\(\int (c+d x)^{-2-2 p} (e+f x) (a+b x^2)^p \, dx\) [179]

Optimal result

Integrand size = 26, antiderivative size = 377 \[ \int (c+d x)^{-2-2 p} (e+f x) \left (a+b x^2\right )^p \, dx=-\frac {f (c+d x)^{-2 p} \left (a+b x^2\right )^p \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{2 d^2 p}-\frac {(d e-c f) \left (\sqrt {-a}-\sqrt {b} x\right ) \left (-\frac {\left (\sqrt {b} c+\sqrt {-a} d\right ) \left (\sqrt {-a}+\sqrt {b} x\right )}{\left (\sqrt {b} c-\sqrt {-a} d\right ) \left (\sqrt {-a}-\sqrt {b} x\right )}\right )^{-p} (c+d x)^{-1-2 p} \left (a+b x^2\right )^p \operatorname {Hypergeometric2F1}\left (-1-2 p,-p,-2 p,\frac {2 \sqrt {-a} \sqrt {b} (c+d x)}{\left (\sqrt {b} c-\sqrt {-a} d\right ) \left (\sqrt {-a}-\sqrt {b} x\right )}\right )}{d \left (\sqrt {b} c+\sqrt {-a} d\right ) (1+2 p)} \] Output:

-1/2*f*(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^2/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*f+ 
d*e)*((-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/(b^(1/2)*c+(-a)^(1/2)*d)/(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]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.88 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {719, 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.

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

\(\Big \downarrow \) 719

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

\(\Big \downarrow \) 489

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

\(\Big \downarrow \) 514

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

\(\Big \downarrow \) 150

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

Input:

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

Output:

-1/2*(f*(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 + (Sq 
rt[-a]*d)/Sqrt[b]))^p) - ((d*e - c*f)*(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]*Sq 
rt[b]*(c + d*x))/((Sqrt[b]*c - Sqrt[-a]*d)*(Sqrt[-a] - Sqrt[b]*x))])/(d*(S 
qrt[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)
 

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 719

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + 
Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, 
d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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