\(\int \frac {(c+d x)^2 (A+B x+C x^2+D x^3)}{(a+b x^2)^2} \, dx\) [34]

Optimal result

Integrand size = 32, antiderivative size = 260 \[ \int \frac {(c+d x)^2 \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^2} \, dx=\frac {d (C d+2 c D) x}{b^2}+\frac {d^2 D x^2}{2 b^2}-\frac {b^2 c (B c+2 A d)+a^2 d^2 D-a b \left (2 c C d+B d^2+c^2 D\right )}{2 b^3 \left (a+b x^2\right )}+\frac {\left (A b \left (b c^2-a d^2\right )-a (b c (c C+2 B d)-a d (C d+2 c D))\right ) x}{2 a b^2 \left (a+b x^2\right )}+\frac {\left (A b \left (b c^2+a d^2\right )+a (b c (c C+2 B d)-3 a d (C d+2 c D))\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{5/2}}-\frac {\left (2 a d^2 D-b \left (2 c C d+B d^2+c^2 D\right )\right ) \log \left (a+b x^2\right )}{2 b^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.86 \[ \int \frac {(c+d x)^2 \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^2} \, dx=\frac {2 b d (C d+2 c D) x+b d^2 D x^2+\frac {-a^3 d^2 D+A b^3 c^2 x-a b^2 \left (c^2 C x+A d (2 c+d x)+B c (c+2 d x)\right )+a^2 b \left (c^2 D+d^2 (B+C x)+2 c d (C+D x)\right )}{a \left (a+b x^2\right )}+\frac {\sqrt {b} \left (A b \left (b c^2+a d^2\right )+a (b c (c C+2 B d)-3 a d (C d+2 c D))\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2}}+\left (-2 a d^2 D+b \left (2 c C d+B d^2+c^2 D\right )\right ) \log \left (a+b x^2\right )}{2 b^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.77, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2176, 25, 2160, 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.

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

\(\Big \downarrow \) 2176

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2160

\(\displaystyle \frac {\int \left (\frac {2 a D x d^2}{b}-\frac {(A b d-3 a C d-4 a c D) d}{b}+\frac {A b \left (b c^2+a d^2\right )+a (b c (c C+2 B d)-3 a d (C d+2 c D))-2 a \left (2 a d^2 D-b \left (D c^2+2 C d c+B d^2\right )\right ) x}{b \left (b x^2+a\right )}\right )dx}{2 a b}-\frac {(c+d x)^2 \left (a \left (B-\frac {a D}{b}\right )-x (A b-a C)\right )}{2 a b \left (a+b x^2\right )}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (A b \left (a d^2+b c^2\right )+a (b c (2 B d+c C)-3 a d (2 c D+C d))\right )}{\sqrt {a} b^{3/2}}-\frac {d x (-4 a c D-3 a C d+A b d)}{b}-\frac {a \log \left (a+b x^2\right ) \left (2 a d^2 D-b \left (B d^2+c^2 D+2 c C d\right )\right )}{b^2}+\frac {a d^2 D x^2}{b}}{2 a b}-\frac {(c+d x)^2 \left (a \left (B-\frac {a D}{b}\right )-x (A b-a C)\right )}{2 a b \left (a+b x^2\right )}\)

Input:

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

Output:

-1/2*((a*(B - (a*D)/b) - (A*b - a*C)*x)*(c + d*x)^2)/(a*b*(a + b*x^2)) + ( 
-((d*(A*b*d - 3*a*C*d - 4*a*c*D)*x)/b) + (a*d^2*D*x^2)/b + ((A*b*(b*c^2 + 
a*d^2) + a*(b*c*(c*C + 2*B*d) - 3*a*d*(C*d + 2*c*D)))*ArcTan[(Sqrt[b]*x)/S 
qrt[a]])/(Sqrt[a]*b^(3/2)) - (a*(2*a*d^2*D - b*(2*c*C*d + B*d^2 + c^2*D))* 
Log[a + b*x^2])/b^2)/(2*a*b)
 

