\(\int \frac {x^2}{(c+d x)^3 (a+b x^2)} \, dx\) [161]

Optimal result

Integrand size = 20, antiderivative size = 183 \[ \int \frac {x^2}{(c+d x)^3 \left (a+b x^2\right )} \, dx=-\frac {c^2}{2 d \left (b c^2+a d^2\right ) (c+d x)^2}+\frac {2 a c d}{\left (b c^2+a d^2\right )^2 (c+d x)}-\frac {\sqrt {a} \sqrt {b} c \left (b c^2-3 a d^2\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\left (b c^2+a d^2\right )^3}-\frac {a d \left (3 b c^2-a d^2\right ) \log (c+d x)}{\left (b c^2+a d^2\right )^3}+\frac {a d \left (3 b c^2-a d^2\right ) \log \left (a+b x^2\right )}{2 \left (b c^2+a d^2\right )^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.80 \[ \int \frac {x^2}{(c+d x)^3 \left (a+b x^2\right )} \, dx=\frac {-\frac {c \left (b c^2+a d^2\right ) \left (b c^3-a d^2 (3 c+4 d x)\right )}{d (c+d x)^2}+2 \sqrt {a} \sqrt {b} c \left (-b c^2+3 a d^2\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )+2 a d \left (-3 b c^2+a d^2\right ) \log (c+d x)+a d \left (3 b c^2-a d^2\right ) \log \left (a+b x^2\right )}{2 \left (b c^2+a d^2\right )^3} \] Input:

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

Output:

(-((c*(b*c^2 + a*d^2)*(b*c^3 - a*d^2*(3*c + 4*d*x)))/(d*(c + d*x)^2)) + 2* 
Sqrt[a]*Sqrt[b]*c*(-(b*c^2) + 3*a*d^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]] + 2*a*d 
*(-3*b*c^2 + a*d^2)*Log[c + d*x] + a*d*(3*b*c^2 - a*d^2)*Log[a + b*x^2])/( 
2*(b*c^2 + a*d^2)^3)
 

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.07, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {603, 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.

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

\(\Big \downarrow \) 603

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 657

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-1/2*c^2/(d*(b*c^2 + a*d^2)*(c + d*x)^2) - (a*((-2*c*d)/((b*c^2 + a*d^2)*( 
c + d*x)) + (Sqrt[b]*c*(b*c^2 - 3*a*d^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqr 
t[a]*(b*c^2 + a*d^2)^2) + (d*(3*b*c^2 - a*d^2)*Log[c + d*x])/(b*c^2 + a*d^ 
2)^2 - (d*(3*b*c^2 - a*d^2)*Log[a + b*x^2])/(2*(b*c^2 + a*d^2)^2)))/(b*c^2 
 + a*d^2)
 

Defintions of rubi rules used

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 603

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol 
] :> With[{Qx = PolynomialQuotient[x^m, c + d*x, x], R = PolynomialRemainde 
r[x^m, c + d*x, x]}, Simp[d*R*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/((n + 
1)*(b*c^2 + a*d^2))), x] + Simp[1/((n + 1)*(b*c^2 + a*d^2))   Int[(c + d*x) 
^(n + 1)*(a + b*x^2)^p*ExpandToSum[(n + 1)*(b*c^2 + a*d^2)*Qx + b*c*R*(n + 
1) - b*d*R*(n + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, p}, x] && IGt 
Q[m, 1] && LtQ[n, -1] && NeQ[b*c^2 + a*d^2, 0]
 

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]
 

Maple [A] (verified)

