\(\int \frac {x}{(c+d x) (a+b x^2)^3} \, dx\) [247]

Optimal result

Integrand size = 18, antiderivative size = 204 \[ \int \frac {x}{(c+d x) \left (a+b x^2\right )^3} \, dx=-\frac {c-d x}{4 \left (b c^2+a d^2\right ) \left (a+b x^2\right )^2}-\frac {d \left (4 a c d+\left (b c^2-3 a d^2\right ) x\right )}{8 a \left (b c^2+a d^2\right )^2 \left (a+b x^2\right )}-\frac {d \left (b^2 c^4+6 a b c^2 d^2-3 a^2 d^4\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{3/2} \sqrt {b} \left (b c^2+a d^2\right )^3}-\frac {c d^4 \log (c+d x)}{\left (b c^2+a d^2\right )^3}+\frac {c d^4 \log \left (a+b x^2\right )}{2 \left (b c^2+a d^2\right )^3} \] Output:

-1/4*(-d*x+c)/(a*d^2+b*c^2)/(b*x^2+a)^2-1/8*d*(4*a*c*d+(-3*a*d^2+b*c^2)*x) 
/a/(a*d^2+b*c^2)^2/(b*x^2+a)-1/8*d*(-3*a^2*d^4+6*a*b*c^2*d^2+b^2*c^4)*arct 
an(b^(1/2)*x/a^(1/2))/a^(3/2)/b^(1/2)/(a*d^2+b*c^2)^3-c*d^4*ln(d*x+c)/(a*d 
^2+b*c^2)^3+1/2*c*d^4*ln(b*x^2+a)/(a*d^2+b*c^2)^3
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.83 \[ \int \frac {x}{(c+d x) \left (a+b x^2\right )^3} \, dx=\frac {-\frac {2 \left (b c^2+a d^2\right )^2 (c-d x)}{\left (a+b x^2\right )^2}-\frac {d \left (b c^2+a d^2\right ) \left (b c^2 x+a d (4 c-3 d x)\right )}{a \left (a+b x^2\right )}+\frac {d \left (-b^2 c^4-6 a b c^2 d^2+3 a^2 d^4\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b}}-8 c d^4 \log (c+d x)+4 c d^4 \log \left (a+b x^2\right )}{8 \left (b c^2+a d^2\right )^3} \] Input:

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

Output:

((-2*(b*c^2 + a*d^2)^2*(c - d*x))/(a + b*x^2)^2 - (d*(b*c^2 + a*d^2)*(b*c^ 
2*x + a*d*(4*c - 3*d*x)))/(a*(a + b*x^2)) + (d*(-(b^2*c^4) - 6*a*b*c^2*d^2 
 + 3*a^2*d^4)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(3/2)*Sqrt[b]) - 8*c*d^4*Log 
[c + d*x] + 4*c*d^4*Log[a + b*x^2])/(8*(b*c^2 + a*d^2)^3)
 

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.17, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {593, 25, 686, 25, 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.

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

\(\Big \downarrow \) 593

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 686

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 657

\(\displaystyle -\frac {d \left (\frac {\int \left (\frac {8 a c d^4}{\left (b c^2+a d^2\right ) (c+d x)}+\frac {b^2 c^4+6 a b d^2 c^2-8 a b d^3 x c-3 a^2 d^4}{\left (b c^2+a d^2\right ) \left (b x^2+a\right )}\right )dx}{2 a \left (a d^2+b c^2\right )}+\frac {x \left (b c^2-3 a d^2\right )+4 a c d}{2 a \left (a+b x^2\right ) \left (a d^2+b c^2\right )}\right )}{4 \left (a d^2+b c^2\right )}-\frac {c-d x}{4 \left (a+b x^2\right )^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {d \left (\frac {\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-3 a^2 d^4+6 a b c^2 d^2+b^2 c^4\right )}{\sqrt {a} \sqrt {b} \left (a d^2+b c^2\right )}-\frac {4 a c d^3 \log \left (a+b x^2\right )}{a d^2+b c^2}+\frac {8 a c d^3 \log (c+d x)}{a d^2+b c^2}}{2 a \left (a d^2+b c^2\right )}+\frac {x \left (b c^2-3 a d^2\right )+4 a c d}{2 a \left (a+b x^2\right ) \left (a d^2+b c^2\right )}\right )}{4 \left (a d^2+b c^2\right )}-\frac {c-d x}{4 \left (a+b x^2\right )^2 \left (a d^2+b c^2\right )}\)

