b + c^2]))^p)
 

Defintions of rubi rules used

rule 1515

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb 
ol] :> Int[ExpandIntegrand[(d + e*x^2)*(a + b*x^2 + c*x^4)^p, x], x] /; Fre 
eQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 
 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(4*(a + b*x**4 + c*x**2)**p*a*p*x + 3*(a + b*x**4 + c*x**2)**p*a*x + 4*(a 
+ b*x**4 + c*x**2)**p*b*p*x**3 + (a + b*x**4 + c*x**2)**p*b*x**3 + 2*(a + 
b*x**4 + c*x**2)**p*c*p*x + 256*int((a + b*x**4 + c*x**2)**p/(16*a*p**2 + 
16*a*p + 3*a + 16*b*p**2*x**4 + 16*b*p*x**4 + 3*b*x**4 + 16*c*p**2*x**2 + 
16*c*p*x**2 + 3*c*x**2),x)*a**2*p**4 + 448*int((a + b*x**4 + c*x**2)**p/(1 
6*a*p**2 + 16*a*p + 3*a + 16*b*p**2*x**4 + 16*b*p*x**4 + 3*b*x**4 + 16*c*p 
**2*x**2 + 16*c*p*x**2 + 3*c*x**2),x)*a**2*p**3 + 240*int((a + b*x**4 + c* 
x**2)**p/(16*a*p**2 + 16*a*p + 3*a + 16*b*p**2*x**4 + 16*b*p*x**4 + 3*b*x* 
*4 + 16*c*p**2*x**2 + 16*c*p*x**2 + 3*c*x**2),x)*a**2*p**2 + 36*int((a + b 
*x**4 + c*x**2)**p/(16*a*p**2 + 16*a*p + 3*a + 16*b*p**2*x**4 + 16*b*p*x** 
4 + 3*b*x**4 + 16*c*p**2*x**2 + 16*c*p*x**2 + 3*c*x**2),x)*a**2*p - 32*int 
((a + b*x**4 + c*x**2)**p/(16*a*p**2 + 16*a*p + 3*a + 16*b*p**2*x**4 + 16* 
b*p*x**4 + 3*b*x**4 + 16*c*p**2*x**2 + 16*c*p*x**2 + 3*c*x**2),x)*a*c*p**3 
 - 32*int((a + b*x**4 + c*x**2)**p/(16*a*p**2 + 16*a*p + 3*a + 16*b*p**2*x 
**4 + 16*b*p*x**4 + 3*b*x**4 + 16*c*p**2*x**2 + 16*c*p*x**2 + 3*c*x**2),x) 
*a*c*p**2 - 6*int((a + b*x**4 + c*x**2)**p/(16*a*p**2 + 16*a*p + 3*a + 16* 
b*p**2*x**4 + 16*b*p*x**4 + 3*b*x**4 + 16*c*p**2*x**2 + 16*c*p*x**2 + 3*c* 
x**2),x)*a*c*p + 256*int(((a + b*x**4 + c*x**2)**p*x**2)/(16*a*p**2 + 16*a 
*p + 3*a + 16*b*p**2*x**4 + 16*b*p*x**4 + 3*b*x**4 + 16*c*p**2*x**2 + 16*c 
*p*x**2 + 3*c*x**2),x)*a*b*p**4 + 320*int(((a + b*x**4 + c*x**2)**p*x**...