Defintions of rubi rules used

rule 25

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

rule 2009

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

rule 2160

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

rule 2176

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

Maple [A] (verified)

Time = 0.99 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.00

method	result	size
default	\(\frac {d \left (\frac {1}{2} D d \,x^{2}+C x d +2 D c x \right )}{b^{2}}+\frac {\frac {-\frac {\left (A a b \,d^{2}-A \,b^{2} c^{2}+2 a b B c d -a^{2} C \,d^{2}+C a b \,c^{2}-2 a^{2} c d D\right ) x}{2 a}-\frac {2 A \,b^{2} c d -B a b \,d^{2}+B \,b^{2} c^{2}-2 a b c d C +a^{2} d^{2} D-D a b \,c^{2}}{2 b}}{b \,x^{2}+a}+\frac {\frac {\left (2 B a b \,d^{2}+4 a b c d C -4 a^{2} d^{2} D+2 D a b \,c^{2}\right ) \ln \left (b \,x^{2}+a \right )}{2 b}+\frac {\left (A a b \,d^{2}+A \,b^{2} c^{2}+2 a b B c d -3 a^{2} C \,d^{2}+C a b \,c^{2}-6 a^{2} c d D\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}}}{2 a}}{b^{2}}\)	\(261\)

Input:

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

Output:

d/b^2*(1/2*D*d*x^2+C*x*d+2*D*c*x)+1/b^2*((-1/2*(A*a*b*d^2-A*b^2*c^2+2*B*a* 
b*c*d-C*a^2*d^2+C*a*b*c^2-2*D*a^2*c*d)/a*x-1/2*(2*A*b^2*c*d-B*a*b*d^2+B*b^ 
2*c^2-2*C*a*b*c*d+D*a^2*d^2-D*a*b*c^2)/b)/(b*x^2+a)+1/2/a*(1/2*(2*B*a*b*d^ 
2+4*C*a*b*c*d-4*D*a^2*d^2+2*D*a*b*c^2)/b*ln(b*x^2+a)+(A*a*b*d^2+A*b^2*c^2+ 
2*B*a*b*c*d-3*C*a^2*d^2+C*a*b*c^2-6*D*a^2*c*d)/(a*b)^(1/2)*arctan(b*x/(a*b 
)^(1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 885, normalized size of antiderivative = 3.40 \[ \int \frac {(c+d x)^2 \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 876 vs. \(2 (248) = 496\).

