\(\int \frac {A+B x+C x^2+D x^3}{(c+d x) (a+b x^2)^2} \, dx\) [37]

Optimal result

Integrand size = 32, antiderivative size = 284 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x) \left (a+b x^2\right )^2} \, dx=-\frac {a (b B c-A b d+a C d-a c D)-\left (A b^2 c-a (b c C-b B d+a d D)\right ) x}{2 a b \left (b c^2+a d^2\right ) \left (a+b x^2\right )}+\frac {\left (A b^2 c \left (b c^2+3 a d^2\right )+a \left (b^2 c^2 (c C-B d)+a^2 d^3 D-a b d \left (c C d-B d^2-3 c^2 D\right )\right )\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{3/2} \left (b c^2+a d^2\right )^2}+\frac {\left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \log (c+d x)}{\left (b c^2+a d^2\right )^2}-\frac {\left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \log \left (a+b x^2\right )}{2 \left (b c^2+a d^2\right )^2} \] Output:

-1/2*(a*(-A*b*d+B*b*c+C*a*d-D*a*c)-(A*b^2*c-a*(-B*b*d+C*b*c+D*a*d))*x)/a/b 
/(a*d^2+b*c^2)/(b*x^2+a)+1/2*(A*b^2*c*(3*a*d^2+b*c^2)+a*(b^2*c^2*(-B*d+C*c 
)+a^2*d^3*D-a*b*d*(-B*d^2+C*c*d-3*D*c^2)))*arctan(b^(1/2)*x/a^(1/2))/a^(3/ 
2)/b^(3/2)/(a*d^2+b*c^2)^2+(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*ln(d*x+c)/(a*d^2+ 
b*c^2)^2-1/2*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*ln(b*x^2+a)/(a*d^2+b*c^2)^2
 

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.87 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x) \left (a+b x^2\right )^2} \, dx=\frac {\frac {\left (b c^2+a d^2\right ) \left (A b^2 c x+a b (-B c+A d-c C x+B d x)-a^2 (C d-c D+d D x)\right )}{a b \left (a+b x^2\right )}+\frac {\left (A b^2 c \left (b c^2+3 a d^2\right )+a \left (b^2 c^2 (c C-B d)+a^2 d^3 D+a b d \left (-c C d+B d^2+3 c^2 D\right )\right )\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} b^{3/2}}+2 \left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \log (c+d x)+\left (-c^2 C d+B c d^2-A d^3+c^3 D\right ) \log \left (a+b x^2\right )}{2 \left (b c^2+a d^2\right )^2} \] Input:

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

Output:

(((b*c^2 + a*d^2)*(A*b^2*c*x + a*b*(-(B*c) + A*d - c*C*x + B*d*x) - a^2*(C 
*d - c*D + d*D*x)))/(a*b*(a + b*x^2)) + ((A*b^2*c*(b*c^2 + 3*a*d^2) + a*(b 
^2*c^2*(c*C - B*d) + a^2*d^3*D + a*b*d*(-(c*C*d) + B*d^2 + 3*c^2*D)))*ArcT 
an[(Sqrt[b]*x)/Sqrt[a]])/(a^(3/2)*b^(3/2)) + 2*(c^2*C*d - B*c*d^2 + A*d^3 
- c^3*D)*Log[c + d*x] + (-(c^2*C*d) + B*c*d^2 - A*d^3 + c^3*D)*Log[a + b*x 
^2])/(2*(b*c^2 + a*d^2)^2)
 

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {2178, 25, 27, 657, 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.

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

\(\Big \downarrow \) 2178

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 657

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (a \left (a^2 d^3 D-a b d \left (-B d^2-3 c^2 D+c C d\right )+b^2 c^2 (c C-B d)\right )+A b^2 c \left (3 a d^2+b c^2\right )\right )}{\sqrt {a} \sqrt {b} \left (a d^2+b c^2\right )}-\frac {a b \log \left (a+b x^2\right ) \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{a d^2+b c^2}+\frac {2 a b \log (c+d x) \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{a d^2+b c^2}}{2 a b \left (a d^2+b c^2\right )}-\frac {a (-a c D+a C d-A b d+b B c)-x \left (A b^2 c-a (a d D-b B d+b c C)\right )}{2 a b \left (a+b x^2\right ) \left (a d^2+b c^2\right )}\)

