\(\int \frac {(a+b x^2)^p (c+d x^2)^q (e+f x^2)}{x^4} \, dx\) [75]

Optimal result

Integrand size = 29, antiderivative size = 384 \[ \int \frac {\left (a+b x^2\right )^p \left (c+d x^2\right )^q \left (e+f x^2\right )}{x^4} \, dx=-\frac {e \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^{1+q}}{3 a c x^3}-\frac {(3 a c f-b c e (1-2 p)-a d e (1-2 q)) \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^{1+q}}{3 a^2 c^2 x}-\frac {\left (b^2 c^2 e \left (1-4 p^2\right )-a^2 d (3 c f-d e (1-2 q)) (1+2 q)-a b c (3 c f (1+2 p)-d e (1-2 p-2 q-8 p q))\right ) x \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (c+d x^2\right )^q \left (1+\frac {d x^2}{c}\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{3 a^2 c^2}+\frac {b d (3 a c f-b c e (1-2 p)-a d e (1-2 q)) (3+2 p+2 q) x^3 \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (c+d x^2\right )^q \left (1+\frac {d x^2}{c}\right )^{-q} \operatorname {AppellF1}\left (\frac {3}{2},-p,-q,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{9 a^2 c^2} \] Output:

-1/3*e*(b*x^2+a)^(p+1)*(d*x^2+c)^(1+q)/a/c/x^3-1/3*(3*a*c*f-b*c*e*(1-2*p)- 
a*d*e*(1-2*q))*(b*x^2+a)^(p+1)*(d*x^2+c)^(1+q)/a^2/c^2/x-1/3*(b^2*c^2*e*(- 
4*p^2+1)-a^2*d*(3*c*f-d*e*(1-2*q))*(1+2*q)-a*b*c*(3*c*f*(1+2*p)-d*e*(-8*p* 
q-2*p-2*q+1)))*x*(b*x^2+a)^p*(d*x^2+c)^q*AppellF1(1/2,-p,-q,3/2,-b*x^2/a,- 
d*x^2/c)/a^2/c^2/((1+b*x^2/a)^p)/((1+d*x^2/c)^q)+1/9*b*d*(3*a*c*f-b*c*e*(1 
-2*p)-a*d*e*(1-2*q))*(3+2*p+2*q)*x^3*(b*x^2+a)^p*(d*x^2+c)^q*AppellF1(3/2, 
-p,-q,5/2,-b*x^2/a,-d*x^2/c)/a^2/c^2/((1+b*x^2/a)^p)/((1+d*x^2/c)^q)
 

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.32 \[ \int \frac {\left (a+b x^2\right )^p \left (c+d x^2\right )^q \left (e+f x^2\right )}{x^4} \, dx=-\frac {\left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (c+d x^2\right )^q \left (1+\frac {d x^2}{c}\right )^{-q} \left (e \operatorname {AppellF1}\left (-\frac {3}{2},-p,-q,-\frac {1}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+3 f x^2 \operatorname {AppellF1}\left (-\frac {1}{2},-p,-q,\frac {1}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )}{3 x^3} \] Input:

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

Output:

-1/3*((a + b*x^2)^p*(c + d*x^2)^q*(e*AppellF1[-3/2, -p, -q, -1/2, -((b*x^2 
)/a), -((d*x^2)/c)] + 3*f*x^2*AppellF1[-1/2, -p, -q, 1/2, -((b*x^2)/a), -( 
(d*x^2)/c)]))/(x^3*(1 + (b*x^2)/a)^p*(1 + (d*x^2)/c)^q)
 

Rubi [A] (verified)

Time = 1.20 (sec) , antiderivative size = 387, normalized size of antiderivative = 1.01, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {445, 25, 445, 406, 334, 334, 333, 395, 395, 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.

