\(\int \frac {(a+b x^2+c x^4)^p}{x^4} \, dx\) [1103]

Optimal result

Integrand size = 18, antiderivative size = 138 \[ \int \frac {\left (a+b x^2+c x^4\right )^p}{x^4} \, dx=-\frac {\left (1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^2+c x^4\right )^p \operatorname {AppellF1}\left (-\frac {3}{2},-p,-p,-\frac {1}{2},-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{3 x^3} \] Output:

-1/3*(c*x^4+b*x^2+a)^p*AppellF1(-3/2,-p,-p,-1/2,-2*c*x^2/(b-(-4*a*c+b^2)^( 
1/2)),-2*c*x^2/(b+(-4*a*c+b^2)^(1/2)))/x^3/((1+2*c*x^2/(b-(-4*a*c+b^2)^(1/ 
2)))^p)/((1+2*c*x^2/(b+(-4*a*c+b^2)^(1/2)))^p)
 

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.20 \[ \int \frac {\left (a+b x^2+c x^4\right )^p}{x^4} \, dx=-\frac {\left (\frac {b-\sqrt {b^2-4 a c}+2 c x^2}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^2+c x^4\right )^p \operatorname {AppellF1}\left (-\frac {3}{2},-p,-p,-\frac {1}{2},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},\frac {2 c x^2}{-b+\sqrt {b^2-4 a c}}\right )}{3 x^3} \] Input:

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

Output:

-1/3*((a + b*x^2 + c*x^4)^p*AppellF1[-3/2, -p, -p, -1/2, (-2*c*x^2)/(b + S 
qrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])/(x^3*((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 = 0.44 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1461, 394}

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.

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

\(\Big \downarrow \) 1461

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

\(\Big \downarrow \) 394

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 394

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/2 
, -p, -q, 1 + (m + 1)/2, (-b)*(x^2/a), (-d)*(x^2/c)], x] /; FreeQ[{a, b, c, 
 d, e, m, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, 1] && (Int 
egerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 1461

Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] 
:> Simp[a^IntPart[p]*((a + b*x^2 + c*x^4)^FracPart[p]/((1 + 2*c*(x^2/(b + R 
t[b^2 - 4*a*c, 2])))^FracPart[p]*(1 + 2*c*(x^2/(b - Rt[b^2 - 4*a*c, 2])))^F 
racPart[p]))   Int[(d*x)^m*(1 + 2*c*(x^2/(b + Sqrt[b^2 - 4*a*c])))^p*(1 + 2 
*c*(x^2/(b - Sqrt[b^2 - 4*a*c])))^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (a+b x^2+c x^4\right )^p}{x^4} \, dx=\frac {-\left (c \,x^{4}+b \,x^{2}+a \right )^{p}+4 \left (\int \frac {\left (c \,x^{4}+b \,x^{2}+a \right )^{p}}{2 c p \,x^{6}-3 c \,x^{6}+2 b p \,x^{4}-3 b \,x^{4}+2 a p \,x^{2}-3 a \,x^{2}}d x \right ) b \,p^{2} x^{3}-6 \left (\int \frac {\left (c \,x^{4}+b \,x^{2}+a \right )^{p}}{2 c p \,x^{6}-3 c \,x^{6}+2 b p \,x^{4}-3 b \,x^{4}+2 a p \,x^{2}-3 a \,x^{2}}d x \right ) b p \,x^{3}+8 \left (\int \frac {\left (c \,x^{4}+b \,x^{2}+a \right )^{p}}{2 c p \,x^{4}-3 c \,x^{4}+2 b p \,x^{2}-3 b \,x^{2}+2 a p -3 a}d x \right ) c \,p^{2} x^{3}-12 \left (\int \frac {\left (c \,x^{4}+b \,x^{2}+a \right )^{p}}{2 c p \,x^{4}-3 c \,x^{4}+2 b p \,x^{2}-3 b \,x^{2}+2 a p -3 a}d x \right ) c p \,x^{3}}{3 x^{3}} \] Input:

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

Output:

( - (a + b*x**2 + c*x**4)**p + 4*int((a + b*x**2 + c*x**4)**p/(2*a*p*x**2 
- 3*a*x**2 + 2*b*p*x**4 - 3*b*x**4 + 2*c*p*x**6 - 3*c*x**6),x)*b*p**2*x**3 
 - 6*int((a + b*x**2 + c*x**4)**p/(2*a*p*x**2 - 3*a*x**2 + 2*b*p*x**4 - 3* 
b*x**4 + 2*c*p*x**6 - 3*c*x**6),x)*b*p*x**3 + 8*int((a + b*x**2 + c*x**4)* 
*p/(2*a*p - 3*a + 2*b*p*x**2 - 3*b*x**2 + 2*c*p*x**4 - 3*c*x**4),x)*c*p**2 
*x**3 - 12*int((a + b*x**2 + c*x**4)**p/(2*a*p - 3*a + 2*b*p*x**2 - 3*b*x* 
*2 + 2*c*p*x**4 - 3*c*x**4),x)*c*p*x**3)/(3*x**3)