\(\int (a+b x^2)^p (c+d x^2)^{-\frac {5}{2}-p} (e+f x^2) \, dx\) [571]

Optimal result

Integrand size = 32, antiderivative size = 197 \[ \int \left (a+b x^2\right )^p \left (c+d x^2\right )^{-\frac {5}{2}-p} \left (e+f x^2\right ) \, dx=-\frac {(b e-a f) x \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^{-\frac {3}{2}-p}}{2 a (b c-a d) (1+p)}-\frac {(a c f+2 a d e (1+p)-b c e (3+2 p)) x \left (a+b x^2\right )^{1+p} \left (\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}\right )^{-1-p} \left (c+d x^2\right )^{-\frac {3}{2}-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-1-p,\frac {3}{2},-\frac {(b c-a d) x^2}{a \left (c+d x^2\right )}\right )}{2 a c (b c-a d) (1+p)} \] Output:

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

Mathematica [F]

\[ \int \left (a+b x^2\right )^p \left (c+d x^2\right )^{-\frac {5}{2}-p} \left (e+f x^2\right ) \, dx=\int \left (a+b x^2\right )^p \left (c+d x^2\right )^{-\frac {5}{2}-p} \left (e+f x^2\right ) \, dx \] Input:

Integrate[(a + b*x^2)^p*(c + d*x^2)^(-5/2 - p)*(e + f*x^2),x]
 

Output:

Integrate[(a + b*x^2)^p*(c + d*x^2)^(-5/2 - p)*(e + f*x^2), x]
 

Rubi [A] (warning: unable to verify)

Time = 0.44 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.41, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {406, 296, 294, 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.

\(\displaystyle \int \left (e+f x^2\right ) \left (a+b x^2\right )^p \left (c+d x^2\right )^{-p-\frac {5}{2}} \, dx\)

\(\Big \downarrow \) 406

\(\displaystyle e \int \left (b x^2+a\right )^p \left (d x^2+c\right )^{-p-\frac {5}{2}}dx+f \int x^2 \left (b x^2+a\right )^p \left (d x^2+c\right )^{-p-\frac {5}{2}}dx\)

\(\Big \downarrow \) 296

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

\(\Big \downarrow \) 294

\(\displaystyle f \int x^2 \left (b x^2+a\right )^p \left (d x^2+c\right )^{-p-\frac {5}{2}}dx+e \left (\frac {x \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{-p-\frac {3}{2}} \left (\frac {b c}{(p+1) (b c-a d)}+2\right ) \left (\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}\right )^{-p-1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p-1,\frac {3}{2},-\frac {(b c-a d) x^2}{a \left (d x^2+c\right )}\right )}{2 a c}-\frac {b x \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{-p-\frac {3}{2}}}{2 a (p+1) (b c-a d)}\right )\)

\(\Big \downarrow \) 395

\(\displaystyle f \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \int x^2 \left (\frac {b x^2}{a}+1\right )^p \left (d x^2+c\right )^{-p-\frac {5}{2}}dx+e \left (\frac {x \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{-p-\frac {3}{2}} \left (\frac {b c}{(p+1) (b c-a d)}+2\right ) \left (\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}\right )^{-p-1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p-1,\frac {3}{2},-\frac {(b c-a d) x^2}{a \left (d x^2+c\right )}\right )}{2 a c}-\frac {b x \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{-p-\frac {3}{2}}}{2 a (p+1) (b c-a d)}\right )\)

\(\Big \downarrow \) 395

\(\displaystyle \frac {f \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (\frac {d x^2}{c}+1\right )^{p+\frac {1}{2}} \left (c+d x^2\right )^{-p-\frac {1}{2}} \int x^2 \left (\frac {b x^2}{a}+1\right )^p \left (\frac {d x^2}{c}+1\right )^{-p-\frac {5}{2}}dx}{c^2}+e \left (\frac {x \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{-p-\frac {3}{2}} \left (\frac {b c}{(p+1) (b c-a d)}+2\right ) \left (\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}\right )^{-p-1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p-1,\frac {3}{2},-\frac {(b c-a d) x^2}{a \left (d x^2+c\right )}\right )}{2 a c}-\frac {b x \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{-p-\frac {3}{2}}}{2 a (p+1) (b c-a d)}\right )\)

