3.8.43 \(\int (d+e x)^{-4-2 p} (a+c x^2)^p \, dx\) [743]

3.8.43.1 Optimal result

Integrand size = 21, antiderivative size = 347 \[ \int (d+e x)^{-4-2 p} \left (a+c x^2\right )^p \, dx=-\frac {e (d+e x)^{-3-2 p} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right ) (3+2 p)}-\frac {c d e (2+p) (d+e x)^{-2 (1+p)} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right )^2 (1+p) (3+2 p)}+\frac {c \left (a e^2-c d^2 (3+2 p)\right ) \left (\sqrt {-a}-\sqrt {c} x\right ) \left (-\frac {\left (\sqrt {c} d+\sqrt {-a} e\right ) \left (\sqrt {-a}+\sqrt {c} x\right )}{\left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\sqrt {-a}-\sqrt {c} x\right )}\right )^{-p} (d+e x)^{-1-2 p} \left (a+c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-1-2 p,-p,-2 p,\frac {2 \sqrt {-a} \sqrt {c} (d+e x)}{\left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\sqrt {-a}-\sqrt {c} x\right )}\right )}{\left (\sqrt {c} d+\sqrt {-a} e\right ) \left (c d^2+a e^2\right )^2 (1+2 p) (3+2 p)} \]

output

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

3.8.43.2 Mathematica [F]

\[ \int (d+e x)^{-4-2 p} \left (a+c x^2\right )^p \, dx=\int (d+e x)^{-4-2 p} \left (a+c x^2\right )^p \, dx \]

input

Integrate[(d + e*x)^(-4 - 2*p)*(a + c*x^2)^p,x]
 

output

Integrate[(d + e*x)^(-4 - 2*p)*(a + c*x^2)^p, x]
 

3.8.43.3 Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {498, 25, 679, 489}

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.

\(\displaystyle \int \left (a+c x^2\right )^p (d+e x)^{-2 p-4} \, dx\)

\(\Big \downarrow \) 498

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 679

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

\(\Big \downarrow \) 489

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

input

Int[(d + e*x)^(-4 - 2*p)*(a + c*x^2)^p,x]
 

output

-((e*(d + e*x)^(-3 - 2*p)*(a + c*x^2)^(1 + p))/((c*d^2 + a*e^2)*(3 + 2*p)) 
) + (c*(-((d*e*(2 + p)*(a + c*x^2)^(1 + p))/((c*d^2 + a*e^2)*(1 + p)*(d + 
e*x)^(2*(1 + p)))) + ((a*e^2 - c*d^2*(3 + 2*p))*(Sqrt[-a] - Sqrt[c]*x)*(d 
+ e*x)^(-1 - 2*p)*(a + c*x^2)^p*Hypergeometric2F1[-1 - 2*p, -p, -2*p, (2*S 
qrt[-a]*Sqrt[c]*(d + e*x))/((Sqrt[c]*d - Sqrt[-a]*e)*(Sqrt[-a] - Sqrt[c]*x 
))])/((Sqrt[c]*d + Sqrt[-a]*e)*(c*d^2 + a*e^2)*(1 + 2*p)*(-(((Sqrt[c]*d + 
Sqrt[-a]*e)*(Sqrt[-a] + Sqrt[c]*x))/((Sqrt[c]*d - Sqrt[-a]*e)*(Sqrt[-a] - 
Sqrt[c]*x))))^p)))/((c*d^2 + a*e^2)*(3 + 2*p))
 

3.8.43.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 498

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
d*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/((n + 1)*(b*c^2 + a*d^2))), x] + S 
imp[b/((n + 1)*(b*c^2 + a*d^2))   Int[(c + d*x)^(n + 1)*(a + b*x^2)^p*(c*(n 
 + 1) - d*(n + 2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[n 
, -1] && ((LtQ[n, -1] && IntQuadraticQ[a, 0, b, c, d, n, p, x]) || (SumSimp 
lerQ[n, 1] && IntegerQ[p]) || ILtQ[Simplify[n + 2*p + 3], 0])
 

rule 679

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1 
)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[(c*d*f + a*e*g)/(c*d^2 + a*e^2) 
 Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, 
 p}, x] && EqQ[Simplify[m + 2*p + 3], 0]
 

3.8.43.4 Maple [F]

\[\int \left (e x +d \right )^{-4-2 p} \left (c \,x^{2}+a \right )^{p}d x\]

input

int((e*x+d)^(-4-2*p)*(c*x^2+a)^p,x)
 

output

int((e*x+d)^(-4-2*p)*(c*x^2+a)^p,x)
 

3.8.43.5 Fricas [F]

\[ \int (d+e x)^{-4-2 p} \left (a+c x^2\right )^p \, dx=\int { {\left (c x^{2} + a\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 4} \,d x } \]

input

integrate((e*x+d)^(-4-2*p)*(c*x^2+a)^p,x, algorithm="fricas")
 

output

integral((c*x^2 + a)^p*(e*x + d)^(-2*p - 4), x)
 

3.8.43.6 Sympy [F(-1)]

Timed out. \[ \int (d+e x)^{-4-2 p} \left (a+c x^2\right )^p \, dx=\text {Timed out} \]

input

integrate((e*x+d)**(-4-2*p)*(c*x**2+a)**p,x)
 

output

Timed out
 

3.8.43.7 Maxima [F]

\[ \int (d+e x)^{-4-2 p} \left (a+c x^2\right )^p \, dx=\int { {\left (c x^{2} + a\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 4} \,d x } \]

input

integrate((e*x+d)^(-4-2*p)*(c*x^2+a)^p,x, algorithm="maxima")
 

output

integrate((c*x^2 + a)^p*(e*x + d)^(-2*p - 4), x)
 

3.8.43.8 Giac [F]

\[ \int (d+e x)^{-4-2 p} \left (a+c x^2\right )^p \, dx=\int { {\left (c x^{2} + a\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 4} \,d x } \]

input

integrate((e*x+d)^(-4-2*p)*(c*x^2+a)^p,x, algorithm="giac")
 

output

integrate((c*x^2 + a)^p*(e*x + d)^(-2*p - 4), x)
 

3.8.43.9 Mupad [F(-1)]

Timed out. \[ \int (d+e x)^{-4-2 p} \left (a+c x^2\right )^p \, dx=\int \frac {{\left (c\,x^2+a\right )}^p}{{\left (d+e\,x\right )}^{2\,p+4}} \,d x \]

input

int((a + c*x^2)^p/(d + e*x)^(2*p + 4),x)
 

output

int((a + c*x^2)^p/(d + e*x)^(2*p + 4), x)