\(\int x^2 (c+d x)^{-3-2 p} (a+b x^2)^p \, dx\) [362]

Optimal result

Integrand size = 24, antiderivative size = 443 \[ \int x^2 (c+d x)^{-3-2 p} \left (a+b x^2\right )^p \, dx=-\frac {c^2 (c+d x)^{-2 (1+p)} \left (a+b x^2\right )^{1+p}}{2 d \left (b c^2+a d^2\right ) (1+p)}-\frac {(c+d x)^{-2 p} \left (a+b x^2\right )^p \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{2 d^3 p}+\frac {c \left (b c^2+2 a d^2\right ) \left (\sqrt {-a}-\sqrt {b} x\right ) \left (-\frac {\left (\sqrt {b} c+\sqrt {-a} d\right ) \left (\sqrt {-a}+\sqrt {b} x\right )}{\left (\sqrt {b} c-\sqrt {-a} d\right ) \left (\sqrt {-a}-\sqrt {b} x\right )}\right )^{-p} (c+d x)^{-1-2 p} \left (a+b x^2\right )^p \operatorname {Hypergeometric2F1}\left (-1-2 p,-p,-2 p,\frac {2 \sqrt {-a} \sqrt {b} (c+d x)}{\left (\sqrt {b} c-\sqrt {-a} d\right ) \left (\sqrt {-a}-\sqrt {b} x\right )}\right )}{d^2 \left (\sqrt {b} c+\sqrt {-a} d\right ) \left (b c^2+a d^2\right ) (1+2 p)} \] Output:

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

Mathematica [F]

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

Integrate[x^2*(c + d*x)^(-3 - 2*p)*(a + b*x^2)^p,x]
 

Output:

Integrate[x^2*(c + d*x)^(-3 - 2*p)*(a + b*x^2)^p, x]
 

Rubi [A] (verified)

Time = 1.09 (sec) , antiderivative size = 649, normalized size of antiderivative = 1.47, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {624, 588, 489, 624, 489, 514, 150}

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.

\(\displaystyle \int x^2 \left (a+b x^2\right )^p (c+d x)^{-2 p-3} \, dx\)

\(\Big \downarrow \) 624

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

\(\Big \downarrow \) 588

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

\(\Big \downarrow \) 489

\(\displaystyle \frac {\int x (c+d x)^{-2 (p+1)} \left (b x^2+a\right )^pdx}{d}-\frac {c \left (\frac {c \left (a+b x^2\right )^{p+1} (c+d x)^{-2 (p+1)}}{2 (p+1) \left (a d^2+b c^2\right )}-\frac {a d \left (\sqrt {-a}-\sqrt {b} x\right ) \left (a+b x^2\right )^p (c+d x)^{-2 p-1} \left (-\frac {\left (\sqrt {-a}+\sqrt {b} x\right ) \left (\sqrt {-a} d+\sqrt {b} c\right )}{\left (\sqrt {-a}-\sqrt {b} x\right ) \left (\sqrt {b} c-\sqrt {-a} d\right )}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-2 p-1,-p,-2 p,\frac {2 \sqrt {-a} \sqrt {b} (c+d x)}{\left (\sqrt {b} c-\sqrt {-a} d\right ) \left (\sqrt {-a}-\sqrt {b} x\right )}\right )}{(2 p+1) \left (\sqrt {-a} d+\sqrt {b} c\right ) \left (a d^2+b c^2\right )}\right )}{d}\)

\(\Big \downarrow \) 624

\(\displaystyle \frac {\frac {\int (c+d x)^{-2 p-1} \left (b x^2+a\right )^pdx}{d}-\frac {c \int (c+d x)^{-2 (p+1)} \left (b x^2+a\right )^pdx}{d}}{d}-\frac {c \left (\frac {c \left (a+b x^2\right )^{p+1} (c+d x)^{-2 (p+1)}}{2 (p+1) \left (a d^2+b c^2\right )}-\frac {a d \left (\sqrt {-a}-\sqrt {b} x\right ) \left (a+b x^2\right )^p (c+d x)^{-2 p-1} \left (-\frac {\left (\sqrt {-a}+\sqrt {b} x\right ) \left (\sqrt {-a} d+\sqrt {b} c\right )}{\left (\sqrt {-a}-\sqrt {b} x\right ) \left (\sqrt {b} c-\sqrt {-a} d\right )}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-2 p-1,-p,-2 p,\frac {2 \sqrt {-a} \sqrt {b} (c+d x)}{\left (\sqrt {b} c-\sqrt {-a} d\right ) \left (\sqrt {-a}-\sqrt {b} x\right )}\right )}{(2 p+1) \left (\sqrt {-a} d+\sqrt {b} c\right ) \left (a d^2+b c^2\right )}\right )}{d}\)

