3.4.76 \(\int \frac {(d+e x)^n}{x^2 (a+c x^2)^2} \, dx\) [376]

3.4.76.1 Optimal result

Integrand size = 20, antiderivative size = 513 \[ \int \frac {(d+e x)^n}{x^2 \left (a+c x^2\right )^2} \, dx=-\frac {c (a e+c d x) (d+e x)^{1+n}}{2 a^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac {c (d+e x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{2 (-a)^{5/2} \left (\sqrt {c} d-\sqrt {-a} e\right ) (1+n)}-\frac {c \left (c d^2+a e^2 (1-n)+\sqrt {-a} \sqrt {c} d e n\right ) (d+e x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{4 (-a)^{5/2} \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (c d^2+a e^2\right ) (1+n)}+\frac {c (d+e x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{2 (-a)^{5/2} \left (\sqrt {c} d+\sqrt {-a} e\right ) (1+n)}+\frac {c \left (c d^2+a e^2 (1-n)-\sqrt {-a} \sqrt {c} d e n\right ) (d+e x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{4 (-a)^{5/2} \left (\sqrt {c} d+\sqrt {-a} e\right ) \left (c d^2+a e^2\right ) (1+n)}+\frac {e (d+e x)^{1+n} \operatorname {Hypergeometric2F1}\left (2,1+n,2+n,1+\frac {e x}{d}\right )}{a^2 d^2 (1+n)} \]

output

-1/2*c*(c*d*x+a*e)*(e*x+d)^(1+n)/a^2/(a*e^2+c*d^2)/(c*x^2+a)+e*(e*x+d)^(1+ 
n)*hypergeom([2, 1+n],[2+n],1+e*x/d)/a^2/d^2/(1+n)-1/2*c*(e*x+d)^(1+n)*hyp 
ergeom([1, 1+n],[2+n],(e*x+d)*c^(1/2)/(-e*(-a)^(1/2)+d*c^(1/2)))/(-a)^(5/2 
)/(1+n)/(-e*(-a)^(1/2)+d*c^(1/2))+1/2*c*(e*x+d)^(1+n)*hypergeom([1, 1+n],[ 
2+n],(e*x+d)*c^(1/2)/(e*(-a)^(1/2)+d*c^(1/2)))/(-a)^(5/2)/(1+n)/(e*(-a)^(1 
/2)+d*c^(1/2))+1/4*c*(e*x+d)^(1+n)*hypergeom([1, 1+n],[2+n],(e*x+d)*c^(1/2 
)/(e*(-a)^(1/2)+d*c^(1/2)))*(c*d^2+a*e^2*(1-n)-d*e*n*(-a)^(1/2)*c^(1/2))/( 
-a)^(5/2)/(a*e^2+c*d^2)/(1+n)/(e*(-a)^(1/2)+d*c^(1/2))-1/4*c*(e*x+d)^(1+n) 
*hypergeom([1, 1+n],[2+n],(e*x+d)*c^(1/2)/(-e*(-a)^(1/2)+d*c^(1/2)))*(c*d^ 
2+a*e^2*(1-n)+d*e*n*(-a)^(1/2)*c^(1/2))/(-a)^(5/2)/(a*e^2+c*d^2)/(1+n)/(-e 
*(-a)^(1/2)+d*c^(1/2))
 

3.4.76.2 Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 437, normalized size of antiderivative = 0.85 \[ \int \frac {(d+e x)^n}{x^2 \left (a+c x^2\right )^2} \, dx=\frac {1}{4} (d+e x)^{1+n} \left (-\frac {2 c (a e+c d x)}{a^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac {2 c \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{(-a)^{5/2} \left (-\sqrt {c} d+\sqrt {-a} e\right ) (1+n)}+\frac {2 c \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{(-a)^{5/2} \left (\sqrt {c} d+\sqrt {-a} e\right ) (1+n)}+\frac {a c \left (\frac {\left (c d^2-a e^2 (-1+n)+\sqrt {-a} \sqrt {c} d e n\right ) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{\sqrt {c} d-\sqrt {-a} e}-\frac {\left (c d^2-a e^2 (-1+n)-\sqrt {-a} \sqrt {c} d e n\right ) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{\sqrt {c} d+\sqrt {-a} e}\right )}{(-a)^{7/2} \left (c d^2+a e^2\right ) (1+n)}+\frac {4 e \operatorname {Hypergeometric2F1}\left (2,1+n,2+n,1+\frac {e x}{d}\right )}{a^2 d^2 (1+n)}\right ) \]

input

Integrate[(d + e*x)^n/(x^2*(a + c*x^2)^2),x]
 

output

((d + e*x)^(1 + n)*((-2*c*(a*e + c*d*x))/(a^2*(c*d^2 + a*e^2)*(a + c*x^2)) 
 + (2*c*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d 
- Sqrt[-a]*e)])/((-a)^(5/2)*(-(Sqrt[c]*d) + Sqrt[-a]*e)*(1 + n)) + (2*c*Hy 
pergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a] 
*e)])/((-a)^(5/2)*(Sqrt[c]*d + Sqrt[-a]*e)*(1 + n)) + (a*c*(((c*d^2 - a*e^ 
2*(-1 + n) + Sqrt[-a]*Sqrt[c]*d*e*n)*Hypergeometric2F1[1, 1 + n, 2 + n, (S 
qrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[-a]*e)])/(Sqrt[c]*d - Sqrt[-a]*e) - (( 
c*d^2 - a*e^2*(-1 + n) - Sqrt[-a]*Sqrt[c]*d*e*n)*Hypergeometric2F1[1, 1 + 
n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)])/(Sqrt[c]*d + Sqrt 
[-a]*e)))/((-a)^(7/2)*(c*d^2 + a*e^2)*(1 + n)) + (4*e*Hypergeometric2F1[2, 
 1 + n, 2 + n, 1 + (e*x)/d])/(a^2*d^2*(1 + n))))/4
 