Time = 27.46 (sec) , antiderivative size = 876, normalized size of antiderivative = 3.37 \[ \int \frac {(c+d x)^2 \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^2} \, dx=\frac {D d^{2} x^{2}}{2 b^{2}} + x \left (\frac {C d^{2}}{b^{2}} + \frac {2 D c d}{b^{2}}\right ) + \left (- \frac {- B b d^{2} - 2 C b c d + 2 D a d^{2} - D b c^{2}}{2 b^{3}} - \frac {\sqrt {- a^{3} b^{7}} \left (- A a b d^{2} - A b^{2} c^{2} - 2 B a b c d + 3 C a^{2} d^{2} - C a b c^{2} + 6 D a^{2} c d\right )}{4 a^{3} b^{6}}\right ) \log {\left (x + \frac {2 B a^{2} b d^{2} + 4 C a^{2} b c d - 4 D a^{3} d^{2} + 2 D a^{2} b c^{2} - 4 a^{2} b^{3} \left (- \frac {- B b d^{2} - 2 C b c d + 2 D a d^{2} - D b c^{2}}{2 b^{3}} - \frac {\sqrt {- a^{3} b^{7}} \left (- A a b d^{2} - A b^{2} c^{2} - 2 B a b c d + 3 C a^{2} d^{2} - C a b c^{2} + 6 D a^{2} c d\right )}{4 a^{3} b^{6}}\right )}{- A a b^{2} d^{2} - A b^{3} c^{2} - 2 B a b^{2} c d + 3 C a^{2} b d^{2} - C a b^{2} c^{2} + 6 D a^{2} b c d} \right )} + \left (- \frac {- B b d^{2} - 2 C b c d + 2 D a d^{2} - D b c^{2}}{2 b^{3}} + \frac {\sqrt {- a^{3} b^{7}} \left (- A a b d^{2} - A b^{2} c^{2} - 2 B a b c d + 3 C a^{2} d^{2} - C a b c^{2} + 6 D a^{2} c d\right )}{4 a^{3} b^{6}}\right ) \log {\left (x + \frac {2 B a^{2} b d^{2} + 4 C a^{2} b c d - 4 D a^{3} d^{2} + 2 D a^{2} b c^{2} - 4 a^{2} b^{3} \left (- \frac {- B b d^{2} - 2 C b c d + 2 D a d^{2} - D b c^{2}}{2 b^{3}} + \frac {\sqrt {- a^{3} b^{7}} \left (- A a b d^{2} - A b^{2} c^{2} - 2 B a b c d + 3 C a^{2} d^{2} - C a b c^{2} + 6 D a^{2} c d\right )}{4 a^{3} b^{6}}\right )}{- A a b^{2} d^{2} - A b^{3} c^{2} - 2 B a b^{2} c d + 3 C a^{2} b d^{2} - C a b^{2} c^{2} + 6 D a^{2} b c d} \right )} + \frac {- 2 A a b^{2} c d + B a^{2} b d^{2} - B a b^{2} c^{2} + 2 C a^{2} b c d - D a^{3} d^{2} + D a^{2} b c^{2} + x \left (- A a b^{2} d^{2} + A b^{3} c^{2} - 2 B a b^{2} c d + C a^{2} b d^{2} - C a b^{2} c^{2} + 2 D a^{2} b c d\right )}{2 a^{2} b^{3} + 2 a b^{4} x^{2}} \] Input:

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

Output:

D*d**2*x**2/(2*b**2) + x*(C*d**2/b**2 + 2*D*c*d/b**2) + (-(-B*b*d**2 - 2*C 
*b*c*d + 2*D*a*d**2 - D*b*c**2)/(2*b**3) - sqrt(-a**3*b**7)*(-A*a*b*d**2 - 
 A*b**2*c**2 - 2*B*a*b*c*d + 3*C*a**2*d**2 - C*a*b*c**2 + 6*D*a**2*c*d)/(4 
*a**3*b**6))*log(x + (2*B*a**2*b*d**2 + 4*C*a**2*b*c*d - 4*D*a**3*d**2 + 2 
*D*a**2*b*c**2 - 4*a**2*b**3*(-(-B*b*d**2 - 2*C*b*c*d + 2*D*a*d**2 - D*b*c 
**2)/(2*b**3) - sqrt(-a**3*b**7)*(-A*a*b*d**2 - A*b**2*c**2 - 2*B*a*b*c*d 
+ 3*C*a**2*d**2 - C*a*b*c**2 + 6*D*a**2*c*d)/(4*a**3*b**6)))/(-A*a*b**2*d* 
*2 - A*b**3*c**2 - 2*B*a*b**2*c*d + 3*C*a**2*b*d**2 - C*a*b**2*c**2 + 6*D* 
a**2*b*c*d)) + (-(-B*b*d**2 - 2*C*b*c*d + 2*D*a*d**2 - D*b*c**2)/(2*b**3) 
+ sqrt(-a**3*b**7)*(-A*a*b*d**2 - A*b**2*c**2 - 2*B*a*b*c*d + 3*C*a**2*d** 
2 - C*a*b*c**2 + 6*D*a**2*c*d)/(4*a**3*b**6))*log(x + (2*B*a**2*b*d**2 + 4 
*C*a**2*b*c*d - 4*D*a**3*d**2 + 2*D*a**2*b*c**2 - 4*a**2*b**3*(-(-B*b*d**2 
 - 2*C*b*c*d + 2*D*a*d**2 - D*b*c**2)/(2*b**3) + sqrt(-a**3*b**7)*(-A*a*b* 
d**2 - A*b**2*c**2 - 2*B*a*b*c*d + 3*C*a**2*d**2 - C*a*b*c**2 + 6*D*a**2*c 
*d)/(4*a**3*b**6)))/(-A*a*b**2*d**2 - A*b**3*c**2 - 2*B*a*b**2*c*d + 3*C*a 
**2*b*d**2 - C*a*b**2*c**2 + 6*D*a**2*b*c*d)) + (-2*A*a*b**2*c*d + B*a**2* 
b*d**2 - B*a*b**2*c**2 + 2*C*a**2*b*c*d - D*a**3*d**2 + D*a**2*b*c**2 + x* 
(-A*a*b**2*d**2 + A*b**3*c**2 - 2*B*a*b**2*c*d + C*a**2*b*d**2 - C*a*b**2* 
c**2 + 2*D*a**2*b*c*d))/(2*a**2*b**3 + 2*a*b**4*x**2)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.05 \[ \int \frac {(c+d x)^2 \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^2} \, dx=\frac {{\left (D a^{2} b - B a b^{2}\right )} c^{2} + 2 \, {\left (C a^{2} b - A a b^{2}\right )} c d - {\left (D a^{3} - B a^{2} b\right )} d^{2} - {\left ({\left (C a b^{2} - A b^{3}\right )} c^{2} - 2 \, {\left (D a^{2} b - B a b^{2}\right )} c d - {\left (C a^{2} b - A a b^{2}\right )} d^{2}\right )} x}{2 \, {\left (a b^{4} x^{2} + a^{2} b^{3}\right )}} + \frac {D d^{2} x^{2} + 2 \, {\left (2 \, D c d + C d^{2}\right )} x}{2 \, b^{2}} + \frac {{\left (D b c^{2} + 2 \, C b c d - {\left (2 \, D a - B b\right )} d^{2}\right )} \log \left (b x^{2} + a\right )}{2 \, b^{3}} + \frac {{\left ({\left (C a b + A b^{2}\right )} c^{2} - 2 \, {\left (3 \, D a^{2} - B a b\right )} c d - {\left (3 \, C a^{2} - A a b\right )} d^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a b^{2}} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.07 \[ \int \frac {(c+d x)^2 \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^2} \, dx=\frac {{\left (D b c^{2} + 2 \, C b c d - 2 \, D a d^{2} + B b d^{2}\right )} \log \left (b x^{2} + a\right )}{2 \, b^{3}} + \frac {{\left (C a b c^{2} + A b^{2} c^{2} - 6 \, D a^{2} c d + 2 \, B a b c d - 3 \, C a^{2} d^{2} + A a b d^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a b^{2}} + \frac {D b^{2} d^{2} x^{2} + 4 \, D b^{2} c d x + 2 \, C b^{2} d^{2} x}{2 \, b^{4}} + \frac {D a^{2} b c^{2} - B a b^{2} c^{2} + 2 \, C a^{2} b c d - 2 \, A a b^{2} c d - D a^{3} d^{2} + B a^{2} b d^{2} - {\left (C a b^{2} c^{2} - A b^{3} c^{2} - 2 \, D a^{2} b c d + 2 \, B a b^{2} c d - C a^{2} b d^{2} + A a b^{2} d^{2}\right )} x}{2 \, {\left (b x^{2} + a\right )} a b^{3}} \] Input:

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

Output:

1/2*(D*b*c^2 + 2*C*b*c*d - 2*D*a*d^2 + B*b*d^2)*log(b*x^2 + a)/b^3 + 1/2*( 
C*a*b*c^2 + A*b^2*c^2 - 6*D*a^2*c*d + 2*B*a*b*c*d - 3*C*a^2*d^2 + A*a*b*d^ 
2)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a*b^2) + 1/2*(D*b^2*d^2*x^2 + 4*D*b^2* 
c*d*x + 2*C*b^2*d^2*x)/b^4 + 1/2*(D*a^2*b*c^2 - B*a*b^2*c^2 + 2*C*a^2*b*c* 
d - 2*A*a*b^2*c*d - D*a^3*d^2 + B*a^2*b*d^2 - (C*a*b^2*c^2 - A*b^3*c^2 - 2 
*D*a^2*b*c*d + 2*B*a*b^2*c*d - C*a^2*b*d^2 + A*a*b^2*d^2)*x)/((b*x^2 + a)* 
a*b^3)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 514, normalized size of antiderivative = 1.98 \[ \int \frac {(c+d x)^2 \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^2} \, dx=\frac {-2 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{3} d^{3}-9 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a b c \,d^{2} x^{2}+\mathrm {log}\left (b \,x^{2}+a \right ) a^{2} b^{2} d^{2}-a^{2} b^{2} d^{2} x +a \,b^{3} c^{2} x -a \,b^{3} d^{2} x^{2}-a \,b^{2} c^{3} x +a \,b^{2} d^{3} x^{4}+\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b^{2} c^{3} x^{2}+\mathrm {log}\left (b \,x^{2}+a \right ) a \,b^{3} d^{2} x^{2}-9 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} c \,d^{2}+3 \,\mathrm {log}\left (b \,x^{2}+a \right ) a \,b^{2} c^{2} d \,x^{2}+b^{4} c^{2} x^{2}+2 a^{2} b \,d^{3} x^{2}+\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{2} d^{2} x^{2}+2 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{2} c d +2 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b^{3} c d \,x^{2}+3 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{2} b \,c^{2} d -2 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{2} b \,d^{3} x^{2}+9 a^{2} b c \,d^{2} x +2 a \,b^{3} c d \,x^{2}-2 a \,b^{3} c d x -3 a \,b^{2} c^{2} d \,x^{2}+6 a \,b^{2} c \,d^{2} x^{3}+\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b \,d^{2}+\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{2} c^{2}+\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b^{3} c^{2} x^{2}+\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a b \,c^{3}}{2 a \,b^{3} \left (b \,x^{2}+a \right )} \] Input:

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

Output:

(sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*b*d**2 - 9*sqrt(b)*sqr 
t(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*c*d**2 + sqrt(b)*sqrt(a)*atan((b*x 
)/(sqrt(b)*sqrt(a)))*a*b**2*c**2 + 2*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*s 
qrt(a)))*a*b**2*c*d + sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b**2 
*d**2*x**2 + sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b*c**3 - 9*sq 
rt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b*c*d**2*x**2 + sqrt(b)*sqrt 
(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**3*c**2*x**2 + 2*sqrt(b)*sqrt(a)*atan( 
(b*x)/(sqrt(b)*sqrt(a)))*b**3*c*d*x**2 + sqrt(b)*sqrt(a)*atan((b*x)/(sqrt( 
b)*sqrt(a)))*b**2*c**3*x**2 - 2*log(a + b*x**2)*a**3*d**3 + log(a + b*x**2 
)*a**2*b**2*d**2 + 3*log(a + b*x**2)*a**2*b*c**2*d - 2*log(a + b*x**2)*a** 
2*b*d**3*x**2 + log(a + b*x**2)*a*b**3*d**2*x**2 + 3*log(a + b*x**2)*a*b** 
2*c**2*d*x**2 - a**2*b**2*d**2*x + 9*a**2*b*c*d**2*x + 2*a**2*b*d**3*x**2 
+ a*b**3*c**2*x + 2*a*b**3*c*d*x**2 - 2*a*b**3*c*d*x - a*b**3*d**2*x**2 - 
a*b**2*c**3*x - 3*a*b**2*c**2*d*x**2 + 6*a*b**2*c*d**2*x**3 + a*b**2*d**3* 
x**4 + b**4*c**2*x**2)/(2*a*b**3*(a + b*x**2))