\(\Big \downarrow \) 489

\(\displaystyle \frac {\frac {\int (c+d x)^{-2 p-1} \left (b x^2+a\right )^pdx}{d}+\frac {c \left (\sqrt {-a}-\sqrt {b} x\right ) \left (a+b x^2\right )^p (c+d x)^{-2 p-1} \left (-\frac {\left (\sqrt {-a}+\sqrt {b} x\right ) \left (\sqrt {-a} d+\sqrt {b} c\right )}{\left (\sqrt {-a}-\sqrt {b} x\right ) \left (\sqrt {b} c-\sqrt {-a} d\right )}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-2 p-1,-p,-2 p,\frac {2 \sqrt {-a} \sqrt {b} (c+d x)}{\left (\sqrt {b} c-\sqrt {-a} d\right ) \left (\sqrt {-a}-\sqrt {b} x\right )}\right )}{d (2 p+1) \left (\sqrt {-a} d+\sqrt {b} c\right )}}{d}-\frac {c \left (\frac {c \left (a+b x^2\right )^{p+1} (c+d x)^{-2 (p+1)}}{2 (p+1) \left (a d^2+b c^2\right )}-\frac {a d \left (\sqrt {-a}-\sqrt {b} x\right ) \left (a+b x^2\right )^p (c+d x)^{-2 p-1} \left (-\frac {\left (\sqrt {-a}+\sqrt {b} x\right ) \left (\sqrt {-a} d+\sqrt {b} c\right )}{\left (\sqrt {-a}-\sqrt {b} x\right ) \left (\sqrt {b} c-\sqrt {-a} d\right )}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-2 p-1,-p,-2 p,\frac {2 \sqrt {-a} \sqrt {b} (c+d x)}{\left (\sqrt {b} c-\sqrt {-a} d\right ) \left (\sqrt {-a}-\sqrt {b} x\right )}\right )}{(2 p+1) \left (\sqrt {-a} d+\sqrt {b} c\right ) \left (a d^2+b c^2\right )}\right )}{d}\)

\(\Big \downarrow \) 514

\(\displaystyle \frac {\frac {\left (a+b x^2\right )^p \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{\frac {\sqrt {-a} d}{\sqrt {b}}+c}\right )^{-p} \int (c+d x)^{-2 p-1} \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^p \left (1-\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^pd(c+d x)}{d^2}+\frac {c \left (\sqrt {-a}-\sqrt {b} x\right ) \left (a+b x^2\right )^p (c+d x)^{-2 p-1} \left (-\frac {\left (\sqrt {-a}+\sqrt {b} x\right ) \left (\sqrt {-a} d+\sqrt {b} c\right )}{\left (\sqrt {-a}-\sqrt {b} x\right ) \left (\sqrt {b} c-\sqrt {-a} d\right )}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-2 p-1,-p,-2 p,\frac {2 \sqrt {-a} \sqrt {b} (c+d x)}{\left (\sqrt {b} c-\sqrt {-a} d\right ) \left (\sqrt {-a}-\sqrt {b} x\right )}\right )}{d (2 p+1) \left (\sqrt {-a} d+\sqrt {b} c\right )}}{d}-\frac {c \left (\frac {c \left (a+b x^2\right )^{p+1} (c+d x)^{-2 (p+1)}}{2 (p+1) \left (a d^2+b c^2\right )}-\frac {a d \left (\sqrt {-a}-\sqrt {b} x\right ) \left (a+b x^2\right )^p (c+d x)^{-2 p-1} \left (-\frac {\left (\sqrt {-a}+\sqrt {b} x\right ) \left (\sqrt {-a} d+\sqrt {b} c\right )}{\left (\sqrt {-a}-\sqrt {b} x\right ) \left (\sqrt {b} c-\sqrt {-a} d\right )}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-2 p-1,-p,-2 p,\frac {2 \sqrt {-a} \sqrt {b} (c+d x)}{\left (\sqrt {b} c-\sqrt {-a} d\right ) \left (\sqrt {-a}-\sqrt {b} x\right )}\right )}{(2 p+1) \left (\sqrt {-a} d+\sqrt {b} c\right ) \left (a d^2+b c^2\right )}\right )}{d}\)

