\(\int \frac {x^6}{(c+d x)^2 (a+b x^2)^2} \, dx\) [198]

Optimal result

Integrand size = 20, antiderivative size = 253 \[ \int \frac {x^6}{(c+d x)^2 \left (a+b x^2\right )^2} \, dx=\frac {x}{b^2 d^2}-\frac {c^6}{d^3 \left (b c^2+a d^2\right )^2 (c+d x)}-\frac {a^2 \left (2 a c d \left (b c^2+a d^2\right )+\left (b^2 c^4-a^2 d^4\right ) x\right )}{2 b^2 \left (b c^2+a d^2\right )^3 \left (a+b x^2\right )}+\frac {a^{3/2} \left (5 b^2 c^4-6 a b c^2 d^2-3 a^2 d^4\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{5/2} \left (b c^2+a d^2\right )^3}-\frac {2 c^5 \left (b c^2+3 a d^2\right ) \log (c+d x)}{d^3 \left (b c^2+a d^2\right )^3}-\frac {a^2 c d \left (3 b c^2+a d^2\right ) \log \left (a+b x^2\right )}{b^2 \left (b c^2+a d^2\right )^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.92 \[ \int \frac {x^6}{(c+d x)^2 \left (a+b x^2\right )^2} \, dx=\frac {x}{b^2 d^2}-\frac {c^6}{d^3 \left (b c^2+a d^2\right )^2 (c+d x)}+\frac {a^2 \left (-b c^2 x+a d (-2 c+d x)\right )}{2 b^2 \left (b c^2+a d^2\right )^2 \left (a+b x^2\right )}-\frac {a^{3/2} \left (-5 b^2 c^4+6 a b c^2 d^2+3 a^2 d^4\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{5/2} \left (b c^2+a d^2\right )^3}-\frac {2 \left (b c^7+3 a c^5 d^2\right ) \log (c+d x)}{\left (b c^2 d+a d^3\right )^3}-\frac {a^2 c d \left (3 b c^2+a d^2\right ) \log \left (a+b x^2\right )}{b^2 \left (b c^2+a d^2\right )^3} \] Input:

Integrate[x^6/((c + d*x)^2*(a + b*x^2)^2),x]
 

Output:

x/(b^2*d^2) - c^6/(d^3*(b*c^2 + a*d^2)^2*(c + d*x)) + (a^2*(-(b*c^2*x) + a 
*d*(-2*c + d*x)))/(2*b^2*(b*c^2 + a*d^2)^2*(a + b*x^2)) - (a^(3/2)*(-5*b^2 
*c^4 + 6*a*b*c^2*d^2 + 3*a^2*d^4)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(5/2)* 
(b*c^2 + a*d^2)^3) - (2*(b*c^7 + 3*a*c^5*d^2)*Log[c + d*x])/(b*c^2*d + a*d 
^3)^3 - (a^2*c*d*(3*b*c^2 + a*d^2)*Log[a + b*x^2])/(b^2*(b*c^2 + a*d^2)^3)
 

Rubi [A] (verified)

Time = 1.51 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {601, 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.

\(\displaystyle \int \frac {x^6}{\left (a+b x^2\right )^2 (c+d x)^2} \, dx\)