Input:

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

Output:

-1/4*(c - d*x)/((b*c^2 + a*d^2)*(a + b*x^2)^2) - (d*((4*a*c*d + (b*c^2 - 3 
*a*d^2)*x)/(2*a*(b*c^2 + a*d^2)*(a + b*x^2)) + (((b^2*c^4 + 6*a*b*c^2*d^2 
- 3*a^2*d^4)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*(b*c^2 + a*d^2) 
) + (8*a*c*d^3*Log[c + d*x])/(b*c^2 + a*d^2) - (4*a*c*d^3*Log[a + b*x^2])/ 
(b*c^2 + a*d^2))/(2*a*(b*c^2 + a*d^2))))/(4*(b*c^2 + a*d^2))
 

Defintions of rubi rules used

rule 25

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

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 593

Int[(x_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
 Simp[(c + d*x)^(n + 1)*(c - d*x)*((a + b*x^2)^(p + 1)/(2*(p + 1)*(b*c^2 + 
a*d^2))), x] - Simp[d/(2*(p + 1)*(b*c^2 + a*d^2))   Int[(c + d*x)^n*(a + b* 
x^2)^(p + 1)*(c*n - d*(n + 2*p + 4)*x), x], x] /; FreeQ[{a, b, c, d, n}, x] 
 && LtQ[p, -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 686

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + 
a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[ 
1/(2*a*c*(p + 1)*(c*d^2 + a*e^2))   Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Sim 
p[f*(c^2*d^2*(2*p + 3) + a*c*e^2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f 
+ a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ 
[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.15

