\(\int \frac {(A+B x) (d+e x)^3}{(a+c x^2)^3} \, dx\) [97]

Optimal result

Integrand size = 22, antiderivative size = 132 \[ \int \frac {(A+B x) (d+e x)^3}{\left (a+c x^2\right )^3} \, dx=-\frac {(a B-A c x) (d+e x)^3}{4 a c \left (a+c x^2\right )^2}-\frac {3 (A c d+a B e) \left (2 a d e-\left (c d^2-a e^2\right ) x\right )}{8 a^2 c^2 \left (a+c x^2\right )}+\frac {3 (A c d+a B e) \left (c d^2+a e^2\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} c^{5/2}} \] Output:

-1/4*(-A*c*x+B*a)*(e*x+d)^3/a/c/(c*x^2+a)^2-3/8*(A*c*d+B*a*e)*(2*a*d*e-(-a 
*e^2+c*d^2)*x)/a^2/c^2/(c*x^2+a)+3/8*(A*c*d+B*a*e)*(a*e^2+c*d^2)*arctan(c^ 
(1/2)*x/a^(1/2))/a^(5/2)/c^(5/2)
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.41 \[ \int \frac {(A+B x) (d+e x)^3}{\left (a+c x^2\right )^3} \, dx=\frac {3 A c^2 d^3 x+3 a c d e (B d+A e) x-a^2 e^2 (12 B d+4 A e+5 B e x)}{8 a^2 c^2 \left (a+c x^2\right )}+\frac {A c^2 d^3 x+a^2 e^2 (3 B d+A e+B e x)-a c d (3 A e (d+e x)+B d (d+3 e x))}{4 a c^2 \left (a+c x^2\right )^2}+\frac {3 (A c d+a B e) \left (c d^2+a e^2\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} c^{5/2}} \] Input:

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

Output:

(3*A*c^2*d^3*x + 3*a*c*d*e*(B*d + A*e)*x - a^2*e^2*(12*B*d + 4*A*e + 5*B*e 
*x))/(8*a^2*c^2*(a + c*x^2)) + (A*c^2*d^3*x + a^2*e^2*(3*B*d + A*e + B*e*x 
) - a*c*d*(3*A*e*(d + e*x) + B*d*(d + 3*e*x)))/(4*a*c^2*(a + c*x^2)^2) + ( 
3*(A*c*d + a*B*e)*(c*d^2 + a*e^2)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(8*a^(5/2)* 
c^(5/2))
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.96, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {678, 487, 218}

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.

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

\(\Big \downarrow \) 678

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

\(\Big \downarrow \) 487

\(\displaystyle \frac {3 (a B e+A c d) \left (\frac {\left (a e^2+c d^2\right ) \int \frac {1}{c x^2+a}dx}{2 a c}-\frac {(d+e x) (a e-c d x)}{2 a c \left (a+c x^2\right )}\right )}{4 a c}-\frac {(d+e x)^3 (a B-A c x)}{4 a c \left (a+c x^2\right )^2}\)

\(\Big \downarrow \) 218

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

Input:

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

Output:

-1/4*((a*B - A*c*x)*(d + e*x)^3)/(a*c*(a + c*x^2)^2) + (3*(A*c*d + a*B*e)* 
(-1/2*((a*e - c*d*x)*(d + e*x))/(a*c*(a + c*x^2)) + ((c*d^2 + a*e^2)*ArcTa 
n[(Sqrt[c]*x)/Sqrt[a]])/(2*a^(3/2)*c^(3/2))))/(4*a*c)
 

Defintions of rubi rules used

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 487

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
(c + d*x)^(n - 1)*(a*d - b*c*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + 
 Simp[(2*p + 3)*((b*c^2 + a*d^2)/(2*a*b*(p + 1)))   Int[(c + d*x)^(n - 2)*( 
a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[n + 2*p + 2, 0] 
 && LtQ[p, -1]
 

rule 678

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_), x_Symbol] :> Simp[(d + e*x)^m*(a + c*x^2)^(p + 1)*((a*g - c*f*x)/(2*a*c 
*(p + 1))), x] - Simp[m*((c*d*f + a*e*g)/(2*a*c*(p + 1)))   Int[(d + e*x)^( 
m - 1)*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && EqQ[S 
implify[m + 2*p + 3], 0] && LtQ[p, -1]
 

Maple [A] (verified)