\(\Big \downarrow \) 601

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2160

\(\displaystyle \frac {\int \left (\frac {2 a c^6}{d^2 \left (b c^2+a d^2\right )^2 (c+d x)^2}-\frac {4 a \left (b c^2+3 a d^2\right ) c^5}{d^2 \left (b c^2+a d^2\right )^3 (c+d x)}+\frac {a^3 \left (5 b^2 c^4-6 a b d^2 c^2-4 b d \left (3 b c^2+a d^2\right ) x c-3 a^2 d^4\right )}{b^2 \left (b c^2+a d^2\right )^3 \left (b x^2+a\right )}+\frac {2 a}{b^2 d^2}\right )dx}{2 a}-\frac {a^2 \left (x \left (b c^2-a d^2\right )+2 a c d\right )}{2 b^2 \left (a+b x^2\right ) \left (a d^2+b c^2\right )^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {2 a^3 c d \left (a d^2+3 b c^2\right ) \log \left (a+b x^2\right )}{b^2 \left (a d^2+b c^2\right )^3}+\frac {a^{5/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-3 a^2 d^4-6 a b c^2 d^2+5 b^2 c^4\right )}{b^{5/2} \left (a d^2+b c^2\right )^3}+\frac {2 a x}{b^2 d^2}-\frac {2 a c^6}{d^3 (c+d x) \left (a d^2+b c^2\right )^2}-\frac {4 a c^5 \left (3 a d^2+b c^2\right ) \log (c+d x)}{d^3 \left (a d^2+b c^2\right )^3}}{2 a}-\frac {a^2 \left (x \left (b c^2-a d^2\right )+2 a c d\right )}{2 b^2 \left (a+b x^2\right ) \left (a d^2+b c^2\right )^2}\)

Input:

Int[x^6/((c + d*x)^2*(a + b*x^2)^2),x]
 

Output:

-1/2*(a^2*(2*a*c*d + (b*c^2 - a*d^2)*x))/(b^2*(b*c^2 + a*d^2)^2*(a + b*x^2 
)) + ((2*a*x)/(b^2*d^2) - (2*a*c^6)/(d^3*(b*c^2 + a*d^2)^2*(c + d*x)) + (a 
^(5/2)*(5*b^2*c^4 - 6*a*b*c^2*d^2 - 3*a^2*d^4)*ArcTan[(Sqrt[b]*x)/Sqrt[a]] 
)/(b^(5/2)*(b*c^2 + a*d^2)^3) - (4*a*c^5*(b*c^2 + 3*a*d^2)*Log[c + d*x])/( 
d^3*(b*c^2 + a*d^2)^3) - (2*a^3*c*d*(3*b*c^2 + a*d^2)*Log[a + b*x^2])/(b^2 
*(b*c^2 + a*d^2)^3))/(2*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]
 

