\(\int \frac {x^6}{(c+d x) (a+b x^2)^2} \, dx\) [187]

Optimal result

Integrand size = 20, antiderivative size = 189 \[ \int \frac {x^6}{(c+d x) \left (a+b x^2\right )^2} \, dx=-\frac {c x}{b^2 d^2}+\frac {x^2}{2 b^2 d}-\frac {a^2 (a d+b c x)}{2 b^3 \left (b c^2+a d^2\right ) \left (a+b x^2\right )}+\frac {a^{3/2} c \left (5 b c^2+3 a d^2\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{5/2} \left (b c^2+a d^2\right )^2}+\frac {c^6 \log (c+d x)}{d^3 \left (b c^2+a d^2\right )^2}-\frac {a^2 d \left (3 b c^2+2 a d^2\right ) \log \left (a+b x^2\right )}{2 b^3 \left (b c^2+a d^2\right )^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.97 \[ \int \frac {x^6}{(c+d x) \left (a+b x^2\right )^2} \, dx=\frac {1}{2} \left (-\frac {2 c x}{b^2 d^2}+\frac {x^2}{b^2 d}-\frac {a^2 (a d+b c x)}{b^3 \left (b c^2+a d^2\right ) \left (a+b x^2\right )}+\frac {a^{3/2} c \left (5 b c^2+3 a d^2\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{5/2} \left (b c^2+a d^2\right )^2}+\frac {2 c^6 \log (c+d x)}{d^3 \left (b c^2+a d^2\right )^2}-\frac {a^2 d \left (3 b c^2+2 a d^2\right ) \log \left (a+b x^2\right )}{b^3 \left (b c^2+a d^2\right )^2}\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.08 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.02, 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.

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

\(\Big \downarrow \) 601

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2160

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-1/2*(a^2*(a*d + b*c*x))/(b^3*(b*c^2 + a*d^2)*(a + b*x^2)) + ((-2*a*c*x)/( 
b^2*d^2) + (a*x^2)/(b^2*d) + (a^(5/2)*c*(5*b*c^2 + 3*a*d^2)*ArcTan[(Sqrt[b 
]*x)/Sqrt[a]])/(b^(5/2)*(b*c^2 + a*d^2)^2) + (2*a*c^6*Log[c + d*x])/(d^3*( 
b*c^2 + a*d^2)^2) - (a^3*d*(3*b*c^2 + 2*a*d^2)*Log[a + b*x^2])/(b^3*(b*c^2 
 + a*d^2)^2))/(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 = 0.31 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.89

