\(\int (d+e x)^{-1-2 p} (e+f x) (a+c x^2)^p \, dx\) [307]

Optimal result

Integrand size = 26, antiderivative size = 325 \[ \int (d+e x)^{-1-2 p} (e+f x) \left (a+c x^2\right )^p \, dx=\frac {f (d+e x)^{1-2 p} \left (a+c x^2\right )^p \left (1-\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{-p} \left (1-\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{-p} \operatorname {AppellF1}\left (1-2 p,-p,-p,2-2 p,\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}},\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )}{e^2 (1-2 p)}-\frac {\left (e^2-d f\right ) (d+e x)^{-2 p} \left (a+c x^2\right )^p \left (1-\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{-p} \left (1-\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{-p} \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}},\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )}{2 e^2 p} \] Output:

f*(e*x+d)^(1-2*p)*(c*x^2+a)^p*AppellF1(1-2*p,-p,-p,2-2*p,(e*x+d)/(d-(-a)^( 
1/2)*e/c^(1/2)),(e*x+d)/(d+(-a)^(1/2)*e/c^(1/2)))/e^2/(1-2*p)/((1-(e*x+d)/ 
(d-(-a)^(1/2)*e/c^(1/2)))^p)/((1-(e*x+d)/(d+(-a)^(1/2)*e/c^(1/2)))^p)-1/2* 
(-d*f+e^2)*(c*x^2+a)^p*AppellF1(-2*p,-p,-p,1-2*p,(e*x+d)/(d-(-a)^(1/2)*e/c 
^(1/2)),(e*x+d)/(d+(-a)^(1/2)*e/c^(1/2)))/e^2/p/((e*x+d)^(2*p))/((1-(e*x+d 
)/(d-(-a)^(1/2)*e/c^(1/2)))^p)/((1-(e*x+d)/(d+(-a)^(1/2)*e/c^(1/2)))^p)
 

Mathematica [A] (warning: unable to verify)

Time = 0.29 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.78 \[ \int (d+e x)^{-1-2 p} (e+f x) \left (a+c x^2\right )^p \, dx=-\frac {\left (\frac {e \left (\sqrt {-\frac {a}{c}}-x\right )}{d+\sqrt {-\frac {a}{c}} e}\right )^{-p} \left (\frac {e \left (\sqrt {-\frac {a}{c}}+x\right )}{-d+\sqrt {-\frac {a}{c}} e}\right )^{-p} (d+e x)^{-2 p} \left (a+c x^2\right )^p \left (2 f p (d+e x) \operatorname {AppellF1}\left (1-2 p,-p,-p,2-2 p,\frac {d+e x}{d-\sqrt {-\frac {a}{c}} e},\frac {d+e x}{d+\sqrt {-\frac {a}{c}} e}\right )+\left (e^2-d f\right ) (-1+2 p) \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,\frac {d+e x}{d-\sqrt {-\frac {a}{c}} e},\frac {d+e x}{d+\sqrt {-\frac {a}{c}} e}\right )\right )}{2 e^2 p (-1+2 p)} \] Input:

Integrate[(d + e*x)^(-1 - 2*p)*(e + f*x)*(a + c*x^2)^p,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {719, 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.

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

\(\Big \downarrow \) 719

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

\(\Big \downarrow \) 514

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

\(\Big \downarrow \) 150

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

Input:

Int[(d + e*x)^(-1 - 2*p)*(e + f*x)*(a + c*x^2)^p,x]
 

Output:

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

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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