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.86

method	result	size
default	\(\frac {x}{b^{2} d^{2}}-\frac {a^{2} \left (\frac {\left (-\frac {a^{2} d^{4}}{2}+\frac {b^{2} c^{4}}{2}\right ) x +a c d \left (a \,d^{2}+b \,c^{2}\right )}{b \,x^{2}+a}+\frac {\left (4 a b c \,d^{3}+12 b^{2} c^{3} d \right ) \ln \left (b \,x^{2}+a \right )}{4 b}+\frac {\left (3 a^{2} d^{4}+6 b \,c^{2} d^{2} a -5 b^{2} c^{4}\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{\left (a \,d^{2}+b \,c^{2}\right )^{3} b^{2}}-\frac {c^{6}}{d^{3} \left (a \,d^{2}+b \,c^{2}\right )^{2} \left (d x +c \right )}-\frac {2 c^{5} \left (3 a \,d^{2}+b \,c^{2}\right ) \ln \left (d x +c \right )}{d^{3} \left (a \,d^{2}+b \,c^{2}\right )^{3}}\)	\(218\)
risch	\(\frac {x}{b^{2} d^{2}}+\frac {\frac {\left (a^{3} d^{6}-a^{2} b \,c^{2} d^{4}-2 b^{3} c^{6}\right ) x^{2}}{2 d \left (a^{2} d^{4}+2 b \,c^{2} d^{2} a +b^{2} c^{4}\right )}-\frac {a^{2} c \,d^{2} x}{2 \left (a \,d^{2}+b \,c^{2}\right )}-\frac {\left (a^{2} d^{4}+b^{2} c^{4}\right ) a \,c^{2}}{d \left (a \,d^{2}+b \,c^{2}\right )^{2}}}{b^{2} d^{2} \left (d x +c \right ) \left (b \,x^{2}+a \right )}-\frac {6 c^{5} \ln \left (d x +c \right ) a}{d \left (a^{3} d^{6}+3 a^{2} b \,c^{2} d^{4}+3 a \,b^{2} c^{4} d^{2}+b^{3} c^{6}\right )}-\frac {2 c^{7} \ln \left (d x +c \right ) b}{d^{3} \left (a^{3} d^{6}+3 a^{2} b \,c^{2} d^{4}+3 a \,b^{2} c^{4} d^{2}+b^{3} c^{6}\right )}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a^{3} b \,d^{6}+3 a^{2} b^{2} c^{2} d^{4}+3 a \,b^{3} c^{4} d^{2}+b^{4} c^{6}\right ) \textit {\_Z}^{2}+\left (8 a^{3} b c \,d^{5}+24 a^{2} b^{2} c^{3} d^{3}\right ) \textit {\_Z} +9 d^{6} a^{4}+25 a^{3} b \,c^{2} d^{4}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (3 a^{5} b \,d^{11}+11 a^{4} b^{2} c^{2} d^{9}+14 a^{3} b^{3} c^{4} d^{7}+6 a^{2} b^{4} c^{6} d^{5}-a \,b^{5} c^{8} d^{3}-b^{6} c^{10} d \right ) \textit {\_R}^{2}+\left (14 a^{5} b c \,d^{10}+58 a^{4} b^{2} c^{3} d^{8}+66 a^{3} b^{3} c^{5} d^{6}+6 a^{2} b^{4} c^{7} d^{4}-24 a \,b^{5} c^{9} d^{2}-8 b^{6} c^{11}\right ) \textit {\_R} +18 a^{6} d^{11}+68 a^{5} b \,c^{2} d^{9}+50 a^{4} b^{2} c^{4} d^{7}-48 a^{3} b^{3} c^{6} d^{5}-16 a^{2} b^{4} c^{8} d^{3}\right ) x +\left (4 a^{5} b c \,d^{10}+16 a^{4} b^{2} c^{3} d^{8}+24 a^{3} b^{3} c^{5} d^{6}+16 a^{2} b^{4} c^{7} d^{4}+4 a \,b^{5} c^{9} d^{2}\right ) \textit {\_R}^{2}+\left (3 a^{6} d^{11}+30 a^{5} b \,c^{2} d^{9}+88 a^{4} b^{2} c^{4} d^{7}+114 a^{3} b^{3} c^{6} d^{5}+69 a^{2} b^{4} c^{8} d^{3}+16 a \,b^{5} c^{10} d \right ) \textit {\_R} +18 a^{6} c \,d^{10}+68 a^{5} b \,c^{3} d^{8}+122 a^{4} b^{2} c^{5} d^{6}+144 a^{3} b^{3} c^{7} d^{4}+40 a^{2} b^{4} c^{9} d^{2}\right )}{4 b^{2} d^{2}}\)	\(803\)

Input:

int(x^6/(d*x+c)^2/(b*x^2+a)^2,x,method=_RETURNVERBOSE)
 

Output:

x/b^2/d^2-a^2/(a*d^2+b*c^2)^3/b^2*(((-1/2*a^2*d^4+1/2*b^2*c^4)*x+a*c*d*(a* 
d^2+b*c^2))/(b*x^2+a)+1/4*(4*a*b*c*d^3+12*b^2*c^3*d)/b*ln(b*x^2+a)+1/2*(3* 
a^2*d^4+6*a*b*c^2*d^2-5*b^2*c^4)/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2)))-c^6/ 
d^3/(a*d^2+b*c^2)^2/(d*x+c)-2*c^5*(3*a*d^2+b*c^2)*ln(d*x+c)/d^3/(a*d^2+b*c 
^2)^3
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 848 vs. \(2 (241) = 482\).