Time = 0.88 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.63

method	result	size
default	\(\frac {\frac {\left (3 A a c d \,e^{2}+3 A \,c^{2} d^{3}-5 B \,e^{3} a^{2}+3 B a c \,d^{2} e \right ) x^{3}}{8 a^{2} c}-\frac {e^{2} \left (A e +3 B d \right ) x^{2}}{2 c}-\frac {\left (3 A a c d \,e^{2}-5 A \,c^{2} d^{3}+3 B \,e^{3} a^{2}+3 B a c \,d^{2} e \right ) x}{8 a \,c^{2}}-\frac {A a \,e^{3}+3 A c \,d^{2} e +3 B a d \,e^{2}+B c \,d^{3}}{4 c^{2}}}{\left (c \,x^{2}+a \right )^{2}}+\frac {3 \left (A a c d \,e^{2}+A \,c^{2} d^{3}+B \,e^{3} a^{2}+B a c \,d^{2} e \right ) \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 a^{2} c^{2} \sqrt {a c}}\)	\(215\)
risch	\(\frac {\frac {\left (3 A a c d \,e^{2}+3 A \,c^{2} d^{3}-5 B \,e^{3} a^{2}+3 B a c \,d^{2} e \right ) x^{3}}{8 a^{2} c}-\frac {e^{2} \left (A e +3 B d \right ) x^{2}}{2 c}-\frac {\left (3 A a c d \,e^{2}-5 A \,c^{2} d^{3}+3 B \,e^{3} a^{2}+3 B a c \,d^{2} e \right ) x}{8 a \,c^{2}}-\frac {A a \,e^{3}+3 A c \,d^{2} e +3 B a d \,e^{2}+B c \,d^{3}}{4 c^{2}}}{\left (c \,x^{2}+a \right )^{2}}-\frac {3 \ln \left (c x +\sqrt {-a c}\right ) A d \,e^{2}}{16 \sqrt {-a c}\, c a}-\frac {3 \ln \left (c x +\sqrt {-a c}\right ) A \,d^{3}}{16 \sqrt {-a c}\, a^{2}}-\frac {3 \ln \left (c x +\sqrt {-a c}\right ) B \,e^{3}}{16 \sqrt {-a c}\, c^{2}}-\frac {3 \ln \left (c x +\sqrt {-a c}\right ) B \,d^{2} e}{16 \sqrt {-a c}\, c a}+\frac {3 \ln \left (-c x +\sqrt {-a c}\right ) A d \,e^{2}}{16 \sqrt {-a c}\, c a}+\frac {3 \ln \left (-c x +\sqrt {-a c}\right ) A \,d^{3}}{16 \sqrt {-a c}\, a^{2}}+\frac {3 \ln \left (-c x +\sqrt {-a c}\right ) B \,e^{3}}{16 \sqrt {-a c}\, c^{2}}+\frac {3 \ln \left (-c x +\sqrt {-a c}\right ) B \,d^{2} e}{16 \sqrt {-a c}\, c a}\)	\(388\)

Input:

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

Output:

(1/8*(3*A*a*c*d*e^2+3*A*c^2*d^3-5*B*a^2*e^3+3*B*a*c*d^2*e)/a^2/c*x^3-1/2*e 
^2*(A*e+3*B*d)*x^2/c-1/8*(3*A*a*c*d*e^2-5*A*c^2*d^3+3*B*a^2*e^3+3*B*a*c*d^ 
2*e)/a/c^2*x-1/4*(A*a*e^3+3*A*c*d^2*e+3*B*a*d*e^2+B*c*d^3)/c^2)/(c*x^2+a)^ 
2+3/8*(A*a*c*d*e^2+A*c^2*d^3+B*a^2*e^3+B*a*c*d^2*e)/a^2/c^2/(a*c)^(1/2)*ar 
ctan(c*x/(a*c)^(1/2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 366 vs. \(2 (118) = 236\).