3.4.76.3 Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 513, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {615, 2009}

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.

\(\displaystyle \int \frac {(d+e x)^n}{x^2 \left (a+c x^2\right )^2} \, dx\)

\(\Big \downarrow \) 615

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {c (d+e x)^{n+1} (a e+c d x)}{2 a^2 \left (a+c x^2\right ) \left (a e^2+c d^2\right )}+\frac {e (d+e x)^{n+1} \operatorname {Hypergeometric2F1}\left (2,n+1,n+2,\frac {e x}{d}+1\right )}{a^2 d^2 (n+1)}-\frac {c (d+e x)^{n+1} \left (\sqrt {-a} \sqrt {c} d e n+a e^2 (1-n)+c d^2\right ) \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{4 (-a)^{5/2} (n+1) \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (a e^2+c d^2\right )}+\frac {c (d+e x)^{n+1} \left (-\sqrt {-a} \sqrt {c} d e n+a e^2 (1-n)+c d^2\right ) \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{4 (-a)^{5/2} (n+1) \left (\sqrt {-a} e+\sqrt {c} d\right ) \left (a e^2+c d^2\right )}-\frac {c (d+e x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{2 (-a)^{5/2} (n+1) \left (\sqrt {c} d-\sqrt {-a} e\right )}+\frac {c (d+e x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{2 (-a)^{5/2} (n+1) \left (\sqrt {-a} e+\sqrt {c} d\right )}\)

input

Int[(d + e*x)^n/(x^2*(a + c*x^2)^2),x]
 

output

-1/2*(c*(a*e + c*d*x)*(d + e*x)^(1 + n))/(a^2*(c*d^2 + a*e^2)*(a + c*x^2)) 
 - (c*(d + e*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e 
*x))/(Sqrt[c]*d - Sqrt[-a]*e)])/(2*(-a)^(5/2)*(Sqrt[c]*d - Sqrt[-a]*e)*(1 
+ n)) - (c*(c*d^2 + a*e^2*(1 - n) + Sqrt[-a]*Sqrt[c]*d*e*n)*(d + e*x)^(1 + 
 n)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sq 
rt[-a]*e)])/(4*(-a)^(5/2)*(Sqrt[c]*d - Sqrt[-a]*e)*(c*d^2 + a*e^2)*(1 + n) 
) + (c*(d + e*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + 
e*x))/(Sqrt[c]*d + Sqrt[-a]*e)])/(2*(-a)^(5/2)*(Sqrt[c]*d + Sqrt[-a]*e)*(1 
 + n)) + (c*(c*d^2 + a*e^2*(1 - n) - Sqrt[-a]*Sqrt[c]*d*e*n)*(d + e*x)^(1 
+ n)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + S 
qrt[-a]*e)])/(4*(-a)^(5/2)*(Sqrt[c]*d + Sqrt[-a]*e)*(c*d^2 + a*e^2)*(1 + n 
)) + (e*(d + e*x)^(1 + n)*Hypergeometric2F1[2, 1 + n, 2 + n, 1 + (e*x)/d]) 
/(a^2*d^2*(1 + n))
 

3.4.76.3.1 Defintions of rubi rules used

rule 615

Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), 
 x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] 
 /; FreeQ[{a, b, c, d, e, m, n}, x] && ILtQ[p, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

3.4.76.4 Maple [F]

\[\int \frac {\left (e x +d \right )^{n}}{x^{2} \left (c \,x^{2}+a \right )^{2}}d x\]

input

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

output

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

3.4.76.5 Fricas [F]

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

input

integrate((e*x+d)^n/x^2/(c*x^2+a)^2,x, algorithm="fricas")
 

output

integral((e*x + d)^n/(c^2*x^6 + 2*a*c*x^4 + a^2*x^2), x)
 

3.4.76.6 Sympy [F(-1)]

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

input

integrate((e*x+d)**n/x**2/(c*x**2+a)**2,x)
 

output

Timed out
 

3.4.76.7 Maxima [F]

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

input

integrate((e*x+d)^n/x^2/(c*x^2+a)^2,x, algorithm="maxima")
 

output

integrate((e*x + d)^n/((c*x^2 + a)^2*x^2), x)
 

3.4.76.8 Giac [F]

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

input

integrate((e*x+d)^n/x^2/(c*x^2+a)^2,x, algorithm="giac")
 

output

integrate((e*x + d)^n/((c*x^2 + a)^2*x^2), x)
 

3.4.76.9 Mupad [F(-1)]

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

input

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

output

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