Input:

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

Output:

-1/2*(a*(b*B*c - A*b*d + a*C*d - a*c*D) - (A*b^2*c - a*(b*c*C - b*B*d + a* 
d*D))*x)/(a*b*(b*c^2 + a*d^2)*(a + b*x^2)) + (((A*b^2*c*(b*c^2 + 3*a*d^2) 
+ a*(b^2*c^2*(c*C - B*d) + a^2*d^3*D - a*b*d*(c*C*d - B*d^2 - 3*c^2*D)))*A 
rcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*(b*c^2 + a*d^2)) + (2*a*b*(c^ 
2*C*d - B*c*d^2 + A*d^3 - c^3*D)*Log[c + d*x])/(b*c^2 + a*d^2) - (a*b*(c^2 
*C*d - B*c*d^2 + A*d^3 - c^3*D)*Log[a + b*x^2])/(b*c^2 + a*d^2))/(2*a*b*(b 
*c^2 + a*d^2))
 

Defintions of rubi rules used

rule 25

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

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 657

Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_) + (c_.)*( 
x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g*x)^n/(a + c*x^ 
2)), x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IntegersQ[n]
 

rule 2009

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

rule 2178

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{Qx = PolynomialQuotient[(d + e*x)^m*Pq, a + b*x^2, x], R = Coeff[Po 
lynomialRemainder[(d + e*x)^m*Pq, a + b*x^2, x], x, 0], S = Coeff[Polynomia 
lRemainder[(d + e*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*S - b*R*x)*((a + 
b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*b*(p + 1))   Int[(d + e*x 
)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*b*(p + 1)*Qx)/(d + e*x)^m + (b*R*( 
2*p + 3))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, d, e}, x] && PolyQ[Pq, x 
] && NeQ[b*d^2 + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]
 

Maple [A] (verified)

Time = 1.23 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.39

method	result	size
default	\(\frac {\frac {\frac {\left (A a \,b^{2} c \,d^{2}+A \,b^{3} c^{3}+B \,a^{2} b \,d^{3}+B a \,b^{2} c^{2} d -C \,a^{2} b c \,d^{2}-C a \,b^{2} c^{3}-D a^{3} d^{3}-D a^{2} b \,c^{2} d \right ) x}{2 a b}+\frac {A a b \,d^{3}+A \,b^{2} c^{2} d -B a b c \,d^{2}-B \,b^{2} c^{3}-C \,a^{2} d^{3}-C a b \,c^{2} d +D a^{2} c \,d^{2}+D a b \,c^{3}}{2 b}}{b \,x^{2}+a}+\frac {\frac {\left (-2 A \,d^{3} a \,b^{2}+2 B a \,b^{2} c \,d^{2}-2 C a \,b^{2} c^{2} d +2 D a \,b^{2} c^{3}\right ) \ln \left (b \,x^{2}+a \right )}{2 b}+\frac {\left (3 A a \,b^{2} c \,d^{2}+A \,b^{3} c^{3}+B \,a^{2} b \,d^{3}-B a \,b^{2} c^{2} d -C \,a^{2} b c \,d^{2}+C a \,b^{2} c^{3}+D a^{3} d^{3}+3 D a^{2} b \,c^{2} d \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}}}{2 b a}}{\left (a \,d^{2}+b \,c^{2}\right )^{2}}+\frac {\left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right ) \ln \left (d x +c \right )}{\left (a \,d^{2}+b \,c^{2}\right )^{2}}\)	\(394\)

Input:

int((D*x^3+C*x^2+B*x+A)/(d*x+c)/(b*x^2+a)^2,x,method=_RETURNVERBOSE)
 

Output:

1/(a*d^2+b*c^2)^2*((1/2*(A*a*b^2*c*d^2+A*b^3*c^3+B*a^2*b*d^3+B*a*b^2*c^2*d 
-C*a^2*b*c*d^2-C*a*b^2*c^3-D*a^3*d^3-D*a^2*b*c^2*d)/a/b*x+1/2*(A*a*b*d^3+A 
*b^2*c^2*d-B*a*b*c*d^2-B*b^2*c^3-C*a^2*d^3-C*a*b*c^2*d+D*a^2*c*d^2+D*a*b*c 
^3)/b)/(b*x^2+a)+1/2/b/a*(1/2*(-2*A*a*b^2*d^3+2*B*a*b^2*c*d^2-2*C*a*b^2*c^ 
2*d+2*D*a*b^2*c^3)/b*ln(b*x^2+a)+(3*A*a*b^2*c*d^2+A*b^3*c^3+B*a^2*b*d^3-B* 
a*b^2*c^2*d-C*a^2*b*c*d^2+C*a*b^2*c^3+D*a^3*d^3+3*D*a^2*b*c^2*d)/(a*b)^(1/ 
2)*arctan(b*x/(a*b)^(1/2))))+(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*ln(d*x+c)/(a*d^ 
2+b*c^2)^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 659 vs. \(2 (269) = 538\).

Time = 135.15 (sec) , antiderivative size = 1342, normalized size of antiderivative = 4.73 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x) \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/4*(2*(D*a^3*b^2 - B*a^2*b^3)*c^3 - 2*(C*a^3*b^2 - A*a^2*b^3)*c^2*d + 2* 
(D*a^4*b - B*a^3*b^2)*c*d^2 - 2*(C*a^4*b - A*a^3*b^2)*d^3 - ((C*a^2*b^2 + 
A*a*b^3)*c^3 + (3*D*a^3*b - B*a^2*b^2)*c^2*d - (C*a^3*b - 3*A*a^2*b^2)*c*d 
^2 + (D*a^4 + B*a^3*b)*d^3 + ((C*a*b^3 + A*b^4)*c^3 + (3*D*a^2*b^2 - B*a*b 
^3)*c^2*d - (C*a^2*b^2 - 3*A*a*b^3)*c*d^2 + (D*a^3*b + B*a^2*b^2)*d^3)*x^2 
)*sqrt(-a*b)*log((b*x^2 - 2*sqrt(-a*b)*x - a)/(b*x^2 + a)) - 2*((C*a^2*b^3 
 - A*a*b^4)*c^3 + (D*a^3*b^2 - B*a^2*b^3)*c^2*d + (C*a^3*b^2 - A*a^2*b^3)* 
c*d^2 + (D*a^4*b - B*a^3*b^2)*d^3)*x + 2*(D*a^3*b^2*c^3 - C*a^3*b^2*c^2*d 
+ B*a^3*b^2*c*d^2 - A*a^3*b^2*d^3 + (D*a^2*b^3*c^3 - C*a^2*b^3*c^2*d + B*a 
^2*b^3*c*d^2 - A*a^2*b^3*d^3)*x^2)*log(b*x^2 + a) - 4*(D*a^3*b^2*c^3 - C*a 
^3*b^2*c^2*d + B*a^3*b^2*c*d^2 - A*a^3*b^2*d^3 + (D*a^2*b^3*c^3 - C*a^2*b^ 
3*c^2*d + B*a^2*b^3*c*d^2 - A*a^2*b^3*d^3)*x^2)*log(d*x + c))/(a^3*b^4*c^4 
 + 2*a^4*b^3*c^2*d^2 + a^5*b^2*d^4 + (a^2*b^5*c^4 + 2*a^3*b^4*c^2*d^2 + a^ 
4*b^3*d^4)*x^2), 1/2*((D*a^3*b^2 - B*a^2*b^3)*c^3 - (C*a^3*b^2 - A*a^2*b^3 
)*c^2*d + (D*a^4*b - B*a^3*b^2)*c*d^2 - (C*a^4*b - A*a^3*b^2)*d^3 + ((C*a^ 
2*b^2 + A*a*b^3)*c^3 + (3*D*a^3*b - B*a^2*b^2)*c^2*d - (C*a^3*b - 3*A*a^2* 
b^2)*c*d^2 + (D*a^4 + B*a^3*b)*d^3 + ((C*a*b^3 + A*b^4)*c^3 + (3*D*a^2*b^2 
 - B*a*b^3)*c^2*d - (C*a^2*b^2 - 3*A*a*b^3)*c*d^2 + (D*a^3*b + B*a^2*b^2)* 
d^3)*x^2)*sqrt(a*b)*arctan(sqrt(a*b)*x/a) - ((C*a^2*b^3 - A*a*b^4)*c^3 + ( 
D*a^3*b^2 - B*a^2*b^3)*c^2*d + (C*a^3*b^2 - A*a^2*b^3)*c*d^2 + (D*a^4*b...
                                                                                    