\(\Big \downarrow \) 394

\(\displaystyle \frac {f x^3 \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (\frac {d x^2}{c}+1\right )^{p-1} \left (c+d x^2\right )^{-p-\frac {1}{2}} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-p,\frac {5}{2},-\frac {c \left (\frac {b x^2}{a}-\frac {d x^2}{c}\right )}{d x^2+c}\right )}{3 c^2}+e \left (\frac {x \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{-p-\frac {3}{2}} \left (\frac {b c}{(p+1) (b c-a d)}+2\right ) \left (\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}\right )^{-p-1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p-1,\frac {3}{2},-\frac {(b c-a d) x^2}{a \left (d x^2+c\right )}\right )}{2 a c}-\frac {b x \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{-p-\frac {3}{2}}}{2 a (p+1) (b c-a d)}\right )\)

Input:

Int[(a + b*x^2)^p*(c + d*x^2)^(-5/2 - p)*(e + f*x^2),x]
 

Output:

e*(-1/2*(b*x*(a + b*x^2)^(1 + p)*(c + d*x^2)^(-3/2 - p))/(a*(b*c - a*d)*(1 
 + p)) + ((2 + (b*c)/((b*c - a*d)*(1 + p)))*x*(a + b*x^2)^(1 + p)*((c*(a + 
 b*x^2))/(a*(c + d*x^2)))^(-1 - p)*(c + d*x^2)^(-3/2 - p)*Hypergeometric2F 
1[1/2, -1 - p, 3/2, -(((b*c - a*d)*x^2)/(a*(c + d*x^2)))])/(2*a*c)) + (f*x 
^3*(a + b*x^2)^p*(c + d*x^2)^(-1/2 - p)*(1 + (d*x^2)/c)^(-1 + p)*Hypergeom 
etric2F1[3/2, -p, 5/2, -((c*((b*x^2)/a - (d*x^2)/c))/(c + d*x^2))])/(3*c^2 
*(1 + (b*x^2)/a)^p)
 

Defintions of rubi rules used

rule 294

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[x*((a + b*x^2)^p/(c*(c*((a + b*x^2)/(a*(c + d*x^2))))^p*(c + d*x^2)^(1/2 
+ p)))*Hypergeometric2F1[1/2, -p, 3/2, (-(b*c - a*d))*(x^2/(a*(c + d*x^2))) 
], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[2*(p + q 
+ 1) + 1, 0]
 

rule 296

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

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]
 

Maple [F]

\[\int \left (b \,x^{2}+a \right )^{p} \left (x^{2} d +c \right )^{-\frac {5}{2}-p} \left (f \,x^{2}+e \right )d x\]

Input:

int((b*x^2+a)^p*(d*x^2+c)^(-5/2-p)*(f*x^2+e),x)
 

Output:

int((b*x^2+a)^p*(d*x^2+c)^(-5/2-p)*(f*x^2+e),x)
 

Fricas [F]

\[ \int \left (a+b x^2\right )^p \left (c+d x^2\right )^{-\frac {5}{2}-p} \left (e+f x^2\right ) \, dx=\int { {\left (f x^{2} + e\right )} {\left (b x^{2} + a\right )}^{p} {\left (d x^{2} + c\right )}^{-p - \frac {5}{2}} \,d x } \] Input:

integrate((b*x^2+a)^p*(d*x^2+c)^(-5/2-p)*(f*x^2+e),x, algorithm="fricas")
 

Output:

integral((f*x^2 + e)*(b*x^2 + a)^p*(d*x^2 + c)^(-p - 5/2), x)
 