method	result	size
default	\(-\frac {-\frac {1}{2} d \,x^{2}+c x}{b^{2} d^{2}}+\frac {a^{2} \left (\frac {\left (-\frac {1}{2} a \,d^{2} c -\frac {1}{2} b \,c^{3}\right ) x -\frac {a d \left (a \,d^{2}+b \,c^{2}\right )}{2 b}}{b \,x^{2}+a}+\frac {\left (-4 a \,d^{3}-6 b \,c^{2} d \right ) \ln \left (b \,x^{2}+a \right )}{4 b}+\frac {\left (3 a \,d^{2} c +5 b \,c^{3}\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{\left (a \,d^{2}+b \,c^{2}\right )^{2} b^{2}}+\frac {c^{6} \ln \left (d x +c \right )}{d^{3} \left (a \,d^{2}+b \,c^{2}\right )^{2}}\)	\(169\)
risch	\(\frac {x^{2}}{2 b^{2} d}-\frac {c x}{b^{2} d^{2}}+\frac {-\frac {a^{2} c \,d^{2} x}{2 \left (a \,d^{2}+b \,c^{2}\right )}-\frac {a^{3} d^{3}}{2 \left (a \,d^{2}+b \,c^{2}\right ) b}}{b^{2} d^{2} \left (b \,x^{2}+a \right )}+\frac {c^{6} \ln \left (d x +c \right )}{d^{3} \left (a^{2} d^{4}+2 b \,c^{2} d^{2} a +b^{2} c^{4}\right )}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a^{2} b^{2} d^{4}+2 a \,b^{3} c^{2} d^{2}+c^{4} b^{4}\right ) \textit {\_Z}^{2}+\left (8 a^{3} b \,d^{5}+12 a^{2} b^{2} c^{2} d^{3}\right ) \textit {\_Z} +16 d^{6} a^{4}+25 a^{3} b \,c^{2} d^{4}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (3 a^{3} b^{2} d^{7}+5 a^{2} b^{3} c^{2} d^{5}+a \,b^{4} c^{4} d^{3}-b^{5} c^{6} d \right ) \textit {\_R}^{2}+\left (20 a^{4} b \,d^{8}+32 a^{3} b^{2} c^{2} d^{6}+4 a^{2} b^{3} c^{4} d^{4}-4 a \,b^{4} c^{6} d^{2}+4 b^{5} c^{8}\right ) \textit {\_R} +32 a^{5} d^{9}+50 a^{4} b \,c^{2} d^{7}-16 a^{2} b^{3} c^{6} d^{3}\right ) x +\left (4 a^{3} b^{2} c \,d^{6}+8 a^{2} b^{3} c^{3} d^{4}+4 a \,b^{4} c^{5} d^{2}\right ) \textit {\_R}^{2}+\left (21 a^{4} b c \,d^{7}+34 a^{3} b^{2} c^{3} d^{5}+5 a^{2} b^{3} c^{5} d^{3}-8 a \,b^{4} c^{7} d \right ) \textit {\_R} +32 a^{5} c \,d^{8}+50 a^{4} b \,c^{3} d^{6}-20 a^{2} b^{3} c^{7} d^{2}\right )}{4 b^{2} d^{2}}\)	\(500\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 409 vs. \(2 (173) = 346\).

