\(\int \frac {x^2}{(c+d x)^3 (a+b x^2)^3} \, dx\) [271]

Optimal result

Integrand size = 20, antiderivative size = 403 \[ \int \frac {x^2}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx=-\frac {c^2 d^3}{2 \left (b c^2+a d^2\right )^3 (c+d x)^2}-\frac {2 c d^3 \left (2 b c^2-a d^2\right )}{\left (b c^2+a d^2\right )^4 (c+d x)}-\frac {a d \left (3 b c^2-a d^2\right )+b c \left (b c^2-3 a d^2\right ) x}{4 \left (b c^2+a d^2\right )^3 \left (a+b x^2\right )^2}+\frac {4 a d \left (3 b^2 c^4-8 a b c^2 d^2+a^2 d^4\right )+b c \left (b^2 c^4-26 a b c^2 d^2+21 a^2 d^4\right ) x}{8 a \left (b c^2+a d^2\right )^4 \left (a+b x^2\right )}+\frac {\sqrt {b} c \left (b^3 c^6+23 a b^2 c^4 d^2-125 a^2 b c^2 d^4+45 a^3 d^6\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{3/2} \left (b c^2+a d^2\right )^5}+\frac {d^3 \left (10 b^2 c^4-13 a b c^2 d^2+a^2 d^4\right ) \log (c+d x)}{\left (b c^2+a d^2\right )^5}-\frac {d^3 \left (10 b^2 c^4-13 a b c^2 d^2+a^2 d^4\right ) \log \left (a+b x^2\right )}{2 \left (b c^2+a d^2\right )^5} \] Output:

-1/2*c^2*d^3/(a*d^2+b*c^2)^3/(d*x+c)^2-2*c*d^3*(-a*d^2+2*b*c^2)/(a*d^2+b*c 
^2)^4/(d*x+c)-1/4*(a*d*(-a*d^2+3*b*c^2)+b*c*(-3*a*d^2+b*c^2)*x)/(a*d^2+b*c 
^2)^3/(b*x^2+a)^2+1/8*(4*a*d*(a^2*d^4-8*a*b*c^2*d^2+3*b^2*c^4)+b*c*(21*a^2 
*d^4-26*a*b*c^2*d^2+b^2*c^4)*x)/a/(a*d^2+b*c^2)^4/(b*x^2+a)+1/8*b^(1/2)*c* 
(45*a^3*d^6-125*a^2*b*c^2*d^4+23*a*b^2*c^4*d^2+b^3*c^6)*arctan(b^(1/2)*x/a 
^(1/2))/a^(3/2)/(a*d^2+b*c^2)^5+d^3*(a^2*d^4-13*a*b*c^2*d^2+10*b^2*c^4)*ln 
(d*x+c)/(a*d^2+b*c^2)^5-1/2*d^3*(a^2*d^4-13*a*b*c^2*d^2+10*b^2*c^4)*ln(b*x 
^2+a)/(a*d^2+b*c^2)^5
 

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 351, normalized size of antiderivative = 0.87 \[ \int \frac {x^2}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx=\frac {-\frac {4 d^3 \left (b c^3+a c d^2\right )^2}{(c+d x)^2}-\frac {16 c d^3 \left (2 b c^2-a d^2\right ) \left (b c^2+a d^2\right )}{c+d x}-\frac {2 \left (b c^2+a d^2\right )^2 \left (-a^2 d^3+b^2 c^3 x+3 a b c d (c-d x)\right )}{\left (a+b x^2\right )^2}+\frac {\left (b c^2+a d^2\right ) \left (4 a^3 d^5+b^3 c^5 x+2 a b^2 c^3 d (6 c-13 d x)+a^2 b c d^3 (-32 c+21 d x)\right )}{a \left (a+b x^2\right )}+\frac {\sqrt {b} c \left (b^3 c^6+23 a b^2 c^4 d^2-125 a^2 b c^2 d^4+45 a^3 d^6\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2}}+8 \left (10 b^2 c^4 d^3-13 a b c^2 d^5+a^2 d^7\right ) \log (c+d x)-4 \left (10 b^2 c^4 d^3-13 a b c^2 d^5+a^2 d^7\right ) \log \left (a+b x^2\right )}{8 \left (b c^2+a d^2\right )^5} \] Input:

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

Output:

