\(\int \frac {(c+d x)^n (a+b x^2)}{x^2} \, dx\) [190]

Optimal result

Integrand size = 18, antiderivative size = 56 \[ \int \frac {(c+d x)^n \left (a+b x^2\right )}{x^2} \, dx=\frac {b (c+d x)^{1+n}}{d (1+n)}+\frac {a d (c+d x)^{1+n} \operatorname {Hypergeometric2F1}\left (2,1+n,2+n,1+\frac {d x}{c}\right )}{c^2 (1+n)} \] Output:

b*(d*x+c)^(1+n)/d/(1+n)+a*d*(d*x+c)^(1+n)*hypergeom([2, 1+n],[2+n],1+d*x/c 
)/c^2/(1+n)
 

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.86 \[ \int \frac {(c+d x)^n \left (a+b x^2\right )}{x^2} \, dx=\frac {(c+d x)^{1+n} \left (b c^2+a d^2 \operatorname {Hypergeometric2F1}\left (2,1+n,2+n,1+\frac {d x}{c}\right )\right )}{c^2 d (1+n)} \] Input:

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

Output:

((c + d*x)^(1 + n)*(b*c^2 + a*d^2*Hypergeometric2F1[2, 1 + n, 2 + n, 1 + ( 
d*x)/c]))/(c^2*d*(1 + n))
 

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.46, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {520, 25, 90, 75}

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.

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

\(\Big \downarrow \) 520

\(\displaystyle -\frac {\int -\frac {(a d n+b c x) (c+d x)^n}{x}dx}{c}-\frac {a (c+d x)^{n+1}}{c x}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {(a d n+b c x) (c+d x)^n}{x}dx}{c}-\frac {a (c+d x)^{n+1}}{c x}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {a d n \int \frac {(c+d x)^n}{x}dx+\frac {b c (c+d x)^{n+1}}{d (n+1)}}{c}-\frac {a (c+d x)^{n+1}}{c x}\)

\(\Big \downarrow \) 75

\(\displaystyle \frac {\frac {b c (c+d x)^{n+1}}{d (n+1)}-\frac {a d n (c+d x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {d x}{c}+1\right )}{c (n+1)}}{c}-\frac {a (c+d x)^{n+1}}{c x}\)

Input:

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

Output:

-((a*(c + d*x)^(1 + n))/(c*x)) + ((b*c*(c + d*x)^(1 + n))/(d*(1 + n)) - (a 
*d*n*(c + d*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (d*x)/c])/(c 
*(1 + n)))/c
 

Defintions of rubi rules used

rule 25

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

rule 75

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x 
)^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))*Hypergeometric2F1[-m, n + 1, n + 2, 1 + 
 d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (IntegerQ[m] 
 || GtQ[-d/(b*c), 0])
 

rule 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

rule 520

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), 
 x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2)^p, e*x, x], R = Pol 
ynomialRemainder[(a + b*x^2)^p, e*x, x]}, Simp[R*(e*x)^(m + 1)*((c + d*x)^( 
n + 1)/((m + 1)*(e*c))), x] + Simp[1/((m + 1)*(e*c))   Int[(e*x)^(m + 1)*(c 
 + d*x)^n*ExpandToSum[(m + 1)*(e*c)*Qx - d*R*(m + n + 2), x], x], x]] /; Fr 
eeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[m, -1] &&  !IntegerQ[n]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((b*x^2 + a)*(d*x + c)^n/x^2, x)
 

Sympy [A] (verification not implemented)

Time = 1.65 (sec) , antiderivative size = 173, normalized size of antiderivative = 3.09 \[ \int \frac {(c+d x)^n \left (a+b x^2\right )}{x^2} \, dx=- \frac {a d^{n + 2} n \left (\frac {c}{d} + x\right )^{n + 1} \Gamma \left (n + 1\right )}{c d x \Gamma \left (n + 2\right )} - \frac {a d^{n + 2} \left (\frac {c}{d} + x\right )^{n + 1} \Gamma \left (n + 1\right )}{c d x \Gamma \left (n + 2\right )} - \frac {a d^{n + 2} n^{2} \left (\frac {c}{d} + x\right )^{n + 1} \Phi \left (1 + \frac {d x}{c}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{c^{2} \Gamma \left (n + 2\right )} - \frac {a d^{n + 2} n \left (\frac {c}{d} + x\right )^{n + 1} \Phi \left (1 + \frac {d x}{c}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{c^{2} \Gamma \left (n + 2\right )} + b \left (\begin {cases} c^{n} x & \text {for}\: d = 0 \\\frac {\begin {cases} \frac {\left (c + d x\right )^{n + 1}}{n + 1} & \text {for}\: n \neq -1 \\\log {\left (c + d x \right )} & \text {otherwise} \end {cases}}{d} & \text {otherwise} \end {cases}\right ) \] Input:

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

Output:

-a*d**(n + 2)*n*(c/d + x)**(n + 1)*gamma(n + 1)/(c*d*x*gamma(n + 2)) - a*d 
**(n + 2)*(c/d + x)**(n + 1)*gamma(n + 1)/(c*d*x*gamma(n + 2)) - a*d**(n + 
 2)*n**2*(c/d + x)**(n + 1)*lerchphi(1 + d*x/c, 1, n + 1)*gamma(n + 1)/(c* 
*2*gamma(n + 2)) - a*d**(n + 2)*n*(c/d + x)**(n + 1)*lerchphi(1 + d*x/c, 1 
, n + 1)*gamma(n + 1)/(c**2*gamma(n + 2)) + b*Piecewise((c**n*x, Eq(d, 0)) 
, (Piecewise(((c + d*x)**(n + 1)/(n + 1), Ne(n, -1)), (log(c + d*x), True) 
)/d, True))
 

Maxima [F]

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

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

Output:

a*integrate((d*x + c)^n/x^2, x) + (d*x + c)^(n + 1)*b/(d*(n + 1))
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - (c + d*x)**n*a*d*n - (c + d*x)**n*a*d + (c + d*x)**n*b*c*x + (c + d*x) 
**n*b*d*x**2 + int((c + d*x)**n/(c*x + d*x**2),x)*a*d**2*n**2*x + int((c + 
 d*x)**n/(c*x + d*x**2),x)*a*d**2*n*x)/(d*x*(n + 1))