Time = 0.12 (sec) , antiderivative size = 752, normalized size of antiderivative = 5.70 \[ \int \frac {(A+B x) (d+e x)^3}{\left (a+c x^2\right )^3} \, dx=\left [-\frac {4 \, B a^{3} c^{2} d^{3} + 12 \, A a^{3} c^{2} d^{2} e + 12 \, B a^{4} c d e^{2} + 4 \, A a^{4} c e^{3} - 2 \, {\left (3 \, A a c^{4} d^{3} + 3 \, B a^{2} c^{3} d^{2} e + 3 \, A a^{2} c^{3} d e^{2} - 5 \, B a^{3} c^{2} e^{3}\right )} x^{3} + 8 \, {\left (3 \, B a^{3} c^{2} d e^{2} + A a^{3} c^{2} e^{3}\right )} x^{2} + 3 \, {\left (A a^{2} c^{2} d^{3} + B a^{3} c d^{2} e + A a^{3} c d e^{2} + B a^{4} e^{3} + {\left (A c^{4} d^{3} + B a c^{3} d^{2} e + A a c^{3} d e^{2} + B a^{2} c^{2} e^{3}\right )} x^{4} + 2 \, {\left (A a c^{3} d^{3} + B a^{2} c^{2} d^{2} e + A a^{2} c^{2} d e^{2} + B a^{3} c e^{3}\right )} x^{2}\right )} \sqrt {-a c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right ) - 2 \, {\left (5 \, A a^{2} c^{3} d^{3} - 3 \, B a^{3} c^{2} d^{2} e - 3 \, A a^{3} c^{2} d e^{2} - 3 \, B a^{4} c e^{3}\right )} x}{16 \, {\left (a^{3} c^{5} x^{4} + 2 \, a^{4} c^{4} x^{2} + a^{5} c^{3}\right )}}, -\frac {2 \, B a^{3} c^{2} d^{3} + 6 \, A a^{3} c^{2} d^{2} e + 6 \, B a^{4} c d e^{2} + 2 \, A a^{4} c e^{3} - {\left (3 \, A a c^{4} d^{3} + 3 \, B a^{2} c^{3} d^{2} e + 3 \, A a^{2} c^{3} d e^{2} - 5 \, B a^{3} c^{2} e^{3}\right )} x^{3} + 4 \, {\left (3 \, B a^{3} c^{2} d e^{2} + A a^{3} c^{2} e^{3}\right )} x^{2} - 3 \, {\left (A a^{2} c^{2} d^{3} + B a^{3} c d^{2} e + A a^{3} c d e^{2} + B a^{4} e^{3} + {\left (A c^{4} d^{3} + B a c^{3} d^{2} e + A a c^{3} d e^{2} + B a^{2} c^{2} e^{3}\right )} x^{4} + 2 \, {\left (A a c^{3} d^{3} + B a^{2} c^{2} d^{2} e + A a^{2} c^{2} d e^{2} + B a^{3} c e^{3}\right )} x^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) - {\left (5 \, A a^{2} c^{3} d^{3} - 3 \, B a^{3} c^{2} d^{2} e - 3 \, A a^{3} c^{2} d e^{2} - 3 \, B a^{4} c e^{3}\right )} x}{8 \, {\left (a^{3} c^{5} x^{4} + 2 \, a^{4} c^{4} x^{2} + a^{5} c^{3}\right )}}\right ] \] Input:

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

Output:

[-1/16*(4*B*a^3*c^2*d^3 + 12*A*a^3*c^2*d^2*e + 12*B*a^4*c*d*e^2 + 4*A*a^4* 
c*e^3 - 2*(3*A*a*c^4*d^3 + 3*B*a^2*c^3*d^2*e + 3*A*a^2*c^3*d*e^2 - 5*B*a^3 
*c^2*e^3)*x^3 + 8*(3*B*a^3*c^2*d*e^2 + A*a^3*c^2*e^3)*x^2 + 3*(A*a^2*c^2*d 
^3 + B*a^3*c*d^2*e + A*a^3*c*d*e^2 + B*a^4*e^3 + (A*c^4*d^3 + B*a*c^3*d^2* 
e + A*a*c^3*d*e^2 + B*a^2*c^2*e^3)*x^4 + 2*(A*a*c^3*d^3 + B*a^2*c^2*d^2*e 
+ A*a^2*c^2*d*e^2 + B*a^3*c*e^3)*x^2)*sqrt(-a*c)*log((c*x^2 - 2*sqrt(-a*c) 
*x - a)/(c*x^2 + a)) - 2*(5*A*a^2*c^3*d^3 - 3*B*a^3*c^2*d^2*e - 3*A*a^3*c^ 
2*d*e^2 - 3*B*a^4*c*e^3)*x)/(a^3*c^5*x^4 + 2*a^4*c^4*x^2 + a^5*c^3), -1/8* 
(2*B*a^3*c^2*d^3 + 6*A*a^3*c^2*d^2*e + 6*B*a^4*c*d*e^2 + 2*A*a^4*c*e^3 - ( 
3*A*a*c^4*d^3 + 3*B*a^2*c^3*d^2*e + 3*A*a^2*c^3*d*e^2 - 5*B*a^3*c^2*e^3)*x 
^3 + 4*(3*B*a^3*c^2*d*e^2 + A*a^3*c^2*e^3)*x^2 - 3*(A*a^2*c^2*d^3 + B*a^3* 
c*d^2*e + A*a^3*c*d*e^2 + B*a^4*e^3 + (A*c^4*d^3 + B*a*c^3*d^2*e + A*a*c^3 
*d*e^2 + B*a^2*c^2*e^3)*x^4 + 2*(A*a*c^3*d^3 + B*a^2*c^2*d^2*e + A*a^2*c^2 
*d*e^2 + B*a^3*c*e^3)*x^2)*sqrt(a*c)*arctan(sqrt(a*c)*x/a) - (5*A*a^2*c^3* 
d^3 - 3*B*a^3*c^2*d^2*e - 3*A*a^3*c^2*d*e^2 - 3*B*a^4*c*e^3)*x)/(a^3*c^5*x 
^4 + 2*a^4*c^4*x^2 + a^5*c^3)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 468 vs. \(2 (129) = 258\).

