\(\int \frac {(c x)^m (A+B x+C x^2+D x^3)}{a+b x^2} \, dx\) [139]

Optimal result

Integrand size = 30, antiderivative size = 151 \[ \int \frac {(c x)^m \left (A+B x+C x^2+D x^3\right )}{a+b x^2} \, dx=\frac {C (c x)^{1+m}}{b c (1+m)}+\frac {D (c x)^{2+m}}{b c^2 (2+m)}+\frac {(A b-a C) (c x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )}{a b c (1+m)}+\frac {(b B-a D) (c x)^{2+m} \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},-\frac {b x^2}{a}\right )}{a b c^2 (2+m)} \] Output:

C*(c*x)^(1+m)/b/c/(1+m)+D*(c*x)^(2+m)/b/c^2/(2+m)+(A*b-C*a)*(c*x)^(1+m)*hy 
pergeom([1, 1/2+1/2*m],[3/2+1/2*m],-b*x^2/a)/a/b/c/(1+m)+(B*b-D*a)*(c*x)^( 
2+m)*hypergeom([1, 1+1/2*m],[2+1/2*m],-b*x^2/a)/a/b/c^2/(2+m)
 

Mathematica [A] (verified)

Time = 0.96 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.74 \[ \int \frac {(c x)^m \left (A+B x+C x^2+D x^3\right )}{a+b x^2} \, dx=\frac {x (c x)^m \left (a C (2+m)+a D (1+m) x+(A b-a C) (2+m) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )+(b B-a D) (1+m) x \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},-\frac {b x^2}{a}\right )\right )}{a b (1+m) (2+m)} \] Input:

Integrate[((c*x)^m*(A + B*x + C*x^2 + D*x^3))/(a + b*x^2),x]
 

Output:

(x*(c*x)^m*(a*C*(2 + m) + a*D*(1 + m)*x + (A*b - a*C)*(2 + m)*Hypergeometr 
ic2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)] + (b*B - a*D)*(1 + m)*x*Hyper 
geometric2F1[1, (2 + m)/2, (4 + m)/2, -((b*x^2)/a)]))/(a*b*(1 + m)*(2 + m) 
)
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2333, 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.

\(\displaystyle \int \frac {(c x)^m \left (A+B x+C x^2+D x^3\right )}{a+b x^2} \, dx\)

\(\Big \downarrow \) 2333

\(\displaystyle \int \left (\frac {(c x)^m (x (b B-a D)-a C+A b)}{b \left (a+b x^2\right )}+\frac {C (c x)^m}{b}+\frac {D (c x)^{m+1}}{b c}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {(c x)^{m+1} (A b-a C) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\frac {b x^2}{a}\right )}{a b c (m+1)}+\frac {(c x)^{m+2} (b B-a D) \operatorname {Hypergeometric2F1}\left (1,\frac {m+2}{2},\frac {m+4}{2},-\frac {b x^2}{a}\right )}{a b c^2 (m+2)}+\frac {D (c x)^{m+2}}{b c^2 (m+2)}+\frac {C (c x)^{m+1}}{b c (m+1)}\)

Input:

Int[((c*x)^m*(A + B*x + C*x^2 + D*x^3))/(a + b*x^2),x]
 

Output:

(C*(c*x)^(1 + m))/(b*c*(1 + m)) + (D*(c*x)^(2 + m))/(b*c^2*(2 + m)) + ((A* 
b - a*C)*(c*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2 
)/a)])/(a*b*c*(1 + m)) + ((b*B - a*D)*(c*x)^(2 + m)*Hypergeometric2F1[1, ( 
2 + m)/2, (4 + m)/2, -((b*x^2)/a)])/(a*b*c^2*(2 + m))
 

Defintions of rubi rules used

rule 2009

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

rule 2333

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ 
ExpandIntegrand[(c*x)^m*Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] 
&& PolyQ[Pq, x] && IGtQ[p, -2]
 

Maple [F]

\[\int \frac {\left (c x \right )^{m} \left (D x^{3}+C \,x^{2}+B x +A \right )}{b \,x^{2}+a}d x\]

Input:

int((c*x)^m*(D*x^3+C*x^2+B*x+A)/(b*x^2+a),x)
 