                                                                                    
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.23 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x) \left (a+b x^2\right )^2} \, dx=\frac {{\left (D c^{3} - C c^{2} d + B c d^{2} - A d^{3}\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}\right )}} - \frac {{\left (D c^{3} - C c^{2} d + B c d^{2} - A d^{3}\right )} \log \left (d x + c\right )}{b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}} + \frac {{\left ({\left (C a b^{2} + A b^{3}\right )} c^{3} + {\left (3 \, D a^{2} b - B a b^{2}\right )} c^{2} d - {\left (C a^{2} b - 3 \, A a b^{2}\right )} c d^{2} + {\left (D a^{3} + B a^{2} b\right )} d^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (a b^{3} c^{4} + 2 \, a^{2} b^{2} c^{2} d^{2} + a^{3} b d^{4}\right )} \sqrt {a b}} + \frac {{\left (D a^{2} - B a b\right )} c - {\left (C a^{2} - A a b\right )} d - {\left ({\left (C a b - A b^{2}\right )} c + {\left (D a^{2} - B a b\right )} d\right )} x}{2 \, {\left (a^{2} b^{2} c^{2} + a^{3} b d^{2} + {\left (a b^{3} c^{2} + a^{2} b^{2} d^{2}\right )} x^{2}\right )}} \] Input:

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

Output:

1/2*(D*c^3 - C*c^2*d + B*c*d^2 - A*d^3)*log(b*x^2 + a)/(b^2*c^4 + 2*a*b*c^ 
2*d^2 + a^2*d^4) - (D*c^3 - C*c^2*d + B*c*d^2 - A*d^3)*log(d*x + c)/(b^2*c 
^4 + 2*a*b*c^2*d^2 + a^2*d^4) + 1/2*((C*a*b^2 + A*b^3)*c^3 + (3*D*a^2*b - 
B*a*b^2)*c^2*d - (C*a^2*b - 3*A*a*b^2)*c*d^2 + (D*a^3 + B*a^2*b)*d^3)*arct 
an(b*x/sqrt(a*b))/((a*b^3*c^4 + 2*a^2*b^2*c^2*d^2 + a^3*b*d^4)*sqrt(a*b)) 
+ 1/2*((D*a^2 - B*a*b)*c - (C*a^2 - A*a*b)*d - ((C*a*b - A*b^2)*c + (D*a^2 
 - B*a*b)*d)*x)/(a^2*b^2*c^2 + a^3*b*d^2 + (a*b^3*c^2 + a^2*b^2*d^2)*x^2)
 

Giac [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.57 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x) \left (a+b x^2\right )^2} \, dx=\frac {{\left (D c^{3} - C c^{2} d + B c d^{2} - A d^{3}\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}\right )}} - \frac {{\left (D c^{3} d - C c^{2} d^{2} + B c d^{3} - A d^{4}\right )} \log \left ({\left | d x + c \right |}\right )}{b^{2} c^{4} d + 2 \, a b c^{2} d^{3} + a^{2} d^{5}} + \frac {{\left (C a b^{2} c^{3} + A b^{3} c^{3} + 3 \, D a^{2} b c^{2} d - B a b^{2} c^{2} d - C a^{2} b c d^{2} + 3 \, A a b^{2} c d^{2} + D a^{3} d^{3} + B a^{2} b d^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (a b^{3} c^{4} + 2 \, a^{2} b^{2} c^{2} d^{2} + a^{3} b d^{4}\right )} \sqrt {a b}} + \frac {D a^{2} b c^{3} - B a b^{2} c^{3} - C a^{2} b c^{2} d + A a b^{2} c^{2} d + D a^{3} c d^{2} - B a^{2} b c d^{2} - C a^{3} d^{3} + A a^{2} b d^{3} - {\left (C a b^{2} c^{3} - A b^{3} c^{3} + D a^{2} b c^{2} d - B a b^{2} c^{2} d + C a^{2} b c d^{2} - A a b^{2} c d^{2} + D a^{3} d^{3} - B a^{2} b d^{3}\right )} x}{2 \, {\left (b c^{2} + a d^{2}\right )}^{2} {\left (b x^{2} + a\right )} a b} \] Input:

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

Output:

1/2*(D*c^3 - C*c^2*d + B*c*d^2 - A*d^3)*log(b*x^2 + a)/(b^2*c^4 + 2*a*b*c^ 
2*d^2 + a^2*d^4) - (D*c^3*d - C*c^2*d^2 + B*c*d^3 - A*d^4)*log(abs(d*x + c 
))/(b^2*c^4*d + 2*a*b*c^2*d^3 + a^2*d^5) + 1/2*(C*a*b^2*c^3 + A*b^3*c^3 + 
3*D*a^2*b*c^2*d - B*a*b^2*c^2*d - C*a^2*b*c*d^2 + 3*A*a*b^2*c*d^2 + D*a^3* 
d^3 + B*a^2*b*d^3)*arctan(b*x/sqrt(a*b))/((a*b^3*c^4 + 2*a^2*b^2*c^2*d^2 + 
 a^3*b*d^4)*sqrt(a*b)) + 1/2*(D*a^2*b*c^3 - B*a*b^2*c^3 - C*a^2*b*c^2*d + 
A*a*b^2*c^2*d + D*a^3*c*d^2 - B*a^2*b*c*d^2 - C*a^3*d^3 + A*a^2*b*d^3 - (C 
*a*b^2*c^3 - A*b^3*c^3 + D*a^2*b*c^2*d - B*a*b^2*c^2*d + C*a^2*b*c*d^2 - A 
*a*b^2*c*d^2 + D*a^3*d^3 - B*a^2*b*d^3)*x)/((b*c^2 + a*d^2)^2*(b*x^2 + a)* 
a*b)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 728, normalized size of antiderivative = 2.56 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x) \left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3*d**4 + 3*sqrt(b)*sqrt( 
a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*b**2*c*d**2 + sqrt(b)*sqrt(a)*atan(( 
b*x)/(sqrt(b)*sqrt(a)))*a**2*b**2*d**3 + 2*sqrt(b)*sqrt(a)*atan((b*x)/(sqr 
t(b)*sqrt(a)))*a**2*b*c**2*d**2 + sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt 
(a)))*a**2*b*d**4*x**2 + sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b 
**3*c**3 - sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b**3*c**2*d + 3 
*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b**3*c*d**2*x**2 + sqrt(b 
)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b**3*d**3*x**2 + sqrt(b)*sqrt(a) 
*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b**2*c**4 + 2*sqrt(b)*sqrt(a)*atan((b*x)/ 
(sqrt(b)*sqrt(a)))*a*b**2*c**2*d**2*x**2 + sqrt(b)*sqrt(a)*atan((b*x)/(sqr 
t(b)*sqrt(a)))*b**4*c**3*x**2 - sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a 
)))*b**4*c**2*d*x**2 + sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**3* 
c**4*x**2 - log(a + b*x**2)*a**3*b**2*d**3 + log(a + b*x**2)*a**2*b**3*c*d 
**2 - log(a + b*x**2)*a**2*b**3*d**3*x**2 + log(a + b*x**2)*a*b**4*c*d**2* 
x**2 + 2*log(c + d*x)*a**3*b**2*d**3 - 2*log(c + d*x)*a**2*b**3*c*d**2 + 2 
*log(c + d*x)*a**2*b**3*d**3*x**2 - 2*log(c + d*x)*a*b**4*c*d**2*x**2 - a* 
*3*b*d**4*x + a**2*b**3*c*d**2*x - a**2*b**3*d**3*x**2 + a**2*b**3*d**3*x 
- 2*a**2*b**2*c**2*d**2*x + a*b**4*c**3*x - a*b**4*c**2*d*x**2 + a*b**4*c* 
*2*d*x + a*b**4*c*d**2*x**2 - a*b**3*c**4*x + b**5*c**3*x**2)/(2*a*b**2*(a 
**3*d**4 + 2*a**2*b*c**2*d**2 + a**2*b*d**4*x**2 + a*b**2*c**4 + 2*a*b*...