Time = 8.58 (sec) , antiderivative size = 468, normalized size of antiderivative = 3.55 \[ \int \frac {(A+B x) (d+e x)^3}{\left (a+c x^2\right )^3} \, dx=- \frac {3 \sqrt {- \frac {1}{a^{5} c^{5}}} \left (a e^{2} + c d^{2}\right ) \left (A c d + B a e\right ) \log {\left (- \frac {3 a^{3} c^{2} \sqrt {- \frac {1}{a^{5} c^{5}}} \left (a e^{2} + c d^{2}\right ) \left (A c d + B a e\right )}{3 A a c d e^{2} + 3 A c^{2} d^{3} + 3 B a^{2} e^{3} + 3 B a c d^{2} e} + x \right )}}{16} + \frac {3 \sqrt {- \frac {1}{a^{5} c^{5}}} \left (a e^{2} + c d^{2}\right ) \left (A c d + B a e\right ) \log {\left (\frac {3 a^{3} c^{2} \sqrt {- \frac {1}{a^{5} c^{5}}} \left (a e^{2} + c d^{2}\right ) \left (A c d + B a e\right )}{3 A a c d e^{2} + 3 A c^{2} d^{3} + 3 B a^{2} e^{3} + 3 B a c d^{2} e} + x \right )}}{16} + \frac {- 2 A a^{3} e^{3} - 6 A a^{2} c d^{2} e - 6 B a^{3} d e^{2} - 2 B a^{2} c d^{3} + x^{3} \cdot \left (3 A a c^{2} d e^{2} + 3 A c^{3} d^{3} - 5 B a^{2} c e^{3} + 3 B a c^{2} d^{2} e\right ) + x^{2} \left (- 4 A a^{2} c e^{3} - 12 B a^{2} c d e^{2}\right ) + x \left (- 3 A a^{2} c d e^{2} + 5 A a c^{2} d^{3} - 3 B a^{3} e^{3} - 3 B a^{2} c d^{2} e\right )}{8 a^{4} c^{2} + 16 a^{3} c^{3} x^{2} + 8 a^{2} c^{4} x^{4}} \] Input:

integrate((B*x+A)*(e*x+d)**3/(c*x**2+a)**3,x)
 

Output:

-3*sqrt(-1/(a**5*c**5))*(a*e**2 + c*d**2)*(A*c*d + B*a*e)*log(-3*a**3*c**2 
*sqrt(-1/(a**5*c**5))*(a*e**2 + c*d**2)*(A*c*d + B*a*e)/(3*A*a*c*d*e**2 + 
3*A*c**2*d**3 + 3*B*a**2*e**3 + 3*B*a*c*d**2*e) + x)/16 + 3*sqrt(-1/(a**5* 
c**5))*(a*e**2 + c*d**2)*(A*c*d + B*a*e)*log(3*a**3*c**2*sqrt(-1/(a**5*c** 
5))*(a*e**2 + c*d**2)*(A*c*d + B*a*e)/(3*A*a*c*d*e**2 + 3*A*c**2*d**3 + 3* 
B*a**2*e**3 + 3*B*a*c*d**2*e) + x)/16 + (-2*A*a**3*e**3 - 6*A*a**2*c*d**2* 
e - 6*B*a**3*d*e**2 - 2*B*a**2*c*d**3 + x**3*(3*A*a*c**2*d*e**2 + 3*A*c**3 
*d**3 - 5*B*a**2*c*e**3 + 3*B*a*c**2*d**2*e) + x**2*(-4*A*a**2*c*e**3 - 12 
*B*a**2*c*d*e**2) + x*(-3*A*a**2*c*d*e**2 + 5*A*a*c**2*d**3 - 3*B*a**3*e** 
3 - 3*B*a**2*c*d**2*e))/(8*a**4*c**2 + 16*a**3*c**3*x**2 + 8*a**2*c**4*x** 
4)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (118) = 236\).

Time = 0.15 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.88 \[ \int \frac {(A+B x) (d+e x)^3}{\left (a+c x^2\right )^3} \, dx=-\frac {2 \, B a^{2} c d^{3} + 6 \, A a^{2} c d^{2} e + 6 \, B a^{3} d e^{2} + 2 \, A a^{3} e^{3} - {\left (3 \, A c^{3} d^{3} + 3 \, B a c^{2} d^{2} e + 3 \, A a c^{2} d e^{2} - 5 \, B a^{2} c e^{3}\right )} x^{3} + 4 \, {\left (3 \, B a^{2} c d e^{2} + A a^{2} c e^{3}\right )} x^{2} - {\left (5 \, A a c^{2} d^{3} - 3 \, B a^{2} c d^{2} e - 3 \, A a^{2} c d e^{2} - 3 \, B a^{3} e^{3}\right )} x}{8 \, {\left (a^{2} c^{4} x^{4} + 2 \, a^{3} c^{3} x^{2} + a^{4} c^{2}\right )}} + \frac {3 \, {\left (A c^{2} d^{3} + B a c d^{2} e + A a c d e^{2} + B a^{2} e^{3}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {a c} a^{2} c^{2}} \] Input:

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

Output:

-1/8*(2*B*a^2*c*d^3 + 6*A*a^2*c*d^2*e + 6*B*a^3*d*e^2 + 2*A*a^3*e^3 - (3*A 
*c^3*d^3 + 3*B*a*c^2*d^2*e + 3*A*a*c^2*d*e^2 - 5*B*a^2*c*e^3)*x^3 + 4*(3*B 
*a^2*c*d*e^2 + A*a^2*c*e^3)*x^2 - (5*A*a*c^2*d^3 - 3*B*a^2*c*d^2*e - 3*A*a 
^2*c*d*e^2 - 3*B*a^3*e^3)*x)/(a^2*c^4*x^4 + 2*a^3*c^3*x^2 + a^4*c^2) + 3/8 
*(A*c^2*d^3 + B*a*c*d^2*e + A*a*c*d*e^2 + B*a^2*e^3)*arctan(c*x/sqrt(a*c)) 
/(sqrt(a*c)*a^2*c^2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 239 vs. \(2 (118) = 236\).

Time = 0.13 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.81 \[ \int \frac {(A+B x) (d+e x)^3}{\left (a+c x^2\right )^3} \, dx=\frac {3 \, {\left (A c^{2} d^{3} + B a c d^{2} e + A a c d e^{2} + B a^{2} e^{3}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {a c} a^{2} c^{2}} + \frac {3 \, A c^{3} d^{3} x^{3} + 3 \, B a c^{2} d^{2} e x^{3} + 3 \, A a c^{2} d e^{2} x^{3} - 5 \, B a^{2} c e^{3} x^{3} - 12 \, B a^{2} c d e^{2} x^{2} - 4 \, A a^{2} c e^{3} x^{2} + 5 \, A a c^{2} d^{3} x - 3 \, B a^{2} c d^{2} e x - 3 \, A a^{2} c d e^{2} x - 3 \, B a^{3} e^{3} x - 2 \, B a^{2} c d^{3} - 6 \, A a^{2} c d^{2} e - 6 \, B a^{3} d e^{2} - 2 \, A a^{3} e^{3}}{8 \, {\left (c x^{2} + a\right )}^{2} a^{2} c^{2}} \] Input:

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

Output:

3/8*(A*c^2*d^3 + B*a*c*d^2*e + A*a*c*d*e^2 + B*a^2*e^3)*arctan(c*x/sqrt(a* 
c))/(sqrt(a*c)*a^2*c^2) + 1/8*(3*A*c^3*d^3*x^3 + 3*B*a*c^2*d^2*e*x^3 + 3*A 
*a*c^2*d*e^2*x^3 - 5*B*a^2*c*e^3*x^3 - 12*B*a^2*c*d*e^2*x^2 - 4*A*a^2*c*e^ 
3*x^2 + 5*A*a*c^2*d^3*x - 3*B*a^2*c*d^2*e*x - 3*A*a^2*c*d*e^2*x - 3*B*a^3* 
e^3*x - 2*B*a^2*c*d^3 - 6*A*a^2*c*d^2*e - 6*B*a^3*d*e^2 - 2*A*a^3*e^3)/((c 
*x^2 + a)^2*a^2*c^2)
 

Mupad [B] (verification not implemented)

Time = 6.27 (sec) , antiderivative size = 265, normalized size of antiderivative = 2.01 \[ \int \frac {(A+B x) (d+e x)^3}{\left (a+c x^2\right )^3} \, dx=\frac {3\,\mathrm {atan}\left (\frac {\sqrt {c}\,x\,\left (A\,c\,d+B\,a\,e\right )\,\left (c\,d^2+a\,e^2\right )}{\sqrt {a}\,\left (B\,a^2\,e^3+B\,a\,c\,d^2\,e+A\,a\,c\,d\,e^2+A\,c^2\,d^3\right )}\right )\,\left (A\,c\,d+B\,a\,e\right )\,\left (c\,d^2+a\,e^2\right )}{8\,a^{5/2}\,c^{5/2}}-\frac {\frac {B\,c\,d^3+3\,A\,c\,d^2\,e+3\,B\,a\,d\,e^2+A\,a\,e^3}{4\,c^2}+\frac {x^2\,\left (A\,e^3+3\,B\,d\,e^2\right )}{2\,c}+\frac {x\,\left (3\,B\,a^2\,e^3+3\,B\,a\,c\,d^2\,e+3\,A\,a\,c\,d\,e^2-5\,A\,c^2\,d^3\right )}{8\,a\,c^2}-\frac {x^3\,\left (-5\,B\,a^2\,e^3+3\,B\,a\,c\,d^2\,e+3\,A\,a\,c\,d\,e^2+3\,A\,c^2\,d^3\right )}{8\,a^2\,c}}{a^2+2\,a\,c\,x^2+c^2\,x^4} \] Input:

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

Output:

(3*atan((c^(1/2)*x*(A*c*d + B*a*e)*(a*e^2 + c*d^2))/(a^(1/2)*(A*c^2*d^3 + 
B*a^2*e^3 + A*a*c*d*e^2 + B*a*c*d^2*e)))*(A*c*d + B*a*e)*(a*e^2 + c*d^2))/ 
(8*a^(5/2)*c^(5/2)) - ((A*a*e^3 + B*c*d^3 + 3*B*a*d*e^2 + 3*A*c*d^2*e)/(4* 
c^2) + (x^2*(A*e^3 + 3*B*d*e^2))/(2*c) + (x*(3*B*a^2*e^3 - 5*A*c^2*d^3 + 3 
*A*a*c*d*e^2 + 3*B*a*c*d^2*e))/(8*a*c^2) - (x^3*(3*A*c^2*d^3 - 5*B*a^2*e^3 
 + 3*A*a*c*d*e^2 + 3*B*a*c*d^2*e))/(8*a^2*c))/(a^2 + c^2*x^4 + 2*a*c*x^2)
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 524, normalized size of antiderivative = 3.97 \[ \int \frac {(A+B x) (d+e x)^3}{\left (a+c x^2\right )^3} \, dx=\frac {3 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c x}{\sqrt {c}\, \sqrt {a}}\right ) a^{3} b \,e^{3}+3 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c x}{\sqrt {c}\, \sqrt {a}}\right ) a^{3} c d \,e^{2}+3 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c x}{\sqrt {c}\, \sqrt {a}}\right ) a^{2} b c \,d^{2} e +6 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c x}{\sqrt {c}\, \sqrt {a}}\right ) a^{2} b c \,e^{3} x^{2}+3 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c x}{\sqrt {c}\, \sqrt {a}}\right ) a^{2} c^{2} d^{3}+6 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c x}{\sqrt {c}\, \sqrt {a}}\right ) a^{2} c^{2} d \,e^{2} x^{2}+6 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c x}{\sqrt {c}\, \sqrt {a}}\right ) a b \,c^{2} d^{2} e \,x^{2}+3 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c x}{\sqrt {c}\, \sqrt {a}}\right ) a b \,c^{2} e^{3} x^{4}+6 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c x}{\sqrt {c}\, \sqrt {a}}\right ) a \,c^{3} d^{3} x^{2}+3 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c x}{\sqrt {c}\, \sqrt {a}}\right ) a \,c^{3} d \,e^{2} x^{4}+3 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c x}{\sqrt {c}\, \sqrt {a}}\right ) b \,c^{3} d^{2} e \,x^{4}+3 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c x}{\sqrt {c}\, \sqrt {a}}\right ) c^{4} d^{3} x^{4}-3 a^{3} b c \,e^{3} x -6 a^{3} c^{2} d^{2} e -3 a^{3} c^{2} d \,e^{2} x -2 a^{2} b \,c^{2} d^{3}-3 a^{2} b \,c^{2} d^{2} e x -5 a^{2} b \,c^{2} e^{3} x^{3}+5 a^{2} c^{3} d^{3} x +3 a^{2} c^{3} d \,e^{2} x^{3}+2 a^{2} c^{3} e^{3} x^{4}+3 a b \,c^{3} d^{2} e \,x^{3}+6 a b \,c^{3} d \,e^{2} x^{4}+3 a \,c^{4} d^{3} x^{3}}{8 a^{2} c^{3} \left (c^{2} x^{4}+2 a c \,x^{2}+a^{2}\right )} \] Input:

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

Output:

(3*sqrt(c)*sqrt(a)*atan((c*x)/(sqrt(c)*sqrt(a)))*a**3*b*e**3 + 3*sqrt(c)*s 
qrt(a)*atan((c*x)/(sqrt(c)*sqrt(a)))*a**3*c*d*e**2 + 3*sqrt(c)*sqrt(a)*ata 
n((c*x)/(sqrt(c)*sqrt(a)))*a**2*b*c*d**2*e + 6*sqrt(c)*sqrt(a)*atan((c*x)/ 
(sqrt(c)*sqrt(a)))*a**2*b*c*e**3*x**2 + 3*sqrt(c)*sqrt(a)*atan((c*x)/(sqrt 
(c)*sqrt(a)))*a**2*c**2*d**3 + 6*sqrt(c)*sqrt(a)*atan((c*x)/(sqrt(c)*sqrt( 
a)))*a**2*c**2*d*e**2*x**2 + 6*sqrt(c)*sqrt(a)*atan((c*x)/(sqrt(c)*sqrt(a) 
))*a*b*c**2*d**2*e*x**2 + 3*sqrt(c)*sqrt(a)*atan((c*x)/(sqrt(c)*sqrt(a)))* 
a*b*c**2*e**3*x**4 + 6*sqrt(c)*sqrt(a)*atan((c*x)/(sqrt(c)*sqrt(a)))*a*c** 
3*d**3*x**2 + 3*sqrt(c)*sqrt(a)*atan((c*x)/(sqrt(c)*sqrt(a)))*a*c**3*d*e** 
2*x**4 + 3*sqrt(c)*sqrt(a)*atan((c*x)/(sqrt(c)*sqrt(a)))*b*c**3*d**2*e*x** 
4 + 3*sqrt(c)*sqrt(a)*atan((c*x)/(sqrt(c)*sqrt(a)))*c**4*d**3*x**4 - 3*a** 
3*b*c*e**3*x - 6*a**3*c**2*d**2*e - 3*a**3*c**2*d*e**2*x - 2*a**2*b*c**2*d 
**3 - 3*a**2*b*c**2*d**2*e*x - 5*a**2*b*c**2*e**3*x**3 + 5*a**2*c**3*d**3* 
x + 3*a**2*c**3*d*e**2*x**3 + 2*a**2*c**3*e**3*x**4 + 3*a*b*c**3*d**2*e*x* 
*3 + 6*a*b*c**3*d*e**2*x**4 + 3*a*c**4*d**3*x**3)/(8*a**2*c**3*(a**2 + 2*a 
*c*x**2 + c**2*x**4))