Output:

int((c*x)^m*(D*x^3+C*x^2+B*x+A)/(b*x^2+a),x)
 

Fricas [F]

\[ \int \frac {(c x)^m \left (A+B x+C x^2+D x^3\right )}{a+b x^2} \, dx=\int { \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} \left (c x\right )^{m}}{b x^{2} + a} \,d x } \] Input:

integrate((c*x)^m*(D*x^3+C*x^2+B*x+A)/(b*x^2+a),x, algorithm="fricas")
 

Output:

integral((D*x^3 + C*x^2 + B*x + A)*(c*x)^m/(b*x^2 + a), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 3.97 (sec) , antiderivative size = 379, normalized size of antiderivative = 2.51 \[ \int \frac {(c x)^m \left (A+B x+C x^2+D x^3\right )}{a+b x^2} \, dx=\frac {A c^{m} m x^{m + 1} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{4 a \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {A c^{m} x^{m + 1} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{4 a \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {B c^{m} m x^{m + 2} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + 1\right ) \Gamma \left (\frac {m}{2} + 1\right )}{4 a \Gamma \left (\frac {m}{2} + 2\right )} + \frac {B c^{m} x^{m + 2} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + 1\right ) \Gamma \left (\frac {m}{2} + 1\right )}{2 a \Gamma \left (\frac {m}{2} + 2\right )} + \frac {C c^{m} m x^{m + 3} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + \frac {3}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}{4 a \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {3 C c^{m} x^{m + 3} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + \frac {3}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}{4 a \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {D c^{m} m x^{m + 4} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + 2\right ) \Gamma \left (\frac {m}{2} + 2\right )}{4 a \Gamma \left (\frac {m}{2} + 3\right )} + \frac {D c^{m} x^{m + 4} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + 2\right ) \Gamma \left (\frac {m}{2} + 2\right )}{a \Gamma \left (\frac {m}{2} + 3\right )} \] Input:

integrate((c*x)**m*(D*x**3+C*x**2+B*x+A)/(b*x**2+a),x)
 

Output:

A*c**m*m*x**(m + 1)*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 1/2)*gamma 
(m/2 + 1/2)/(4*a*gamma(m/2 + 3/2)) + A*c**m*x**(m + 1)*lerchphi(b*x**2*exp 
_polar(I*pi)/a, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(4*a*gamma(m/2 + 3/2)) + B* 
c**m*m*x**(m + 2)*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 1)*gamma(m/2 
 + 1)/(4*a*gamma(m/2 + 2)) + B*c**m*x**(m + 2)*lerchphi(b*x**2*exp_polar(I 
*pi)/a, 1, m/2 + 1)*gamma(m/2 + 1)/(2*a*gamma(m/2 + 2)) + C*c**m*m*x**(m + 
 3)*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(4*a 
*gamma(m/2 + 5/2)) + 3*C*c**m*x**(m + 3)*lerchphi(b*x**2*exp_polar(I*pi)/a 
, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(4*a*gamma(m/2 + 5/2)) + D*c**m*m*x**(m + 
 4)*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 2)*gamma(m/2 + 2)/(4*a*gam 
ma(m/2 + 3)) + D*c**m*x**(m + 4)*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 
 + 2)*gamma(m/2 + 2)/(a*gamma(m/2 + 3))
 

Maxima [F]

\[ \int \frac {(c x)^m \left (A+B x+C x^2+D x^3\right )}{a+b x^2} \, dx=\int { \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} \left (c x\right )^{m}}{b x^{2} + a} \,d x } \] Input:

integrate((c*x)^m*(D*x^3+C*x^2+B*x+A)/(b*x^2+a),x, algorithm="maxima")
 

Output:

integrate((D*x^3 + C*x^2 + B*x + A)*(c*x)^m/(b*x^2 + a), x)
 

Giac [F]

\[ \int \frac {(c x)^m \left (A+B x+C x^2+D x^3\right )}{a+b x^2} \, dx=\int { \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} \left (c x\right )^{m}}{b x^{2} + a} \,d x } \] Input:

integrate((c*x)^m*(D*x^3+C*x^2+B*x+A)/(b*x^2+a),x, algorithm="giac")
 