Time = 4.23 (sec) , antiderivative size = 1716, normalized size of antiderivative = 6.78 \[ \int \frac {x^6}{(c+d x)^2 \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(x^6/(d*x+c)^2/(b*x^2+a)^2,x, algorithm="fricas")
 

Output:

[-1/4*(4*a*b^3*c^8 + 4*a^2*b^2*c^6*d^2 + 4*a^3*b*c^4*d^4 + 4*a^4*c^2*d^6 - 
 4*(b^4*c^6*d^2 + 3*a*b^3*c^4*d^4 + 3*a^2*b^2*c^2*d^6 + a^3*b*d^8)*x^4 - 4 
*(b^4*c^7*d + 3*a*b^3*c^5*d^3 + 3*a^2*b^2*c^3*d^5 + a^3*b*c*d^7)*x^3 + 2*( 
2*b^4*c^8 - 5*a^2*b^2*c^4*d^4 - 6*a^3*b*c^2*d^6 - 3*a^4*d^8)*x^2 + (5*a^2* 
b^2*c^5*d^3 - 6*a^3*b*c^3*d^5 - 3*a^4*c*d^7 + (5*a*b^3*c^4*d^4 - 6*a^2*b^2 
*c^2*d^6 - 3*a^3*b*d^8)*x^3 + (5*a*b^3*c^5*d^3 - 6*a^2*b^2*c^3*d^5 - 3*a^3 
*b*c*d^7)*x^2 + (5*a^2*b^2*c^4*d^4 - 6*a^3*b*c^2*d^6 - 3*a^4*d^8)*x)*sqrt( 
-a/b)*log((b*x^2 - 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) - 2*(2*a*b^3*c^7*d + 
 5*a^2*b^2*c^5*d^3 + 4*a^3*b*c^3*d^5 + a^4*c*d^7)*x + 4*(3*a^3*b*c^4*d^4 + 
 a^4*c^2*d^6 + (3*a^2*b^2*c^3*d^5 + a^3*b*c*d^7)*x^3 + (3*a^2*b^2*c^4*d^4 
+ a^3*b*c^2*d^6)*x^2 + (3*a^3*b*c^3*d^5 + a^4*c*d^7)*x)*log(b*x^2 + a) + 8 
*(a*b^3*c^8 + 3*a^2*b^2*c^6*d^2 + (b^4*c^7*d + 3*a*b^3*c^5*d^3)*x^3 + (b^4 
*c^8 + 3*a*b^3*c^6*d^2)*x^2 + (a*b^3*c^7*d + 3*a^2*b^2*c^5*d^3)*x)*log(d*x 
 + c))/(a*b^5*c^7*d^3 + 3*a^2*b^4*c^5*d^5 + 3*a^3*b^3*c^3*d^7 + a^4*b^2*c* 
d^9 + (b^6*c^6*d^4 + 3*a*b^5*c^4*d^6 + 3*a^2*b^4*c^2*d^8 + a^3*b^3*d^10)*x 
^3 + (b^6*c^7*d^3 + 3*a*b^5*c^5*d^5 + 3*a^2*b^4*c^3*d^7 + a^3*b^3*c*d^9)*x 
^2 + (a*b^5*c^6*d^4 + 3*a^2*b^4*c^4*d^6 + 3*a^3*b^3*c^2*d^8 + a^4*b^2*d^10 
)*x), -1/2*(2*a*b^3*c^8 + 2*a^2*b^2*c^6*d^2 + 2*a^3*b*c^4*d^4 + 2*a^4*c^2* 
d^6 - 2*(b^4*c^6*d^2 + 3*a*b^3*c^4*d^4 + 3*a^2*b^2*c^2*d^6 + a^3*b*d^8)*x^ 
4 - 2*(b^4*c^7*d + 3*a*b^3*c^5*d^3 + 3*a^2*b^2*c^3*d^5 + a^3*b*c*d^7)*x...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^6}{(c+d x)^2 \left (a+b x^2\right )^2} \, dx=\text {Timed out} \] Input:

integrate(x**6/(d*x+c)**2/(b*x**2+a)**2,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.87 \[ \int \frac {x^6}{(c+d x)^2 \left (a+b x^2\right )^2} \, dx=-\frac {{\left (3 \, a^{2} b c^{3} d + a^{3} c d^{3}\right )} \log \left (b x^{2} + a\right )}{b^{5} c^{6} + 3 \, a b^{4} c^{4} d^{2} + 3 \, a^{2} b^{3} c^{2} d^{4} + a^{3} b^{2} d^{6}} - \frac {2 \, {\left (b c^{7} + 3 \, a c^{5} d^{2}\right )} \log \left (d x + c\right )}{b^{3} c^{6} d^{3} + 3 \, a b^{2} c^{4} d^{5} + 3 \, a^{2} b c^{2} d^{7} + a^{3} d^{9}} + \frac {{\left (5 \, a^{2} b^{2} c^{4} - 6 \, a^{3} b c^{2} d^{2} - 3 \, a^{4} d^{4}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (b^{5} c^{6} + 3 \, a b^{4} c^{4} d^{2} + 3 \, a^{2} b^{3} c^{2} d^{4} + a^{3} b^{2} d^{6}\right )} \sqrt {a b}} - \frac {2 \, a b^{2} c^{6} + 2 \, a^{3} c^{2} d^{4} + {\left (2 \, b^{3} c^{6} + a^{2} b c^{2} d^{4} - a^{3} d^{6}\right )} x^{2} + {\left (a^{2} b c^{3} d^{3} + a^{3} c d^{5}\right )} x}{2 \, {\left (a b^{4} c^{5} d^{3} + 2 \, a^{2} b^{3} c^{3} d^{5} + a^{3} b^{2} c d^{7} + {\left (b^{5} c^{4} d^{4} + 2 \, a b^{4} c^{2} d^{6} + a^{2} b^{3} d^{8}\right )} x^{3} + {\left (b^{5} c^{5} d^{3} + 2 \, a b^{4} c^{3} d^{5} + a^{2} b^{3} c d^{7}\right )} x^{2} + {\left (a b^{4} c^{4} d^{4} + 2 \, a^{2} b^{3} c^{2} d^{6} + a^{3} b^{2} d^{8}\right )} x\right )}} + \frac {x}{b^{2} d^{2}} \] Input:

integrate(x^6/(d*x+c)^2/(b*x^2+a)^2,x, algorithm="maxima")
 

Output:

-(3*a^2*b*c^3*d + a^3*c*d^3)*log(b*x^2 + a)/(b^5*c^6 + 3*a*b^4*c^4*d^2 + 3 
*a^2*b^3*c^2*d^4 + a^3*b^2*d^6) - 2*(b*c^7 + 3*a*c^5*d^2)*log(d*x + c)/(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) + 1/2*(5*a^2*b^2* 
c^4 - 6*a^3*b*c^2*d^2 - 3*a^4*d^4)*arctan(b*x/sqrt(a*b))/((b^5*c^6 + 3*a*b 
^4*c^4*d^2 + 3*a^2*b^3*c^2*d^4 + a^3*b^2*d^6)*sqrt(a*b)) - 1/2*(2*a*b^2*c^ 
6 + 2*a^3*c^2*d^4 + (2*b^3*c^6 + a^2*b*c^2*d^4 - a^3*d^6)*x^2 + (a^2*b*c^3 
*d^3 + a^3*c*d^5)*x)/(a*b^4*c^5*d^3 + 2*a^2*b^3*c^3*d^5 + a^3*b^2*c*d^7 + 
(b^5*c^4*d^4 + 2*a*b^4*c^2*d^6 + a^2*b^3*d^8)*x^3 + (b^5*c^5*d^3 + 2*a*b^4 
*c^3*d^5 + a^2*b^3*c*d^7)*x^2 + (a*b^4*c^4*d^4 + 2*a^2*b^3*c^2*d^6 + a^3*b 
^2*d^8)*x) + x/(b^2*d^2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 545 vs. \(2 (241) = 482\).