method	result	size
default	\(\frac {\frac {\frac {b d \left (3 a^{2} d^{4}+2 b \,c^{2} d^{2} a -b^{2} c^{4}\right ) x^{3}}{8 a}+\left (-\frac {1}{2} a c \,d^{4} b -\frac {1}{2} b^{2} c^{3} d^{2}\right ) x^{2}+\frac {d \left (5 a^{2} d^{4}+6 b \,c^{2} d^{2} a +b^{2} c^{4}\right ) x}{8}-\frac {3 a^{2} c \,d^{4}}{4}-a \,c^{3} d^{2} b -\frac {c^{5} b^{2}}{4}}{\left (b \,x^{2}+a \right )^{2}}+\frac {d \left (4 a c \,d^{3} \ln \left (b \,x^{2}+a \right )+\frac {\left (3 a^{2} d^{4}-6 b \,c^{2} d^{2} a -b^{2} c^{4}\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}}\right )}{8 a}}{\left (a \,d^{2}+b \,c^{2}\right )^{3}}-\frac {c \,d^{4} \ln \left (d x +c \right )}{\left (a \,d^{2}+b \,c^{2}\right )^{3}}\)	\(235\)
risch	\(\frac {\frac {d b \left (3 a \,d^{2}-b \,c^{2}\right ) x^{3}}{8 \left (a^{2} d^{4}+2 b \,c^{2} d^{2} a +b^{2} c^{4}\right ) a}-\frac {b c \,d^{2} x^{2}}{2 \left (a^{2} d^{4}+2 b \,c^{2} d^{2} a +b^{2} c^{4}\right )}+\frac {d \left (5 a \,d^{2}+b \,c^{2}\right ) x}{8 a^{2} d^{4}+16 b \,c^{2} d^{2} a +8 b^{2} c^{4}}-\frac {\left (3 a \,d^{2}+b \,c^{2}\right ) c}{4 \left (a^{2} d^{4}+2 b \,c^{2} d^{2} a +b^{2} c^{4}\right )}}{\left (b \,x^{2}+a \right )^{2}}-\frac {c \,d^{4} \ln \left (d x +c \right )}{a^{3} d^{6}+3 a^{2} b \,c^{2} d^{4}+3 a \,b^{2} c^{4} d^{2}+b^{3} c^{6}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a^{6} b \,d^{6}+3 a^{5} b^{2} c^{2} d^{4}+3 a^{4} b^{3} c^{4} d^{2}+a^{3} b^{4} c^{6}\right ) \textit {\_Z}^{2}-16 a^{3} b c \,d^{4} \textit {\_Z} +9 a \,d^{4}+b \,c^{2} d^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (3 a^{7} b \,d^{10}+11 a^{6} b^{2} c^{2} d^{8}+14 a^{5} b^{3} c^{4} d^{6}+6 a^{4} b^{4} c^{6} d^{4}-a^{3} b^{5} c^{8} d^{2}-a^{2} b^{6} c^{10}\right ) \textit {\_R}^{2}+\left (-18 a^{4} b c \,d^{8}-38 a^{3} b^{2} c^{3} d^{6}-22 a^{2} b^{3} c^{5} d^{4}-2 a \,b^{4} c^{7} d^{2}\right ) \textit {\_R} +18 a^{2} d^{8}-12 a b \,c^{2} d^{6}+2 b^{2} c^{4} d^{4}\right ) x +\left (4 a^{7} b c \,d^{9}+16 a^{6} b^{2} c^{3} d^{7}+24 a^{5} b^{3} c^{5} d^{5}+16 a^{4} b^{4} c^{7} d^{3}+4 a^{3} b^{5} c^{9} d \right ) \textit {\_R}^{2}+\left (-3 a^{5} d^{9}-10 a^{4} b \,c^{2} d^{7}-12 a^{3} b^{2} c^{4} d^{5}-6 a^{2} b^{3} c^{6} d^{3}-a \,b^{4} c^{8} d \right ) \textit {\_R} -30 a^{2} c \,d^{7}+4 a b \,c^{3} d^{5}+2 b^{2} c^{5} d^{3}\right )\right )}{16}\)	\(651\)

Input:

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

Output:

1/(a*d^2+b*c^2)^3*((1/8*b*d*(3*a^2*d^4+2*a*b*c^2*d^2-b^2*c^4)/a*x^3+(-1/2* 
a*c*d^4*b-1/2*b^2*c^3*d^2)*x^2+1/8*d*(5*a^2*d^4+6*a*b*c^2*d^2+b^2*c^4)*x-3 
/4*a^2*c*d^4-a*c^3*d^2*b-1/4*c^5*b^2)/(b*x^2+a)^2+1/8*d/a*(4*a*c*d^3*ln(b* 
x^2+a)+(3*a^2*d^4-6*a*b*c^2*d^2-b^2*c^4)/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2 
))))-c*d^4*ln(d*x+c)/(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. 520 vs. \(2 (189) = 378\).

Time = 1.58 (sec) , antiderivative size = 1062, normalized size of antiderivative = 5.21 \[ \int \frac {x}{(c+d x) \left (a+b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

