3.4.75 \(\int \frac {(d+e x)^n}{x (a+c x^2)^2} \, dx\) [375]

3.4.75.1 Optimal result

Integrand size = 20, antiderivative size = 489 \[ \int \frac {(d+e x)^n}{x \left (a+c x^2\right )^2} \, dx=\frac {c (d-e x) (d+e x)^{1+n}}{2 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac {\sqrt {c} (d+e x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{2 a^2 \left (\sqrt {c} d-\sqrt {-a} e\right ) (1+n)}+\frac {\sqrt {c} e \left (\sqrt {c} d+\sqrt {-a} e\right ) n (d+e x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{4 (-a)^{3/2} \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (c d^2+a e^2\right ) (1+n)}+\frac {\sqrt {c} (d+e x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{2 a^2 \left (\sqrt {c} d+\sqrt {-a} e\right ) (1+n)}-\frac {\sqrt {c} e \left (\sqrt {-a} \sqrt {c} d+a e\right ) n (d+e x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{4 a^2 \left (\sqrt {c} d+\sqrt {-a} e\right ) \left (c d^2+a e^2\right ) (1+n)}-\frac {(d+e x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {e x}{d}\right )}{a^2 d (1+n)} \]

output

1/2*c*(-e*x+d)*(e*x+d)^(1+n)/a/(a*e^2+c*d^2)/(c*x^2+a)-(e*x+d)^(1+n)*hyper 
geom([1, 1+n],[2+n],1+e*x/d)/a^2/d/(1+n)+1/2*(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^(1/2)/a^2/(1+n)/(-e 
*(-a)^(1/2)+d*c^(1/2))+1/2*(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^(1/2)/a^2/(1+n)/(e*(-a)^(1/2)+d*c^(1/2 
))+1/4*e*n*(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^(1/2)*(e*(-a)^(1/2)+d*c^(1/2))/(-a)^(3/2)/(a*e^2+c*d^ 
2)/(1+n)/(-e*(-a)^(1/2)+d*c^(1/2))-1/4*e*n*(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^(1/2)*(a*e+d*(-a)^(1/2 
)*c^(1/2))/a^2/(a*e^2+c*d^2)/(1+n)/(e*(-a)^(1/2)+d*c^(1/2))
 

3.4.75.2 Mathematica [A] (verified)

Time = 0.59 (sec) , antiderivative size = 391, normalized size of antiderivative = 0.80 \[ \int \frac {(d+e x)^n}{x \left (a+c x^2\right )^2} \, dx=\frac {(d+e x)^{1+n} \left (\frac {2 a c (d-e x)}{\left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac {4 \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {d+e x}{d}\right )}{d+d n}+\frac {2 \sqrt {c} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{\left (\sqrt {c} d-\sqrt {-a} e\right ) (1+n)}+\frac {2 \sqrt {c} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{\left (\sqrt {c} d+\sqrt {-a} e\right ) (1+n)}+\frac {\sqrt {c} e n \left (\left (\sqrt {-a} c d^2-2 a \sqrt {c} d e+(-a)^{3/2} e^2\right ) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )+\left (-\sqrt {-a} c d^2-2 a \sqrt {c} d e+\sqrt {-a} a e^2\right ) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )\right )}{\left (c d^2+a e^2\right )^2 (1+n)}\right )}{4 a^2} \]

input

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

output

((d + e*x)^(1 + n)*((2*a*c*(d - e*x))/((c*d^2 + a*e^2)*(a + c*x^2)) - (4*H 
ypergeometric2F1[1, 1 + n, 2 + n, (d + e*x)/d])/(d + d*n) + (2*Sqrt[c]*Hyp 
ergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[-a]* 
e)])/((Sqrt[c]*d - Sqrt[-a]*e)*(1 + n)) + (2*Sqrt[c]*Hypergeometric2F1[1, 
1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)])/((Sqrt[c]*d + 
 Sqrt[-a]*e)*(1 + n)) + (Sqrt[c]*e*n*((Sqrt[-a]*c*d^2 - 2*a*Sqrt[c]*d*e + 
(-a)^(3/2)*e^2)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sq 
rt[c]*d - Sqrt[-a]*e)] + (-(Sqrt[-a]*c*d^2) - 2*a*Sqrt[c]*d*e + Sqrt[-a]*a 
*e^2)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + 
Sqrt[-a]*e)]))/((c*d^2 + a*e^2)^2*(1 + n))))/(4*a^2)
 

3.4.75.3 Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 484, normalized size of antiderivative = 0.99, 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.

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

\(\Big \downarrow \) 615

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\sqrt {c} e n \left (\sqrt {-a} \sqrt {c} d+a e\right ) (d+e x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{4 a^2 (n+1) \left (\sqrt {-a} e+\sqrt {c} d\right ) \left (a e^2+c d^2\right )}+\frac {\sqrt {c} (d+e x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{2 a^2 (n+1) \left (\sqrt {c} d-\sqrt {-a} e\right )}+\frac {\sqrt {c} (d+e x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{2 a^2 (n+1) \left (\sqrt {-a} e+\sqrt {c} d\right )}-\frac {(d+e x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {e x}{d}+1\right )}{a^2 d (n+1)}+\frac {c e n \left (\frac {\sqrt {-a} e}{\sqrt {c}}+d\right ) (d+e x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{4 (-a)^{3/2} (n+1) \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (a e^2+c d^2\right )}+\frac {c (d-e x) (d+e x)^{n+1}}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )}\)

input

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

output

(c*(d - e*x)*(d + e*x)^(1 + n))/(2*a*(c*d^2 + a*e^2)*(a + c*x^2)) + (Sqrt[ 
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^2*(Sqrt[c]*d - Sqrt[-a]*e)*(1 + n)) + (c 
*e*(d + (Sqrt[-a]*e)/Sqrt[c])*n*(d + e*x)^(1 + n)*Hypergeometric2F1[1, 1 + 
 n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[-a]*e)])/(4*(-a)^(3/2)*(S 
qrt[c]*d - Sqrt[-a]*e)*(c*d^2 + a*e^2)*(1 + n)) + (Sqrt[c]*(d + e*x)^(1 + 
n)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqr 
t[-a]*e)])/(2*a^2*(Sqrt[c]*d + Sqrt[-a]*e)*(1 + n)) - (Sqrt[c]*e*(Sqrt[-a] 
*Sqrt[c]*d + a*e)*n*(d + e*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, ( 
Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)])/(4*a^2*(Sqrt[c]*d + Sqrt[-a] 
*e)*(c*d^2 + a*e^2)*(1 + n)) - ((d + e*x)^(1 + n)*Hypergeometric2F1[1, 1 + 
 n, 2 + n, 1 + (e*x)/d])/(a^2*d*(1 + n))
 

3.4.75.3.1 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]
 

3.4.75.4 Maple [F]

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

input

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

output

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

3.4.75.5 Fricas [F]

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

input

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

output

integral((e*x + d)^n/(c^2*x^5 + 2*a*c*x^3 + a^2*x), x)
 

3.4.75.6 Sympy [F]

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

input

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

output

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

3.4.75.7 Maxima [F]

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

input

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

output

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

3.4.75.8 Giac [F]

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

input

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

output

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

3.4.75.9 Mupad [F(-1)]

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

input

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

output

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