\(\int (c+d x)^{-1-2 p} (a+b x^2)^p \, dx\) [364]

Optimal result

Integrand size = 21, antiderivative size = 155 \[ \int (c+d x)^{-1-2 p} \left (a+b x^2\right )^p \, dx=-\frac {(c+d x)^{-2 p} \left (a+b x^2\right )^p \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{2 d p} \] Output:

-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/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)
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.03 \[ \int (c+d x)^{-1-2 p} \left (a+b x^2\right )^p \, dx=-\frac {\left (\frac {d \left (\sqrt {-\frac {a}{b}}-x\right )}{c+\sqrt {-\frac {a}{b}} d}\right )^{-p} \left (\frac {d \left (\sqrt {-\frac {a}{b}}+x\right )}{-c+\sqrt {-\frac {a}{b}} d}\right )^{-p} (c+d x)^{-2 p} \left (a+b x^2\right )^p \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,\frac {c+d x}{c-\sqrt {-\frac {a}{b}} d},\frac {c+d x}{c+\sqrt {-\frac {a}{b}} d}\right )}{2 d p} \] Input:

Integrate[(c + d*x)^(-1 - 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/b)]*d), (c + d*x)/(c + Sqrt[-(a/b)]*d)])/(d*p*((d*(Sqrt[-(a/b)] - x))/(c 
 + Sqrt[-(a/b)]*d))^p*((d*(Sqrt[-(a/b)] + x))/(-c + Sqrt[-(a/b)]*d))^p*(c 
+ d*x)^(2*p))
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {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.

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

\(\Big \downarrow \) 514

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

\(\Big \downarrow \) 150

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

Input:

Int[(c + d*x)^(-1 - 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*p*(c + d*x)^(2*p 
)*(1 - (c + d*x)/(c - (Sqrt[-a]*d)/Sqrt[b]))^p*(1 - (c + d*x)/(c + (Sqrt[- 
a]*d)/Sqrt[b]))^p)
 

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 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]
 

Maple [F]

\[\int \left (d x +c \right )^{-1-2 p} \left (b \,x^{2}+a \right )^{p}d x\]

Input:

int((d*x+c)^(-1-2*p)*(b*x^2+a)^p,x)
 

Output:

int((d*x+c)^(-1-2*p)*(b*x^2+a)^p,x)
 

Fricas [F]

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

integrate((d*x+c)^(-1-2*p)*(b*x^2+a)^p,x, algorithm="fricas")
 

Output:

integral((b*x^2 + a)^p*(d*x + c)^(-2*p - 1), x)
 

Sympy [F]

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

integrate((d*x+c)**(-1-2*p)*(b*x**2+a)**p,x)
 

Output:

Integral((a + b*x**2)**p*(c + d*x)**(-2*p - 1), x)
 

Maxima [F]

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

integrate((d*x+c)^(-1-2*p)*(b*x^2+a)^p,x, algorithm="maxima")
 

Output:

integrate((b*x^2 + a)^p*(d*x + c)^(-2*p - 1), x)
 

Giac [F]

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

integrate((d*x+c)^(-1-2*p)*(b*x^2+a)^p,x, algorithm="giac")
 

Output:

integrate((b*x^2 + a)^p*(d*x + c)^(-2*p - 1), x)
 

Mupad [F(-1)]

Timed out. \[ \int (c+d x)^{-1-2 p} \left (a+b x^2\right )^p \, dx=\int \frac {{\left (b\,x^2+a\right )}^p}{{\left (c+d\,x\right )}^{2\,p+1}} \,d x \] Input:

int((a + b*x^2)^p/(c + d*x)^(2*p + 1),x)
 

Output:

int((a + b*x^2)^p/(c + d*x)^(2*p + 1), x)
 

Reduce [F]

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

int((d*x+c)^(-1-2*p)*(b*x^2+a)^p,x)
 

Output:

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