Time = 0.31 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.89

method	result	size
default	\(\frac {a b \left (\frac {\left (-a \,d^{3}+3 b \,c^{2} d \right ) \ln \left (b \,x^{2}+a \right )}{2 b}+\frac {\left (3 a \,d^{2} c -b \,c^{3}\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}}\right )}{\left (a \,d^{2}+b \,c^{2}\right )^{3}}-\frac {c^{2}}{2 d \left (a \,d^{2}+b \,c^{2}\right ) \left (d x +c \right )^{2}}+\frac {a d \left (a \,d^{2}-3 b \,c^{2}\right ) \ln \left (d x +c \right )}{\left (a \,d^{2}+b \,c^{2}\right )^{3}}+\frac {2 a c d}{\left (a \,d^{2}+b \,c^{2}\right )^{2} \left (d x +c \right )}\)	\(162\)
risch	\(\frac {\frac {2 a \,d^{2} c x}{a^{2} d^{4}+2 b \,c^{2} d^{2} a +b^{2} c^{4}}+\frac {c^{2} \left (3 a \,d^{2}-b \,c^{2}\right )}{2 d \left (a^{2} d^{4}+2 b \,c^{2} d^{2} a +b^{2} c^{4}\right )}}{\left (d x +c \right )^{2}}+\frac {a^{2} d^{3} \ln \left (d x +c \right )}{a^{3} d^{6}+3 a^{2} b \,c^{2} d^{4}+3 a \,b^{2} c^{4} d^{2}+b^{3} c^{6}}-\frac {3 a d \ln \left (d x +c \right ) b \,c^{2}}{a^{3} d^{6}+3 a^{2} b \,c^{2} d^{4}+3 a \,b^{2} c^{4} d^{2}+b^{3} c^{6}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a^{3} d^{6}+3 a^{2} b \,c^{2} d^{4}+3 a \,b^{2} c^{4} d^{2}+b^{3} c^{6}\right ) \textit {\_Z}^{2}+\left (2 a^{2} d^{3}-6 a b \,c^{2} d \right ) \textit {\_Z} +a \right )}{\sum }\textit {\_R} \ln \left (\left (\left (3 a^{5} d^{10}+11 a^{4} b \,c^{2} d^{8}+14 a^{3} b^{2} c^{4} d^{6}+6 a^{2} b^{3} c^{6} d^{4}-a \,b^{4} c^{8} d^{2}-b^{5} c^{10}\right ) \textit {\_R}^{2}+\left (3 a^{4} d^{7}+a^{3} b \,c^{2} d^{5}-7 a^{2} b^{2} c^{4} d^{3}-5 a \,b^{3} c^{6} d \right ) \textit {\_R} +8 a^{2} b \,c^{2} d^{2}\right ) x +\left (4 a^{5} c \,d^{9}+16 a^{4} b \,c^{3} d^{7}+24 a^{3} b^{2} c^{5} d^{5}+16 a^{2} b^{3} c^{7} d^{3}+4 a \,b^{4} c^{9} d \right ) \textit {\_R}^{2}+\left (-a^{4} c \,d^{6}-3 a^{3} b \,c^{3} d^{4}-3 a^{2} b^{2} c^{5} d^{2}-a \,b^{3} c^{7}\right ) \textit {\_R} +4 d^{3} c \,a^{3}-4 a^{2} b \,c^{3} d \right )\right )}{2}\)	\(541\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 418 vs. \(2 (171) = 342\).

Time = 0.69 (sec) , antiderivative size = 855, normalized size of antiderivative = 4.67 \[ \int \frac {x^2}{(c+d x)^3 \left (a+b x^2\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