Time = 1.39 (sec) , antiderivative size = 842, normalized size of antiderivative = 4.46 \[ \int \frac {x^6}{(c+d x) \left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.29 \[ \int \frac {x^6}{(c+d x) \left (a+b x^2\right )^2} \, dx=\frac {c^{6} \log \left (d x + c\right )}{b^{2} c^{4} d^{3} + 2 \, a b c^{2} d^{5} + a^{2} d^{7}} - \frac {{\left (3 \, a^{2} b c^{2} d + 2 \, a^{3} d^{3}\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (b^{5} c^{4} + 2 \, a b^{4} c^{2} d^{2} + a^{2} b^{3} d^{4}\right )}} + \frac {{\left (5 \, a^{2} b c^{3} + 3 \, a^{3} c d^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (b^{4} c^{4} + 2 \, a b^{3} c^{2} d^{2} + a^{2} b^{2} d^{4}\right )} \sqrt {a b}} - \frac {a^{2} b c x + a^{3} d}{2 \, {\left (a b^{4} c^{2} + a^{2} b^{3} d^{2} + {\left (b^{5} c^{2} + a b^{4} d^{2}\right )} x^{2}\right )}} + \frac {d x^{2} - 2 \, c x}{2 \, b^{2} d^{2}} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.38 \[ \int \frac {x^6}{(c+d x) \left (a+b x^2\right )^2} \, dx=\frac {c^{6} \log \left ({\left | d x + c \right |}\right )}{b^{2} c^{4} d^{3} + 2 \, a b c^{2} d^{5} + a^{2} d^{7}} - \frac {{\left (3 \, a^{2} b c^{2} d + 2 \, a^{3} d^{3}\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (b^{5} c^{4} + 2 \, a b^{4} c^{2} d^{2} + a^{2} b^{3} d^{4}\right )}} + \frac {{\left (5 \, a^{2} b c^{3} + 3 \, a^{3} c d^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (b^{4} c^{4} + 2 \, a b^{3} c^{2} d^{2} + a^{2} b^{2} d^{4}\right )} \sqrt {a b}} - \frac {a^{3} b c^{2} d + a^{4} d^{3} + {\left (a^{2} b^{2} c^{3} + a^{3} b c d^{2}\right )} x}{2 \, {\left (b c^{2} + a d^{2}\right )}^{2} {\left (b x^{2} + a\right )} b^{3}} + \frac {b^{2} d x^{2} - 2 \, b^{2} c x}{2 \, b^{4} d^{2}} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 8.22 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.81 \[ \int \frac {x^6}{(c+d x) \left (a+b x^2\right )^2} \, dx=\frac {x^2}{2\,b^2\,d}-\frac {\ln \left (\sqrt {-a^3\,b^7}+a\,b^4\,x\right )\,\left (4\,a^3\,b^3\,d^3-5\,b\,c^3\,\sqrt {-a^3\,b^7}+6\,a^2\,b^4\,c^2\,d-3\,a\,c\,d^2\,\sqrt {-a^3\,b^7}\right )}{4\,\left (a^2\,b^6\,d^4+2\,a\,b^7\,c^2\,d^2+b^8\,c^4\right )}-\frac {\ln \left (\sqrt {-a^3\,b^7}-a\,b^4\,x\right )\,\left (4\,a^3\,b^3\,d^3+5\,b\,c^3\,\sqrt {-a^3\,b^7}+6\,a^2\,b^4\,c^2\,d+3\,a\,c\,d^2\,\sqrt {-a^3\,b^7}\right )}{4\,\left (a^2\,b^6\,d^4+2\,a\,b^7\,c^2\,d^2+b^8\,c^4\right )}-\frac {\frac {a^3\,d^3}{2\,b\,\left (b\,c^2+a\,d^2\right )}+\frac {a^2\,c\,d^2\,x}{2\,\left (b\,c^2+a\,d^2\right )}}{b^3\,d^2\,x^2+a\,b^2\,d^2}+\frac {c^6\,\ln \left (c+d\,x\right )}{d^3\,{\left (b\,c^2+a\,d^2\right )}^2}-\frac {c\,x}{b^2\,d^2} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 464, normalized size of antiderivative = 2.46 \[ \int \frac {x^6}{(c+d x) \left (a+b x^2\right )^2} \, dx=\frac {3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{3} c \,d^{5}+5 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b \,c^{3} d^{3}+3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b c \,d^{5} x^{2}+5 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{2} c^{3} d^{3} x^{2}-2 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{4} d^{6}-3 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{3} b \,c^{2} d^{4}-2 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{3} b \,d^{6} x^{2}-3 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{2} b^{2} c^{2} d^{4} x^{2}+2 \,\mathrm {log}\left (d x +c \right ) a \,b^{3} c^{6}+2 \,\mathrm {log}\left (d x +c \right ) b^{4} c^{6} x^{2}-3 a^{3} b c \,d^{5} x +2 a^{3} b \,d^{6} x^{2}-5 a^{2} b^{2} c^{3} d^{3} x +3 a^{2} b^{2} c^{2} d^{4} x^{2}-2 a^{2} b^{2} c \,d^{5} x^{3}+a^{2} b^{2} d^{6} x^{4}-2 a \,b^{3} c^{5} d x +a \,b^{3} c^{4} d^{2} x^{2}-4 a \,b^{3} c^{3} d^{3} x^{3}+2 a \,b^{3} c^{2} d^{4} x^{4}-2 b^{4} c^{5} d \,x^{3}+b^{4} c^{4} d^{2} x^{4}}{2 b^{3} d^{3} \left (a^{2} b \,d^{4} x^{2}+2 a \,b^{2} c^{2} d^{2} x^{2}+b^{3} c^{4} x^{2}+a^{3} d^{4}+2 a^{2} b \,c^{2} d^{2}+a \,b^{2} c^{4}\right )} \] Input:

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

Output:

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