\(\Big \downarrow \) 150

\(\displaystyle \frac {\frac {c \left (\sqrt {-a}-\sqrt {b} x\right ) \left (a+b x^2\right )^p (c+d x)^{-2 p-1} \left (-\frac {\left (\sqrt {-a}+\sqrt {b} x\right ) \left (\sqrt {-a} d+\sqrt {b} c\right )}{\left (\sqrt {-a}-\sqrt {b} x\right ) \left (\sqrt {b} c-\sqrt {-a} d\right )}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-2 p-1,-p,-2 p,\frac {2 \sqrt {-a} \sqrt {b} (c+d x)}{\left (\sqrt {b} c-\sqrt {-a} d\right ) \left (\sqrt {-a}-\sqrt {b} x\right )}\right )}{d (2 p+1) \left (\sqrt {-a} d+\sqrt {b} c\right )}-\frac {\left (a+b x^2\right )^p (c+d x)^{-2 p} \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{\frac {\sqrt {-a} d}{\sqrt {b}}+c}\right )^{-p} \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{2 d^2 p}}{d}-\frac {c \left (\frac {c \left (a+b x^2\right )^{p+1} (c+d x)^{-2 (p+1)}}{2 (p+1) \left (a d^2+b c^2\right )}-\frac {a d \left (\sqrt {-a}-\sqrt {b} x\right ) \left (a+b x^2\right )^p (c+d x)^{-2 p-1} \left (-\frac {\left (\sqrt {-a}+\sqrt {b} x\right ) \left (\sqrt {-a} d+\sqrt {b} c\right )}{\left (\sqrt {-a}-\sqrt {b} x\right ) \left (\sqrt {b} c-\sqrt {-a} d\right )}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-2 p-1,-p,-2 p,\frac {2 \sqrt {-a} \sqrt {b} (c+d x)}{\left (\sqrt {b} c-\sqrt {-a} d\right ) \left (\sqrt {-a}-\sqrt {b} x\right )}\right )}{(2 p+1) \left (\sqrt {-a} d+\sqrt {b} c\right ) \left (a d^2+b c^2\right )}\right )}{d}\)

Input:

Int[x^2*(c + d*x)^(-3 - 2*p)*(a + b*x^2)^p,x]
 

Output:

(-1/2*((a + b*x^2)^p*AppellF1[-2*p, -p, -p, 1 - 2*p, (c + d*x)/(c - (Sqrt[ 
-a]*d)/Sqrt[b]), (c + d*x)/(c + (Sqrt[-a]*d)/Sqrt[b])])/(d^2*p*(c + d*x)^( 
2*p)*(1 - (c + d*x)/(c - (Sqrt[-a]*d)/Sqrt[b]))^p*(1 - (c + d*x)/(c + (Sqr 
t[-a]*d)/Sqrt[b]))^p) + (c*(Sqrt[-a] - Sqrt[b]*x)*(c + d*x)^(-1 - 2*p)*(a 
+ b*x^2)^p*Hypergeometric2F1[-1 - 2*p, -p, -2*p, (2*Sqrt[-a]*Sqrt[b]*(c + 
d*x))/((Sqrt[b]*c - Sqrt[-a]*d)*(Sqrt[-a] - Sqrt[b]*x))])/(d*(Sqrt[b]*c + 
Sqrt[-a]*d)*(1 + 2*p)*(-(((Sqrt[b]*c + Sqrt[-a]*d)*(Sqrt[-a] + Sqrt[b]*x)) 
/((Sqrt[b]*c - Sqrt[-a]*d)*(Sqrt[-a] - Sqrt[b]*x))))^p))/d - (c*((c*(a + b 
*x^2)^(1 + p))/(2*(b*c^2 + a*d^2)*(1 + p)*(c + d*x)^(2*(1 + p))) - (a*d*(S 
qrt[-a] - Sqrt[b]*x)*(c + d*x)^(-1 - 2*p)*(a + b*x^2)^p*Hypergeometric2F1[ 
-1 - 2*p, -p, -2*p, (2*Sqrt[-a]*Sqrt[b]*(c + d*x))/((Sqrt[b]*c - Sqrt[-a]* 
d)*(Sqrt[-a] - Sqrt[b]*x))])/((Sqrt[b]*c + Sqrt[-a]*d)*(b*c^2 + a*d^2)*(1 
+ 2*p)*(-(((Sqrt[b]*c + Sqrt[-a]*d)*(Sqrt[-a] + Sqrt[b]*x))/((Sqrt[b]*c - 
Sqrt[-a]*d)*(Sqrt[-a] - Sqrt[b]*x))))^p)))/d
 