((-4*d^3*(b*c^3 + a*c*d^2)^2)/(c + d*x)^2 - (16*c*d^3*(2*b*c^2 - a*d^2)*(b 
*c^2 + a*d^2))/(c + d*x) - (2*(b*c^2 + a*d^2)^2*(-(a^2*d^3) + b^2*c^3*x + 
3*a*b*c*d*(c - d*x)))/(a + b*x^2)^2 + ((b*c^2 + a*d^2)*(4*a^3*d^5 + b^3*c^ 
5*x + 2*a*b^2*c^3*d*(6*c - 13*d*x) + a^2*b*c*d^3*(-32*c + 21*d*x)))/(a*(a 
+ b*x^2)) + (Sqrt[b]*c*(b^3*c^6 + 23*a*b^2*c^4*d^2 - 125*a^2*b*c^2*d^4 + 4 
5*a^3*d^6)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(3/2) + 8*(10*b^2*c^4*d^3 - 13*a 
*b*c^2*d^5 + a^2*d^7)*Log[c + d*x] - 4*(10*b^2*c^4*d^3 - 13*a*b*c^2*d^5 + 
a^2*d^7)*Log[a + b*x^2])/(8*(b*c^2 + a*d^2)^5)
 

Rubi [A] (verified)

Time = 3.52 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {601, 25, 2178, 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.

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

\(\Big \downarrow \) 601

\(\displaystyle -\frac {\int -\frac {\frac {a b \left (b c^2-3 a d^2\right ) c^4}{\left (b c^2+a d^2\right )^3}-\frac {a b d \left (9 b c^2+5 a d^2\right ) x c^3}{\left (b c^2+a d^2\right )^3}-\frac {3 a b d^3 \left (b c^2-3 a d^2\right ) x^3 c}{\left (b c^2+a d^2\right )^3}-\frac {a d^2 \left (9 b^2 c^4-15 a b d^2 c^2-4 a^2 d^4\right ) x^2}{\left (b c^2+a d^2\right )^3}}{(c+d x)^3 \left (b x^2+a\right )^2}dx}{4 a}-\frac {b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )}{4 \left (a+b x^2\right )^2 \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {\frac {a b \left (b c^2-3 a d^2\right ) c^4}{\left (b c^2+a d^2\right )^3}-\frac {a b d \left (9 b c^2+5 a d^2\right ) x c^3}{\left (b c^2+a d^2\right )^3}-\frac {3 a b d^3 \left (b c^2-3 a d^2\right ) x^3 c}{\left (b c^2+a d^2\right )^3}-\frac {a d^2 \left (9 b^2 c^4-15 a b d^2 c^2-4 a^2 d^4\right ) x^2}{\left (b c^2+a d^2\right )^3}}{(c+d x)^3 \left (b x^2+a\right )^2}dx}{4 a}-\frac {b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )}{4 \left (a+b x^2\right )^2 \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 2178

\(\displaystyle \frac {\frac {b c x \left (21 a^2 d^4-26 a b c^2 d^2+b^2 c^4\right )+4 a d \left (a^2 d^4-8 a b c^2 d^2+3 b^2 c^4\right )}{2 \left (a+b x^2\right ) \left (a d^2+b c^2\right )^4}-\frac {\int -\frac {\frac {a b^2 \left (b^2 c^4+22 a b d^2 c^2-27 a^2 d^4\right ) c^4}{\left (b c^2+a d^2\right )^4}+\frac {a b^2 d \left (3 b^2 c^4-14 a b d^2 c^2-65 a^2 d^4\right ) x c^3}{\left (b c^2+a d^2\right )^4}+\frac {a b^2 d^3 \left (b^2 c^4-26 a b d^2 c^2+21 a^2 d^4\right ) x^3 c}{\left (b c^2+a d^2\right )^4}+\frac {a b d^2 \left (3 b^3 c^6-54 a b^2 d^2 c^4-a^2 b d^4 c^2+8 a^3 d^6\right ) x^2}{\left (b c^2+a d^2\right )^4}}{(c+d x)^3 \left (b x^2+a\right )}dx}{2 a b}}{4 a}-\frac {b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )}{4 \left (a+b x^2\right )^2 \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {\frac {a b^2 \left (b^2 c^4+22 a b d^2 c^2-27 a^2 d^4\right ) c^4}{\left (b c^2+a d^2\right )^4}+\frac {a b^2 d \left (3 b^2 c^4-14 a b d^2 c^2-65 a^2 d^4\right ) x c^3}{\left (b c^2+a d^2\right )^4}+\frac {a b^2 d^3 \left (b^2 c^4-26 a b d^2 c^2+21 a^2 d^4\right ) x^3 c}{\left (b c^2+a d^2\right )^4}+\frac {a b d^2 \left (3 b^3 c^6-54 a b^2 d^2 c^4-a^2 b d^4 c^2+8 a^3 d^6\right ) x^2}{\left (b c^2+a d^2\right )^4}}{(c+d x)^3 \left (b x^2+a\right )}dx}{2 a b}+\frac {b c x \left (21 a^2 d^4-26 a b c^2 d^2+b^2 c^4\right )+4 a d \left (a^2 d^4-8 a b c^2 d^2+3 b^2 c^4\right )}{2 \left (a+b x^2\right ) \left (a d^2+b c^2\right )^4}}{4 a}-\frac {b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )}{4 \left (a+b x^2\right )^2 \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 2160