Time = 0.13 (sec) , antiderivative size = 545, normalized size of antiderivative = 2.15 \[ \int \frac {x^6}{(c+d x)^2 \left (a+b x^2\right )^2} \, dx=-\frac {c^{6} d^{5}}{{\left (b^{2} c^{4} d^{8} + 2 \, a b c^{2} d^{10} + a^{2} d^{12}\right )} {\left (d x + c\right )}} - \frac {{\left (3 \, a^{2} b c^{3} d + a^{3} c d^{3}\right )} \log \left (b - \frac {2 \, b c}{d x + c} + \frac {b c^{2}}{{\left (d x + c\right )}^{2}} + \frac {a d^{2}}{{\left (d x + c\right )}^{2}}\right )}{b^{5} c^{6} + 3 \, a b^{4} c^{4} d^{2} + 3 \, a^{2} b^{3} c^{2} d^{4} + a^{3} b^{2} d^{6}} + \frac {{\left (5 \, a^{2} b^{2} c^{4} d^{2} - 6 \, a^{3} b c^{2} d^{4} - 3 \, a^{4} d^{6}\right )} \arctan \left (\frac {b c - \frac {b c^{2}}{d x + c} - \frac {a d^{2}}{d x + c}}{\sqrt {a b} d}\right )}{2 \, {\left (b^{5} c^{6} + 3 \, a b^{4} c^{4} d^{2} + 3 \, a^{2} b^{3} c^{2} d^{4} + a^{3} b^{2} d^{6}\right )} \sqrt {a b} d^{2}} + \frac {2 \, c \log \left (\frac {{\left | d x + c \right |}}{{\left (d x + c\right )}^{2} {\left | d \right |}}\right )}{b^{2} d^{3}} + \frac {{\left (2 \, b^{4} c^{6} + 6 \, a b^{3} c^{4} d^{2} + 6 \, a^{2} b^{2} c^{2} d^{4} + 2 \, a^{3} b d^{6} - \frac {4 \, b^{4} c^{7} d + 12 \, a b^{3} c^{5} d^{3} + 13 \, a^{2} b^{2} c^{3} d^{5} + a^{3} b c d^{7}}{{\left (d x + c\right )} d} + \frac {2 \, b^{4} c^{8} d^{2} + 8 \, a b^{3} c^{6} d^{4} + 13 \, a^{2} b^{2} c^{4} d^{6} + 2 \, a^{3} b c^{2} d^{8} + 3 \, a^{4} d^{10}}{{\left (d x + c\right )}^{2} d^{2}}\right )} {\left (d x + c\right )}}{2 \, {\left (b c^{2} + a d^{2}\right )}^{3} {\left (b - \frac {2 \, b c}{d x + c} + \frac {b c^{2}}{{\left (d x + c\right )}^{2}} + \frac {a d^{2}}{{\left (d x + c\right )}^{2}}\right )} b^{2} d^{3}} \] Input:

integrate(x^6/(d*x+c)^2/(b*x^2+a)^2,x, algorithm="giac")
 

Output:

-c^6*d^5/((b^2*c^4*d^8 + 2*a*b*c^2*d^10 + a^2*d^12)*(d*x + c)) - (3*a^2*b* 
c^3*d + a^3*c*d^3)*log(b - 2*b*c/(d*x + c) + b*c^2/(d*x + c)^2 + a*d^2/(d* 
x + c)^2)/(b^5*c^6 + 3*a*b^4*c^4*d^2 + 3*a^2*b^3*c^2*d^4 + a^3*b^2*d^6) + 
1/2*(5*a^2*b^2*c^4*d^2 - 6*a^3*b*c^2*d^4 - 3*a^4*d^6)*arctan((b*c - b*c^2/ 
(d*x + c) - a*d^2/(d*x + c))/(sqrt(a*b)*d))/((b^5*c^6 + 3*a*b^4*c^4*d^2 + 
3*a^2*b^3*c^2*d^4 + a^3*b^2*d^6)*sqrt(a*b)*d^2) + 2*c*log(abs(d*x + c)/((d 
*x + c)^2*abs(d)))/(b^2*d^3) + 1/2*(2*b^4*c^6 + 6*a*b^3*c^4*d^2 + 6*a^2*b^ 
2*c^2*d^4 + 2*a^3*b*d^6 - (4*b^4*c^7*d + 12*a*b^3*c^5*d^3 + 13*a^2*b^2*c^3 
*d^5 + a^3*b*c*d^7)/((d*x + c)*d) + (2*b^4*c^8*d^2 + 8*a*b^3*c^6*d^4 + 13* 
a^2*b^2*c^4*d^6 + 2*a^3*b*c^2*d^8 + 3*a^4*d^10)/((d*x + c)^2*d^2))*(d*x + 
c)/((b*c^2 + a*d^2)^3*(b - 2*b*c/(d*x + c) + b*c^2/(d*x + c)^2 + a*d^2/(d* 
x + c)^2)*b^2*d^3)
 

