3.5.1 \(\int (c+e x^2)^2 (a+c x^2+b x^4)^p \, dx\) [401]

3.5.1.1 Optimal result

Integrand size = 24, antiderivative size = 358 \[ \int \left (c+e x^2\right )^2 \left (a+c x^2+b x^4\right )^p \, dx=\frac {e^2 x \left (a+c x^2+b x^4\right )^{1+p}}{b (5+4 p)}-\frac {\left (a e^2-b c^2 (5+4 p)\right ) x \left (1+\frac {2 b x^2}{c-\sqrt {-4 a b+c^2}}\right )^{-p} \left (1+\frac {2 b x^2}{c+\sqrt {-4 a b+c^2}}\right )^{-p} \left (a+c x^2+b x^4\right )^p \operatorname {AppellF1}\left (\frac {1}{2},-p,-p,\frac {3}{2},-\frac {2 b x^2}{c-\sqrt {-4 a b+c^2}},-\frac {2 b x^2}{c+\sqrt {-4 a b+c^2}}\right )}{b (5+4 p)}+\frac {c e (10 b-3 e+8 b p-2 e p) x^3 \left (1+\frac {2 b x^2}{c-\sqrt {-4 a b+c^2}}\right )^{-p} \left (1+\frac {2 b x^2}{c+\sqrt {-4 a b+c^2}}\right )^{-p} \left (a+c x^2+b x^4\right )^p \operatorname {AppellF1}\left (\frac {3}{2},-p,-p,\frac {5}{2},-\frac {2 b x^2}{c-\sqrt {-4 a b+c^2}},-\frac {2 b x^2}{c+\sqrt {-4 a b+c^2}}\right )}{3 b (5+4 p)} \]

output

e^2*x*(b*x^4+c*x^2+a)^(p+1)/b/(5+4*p)-(a*e^2-b*c^2*(5+4*p))*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)))/b/(5+4*p)/((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*c*e*(8*b*p-2*e*p+10*b-3*e)*x^3*(b*x^ 
4+c*x^2+a)^p*AppellF1(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)))/b/(5+4*p)/((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)
 

3.5.1.2 Mathematica [A] (verified)

Time = 0.55 (sec) , antiderivative size = 303, normalized size of antiderivative = 0.85 \[ \int \left (c+e x^2\right )^2 \left (a+c x^2+b x^4\right )^p \, dx=\frac {1}{15} x \left (\frac {c-\sqrt {-4 a b+c^2}+2 b x^2}{c-\sqrt {-4 a b+c^2}}\right )^{-p} \left (\frac {c+\sqrt {-4 a b+c^2}+2 b x^2}{c+\sqrt {-4 a b+c^2}}\right )^{-p} \left (a+c x^2+b x^4\right )^p \left (15 c^2 \operatorname {AppellF1}\left (\frac {1}{2},-p,-p,\frac {3}{2},-\frac {2 b x^2}{c+\sqrt {-4 a b+c^2}},\frac {2 b x^2}{-c+\sqrt {-4 a b+c^2}}\right )+e x^2 \left (10 c \operatorname {AppellF1}\left (\frac {3}{2},-p,-p,\frac {5}{2},-\frac {2 b x^2}{c+\sqrt {-4 a b+c^2}},\frac {2 b x^2}{-c+\sqrt {-4 a b+c^2}}\right )+3 e x^2 \operatorname {AppellF1}\left (\frac {5}{2},-p,-p,\frac {7}{2},-\frac {2 b x^2}{c+\sqrt {-4 a b+c^2}},\frac {2 b x^2}{-c+\sqrt {-4 a b+c^2}}\right )\right )\right ) \]

input

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

output

(x*(a + c*x^2 + b*x^4)^p*(15*c^2*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])] + e*x^2*(10*c* 
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*e*x^2*AppellF1[5/2, -p, -p, 7/2, (-2*b*x^2) 
/(c + Sqrt[-4*a*b + c^2]), (2*b*x^2)/(-c + Sqrt[-4*a*b + c^2])])))/(15*((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)
 

3.5.1.3 Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 350, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1518, 25, 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.