\(\displaystyle \frac {\frac {\int \left (\frac {8 a^2 b \left (10 b^2 c^4-13 a b d^2 c^2+a^2 d^4\right ) d^4}{\left (b c^2+a d^2\right )^5 (c+d x)}-\frac {16 a^2 b c \left (a d^2-2 b c^2\right ) d^4}{\left (b c^2+a d^2\right )^4 (c+d x)^2}+\frac {8 a^2 b c^2 d^4}{\left (b c^2+a d^2\right )^3 (c+d x)^3}+\frac {a b^2 \left (c \left (b^3 c^6+23 a b^2 d^2 c^4-125 a^2 b d^4 c^2+45 a^3 d^6\right )-8 a d^3 \left (10 b^2 c^4-13 a b d^2 c^2+a^2 d^4\right ) x\right )}{\left (b c^2+a d^2\right )^5 \left (b x^2+a\right )}\right )dx}{2 a b}+\frac {b c x \left (21 a^2 d^4-26 a b c^2 d^2+b^2 c^4\right )+4 a d \left (a^2 d^4-8 a b c^2 d^2+3 b^2 c^4\right )}{2 \left (a+b x^2\right ) \left (a d^2+b c^2\right )^4}}{4 a}-\frac {b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )}{4 \left (a+b x^2\right )^2 \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {b c x \left (21 a^2 d^4-26 a b c^2 d^2+b^2 c^4\right )+4 a d \left (a^2 d^4-8 a b c^2 d^2+3 b^2 c^4\right )}{2 \left (a+b x^2\right ) \left (a d^2+b c^2\right )^4}+\frac {-\frac {4 a^2 b d^3 \left (a^2 d^4-13 a b c^2 d^2+10 b^2 c^4\right ) \log \left (a+b x^2\right )}{\left (a d^2+b c^2\right )^5}+\frac {8 a^2 b d^3 \left (a^2 d^4-13 a b c^2 d^2+10 b^2 c^4\right ) \log (c+d x)}{\left (a d^2+b c^2\right )^5}-\frac {16 a^2 b c d^3 \left (2 b c^2-a d^2\right )}{(c+d x) \left (a d^2+b c^2\right )^4}-\frac {4 a^2 b c^2 d^3}{(c+d x)^2 \left (a d^2+b c^2\right )^3}+\frac {\sqrt {a} b^{3/2} c \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (45 a^3 d^6-125 a^2 b c^2 d^4+23 a b^2 c^4 d^2+b^3 c^6\right )}{\left (a d^2+b c^2\right )^5}}{2 a b}}{4 a}-\frac {b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )}{4 \left (a+b x^2\right )^2 \left (a d^2+b c^2\right )^3}\)

Input:

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

Output:

-1/4*(a*d*(3*b*c^2 - a*d^2) + b*c*(b*c^2 - 3*a*d^2)*x)/((b*c^2 + a*d^2)^3* 
(a + b*x^2)^2) + ((4*a*d*(3*b^2*c^4 - 8*a*b*c^2*d^2 + a^2*d^4) + b*c*(b^2* 
c^4 - 26*a*b*c^2*d^2 + 21*a^2*d^4)*x)/(2*(b*c^2 + a*d^2)^4*(a + b*x^2)) + 
((-4*a^2*b*c^2*d^3)/((b*c^2 + a*d^2)^3*(c + d*x)^2) - (16*a^2*b*c*d^3*(2*b 
*c^2 - a*d^2))/((b*c^2 + a*d^2)^4*(c + d*x)) + (Sqrt[a]*b^(3/2)*c*(b^3*c^6 
 + 23*a*b^2*c^4*d^2 - 125*a^2*b*c^2*d^4 + 45*a^3*d^6)*ArcTan[(Sqrt[b]*x)/S 
qrt[a]])/(b*c^2 + a*d^2)^5 + (8*a^2*b*d^3*(10*b^2*c^4 - 13*a*b*c^2*d^2 + a 
^2*d^4)*Log[c + d*x])/(b*c^2 + a*d^2)^5 - (4*a^2*b*d^3*(10*b^2*c^4 - 13*a* 
b*c^2*d^2 + a^2*d^4)*Log[a + b*x^2])/(b*c^2 + a*d^2)^5)/(2*a*b))/(4*a)
 

Defintions of rubi rules used

rule 25

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

rule 601

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe 
ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol 
ynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x) 
*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   Int[(c 
+ d*x)^n*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Qx)/(c + d*x)^n + (e* 
(2*p + 3))/(c + d*x)^n, x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 1] 
 && LtQ[p, -1] && ILtQ[n, 0] && NeQ[b*c^2 + a*d^2, 0]
 

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 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 = 0.37 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.09

method	result	size
default	\(\frac {b \left (\frac {\frac {b c \left (21 a^{3} d^{6}-5 a^{2} b \,c^{2} d^{4}-25 a \,b^{2} c^{4} d^{2}+b^{3} c^{6}\right ) x^{3}}{8 a}+\left (\frac {1}{2} a^{3} d^{7}-\frac {7}{2} a^{2} b \,c^{2} d^{5}-\frac {5}{2} a \,b^{2} c^{4} d^{3}+\frac {3}{2} b^{3} c^{6} d \right ) x^{2}+\frac {c \left (27 a^{3} d^{6}+5 a^{2} b \,c^{2} d^{4}-23 a \,b^{2} c^{4} d^{2}-b^{3} c^{6}\right ) x}{8}+\frac {3 a d \left (a^{3} d^{6}-5 a^{2} b \,c^{2} d^{4}-5 a \,b^{2} c^{4} d^{2}+b^{3} c^{6}\right )}{4 b}}{\left (b \,x^{2}+a \right )^{2}}+\frac {\frac {\left (-8 a^{3} d^{7}+104 a^{2} b \,c^{2} d^{5}-80 a \,b^{2} c^{4} d^{3}\right ) \ln \left (b \,x^{2}+a \right )}{2 b}+\frac {\left (45 d^{6} c \,a^{3}-125 a^{2} b \,c^{3} d^{4}+23 a \,b^{2} c^{5} d^{2}+b^{3} c^{7}\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}}}{8 a}\right )}{\left (a \,d^{2}+b \,c^{2}\right )^{5}}-\frac {c^{2} d^{3}}{2 \left (a \,d^{2}+b \,c^{2}\right )^{3} \left (d x +c \right )^{2}}+\frac {d^{3} \left (a^{2} d^{4}-13 b \,c^{2} d^{2} a +10 b^{2} c^{4}\right ) \ln \left (d x +c \right )}{\left (a \,d^{2}+b \,c^{2}\right )^{5}}+\frac {2 c \,d^{3} \left (a \,d^{2}-2 b \,c^{2}\right )}{\left (a \,d^{2}+b \,c^{2}\right )^{4} \left (d x +c \right )}\)	\(439\)
risch	\(\text {Expression too large to display}\)	\(1615\)

Input:

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

Output:

b/(a*d^2+b*c^2)^5*((1/8*b*c*(21*a^3*d^6-5*a^2*b*c^2*d^4-25*a*b^2*c^4*d^2+b 
^3*c^6)/a*x^3+(1/2*a^3*d^7-7/2*a^2*b*c^2*d^5-5/2*a*b^2*c^4*d^3+3/2*b^3*c^6 
*d)*x^2+1/8*c*(27*a^3*d^6+5*a^2*b*c^2*d^4-23*a*b^2*c^4*d^2-b^3*c^6)*x+3/4* 
a*d*(a^3*d^6-5*a^2*b*c^2*d^4-5*a*b^2*c^4*d^2+b^3*c^6)/b)/(b*x^2+a)^2+1/8/a 
*(1/2*(-8*a^3*d^7+104*a^2*b*c^2*d^5-80*a*b^2*c^4*d^3)/b*ln(b*x^2+a)+(45*a^ 
3*c*d^6-125*a^2*b*c^3*d^4+23*a*b^2*c^5*d^2+b^3*c^7)/(a*b)^(1/2)*arctan(b*x 
/(a*b)^(1/2))))-1/2*c^2*d^3/(a*d^2+b*c^2)^3/(d*x+c)^2+d^3*(a^2*d^4-13*a*b* 
c^2*d^2+10*b^2*c^4)*ln(d*x+c)/(a*d^2+b*c^2)^5+2*c*d^3*(a*d^2-2*b*c^2)/(a*d 
^2+b*c^2)^4/(d*x+c)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1977 vs. \(2 (385) = 770\).