Output:

integrate((D*x^3 + C*x^2 + B*x + A)*(c*x)^m/(b*x^2 + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c x)^m \left (A+B x+C x^2+D x^3\right )}{a+b x^2} \, dx=\int \frac {{\left (c\,x\right )}^m\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{b\,x^2+a} \,d x \] Input:

int(((c*x)^m*(A + B*x + C*x^2 + x^3*D))/(a + b*x^2),x)
 

Output:

int(((c*x)^m*(A + B*x + C*x^2 + x^3*D))/(a + b*x^2), x)
 

Reduce [F]

\[ \int \frac {(c x)^m \left (A+B x+C x^2+D x^3\right )}{a+b x^2} \, dx=\frac {c^{m} \left (-x^{m} a d \,m^{2}-3 x^{m} a d m -2 x^{m} a d +x^{m} b^{2} m^{2}+3 x^{m} b^{2} m +2 x^{m} b^{2}+x^{m} b c \,m^{2} x +2 x^{m} b c m x +x^{m} b d \,m^{2} x^{2}+x^{m} b d m \,x^{2}+\left (\int \frac {x^{m}}{b \,x^{3}+a x}d x \right ) a^{2} d \,m^{3}+3 \left (\int \frac {x^{m}}{b \,x^{3}+a x}d x \right ) a^{2} d \,m^{2}+2 \left (\int \frac {x^{m}}{b \,x^{3}+a x}d x \right ) a^{2} d m -\left (\int \frac {x^{m}}{b \,x^{3}+a x}d x \right ) a \,b^{2} m^{3}-3 \left (\int \frac {x^{m}}{b \,x^{3}+a x}d x \right ) a \,b^{2} m^{2}-2 \left (\int \frac {x^{m}}{b \,x^{3}+a x}d x \right ) a \,b^{2} m +\left (\int \frac {x^{m}}{b \,x^{2}+a}d x \right ) a \,b^{2} m^{3}+3 \left (\int \frac {x^{m}}{b \,x^{2}+a}d x \right ) a \,b^{2} m^{2}+2 \left (\int \frac {x^{m}}{b \,x^{2}+a}d x \right ) a \,b^{2} m -\left (\int \frac {x^{m}}{b \,x^{2}+a}d x \right ) a b c \,m^{3}-3 \left (\int \frac {x^{m}}{b \,x^{2}+a}d x \right ) a b c \,m^{2}-2 \left (\int \frac {x^{m}}{b \,x^{2}+a}d x \right ) a b c m \right )}{b^{2} m \left (m^{2}+3 m +2\right )} \] Input:

int((c*x)^m*(D*x^3+C*x^2+B*x+A)/(b*x^2+a),x)
 

Output:

(c**m*( - x**m*a*d*m**2 - 3*x**m*a*d*m - 2*x**m*a*d + x**m*b**2*m**2 + 3*x 
**m*b**2*m + 2*x**m*b**2 + x**m*b*c*m**2*x + 2*x**m*b*c*m*x + x**m*b*d*m** 
2*x**2 + x**m*b*d*m*x**2 + int(x**m/(a*x + b*x**3),x)*a**2*d*m**3 + 3*int( 
x**m/(a*x + b*x**3),x)*a**2*d*m**2 + 2*int(x**m/(a*x + b*x**3),x)*a**2*d*m 
 - int(x**m/(a*x + b*x**3),x)*a*b**2*m**3 - 3*int(x**m/(a*x + b*x**3),x)*a 
*b**2*m**2 - 2*int(x**m/(a*x + b*x**3),x)*a*b**2*m + int(x**m/(a + b*x**2) 
,x)*a*b**2*m**3 + 3*int(x**m/(a + b*x**2),x)*a*b**2*m**2 + 2*int(x**m/(a + 
 b*x**2),x)*a*b**2*m - int(x**m/(a + b*x**2),x)*a*b*c*m**3 - 3*int(x**m/(a 
 + b*x**2),x)*a*b*c*m**2 - 2*int(x**m/(a + b*x**2),x)*a*b*c*m))/(b**2*m*(m 
**2 + 3*m + 2))