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

\(\Big \downarrow \) 1518

\(\displaystyle \frac {\int -\left (\left (-b (4 p+5) c^2-e (8 p b+10 b-3 e-2 e p) x^2 c+a e^2\right ) \left (b x^4+c x^2+a\right )^p\right )dx}{b (4 p+5)}+\frac {e^2 x \left (a+b x^4+c x^2\right )^{p+1}}{b (4 p+5)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {e^2 x \left (a+b x^4+c x^2\right )^{p+1}}{b (4 p+5)}-\frac {\int \left (-b (4 p+5) c^2-e (8 p b+10 b-3 e-2 e p) x^2 c+a e^2\right ) \left (b x^4+c x^2+a\right )^pdx}{b (4 p+5)}\)

\(\Big \downarrow \) 1515

\(\displaystyle \frac {e^2 x \left (a+b x^4+c x^2\right )^{p+1}}{b (4 p+5)}-\frac {\int \left (c e (-8 p b-10 b+3 e+2 e p) x^2 \left (b x^4+c x^2+a\right )^p+a e^2 \left (1-\frac {b c^2 (4 p+5)}{a e^2}\right ) \left (b x^4+c x^2+a\right )^p\right )dx}{b (4 p+5)}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {e^2 x \left (a+b x^4+c x^2\right )^{p+1}}{b (4 p+5)}-\frac {x \left (a e^2-b c^2 (4 p+5)\right ) \left (\frac {2 b x^2}{c-\sqrt {c^2-4 a b}}+1\right )^{-p} \left (\frac {2 b x^2}{\sqrt {c^2-4 a b}+c}+1\right )^{-p} \left (a+b x^4+c x^2\right )^p \operatorname {AppellF1}\left (\frac {1}{2},-p,-p,\frac {3}{2},-\frac {2 b x^2}{c-\sqrt {c^2-4 a b}},-\frac {2 b x^2}{c+\sqrt {c^2-4 a b}}\right )-\frac {1}{3} c e x^3 (8 b p+10 b-2 e p-3 e) \left (\frac {2 b x^2}{c-\sqrt {c^2-4 a b}}+1\right )^{-p} \left (\frac {2 b x^2}{\sqrt {c^2-4 a b}+c}+1\right )^{-p} \left (a+b x^4+c x^2\right )^p \operatorname {AppellF1}\left (\frac {3}{2},-p,-p,\frac {5}{2},-\frac {2 b x^2}{c-\sqrt {c^2-4 a b}},-\frac {2 b x^2}{c+\sqrt {c^2-4 a b}}\right )}{b (4 p+5)}\)

input

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

output

(e^2*x*(a + c*x^2 + b*x^4)^(1 + p))/(b*(5 + 4*p)) - (((a*e^2 - b*c^2*(5 + 
4*p))*x*(a + c*x^2 + b*x^4)^p*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])])/((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) - 
(c*e*(10*b - 3*e + 8*b*p - 2*e*p)*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))/(b*(5 + 4*p))
 

3.5.1.3.1 Defintions of rubi rules used

rule 25

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

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 1518

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x 
_Symbol] :> Simp[e^q*x^(2*q - 3)*((a + b*x^2 + c*x^4)^(p + 1)/(c*(4*p + 2*q 
 + 1))), x] + Simp[1/(c*(4*p + 2*q + 1))   Int[(a + b*x^2 + c*x^4)^p*Expand 
ToSum[c*(4*p + 2*q + 1)*(d + e*x^2)^q - a*(2*q - 3)*e^q*x^(2*q - 4) - b*(2* 
p + 2*q - 1)*e^q*x^(2*q - 2) - c*(4*p + 2*q + 1)*e^q*x^(2*q), x], x], x] /; 
 FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + 
 a*e^2, 0] && IGtQ[q, 1]
 

rule 2009

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

3.5.1.4 Maple [F]

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

input

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

output

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

3.5.1.5 Fricas [F]

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

input

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

output

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

3.5.1.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.5.1.7 Maxima [F]

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

input

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

output

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

3.5.1.8 Giac [F]

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

input

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

output

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

3.5.1.9 Mupad [F(-1)]

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

input

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

output

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