Time = 17.12 (sec) , antiderivative size = 3980, normalized size of antiderivative = 9.88 \[ \int \frac {x^2}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

integrate(x**2/(d*x+c)**3/(b*x**2+a)**3,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1102 vs. \(2 (385) = 770\).

Time = 0.14 (sec) , antiderivative size = 1102, normalized size of antiderivative = 2.73 \[ \int \frac {x^2}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/2*(10*b^2*c^4*d^3 - 13*a*b*c^2*d^5 + a^2*d^7)*log(b*x^2 + a)/(b^5*c^10 
+ 5*a*b^4*c^8*d^2 + 10*a^2*b^3*c^6*d^4 + 10*a^3*b^2*c^4*d^6 + 5*a^4*b*c^2* 
d^8 + a^5*d^10) + (10*b^2*c^4*d^3 - 13*a*b*c^2*d^5 + a^2*d^7)*log(d*x + c) 
/(b^5*c^10 + 5*a*b^4*c^8*d^2 + 10*a^2*b^3*c^6*d^4 + 10*a^3*b^2*c^4*d^6 + 5 
*a^4*b*c^2*d^8 + a^5*d^10) + 1/8*(b^4*c^7 + 23*a*b^3*c^5*d^2 - 125*a^2*b^2 
*c^3*d^4 + 45*a^3*b*c*d^6)*arctan(b*x/sqrt(a*b))/((a*b^5*c^10 + 5*a^2*b^4* 
c^8*d^2 + 10*a^3*b^3*c^6*d^4 + 10*a^4*b^2*c^4*d^6 + 5*a^5*b*c^2*d^8 + a^6* 
d^10)*sqrt(a*b)) + 1/8*(6*a^2*b^2*c^6*d - 72*a^3*b*c^4*d^3 + 18*a^4*c^2*d^ 
5 + (b^4*c^5*d^2 - 58*a*b^3*c^3*d^4 + 37*a^2*b^2*c*d^6)*x^5 + 2*(b^4*c^6*d 
 - 38*a*b^3*c^4*d^3 + 11*a^2*b^2*c^2*d^5 + 2*a^3*b*d^7)*x^4 + (b^4*c^7 - 3 
*a*b^3*c^5*d^2 - 129*a^2*b^2*c^3*d^4 + 67*a^3*b*c*d^6)*x^3 + 2*(5*a*b^3*c^ 
6*d - 71*a^2*b^2*c^4*d^3 + 23*a^3*b*c^2*d^5 + 3*a^4*d^7)*x^2 - (a*b^3*c^7 
+ 10*a^2*b^2*c^5*d^2 + 77*a^3*b*c^3*d^4 - 28*a^4*c*d^6)*x)/(a^3*b^4*c^10 + 
 4*a^4*b^3*c^8*d^2 + 6*a^5*b^2*c^6*d^4 + 4*a^6*b*c^4*d^6 + a^7*c^2*d^8 + ( 
a*b^6*c^8*d^2 + 4*a^2*b^5*c^6*d^4 + 6*a^3*b^4*c^4*d^6 + 4*a^4*b^3*c^2*d^8 
+ a^5*b^2*d^10)*x^6 + 2*(a*b^6*c^9*d + 4*a^2*b^5*c^7*d^3 + 6*a^3*b^4*c^5*d 
^5 + 4*a^4*b^3*c^3*d^7 + a^5*b^2*c*d^9)*x^5 + (a*b^6*c^10 + 6*a^2*b^5*c^8* 
d^2 + 14*a^3*b^4*c^6*d^4 + 16*a^4*b^3*c^4*d^6 + 9*a^5*b^2*c^2*d^8 + 2*a^6* 
b*d^10)*x^4 + 4*(a^2*b^5*c^9*d + 4*a^3*b^4*c^7*d^3 + 6*a^4*b^3*c^5*d^5 + 4 
*a^5*b^2*c^3*d^7 + a^6*b*c*d^9)*x^3 + (2*a^2*b^5*c^10 + 9*a^3*b^4*c^8*d...