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

\(\Big \downarrow \) 445

\(\displaystyle -\frac {\int -\frac {\left (b x^2+a\right )^p \left (d x^2+c\right )^q \left (b d e (2 p+2 q+1) x^2+3 a c f-b c e (1-2 p)-a d (e-2 e q)\right )}{x^2}dx}{3 a c}-\frac {e \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{3 a c x^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {\left (b x^2+a\right )^p \left (d x^2+c\right )^q \left (b d e (2 p+2 q+1) x^2+3 a c f-b c e (1-2 p)-a d e (1-2 q)\right )}{x^2}dx}{3 a c}-\frac {e \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{3 a c x^3}\)

\(\Big \downarrow \) 445

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

\(\Big \downarrow \) 406

\(\displaystyle \frac {-\frac {\left (a^2 (-d) (2 q+1) (3 c f-d e (1-2 q))-a b c (3 c f (2 p+1)-d e (-8 p q-2 p-2 q+1))+b^2 c^2 e \left (1-4 p^2\right )\right ) \int \left (b x^2+a\right )^p \left (d x^2+c\right )^qdx-b d (2 p+2 q+3) (3 a c f-a d e (1-2 q)-b c e (1-2 p)) \int x^2 \left (b x^2+a\right )^p \left (d x^2+c\right )^qdx}{a c}-\frac {\left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1} (3 a c f-a d e (1-2 q)-b c e (1-2 p))}{a c x}}{3 a c}-\frac {e \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{3 a c x^3}\)

\(\Big \downarrow \) 334

\(\displaystyle \frac {-\frac {\left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (a^2 (-d) (2 q+1) (3 c f-d e (1-2 q))-a b c (3 c f (2 p+1)-d e (-8 p q-2 p-2 q+1))+b^2 c^2 e \left (1-4 p^2\right )\right ) \int \left (\frac {b x^2}{a}+1\right )^p \left (d x^2+c\right )^qdx-b d (2 p+2 q+3) (3 a c f-a d e (1-2 q)-b c e (1-2 p)) \int x^2 \left (b x^2+a\right )^p \left (d x^2+c\right )^qdx}{a c}-\frac {\left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1} (3 a c f-a d e (1-2 q)-b c e (1-2 p))}{a c x}}{3 a c}-\frac {e \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{3 a c x^3}\)

\(\Big \downarrow \) 334

\(\displaystyle \frac {-\frac {\left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} \left (a^2 (-d) (2 q+1) (3 c f-d e (1-2 q))-a b c (3 c f (2 p+1)-d e (-8 p q-2 p-2 q+1))+b^2 c^2 e \left (1-4 p^2\right )\right ) \int \left (\frac {b x^2}{a}+1\right )^p \left (\frac {d x^2}{c}+1\right )^qdx-b d (2 p+2 q+3) (3 a c f-a d e (1-2 q)-b c e (1-2 p)) \int x^2 \left (b x^2+a\right )^p \left (d x^2+c\right )^qdx}{a c}-\frac {\left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1} (3 a c f-a d e (1-2 q)-b c e (1-2 p))}{a c x}}{3 a c}-\frac {e \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{3 a c x^3}\)

\(\Big \downarrow \) 333

\(\displaystyle \frac {-\frac {x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right ) \left (a^2 (-d) (2 q+1) (3 c f-d e (1-2 q))-a b c (3 c f (2 p+1)-d e (-8 p q-2 p-2 q+1))+b^2 c^2 e \left (1-4 p^2\right )\right )-b d (2 p+2 q+3) (3 a c f-a d e (1-2 q)-b c e (1-2 p)) \int x^2 \left (b x^2+a\right )^p \left (d x^2+c\right )^qdx}{a c}-\frac {\left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1} (3 a c f-a d e (1-2 q)-b c e (1-2 p))}{a c x}}{3 a c}-\frac {e \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{3 a c x^3}\)

\(\Big \downarrow \) 395

\(\displaystyle \frac {-\frac {x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right ) \left (a^2 (-d) (2 q+1) (3 c f-d e (1-2 q))-a b c (3 c f (2 p+1)-d e (-8 p q-2 p-2 q+1))+b^2 c^2 e \left (1-4 p^2\right )\right )-b d (2 p+2 q+3) \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} (3 a c f-a d e (1-2 q)-b c e (1-2 p)) \int x^2 \left (\frac {b x^2}{a}+1\right )^p \left (d x^2+c\right )^qdx}{a c}-\frac {\left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1} (3 a c f-a d e (1-2 q)-b c e (1-2 p))}{a c x}}{3 a c}-\frac {e \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{3 a c x^3}\)