Defintions of rubi rules used

rule 150

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ 
] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 
, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] &&  !In 
tegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
 

rule 489

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ 
{q = Rt[(-a)*b, 2]}, Simp[(q - b*x)*(c + d*x)^(n + 1)*((a + b*x^2)^p/((n + 
1)*(b*c + d*q)*((b*c + d*q)*((q + b*x)/((b*c - d*q)*(-q + b*x))))^p))*Hyper 
geometric2F1[n + 1, -p, n + 2, 2*b*q*((c + d*x)/((b*c - d*q)*(q - b*x)))], 
x]] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[n + 2*p + 2, 0]
 

rule 514

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ 
{q = Rt[-a/b, 2]}, Simp[(a + b*x^2)^p/(d*(1 - (c + d*x)/(c - d*q))^p*(1 - ( 
c + d*x)/(c + d*q))^p)   Subst[Int[x^n*Simp[1 - x/(c + d*q), x]^p*Simp[1 - 
x/(c - d*q), x]^p, x], x, c + d*x], x]] /; FreeQ[{a, b, c, d, n, p}, x] && 
NeQ[b*c^2 + a*d^2, 0]
 

rule 588

Int[(x_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
 Simp[c*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/(2*(p + 1)*(b*c^2 + a*d^2))) 
, x] + Simp[a*(d/(b*c^2 + a*d^2))   Int[(c + d*x)^(n + 1)*(a + b*x^2)^p, x] 
, x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[Simplify[n + 2*p + 3], 0] && Ne 
Q[b*c^2 + a*d^2, 0]
 

rule 624

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> Simp[1/d   Int[x^(m - 1)*(c + d*x)^(n + 1)*(a + b*x^2)^p, x], x] - Si 
mp[c/d   Int[x^(m - 1)*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, 
 d, n, p}, x] && IGtQ[m, 0]
 

Maple [F]

\[\int x^{2} \left (d x +c \right )^{-3-2 p} \left (b \,x^{2}+a \right )^{p}d x\]

Input:

int(x^2*(d*x+c)^(-3-2*p)*(b*x^2+a)^p,x)
 

Output:

int(x^2*(d*x+c)^(-3-2*p)*(b*x^2+a)^p,x)
 

Fricas [F]

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

integrate(x^2*(d*x+c)^(-3-2*p)*(b*x^2+a)^p,x, algorithm="fricas")
 

Output:

integral((b*x^2 + a)^p*(d*x + c)^(-2*p - 3)*x^2, x)
 

Sympy [F]

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

integrate(x**2*(d*x+c)**(-3-2*p)*(b*x**2+a)**p,x)
 

Output:

Integral(x**2*(a + b*x**2)**p*(c + d*x)**(-2*p - 3), x)
 

Maxima [F]

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

integrate(x^2*(d*x+c)^(-3-2*p)*(b*x^2+a)^p,x, algorithm="maxima")
 

Output:

integrate((b*x^2 + a)^p*(d*x + c)^(-2*p - 3)*x^2, x)
 

Giac [F]

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

integrate(x^2*(d*x+c)^(-3-2*p)*(b*x^2+a)^p,x, algorithm="giac")
 

Output:

integrate((b*x^2 + a)^p*(d*x + c)^(-2*p - 3)*x^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int x^2 (c+d x)^{-3-2 p} \left (a+b x^2\right )^p \, dx=\int \frac {x^2\,{\left (b\,x^2+a\right )}^p}{{\left (c+d\,x\right )}^{2\,p+3}} \,d x \] Input:

int((x^2*(a + b*x^2)^p)/(c + d*x)^(2*p + 3),x)
 

Output:

int((x^2*(a + b*x^2)^p)/(c + d*x)^(2*p + 3), x)
 

Reduce [F]

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

int(x^2*(d*x+c)^(-3-2*p)*(b*x^2+a)^p,x)
 

Output:

int(((a + b*x**2)**p*x**2)/((c + d*x)**(2*p)*c**3 + 3*(c + d*x)**(2*p)*c** 
2*d*x + 3*(c + d*x)**(2*p)*c*d**2*x**2 + (c + d*x)**(2*p)*d**3*x**3),x)