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 724, normalized size of antiderivative = 1.80 \[ \int \frac {x^2}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx=-\frac {{\left (10 \, b^{2} c^{4} d^{3} - 13 \, a b c^{2} d^{5} + a^{2} d^{7}\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (b^{5} c^{10} + 5 \, a b^{4} c^{8} d^{2} + 10 \, a^{2} b^{3} c^{6} d^{4} + 10 \, a^{3} b^{2} c^{4} d^{6} + 5 \, a^{4} b c^{2} d^{8} + a^{5} d^{10}\right )}} + \frac {{\left (10 \, b^{2} c^{4} d^{4} - 13 \, a b c^{2} d^{6} + a^{2} d^{8}\right )} \log \left ({\left | d x + c \right |}\right )}{b^{5} c^{10} d + 5 \, a b^{4} c^{8} d^{3} + 10 \, a^{2} b^{3} c^{6} d^{5} + 10 \, a^{3} b^{2} c^{4} d^{7} + 5 \, a^{4} b c^{2} d^{9} + a^{5} d^{11}} + \frac {{\left (b^{4} c^{7} + 23 \, a b^{3} c^{5} d^{2} - 125 \, a^{2} b^{2} c^{3} d^{4} + 45 \, a^{3} b c d^{6}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, {\left (a b^{5} c^{10} + 5 \, a^{2} b^{4} c^{8} d^{2} + 10 \, a^{3} b^{3} c^{6} d^{4} + 10 \, a^{4} b^{2} c^{4} d^{6} + 5 \, a^{5} b c^{2} d^{8} + a^{6} d^{10}\right )} \sqrt {a b}} + \frac {b^{4} c^{5} d^{2} x^{5} - 58 \, a b^{3} c^{3} d^{4} x^{5} + 37 \, a^{2} b^{2} c d^{6} x^{5} + 2 \, b^{4} c^{6} d x^{4} - 76 \, a b^{3} c^{4} d^{3} x^{4} + 22 \, a^{2} b^{2} c^{2} d^{5} x^{4} + 4 \, a^{3} b d^{7} x^{4} + b^{4} c^{7} x^{3} - 3 \, a b^{3} c^{5} d^{2} x^{3} - 129 \, a^{2} b^{2} c^{3} d^{4} x^{3} + 67 \, a^{3} b c d^{6} x^{3} + 10 \, a b^{3} c^{6} d x^{2} - 142 \, a^{2} b^{2} c^{4} d^{3} x^{2} + 46 \, a^{3} b c^{2} d^{5} x^{2} + 6 \, a^{4} d^{7} x^{2} - a b^{3} c^{7} x - 10 \, a^{2} b^{2} c^{5} d^{2} x - 77 \, a^{3} b c^{3} d^{4} x + 28 \, a^{4} c d^{6} x + 6 \, a^{2} b^{2} c^{6} d - 72 \, a^{3} b c^{4} d^{3} + 18 \, a^{4} c^{2} d^{5}}{8 \, {\left (a b^{4} c^{8} + 4 \, a^{2} b^{3} c^{6} d^{2} + 6 \, a^{3} b^{2} c^{4} d^{4} + 4 \, a^{4} b c^{2} d^{6} + a^{5} d^{8}\right )} {\left (b d x^{3} + b c x^{2} + a d x + a c\right )}^{2}} \] Input:

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

Output:

-1/2*(10*b^2*c^4*d^3 - 13*a*b*c^2*d^5 + a^2*d^7)*log(b*x^2 + a)/(b^5*c^10 
+ 5*a*b^4*c^8*d^2 + 10*a^2*b^3*c^6*d^4 + 10*a^3*b^2*c^4*d^6 + 5*a^4*b*c^2* 
d^8 + a^5*d^10) + (10*b^2*c^4*d^4 - 13*a*b*c^2*d^6 + a^2*d^8)*log(abs(d*x 
+ c))/(b^5*c^10*d + 5*a*b^4*c^8*d^3 + 10*a^2*b^3*c^6*d^5 + 10*a^3*b^2*c^4* 
d^7 + 5*a^4*b*c^2*d^9 + a^5*d^11) + 1/8*(b^4*c^7 + 23*a*b^3*c^5*d^2 - 125* 
a^2*b^2*c^3*d^4 + 45*a^3*b*c*d^6)*arctan(b*x/sqrt(a*b))/((a*b^5*c^10 + 5*a 
^2*b^4*c^8*d^2 + 10*a^3*b^3*c^6*d^4 + 10*a^4*b^2*c^4*d^6 + 5*a^5*b*c^2*d^8 
 + a^6*d^10)*sqrt(a*b)) + 1/8*(b^4*c^5*d^2*x^5 - 58*a*b^3*c^3*d^4*x^5 + 37 
*a^2*b^2*c*d^6*x^5 + 2*b^4*c^6*d*x^4 - 76*a*b^3*c^4*d^3*x^4 + 22*a^2*b^2*c 
^2*d^5*x^4 + 4*a^3*b*d^7*x^4 + b^4*c^7*x^3 - 3*a*b^3*c^5*d^2*x^3 - 129*a^2 
*b^2*c^3*d^4*x^3 + 67*a^3*b*c*d^6*x^3 + 10*a*b^3*c^6*d*x^2 - 142*a^2*b^2*c 
^4*d^3*x^2 + 46*a^3*b*c^2*d^5*x^2 + 6*a^4*d^7*x^2 - a*b^3*c^7*x - 10*a^2*b 
^2*c^5*d^2*x - 77*a^3*b*c^3*d^4*x + 28*a^4*c*d^6*x + 6*a^2*b^2*c^6*d - 72* 
a^3*b*c^4*d^3 + 18*a^4*c^2*d^5)/((a*b^4*c^8 + 4*a^2*b^3*c^6*d^2 + 6*a^3*b^ 
2*c^4*d^4 + 4*a^4*b*c^2*d^6 + a^5*d^8)*(b*d*x^3 + b*c*x^2 + a*d*x + a*c)^2 
)
 

Mupad [B] (verification not implemented)

Time = 8.68 (sec) , antiderivative size = 1799, normalized size of antiderivative = 4.46 \[ \int \frac {x^2}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