\(\Big \downarrow \) 395

\(\displaystyle \frac {-\frac {x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right ) \left (a^2 (-d) (2 q+1) (3 c f-d e (1-2 q))-a b c (3 c f (2 p+1)-d e (-8 p q-2 p-2 q+1))+b^2 c^2 e \left (1-4 p^2\right )\right )-b d (2 p+2 q+3) \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} (3 a c f-a d e (1-2 q)-b c e (1-2 p)) \int x^2 \left (\frac {b x^2}{a}+1\right )^p \left (\frac {d x^2}{c}+1\right )^qdx}{a c}-\frac {\left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1} (3 a c f-a d e (1-2 q)-b c e (1-2 p))}{a c x}}{3 a c}-\frac {e \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{3 a c x^3}\)

\(\Big \downarrow \) 394

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

Input:

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

Output:

-1/3*(e*(a + b*x^2)^(1 + p)*(c + d*x^2)^(1 + q))/(a*c*x^3) + (-(((3*a*c*f 
- b*c*e*(1 - 2*p) - a*d*e*(1 - 2*q))*(a + b*x^2)^(1 + p)*(c + d*x^2)^(1 + 
q))/(a*c*x)) - (((b^2*c^2*e*(1 - 4*p^2) - a^2*d*(3*c*f - d*e*(1 - 2*q))*(1 
 + 2*q) - a*b*c*(3*c*f*(1 + 2*p) - d*e*(1 - 2*p - 2*q - 8*p*q)))*x*(a + b* 
x^2)^p*(c + d*x^2)^q*AppellF1[1/2, -p, -q, 3/2, -((b*x^2)/a), -((d*x^2)/c) 
])/((1 + (b*x^2)/a)^p*(1 + (d*x^2)/c)^q) - (b*d*(3*a*c*f - b*c*e*(1 - 2*p) 
 - a*d*e*(1 - 2*q))*(3 + 2*p + 2*q)*x^3*(a + b*x^2)^p*(c + d*x^2)^q*Appell 
F1[3/2, -p, -q, 5/2, -((b*x^2)/a), -((d*x^2)/c)])/(3*(1 + (b*x^2)/a)^p*(1 
+ (d*x^2)/c)^q))/(a*c))/(3*a*c)
 

Defintions of rubi rules used

rule 25

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

rule 333

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[a^p*c^q*x*AppellF1[1/2, -p, -q, 3/2, (-b)*(x^2/a), (-d)*(x^2/c)], x] /; F 
reeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[p] || GtQ[a, 
0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 334

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[a^IntPart[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p])   Int[ 
(1 + b*(x^2/a))^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, p, q}, x] && 
NeQ[b*c - a*d, 0] &&  !(IntegerQ[p] || GtQ[a, 0])
 

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 395

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

rule 406

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( 
x_)^2), x_Symbol] :> Simp[e   Int[(a + b*x^2)^p*(c + d*x^2)^q, x], x] + Sim 
p[f   Int[x^2*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, 
f, p, q}, x]
 

rule 445

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

Maple [F]

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

Input:

int((b*x^2+a)^p*(d*x^2+c)^q*(f*x^2+e)/x^4,x)
 

Output:

int((b*x^2+a)^p*(d*x^2+c)^q*(f*x^2+e)/x^4,x)
 

Fricas [F]

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

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

Output:

integral((f*x^2 + e)*(b*x^2 + a)^p*(d*x^2 + c)^q/x^4, x)
 

Sympy [F(-1)]

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

integrate((b*x**2+a)**p*(d*x**2+c)**q*(f*x**2+e)/x**4,x)
 

Output:

Timed out
 

Maxima [F]

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

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

Output:

integrate((f*x^2 + e)*(b*x^2 + a)^p*(d*x^2 + c)^q/x^4, x)
 

Giac [F]

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

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

Output:

integrate((f*x^2 + e)*(b*x^2 + a)^p*(d*x^2 + c)^q/x^4, x)
 

Mupad [F(-1)]

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

int(((a + b*x^2)^p*(c + d*x^2)^q*(e + f*x^2))/x^4,x)
 

Output:

int(((a + b*x^2)^p*(c + d*x^2)^q*(e + f*x^2))/x^4, x)
 

Reduce [F]

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

int((b*x^2+a)^p*(d*x^2+c)^q*(f*x^2+e)/x^4,x)
 

Output:

int((b*x^2+a)^p*(d*x^2+c)^q*(f*x^2+e)/x^4,x)