Sympy [F(-1)]

Timed out. \[ \int \left (a+b x^2\right )^p \left (c+d x^2\right )^{-\frac {5}{2}-p} \left (e+f x^2\right ) \, dx=\text {Timed out} \] Input:

integrate((b*x**2+a)**p*(d*x**2+c)**(-5/2-p)*(f*x**2+e),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \left (a+b x^2\right )^p \left (c+d x^2\right )^{-\frac {5}{2}-p} \left (e+f x^2\right ) \, dx=\int { {\left (f x^{2} + e\right )} {\left (b x^{2} + a\right )}^{p} {\left (d x^{2} + c\right )}^{-p - \frac {5}{2}} \,d x } \] Input:

integrate((b*x^2+a)^p*(d*x^2+c)^(-5/2-p)*(f*x^2+e),x, algorithm="maxima")
 

Output:

integrate((f*x^2 + e)*(b*x^2 + a)^p*(d*x^2 + c)^(-p - 5/2), x)
 

Giac [F]

\[ \int \left (a+b x^2\right )^p \left (c+d x^2\right )^{-\frac {5}{2}-p} \left (e+f x^2\right ) \, dx=\int { {\left (f x^{2} + e\right )} {\left (b x^{2} + a\right )}^{p} {\left (d x^{2} + c\right )}^{-p - \frac {5}{2}} \,d x } \] Input:

integrate((b*x^2+a)^p*(d*x^2+c)^(-5/2-p)*(f*x^2+e),x, algorithm="giac")
 

Output:

integrate((f*x^2 + e)*(b*x^2 + a)^p*(d*x^2 + c)^(-p - 5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \left (a+b x^2\right )^p \left (c+d x^2\right )^{-\frac {5}{2}-p} \left (e+f x^2\right ) \, dx=\int \frac {{\left (b\,x^2+a\right )}^p\,\left (f\,x^2+e\right )}{{\left (d\,x^2+c\right )}^{p+\frac {5}{2}}} \,d x \] Input:

int(((a + b*x^2)^p*(e + f*x^2))/(c + d*x^2)^(p + 5/2),x)
 

Output:

int(((a + b*x^2)^p*(e + f*x^2))/(c + d*x^2)^(p + 5/2), x)
 

Reduce [F]

\[ \int \left (a+b x^2\right )^p \left (c+d x^2\right )^{-\frac {5}{2}-p} \left (e+f x^2\right ) \, dx=\left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{\left (d \,x^{2}+c \right )^{p +\frac {1}{2}} c^{2}+2 \left (d \,x^{2}+c \right )^{p +\frac {1}{2}} c d \,x^{2}+\left (d \,x^{2}+c \right )^{p +\frac {1}{2}} d^{2} x^{4}}d x \right ) e +\left (\int \frac {\left (b \,x^{2}+a \right )^{p} x^{2}}{\left (d \,x^{2}+c \right )^{p +\frac {1}{2}} c^{2}+2 \left (d \,x^{2}+c \right )^{p +\frac {1}{2}} c d \,x^{2}+\left (d \,x^{2}+c \right )^{p +\frac {1}{2}} d^{2} x^{4}}d x \right ) f \] Input:

int((b*x^2+a)^p*(d*x^2+c)^(-5/2-p)*(f*x^2+e),x)
 

Output:

int((a + b*x**2)**p/((c + d*x**2)**((2*p + 1)/2)*c**2 + 2*(c + d*x**2)**(( 
2*p + 1)/2)*c*d*x**2 + (c + d*x**2)**((2*p + 1)/2)*d**2*x**4),x)*e + int(( 
(a + b*x**2)**p*x**2)/((c + d*x**2)**((2*p + 1)/2)*c**2 + 2*(c + d*x**2)** 
((2*p + 1)/2)*c*d*x**2 + (c + d*x**2)**((2*p + 1)/2)*d**2*x**4),x)*f