log(c + d*x)*(d^3/(a*d^2 + b*c^2)^3 - (15*b*c^2*d^3)/(a*d^2 + b*c^2)^4 + ( 
24*b^2*c^4*d^3)/(a*d^2 + b*c^2)^5) + ((3*(3*a^3*c^2*d^5 - 12*a^2*b*c^4*d^3 
 + a*b^2*c^6*d))/(4*(a^4*d^8 + b^4*c^8 + 4*a*b^3*c^6*d^2 + 4*a^3*b*c^2*d^6 
 + 6*a^2*b^2*c^4*d^4)) + (x^2*(3*a^3*d^7 + 5*b^3*c^6*d - 71*a*b^2*c^4*d^3 
+ 23*a^2*b*c^2*d^5))/(4*(a^4*d^8 + b^4*c^8 + 4*a*b^3*c^6*d^2 + 4*a^3*b*c^2 
*d^6 + 6*a^2*b^2*c^4*d^4)) + (x^5*(b^4*c^5*d^2 - 58*a*b^3*c^3*d^4 + 37*a^2 
*b^2*c*d^6))/(8*a*(a^4*d^8 + b^4*c^8 + 4*a*b^3*c^6*d^2 + 4*a^3*b*c^2*d^6 + 
 6*a^2*b^2*c^4*d^4)) + (x^3*(b^4*c^7 - 3*a*b^3*c^5*d^2 - 129*a^2*b^2*c^3*d 
^4 + 67*a^3*b*c*d^6))/(8*a*(a^4*d^8 + b^4*c^8 + 4*a*b^3*c^6*d^2 + 4*a^3*b* 
c^2*d^6 + 6*a^2*b^2*c^4*d^4)) + (x^4*(2*a^3*b*d^7 + b^4*c^6*d - 38*a*b^3*c 
^4*d^3 + 11*a^2*b^2*c^2*d^5))/(4*a*(a^4*d^8 + b^4*c^8 + 4*a*b^3*c^6*d^2 + 
4*a^3*b*c^2*d^6 + 6*a^2*b^2*c^4*d^4)) - (c*x*(b^3*c^6 - 28*a^3*d^6 + 10*a* 
b^2*c^4*d^2 + 77*a^2*b*c^2*d^4))/(8*(a^4*d^8 + b^4*c^8 + 4*a*b^3*c^6*d^2 + 
 4*a^3*b*c^2*d^6 + 6*a^2*b^2*c^4*d^4)))/(x^2*(a^2*d^2 + 2*a*b*c^2) + x^4*( 
b^2*c^2 + 2*a*b*d^2) + a^2*c^2 + b^2*d^2*x^6 + 2*a^2*c*d*x + 2*b^2*c*d*x^5 
 + 4*a*b*c*d*x^3) - (log(b^8*c^18*(-a^3*b)^(3/2) - 576*a^12*d^18*(-a^3*b)^ 
(1/2) + 50212*c^6*d^12*(-a^3*b)^(7/2) + a^4*b^10*c^18*x - 72288*a^4*c^4*d^ 
14*(-a^3*b)^(5/2) - 11799*a^8*c^2*d^16*(-a^3*b)^(3/2) - 52544*b^4*c^12*d^6 
*(-a^3*b)^(5/2) + 576*a^13*b*d^18*x + 114576*a^2*b^2*c^8*d^10*(-a^3*b)^(5/ 
2) + 372*a^2*b^6*c^14*d^4*(-a^3*b)^(3/2) + 48*a^5*b^9*c^16*d^2*x + 372*...
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 2974, normalized size of antiderivative = 7.38 \[ \int \frac {x^2}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(90*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**5*c**3*d**6 + 180*sqr 
t(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**5*c**2*d**7*x + 90*sqrt(b)*s 
qrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**5*c*d**8*x**2 - 250*sqrt(b)*sqrt(a 
)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**4*b*c**5*d**4 - 500*sqrt(b)*sqrt(a)*ata 
n((b*x)/(sqrt(b)*sqrt(a)))*a**4*b*c**4*d**5*x - 70*sqrt(b)*sqrt(a)*atan((b 
*x)/(sqrt(b)*sqrt(a)))*a**4*b*c**3*d**6*x**2 + 360*sqrt(b)*sqrt(a)*atan((b 
*x)/(sqrt(b)*sqrt(a)))*a**4*b*c**2*d**7*x**3 + 180*sqrt(b)*sqrt(a)*atan((b 
*x)/(sqrt(b)*sqrt(a)))*a**4*b*c*d**8*x**4 + 46*sqrt(b)*sqrt(a)*atan((b*x)/ 
(sqrt(b)*sqrt(a)))*a**3*b**2*c**7*d**2 + 92*sqrt(b)*sqrt(a)*atan((b*x)/(sq 
rt(b)*sqrt(a)))*a**3*b**2*c**6*d**3*x - 454*sqrt(b)*sqrt(a)*atan((b*x)/(sq 
rt(b)*sqrt(a)))*a**3*b**2*c**5*d**4*x**2 - 1000*sqrt(b)*sqrt(a)*atan((b*x) 
/(sqrt(b)*sqrt(a)))*a**3*b**2*c**4*d**5*x**3 - 410*sqrt(b)*sqrt(a)*atan((b 
*x)/(sqrt(b)*sqrt(a)))*a**3*b**2*c**3*d**6*x**4 + 180*sqrt(b)*sqrt(a)*atan 
((b*x)/(sqrt(b)*sqrt(a)))*a**3*b**2*c**2*d**7*x**5 + 90*sqrt(b)*sqrt(a)*at 
an((b*x)/(sqrt(b)*sqrt(a)))*a**3*b**2*c*d**8*x**6 + 2*sqrt(b)*sqrt(a)*atan 
((b*x)/(sqrt(b)*sqrt(a)))*a**2*b**3*c**9 + 4*sqrt(b)*sqrt(a)*atan((b*x)/(s 
qrt(b)*sqrt(a)))*a**2*b**3*c**8*d*x + 94*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt( 
b)*sqrt(a)))*a**2*b**3*c**7*d**2*x**2 + 184*sqrt(b)*sqrt(a)*atan((b*x)/(sq 
rt(b)*sqrt(a)))*a**2*b**3*c**6*d**3*x**3 - 158*sqrt(b)*sqrt(a)*atan((b*x)/ 
(sqrt(b)*sqrt(a)))*a**2*b**3*c**5*d**4*x**4 - 500*sqrt(b)*sqrt(a)*atan(...