[-1/2*(b^2*c^6 - 2*a*b*c^4*d^2 - 3*a^2*c^2*d^4 + (b*c^5*d - 3*a*c^3*d^3 + 
(b*c^3*d^3 - 3*a*c*d^5)*x^2 + 2*(b*c^4*d^2 - 3*a*c^2*d^4)*x)*sqrt(-a*b)*lo 
g((b*x^2 + 2*sqrt(-a*b)*x - a)/(b*x^2 + a)) - 4*(a*b*c^3*d^3 + a^2*c*d^5)* 
x - (3*a*b*c^4*d^2 - a^2*c^2*d^4 + (3*a*b*c^2*d^4 - a^2*d^6)*x^2 + 2*(3*a* 
b*c^3*d^3 - a^2*c*d^5)*x)*log(b*x^2 + a) + 2*(3*a*b*c^4*d^2 - a^2*c^2*d^4 
+ (3*a*b*c^2*d^4 - a^2*d^6)*x^2 + 2*(3*a*b*c^3*d^3 - a^2*c*d^5)*x)*log(d*x 
 + c))/(b^3*c^8*d + 3*a*b^2*c^6*d^3 + 3*a^2*b*c^4*d^5 + a^3*c^2*d^7 + (b^3 
*c^6*d^3 + 3*a*b^2*c^4*d^5 + 3*a^2*b*c^2*d^7 + a^3*d^9)*x^2 + 2*(b^3*c^7*d 
^2 + 3*a*b^2*c^5*d^4 + 3*a^2*b*c^3*d^6 + a^3*c*d^8)*x), -1/2*(b^2*c^6 - 2* 
a*b*c^4*d^2 - 3*a^2*c^2*d^4 + 2*(b*c^5*d - 3*a*c^3*d^3 + (b*c^3*d^3 - 3*a* 
c*d^5)*x^2 + 2*(b*c^4*d^2 - 3*a*c^2*d^4)*x)*sqrt(a*b)*arctan(sqrt(a*b)*x/a 
) - 4*(a*b*c^3*d^3 + a^2*c*d^5)*x - (3*a*b*c^4*d^2 - a^2*c^2*d^4 + (3*a*b* 
c^2*d^4 - a^2*d^6)*x^2 + 2*(3*a*b*c^3*d^3 - a^2*c*d^5)*x)*log(b*x^2 + a) + 
 2*(3*a*b*c^4*d^2 - a^2*c^2*d^4 + (3*a*b*c^2*d^4 - a^2*d^6)*x^2 + 2*(3*a*b 
*c^3*d^3 - a^2*c*d^5)*x)*log(d*x + c))/(b^3*c^8*d + 3*a*b^2*c^6*d^3 + 3*a^ 
2*b*c^4*d^5 + a^3*c^2*d^7 + (b^3*c^6*d^3 + 3*a*b^2*c^4*d^5 + 3*a^2*b*c^2*d 
^7 + a^3*d^9)*x^2 + 2*(b^3*c^7*d^2 + 3*a*b^2*c^5*d^4 + 3*a^2*b*c^3*d^6 + a 
^3*c*d^8)*x)]
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.81 \[ \int \frac {x^2}{(c+d x)^3 \left (a+b x^2\right )} \, dx=\frac {{\left (3 \, a b c^{2} d - a^{2} d^{3}\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (b^{3} c^{6} + 3 \, a b^{2} c^{4} d^{2} + 3 \, a^{2} b c^{2} d^{4} + a^{3} d^{6}\right )}} - \frac {{\left (3 \, a b c^{2} d - a^{2} d^{3}\right )} \log \left (d x + c\right )}{b^{3} c^{6} + 3 \, a b^{2} c^{4} d^{2} + 3 \, a^{2} b c^{2} d^{4} + a^{3} d^{6}} - \frac {{\left (a b^{2} c^{3} - 3 \, a^{2} b c d^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{{\left (b^{3} c^{6} + 3 \, a b^{2} c^{4} d^{2} + 3 \, a^{2} b c^{2} d^{4} + a^{3} d^{6}\right )} \sqrt {a b}} + \frac {4 \, a c d^{3} x - b c^{4} + 3 \, a c^{2} d^{2}}{2 \, {\left (b^{2} c^{6} d + 2 \, a b c^{4} d^{3} + a^{2} c^{2} d^{5} + {\left (b^{2} c^{4} d^{3} + 2 \, a b c^{2} d^{5} + a^{2} d^{7}\right )} x^{2} + 2 \, {\left (b^{2} c^{5} d^{2} + 2 \, a b c^{3} d^{4} + a^{2} c d^{6}\right )} x\right )}} \] Input:

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

Output:

1/2*(3*a*b*c^2*d - a^2*d^3)*log(b*x^2 + a)/(b^3*c^6 + 3*a*b^2*c^4*d^2 + 3* 
a^2*b*c^2*d^4 + a^3*d^6) - (3*a*b*c^2*d - a^2*d^3)*log(d*x + c)/(b^3*c^6 + 
 3*a*b^2*c^4*d^2 + 3*a^2*b*c^2*d^4 + a^3*d^6) - (a*b^2*c^3 - 3*a^2*b*c*d^2 
)*arctan(b*x/sqrt(a*b))/((b^3*c^6 + 3*a*b^2*c^4*d^2 + 3*a^2*b*c^2*d^4 + a^ 
3*d^6)*sqrt(a*b)) + 1/2*(4*a*c*d^3*x - b*c^4 + 3*a*c^2*d^2)/(b^2*c^6*d + 2 
*a*b*c^4*d^3 + a^2*c^2*d^5 + (b^2*c^4*d^3 + 2*a*b*c^2*d^5 + a^2*d^7)*x^2 + 
 2*(b^2*c^5*d^2 + 2*a*b*c^3*d^4 + a^2*c*d^6)*x)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.58 \[ \int \frac {x^2}{(c+d x)^3 \left (a+b x^2\right )} \, dx=\frac {{\left (3 \, a b c^{2} d - a^{2} d^{3}\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (b^{3} c^{6} + 3 \, a b^{2} c^{4} d^{2} + 3 \, a^{2} b c^{2} d^{4} + a^{3} d^{6}\right )}} - \frac {{\left (3 \, a b c^{2} d^{2} - a^{2} d^{4}\right )} \log \left ({\left | d x + c \right |}\right )}{b^{3} c^{6} d + 3 \, a b^{2} c^{4} d^{3} + 3 \, a^{2} b c^{2} d^{5} + a^{3} d^{7}} - \frac {{\left (a b^{2} c^{3} - 3 \, a^{2} b c d^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{{\left (b^{3} c^{6} + 3 \, a b^{2} c^{4} d^{2} + 3 \, a^{2} b c^{2} d^{4} + a^{3} d^{6}\right )} \sqrt {a b}} - \frac {b^{2} c^{6} - 2 \, a b c^{4} d^{2} - 3 \, a^{2} c^{2} d^{4} - 4 \, {\left (a b c^{3} d^{3} + a^{2} c d^{5}\right )} x}{2 \, {\left (b c^{2} + a d^{2}\right )}^{3} {\left (d x + c\right )}^{2} d} \] Input:

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

Output:

1/2*(3*a*b*c^2*d - a^2*d^3)*log(b*x^2 + a)/(b^3*c^6 + 3*a*b^2*c^4*d^2 + 3* 
a^2*b*c^2*d^4 + a^3*d^6) - (3*a*b*c^2*d^2 - a^2*d^4)*log(abs(d*x + c))/(b^ 
3*c^6*d + 3*a*b^2*c^4*d^3 + 3*a^2*b*c^2*d^5 + a^3*d^7) - (a*b^2*c^3 - 3*a^ 
2*b*c*d^2)*arctan(b*x/sqrt(a*b))/((b^3*c^6 + 3*a*b^2*c^4*d^2 + 3*a^2*b*c^2 
*d^4 + a^3*d^6)*sqrt(a*b)) - 1/2*(b^2*c^6 - 2*a*b*c^4*d^2 - 3*a^2*c^2*d^4 
- 4*(a*b*c^3*d^3 + a^2*c*d^5)*x)/((b*c^2 + a*d^2)^3*(d*x + c)^2*d)
 

Mupad [B] (verification not implemented)

Time = 8.85 (sec) , antiderivative size = 688, normalized size of antiderivative = 3.76 \[ \int \frac {x^2}{(c+d x)^3 \left (a+b x^2\right )} \, dx=\frac {\ln \left (c+d\,x\right )\,\left (a^2\,d^3-3\,a\,b\,c^2\,d\right )}{a^3\,d^6+3\,a^2\,b\,c^2\,d^4+3\,a\,b^2\,c^4\,d^2+b^3\,c^6}-\frac {\ln \left (b^4\,c^{10}\,{\left (-a\,b\right )}^{3/2}-9\,a^6\,d^{10}\,\sqrt {-a\,b}+106\,c^6\,d^4\,{\left (-a\,b\right )}^{7/2}+6\,a^2\,c^4\,d^6\,{\left (-a\,b\right )}^{5/2}-27\,a^4\,c^2\,d^8\,{\left (-a\,b\right )}^{3/2}-77\,b^2\,c^8\,d^2\,{\left (-a\,b\right )}^{5/2}+a\,b^6\,c^{10}\,x+9\,a^6\,b\,d^{10}\,x+77\,a^2\,b^5\,c^8\,d^2\,x+106\,a^3\,b^4\,c^6\,d^4\,x-6\,a^4\,b^3\,c^4\,d^6\,x-27\,a^5\,b^2\,c^2\,d^8\,x\right )\,\left (\frac {a^2\,d^3}{2}-b\,\left (\frac {c^3\,\sqrt {-a\,b}}{2}+\frac {3\,a\,c^2\,d}{2}\right )+\frac {3\,a\,c\,d^2\,\sqrt {-a\,b}}{2}\right )}{a^3\,d^6+3\,a^2\,b\,c^2\,d^4+3\,a\,b^2\,c^4\,d^2+b^3\,c^6}-\frac {\ln \left (9\,a^6\,d^{10}\,\sqrt {-a\,b}-b^4\,c^{10}\,{\left (-a\,b\right )}^{3/2}-106\,c^6\,d^4\,{\left (-a\,b\right )}^{7/2}-6\,a^2\,c^4\,d^6\,{\left (-a\,b\right )}^{5/2}+27\,a^4\,c^2\,d^8\,{\left (-a\,b\right )}^{3/2}+77\,b^2\,c^8\,d^2\,{\left (-a\,b\right )}^{5/2}+a\,b^6\,c^{10}\,x+9\,a^6\,b\,d^{10}\,x+77\,a^2\,b^5\,c^8\,d^2\,x+106\,a^3\,b^4\,c^6\,d^4\,x-6\,a^4\,b^3\,c^4\,d^6\,x-27\,a^5\,b^2\,c^2\,d^8\,x\right )\,\left (b\,\left (\frac {c^3\,\sqrt {-a\,b}}{2}-\frac {3\,a\,c^2\,d}{2}\right )+\frac {a^2\,d^3}{2}-\frac {3\,a\,c\,d^2\,\sqrt {-a\,b}}{2}\right )}{a^3\,d^6+3\,a^2\,b\,c^2\,d^4+3\,a\,b^2\,c^4\,d^2+b^3\,c^6}-\frac {\frac {b\,c^4-3\,a\,c^2\,d^2}{2\,d\,\left (a^2\,d^4+2\,a\,b\,c^2\,d^2+b^2\,c^4\right )}-\frac {2\,a\,c\,d^2\,x}{a^2\,d^4+2\,a\,b\,c^2\,d^2+b^2\,c^4}}{c^2+2\,c\,d\,x+d^2\,x^2} \] Input:

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

Output:

(log(c + d*x)*(a^2*d^3 - 3*a*b*c^2*d))/(a^3*d^6 + b^3*c^6 + 3*a*b^2*c^4*d^ 
2 + 3*a^2*b*c^2*d^4) - (log(b^4*c^10*(-a*b)^(3/2) - 9*a^6*d^10*(-a*b)^(1/2 
) + 106*c^6*d^4*(-a*b)^(7/2) + 6*a^2*c^4*d^6*(-a*b)^(5/2) - 27*a^4*c^2*d^8 
*(-a*b)^(3/2) - 77*b^2*c^8*d^2*(-a*b)^(5/2) + a*b^6*c^10*x + 9*a^6*b*d^10* 
x + 77*a^2*b^5*c^8*d^2*x + 106*a^3*b^4*c^6*d^4*x - 6*a^4*b^3*c^4*d^6*x - 2 
7*a^5*b^2*c^2*d^8*x)*((a^2*d^3)/2 - b*((c^3*(-a*b)^(1/2))/2 + (3*a*c^2*d)/ 
2) + (3*a*c*d^2*(-a*b)^(1/2))/2))/(a^3*d^6 + b^3*c^6 + 3*a*b^2*c^4*d^2 + 3 
*a^2*b*c^2*d^4) - (log(9*a^6*d^10*(-a*b)^(1/2) - b^4*c^10*(-a*b)^(3/2) - 1 
06*c^6*d^4*(-a*b)^(7/2) - 6*a^2*c^4*d^6*(-a*b)^(5/2) + 27*a^4*c^2*d^8*(-a* 
b)^(3/2) + 77*b^2*c^8*d^2*(-a*b)^(5/2) + a*b^6*c^10*x + 9*a^6*b*d^10*x + 7 
7*a^2*b^5*c^8*d^2*x + 106*a^3*b^4*c^6*d^4*x - 6*a^4*b^3*c^4*d^6*x - 27*a^5 
*b^2*c^2*d^8*x)*(b*((c^3*(-a*b)^(1/2))/2 - (3*a*c^2*d)/2) + (a^2*d^3)/2 - 
(3*a*c*d^2*(-a*b)^(1/2))/2))/(a^3*d^6 + b^3*c^6 + 3*a*b^2*c^4*d^2 + 3*a^2* 
b*c^2*d^4) - ((b*c^4 - 3*a*c^2*d^2)/(2*d*(a^2*d^4 + b^2*c^4 + 2*a*b*c^2*d^ 
2)) - (2*a*c*d^2*x)/(a^2*d^4 + b^2*c^4 + 2*a*b*c^2*d^2))/(c^2 + d^2*x^2 + 
2*c*d*x)
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 561, normalized size of antiderivative = 3.07 \[ \int \frac {x^2}{(c+d x)^3 \left (a+b x^2\right )} \, dx=\frac {6 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,c^{3} d^{3}+12 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,c^{2} d^{4} x +6 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a c \,d^{5} x^{2}-2 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b \,c^{5} d -4 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b \,c^{4} d^{2} x -2 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b \,c^{3} d^{3} x^{2}-\mathrm {log}\left (b \,x^{2}+a \right ) a^{2} c^{2} d^{4}-2 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{2} c \,d^{5} x -\mathrm {log}\left (b \,x^{2}+a \right ) a^{2} d^{6} x^{2}+3 \,\mathrm {log}\left (b \,x^{2}+a \right ) a b \,c^{4} d^{2}+6 \,\mathrm {log}\left (b \,x^{2}+a \right ) a b \,c^{3} d^{3} x +3 \,\mathrm {log}\left (b \,x^{2}+a \right ) a b \,c^{2} d^{4} x^{2}+2 \,\mathrm {log}\left (d x +c \right ) a^{2} c^{2} d^{4}+4 \,\mathrm {log}\left (d x +c \right ) a^{2} c \,d^{5} x +2 \,\mathrm {log}\left (d x +c \right ) a^{2} d^{6} x^{2}-6 \,\mathrm {log}\left (d x +c \right ) a b \,c^{4} d^{2}-12 \,\mathrm {log}\left (d x +c \right ) a b \,c^{3} d^{3} x -6 \,\mathrm {log}\left (d x +c \right ) a b \,c^{2} d^{4} x^{2}+a^{2} c^{2} d^{4}-2 a^{2} d^{6} x^{2}-2 a b \,c^{2} d^{4} x^{2}-b^{2} c^{6}}{2 d \left (a^{3} d^{8} x^{2}+3 a^{2} b \,c^{2} d^{6} x^{2}+3 a \,b^{2} c^{4} d^{4} x^{2}+b^{3} c^{6} d^{2} x^{2}+2 a^{3} c \,d^{7} x +6 a^{2} b \,c^{3} d^{5} x +6 a \,b^{2} c^{5} d^{3} x +2 b^{3} c^{7} d x +a^{3} c^{2} d^{6}+3 a^{2} b \,c^{4} d^{4}+3 a \,b^{2} c^{6} d^{2}+b^{3} c^{8}\right )} \] Input:

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

Output:

(6*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*c**3*d**3 + 12*sqrt(b)* 
sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*c**2*d**4*x + 6*sqrt(b)*sqrt(a)*at 
an((b*x)/(sqrt(b)*sqrt(a)))*a*c*d**5*x**2 - 2*sqrt(b)*sqrt(a)*atan((b*x)/( 
sqrt(b)*sqrt(a)))*b*c**5*d - 4*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a) 
))*b*c**4*d**2*x - 2*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b*c**3* 
d**3*x**2 - log(a + b*x**2)*a**2*c**2*d**4 - 2*log(a + b*x**2)*a**2*c*d**5 
*x - log(a + b*x**2)*a**2*d**6*x**2 + 3*log(a + b*x**2)*a*b*c**4*d**2 + 6* 
log(a + b*x**2)*a*b*c**3*d**3*x + 3*log(a + b*x**2)*a*b*c**2*d**4*x**2 + 2 
*log(c + d*x)*a**2*c**2*d**4 + 4*log(c + d*x)*a**2*c*d**5*x + 2*log(c + d* 
x)*a**2*d**6*x**2 - 6*log(c + d*x)*a*b*c**4*d**2 - 12*log(c + d*x)*a*b*c** 
3*d**3*x - 6*log(c + d*x)*a*b*c**2*d**4*x**2 + a**2*c**2*d**4 - 2*a**2*d** 
6*x**2 - 2*a*b*c**2*d**4*x**2 - b**2*c**6)/(2*d*(a**3*c**2*d**6 + 2*a**3*c 
*d**7*x + a**3*d**8*x**2 + 3*a**2*b*c**4*d**4 + 6*a**2*b*c**3*d**5*x + 3*a 
**2*b*c**2*d**6*x**2 + 3*a*b**2*c**6*d**2 + 6*a*b**2*c**5*d**3*x + 3*a*b** 
2*c**4*d**4*x**2 + b**3*c**8 + 2*b**3*c**7*d*x + b**3*c**6*d**2*x**2))