Mupad [B] (verification not implemented)

Time = 7.71 (sec) , antiderivative size = 544, normalized size of antiderivative = 2.15 \[ \int \frac {x^6}{(c+d x)^2 \left (a+b x^2\right )^2} \, dx=\frac {x}{b^2\,d^2}-\frac {\ln \left (c+d\,x\right )\,\left (2\,b\,c^7+6\,a\,c^5\,d^2\right )}{a^3\,d^9+3\,a^2\,b\,c^2\,d^7+3\,a\,b^2\,c^4\,d^5+b^3\,c^6\,d^3}-\frac {\frac {x^2\,\left (-a^3\,d^6+a^2\,b\,c^2\,d^4+2\,b^3\,c^6\right )}{2\,d\,\left (a^2\,d^4+2\,a\,b\,c^2\,d^2+b^2\,c^4\right )}+\frac {a\,c^2\,\left (a^2\,d^4+b^2\,c^4\right )}{d\,{\left (b\,c^2+a\,d^2\right )}^2}+\frac {a^2\,c\,d^2\,x}{2\,\left (b\,c^2+a\,d^2\right )}}{b^3\,d^3\,x^3+c\,b^3\,d^2\,x^2+a\,b^2\,d^3\,x+a\,c\,b^2\,d^2}-\frac {\ln \left (\sqrt {-a^3\,b^5}+a\,b^3\,x\right )\,\left (3\,a^2\,d^4\,\sqrt {-a^3\,b^5}-5\,b^2\,c^4\,\sqrt {-a^3\,b^5}+12\,a^2\,b^4\,c^3\,d+4\,a^3\,b^3\,c\,d^3+6\,a\,b\,c^2\,d^2\,\sqrt {-a^3\,b^5}\right )}{4\,\left (a^3\,b^5\,d^6+3\,a^2\,b^6\,c^2\,d^4+3\,a\,b^7\,c^4\,d^2+b^8\,c^6\right )}-\frac {\ln \left (\sqrt {-a^3\,b^5}-a\,b^3\,x\right )\,\left (5\,b^2\,c^4\,\sqrt {-a^3\,b^5}-3\,a^2\,d^4\,\sqrt {-a^3\,b^5}+12\,a^2\,b^4\,c^3\,d+4\,a^3\,b^3\,c\,d^3-6\,a\,b\,c^2\,d^2\,\sqrt {-a^3\,b^5}\right )}{4\,\left (a^3\,b^5\,d^6+3\,a^2\,b^6\,c^2\,d^4+3\,a\,b^7\,c^4\,d^2+b^8\,c^6\right )} \] Input:

int(x^6/((a + b*x^2)^2*(c + d*x)^2),x)
 

Output:

x/(b^2*d^2) - (log(c + d*x)*(2*b*c^7 + 6*a*c^5*d^2))/(a^3*d^9 + b^3*c^6*d^ 
3 + 3*a*b^2*c^4*d^5 + 3*a^2*b*c^2*d^7) - ((x^2*(2*b^3*c^6 - a^3*d^6 + a^2* 
b*c^2*d^4))/(2*d*(a^2*d^4 + b^2*c^4 + 2*a*b*c^2*d^2)) + (a*c^2*(a^2*d^4 + 
b^2*c^4))/(d*(a*d^2 + b*c^2)^2) + (a^2*c*d^2*x)/(2*(a*d^2 + b*c^2)))/(b^3* 
d^3*x^3 + b^3*c*d^2*x^2 + a*b^2*c*d^2 + a*b^2*d^3*x) - (log((-a^3*b^5)^(1/ 
2) + a*b^3*x)*(3*a^2*d^4*(-a^3*b^5)^(1/2) - 5*b^2*c^4*(-a^3*b^5)^(1/2) + 1 
2*a^2*b^4*c^3*d + 4*a^3*b^3*c*d^3 + 6*a*b*c^2*d^2*(-a^3*b^5)^(1/2)))/(4*(b 
^8*c^6 + a^3*b^5*d^6 + 3*a*b^7*c^4*d^2 + 3*a^2*b^6*c^2*d^4)) - (log((-a^3* 
b^5)^(1/2) - a*b^3*x)*(5*b^2*c^4*(-a^3*b^5)^(1/2) - 3*a^2*d^4*(-a^3*b^5)^( 
1/2) + 12*a^2*b^4*c^3*d + 4*a^3*b^3*c*d^3 - 6*a*b*c^2*d^2*(-a^3*b^5)^(1/2) 
))/(4*(b^8*c^6 + a^3*b^5*d^6 + 3*a*b^7*c^4*d^2 + 3*a^2*b^6*c^2*d^4))
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 1151, normalized size of antiderivative = 4.55 \[ \int \frac {x^6}{(c+d x)^2 \left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

int(x^6/(d*x+c)^2/(b*x^2+a)^2,x)
 

Output:

( - 3*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**4*c**2*d**7 - 3*sqr 
t(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**4*c*d**8*x - 6*sqrt(b)*sqrt( 
a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3*b*c**4*d**5 - 6*sqrt(b)*sqrt(a)*atan 
((b*x)/(sqrt(b)*sqrt(a)))*a**3*b*c**3*d**6*x - 3*sqrt(b)*sqrt(a)*atan((b*x 
)/(sqrt(b)*sqrt(a)))*a**3*b*c**2*d**7*x**2 - 3*sqrt(b)*sqrt(a)*atan((b*x)/ 
(sqrt(b)*sqrt(a)))*a**3*b*c*d**8*x**3 + 5*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt 
(b)*sqrt(a)))*a**2*b**2*c**6*d**3 + 5*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)* 
sqrt(a)))*a**2*b**2*c**5*d**4*x - 6*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sq 
rt(a)))*a**2*b**2*c**4*d**5*x**2 - 6*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*s 
qrt(a)))*a**2*b**2*c**3*d**6*x**3 + 5*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)* 
sqrt(a)))*a*b**3*c**6*d**3*x**2 + 5*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sq 
rt(a)))*a*b**3*c**5*d**4*x**3 - 2*log(a + b*x**2)*a**4*b*c**3*d**6 - 2*log 
(a + b*x**2)*a**4*b*c**2*d**7*x - 6*log(a + b*x**2)*a**3*b**2*c**5*d**4 - 
6*log(a + b*x**2)*a**3*b**2*c**4*d**5*x - 2*log(a + b*x**2)*a**3*b**2*c**3 
*d**6*x**2 - 2*log(a + b*x**2)*a**3*b**2*c**2*d**7*x**3 - 6*log(a + b*x**2 
)*a**2*b**3*c**5*d**4*x**2 - 6*log(a + b*x**2)*a**2*b**3*c**4*d**5*x**3 - 
12*log(c + d*x)*a**2*b**3*c**7*d**2 - 12*log(c + d*x)*a**2*b**3*c**6*d**3* 
x - 4*log(c + d*x)*a*b**4*c**9 - 4*log(c + d*x)*a*b**4*c**8*d*x - 12*log(c 
 + d*x)*a*b**4*c**7*d**2*x**2 - 12*log(c + d*x)*a*b**4*c**6*d**3*x**3 - 4* 
log(c + d*x)*b**5*c**9*x**2 - 4*log(c + d*x)*b**5*c**8*d*x**3 - 3*a**5*...