[-1/16*(4*a^2*b^3*c^5 + 16*a^3*b^2*c^3*d^2 + 12*a^4*b*c*d^4 + 2*(a*b^4*c^4 
*d - 2*a^2*b^3*c^2*d^3 - 3*a^3*b^2*d^5)*x^3 + 8*(a^2*b^3*c^3*d^2 + a^3*b^2 
*c*d^4)*x^2 - (a^2*b^2*c^4*d + 6*a^3*b*c^2*d^3 - 3*a^4*d^5 + (b^4*c^4*d + 
6*a*b^3*c^2*d^3 - 3*a^2*b^2*d^5)*x^4 + 2*(a*b^3*c^4*d + 6*a^2*b^2*c^2*d^3 
- 3*a^3*b*d^5)*x^2)*sqrt(-a*b)*log((b*x^2 - 2*sqrt(-a*b)*x - a)/(b*x^2 + a 
)) - 2*(a^2*b^3*c^4*d + 6*a^3*b^2*c^2*d^3 + 5*a^4*b*d^5)*x - 8*(a^2*b^3*c* 
d^4*x^4 + 2*a^3*b^2*c*d^4*x^2 + a^4*b*c*d^4)*log(b*x^2 + a) + 16*(a^2*b^3* 
c*d^4*x^4 + 2*a^3*b^2*c*d^4*x^2 + a^4*b*c*d^4)*log(d*x + c))/(a^4*b^4*c^6 
+ 3*a^5*b^3*c^4*d^2 + 3*a^6*b^2*c^2*d^4 + a^7*b*d^6 + (a^2*b^6*c^6 + 3*a^3 
*b^5*c^4*d^2 + 3*a^4*b^4*c^2*d^4 + a^5*b^3*d^6)*x^4 + 2*(a^3*b^5*c^6 + 3*a 
^4*b^4*c^4*d^2 + 3*a^5*b^3*c^2*d^4 + a^6*b^2*d^6)*x^2), -1/8*(2*a^2*b^3*c^ 
5 + 8*a^3*b^2*c^3*d^2 + 6*a^4*b*c*d^4 + (a*b^4*c^4*d - 2*a^2*b^3*c^2*d^3 - 
 3*a^3*b^2*d^5)*x^3 + 4*(a^2*b^3*c^3*d^2 + a^3*b^2*c*d^4)*x^2 + (a^2*b^2*c 
^4*d + 6*a^3*b*c^2*d^3 - 3*a^4*d^5 + (b^4*c^4*d + 6*a*b^3*c^2*d^3 - 3*a^2* 
b^2*d^5)*x^4 + 2*(a*b^3*c^4*d + 6*a^2*b^2*c^2*d^3 - 3*a^3*b*d^5)*x^2)*sqrt 
(a*b)*arctan(sqrt(a*b)*x/a) - (a^2*b^3*c^4*d + 6*a^3*b^2*c^2*d^3 + 5*a^4*b 
*d^5)*x - 4*(a^2*b^3*c*d^4*x^4 + 2*a^3*b^2*c*d^4*x^2 + a^4*b*c*d^4)*log(b* 
x^2 + a) + 8*(a^2*b^3*c*d^4*x^4 + 2*a^3*b^2*c*d^4*x^2 + a^4*b*c*d^4)*log(d 
*x + c))/(a^4*b^4*c^6 + 3*a^5*b^3*c^4*d^2 + 3*a^6*b^2*c^2*d^4 + a^7*b*d^6 
+ (a^2*b^6*c^6 + 3*a^3*b^5*c^4*d^2 + 3*a^4*b^4*c^2*d^4 + a^5*b^3*d^6)*x...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.82 \[ \int \frac {x}{(c+d x) \left (a+b x^2\right )^3} \, dx=\frac {c d^{4} \log \left (b x^{2} + a\right )}{2 \, {\left (b^{3} c^{6} + 3 \, a b^{2} c^{4} d^{2} + 3 \, a^{2} b c^{2} d^{4} + a^{3} d^{6}\right )}} - \frac {c d^{4} \log \left (d x + c\right )}{b^{3} c^{6} + 3 \, a b^{2} c^{4} d^{2} + 3 \, a^{2} b c^{2} d^{4} + a^{3} d^{6}} - \frac {{\left (b^{2} c^{4} d + 6 \, a b c^{2} d^{3} - 3 \, a^{2} d^{5}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, {\left (a b^{3} c^{6} + 3 \, a^{2} b^{2} c^{4} d^{2} + 3 \, a^{3} b c^{2} d^{4} + a^{4} d^{6}\right )} \sqrt {a b}} - \frac {4 \, a b c d^{2} x^{2} + 2 \, a b c^{3} + 6 \, a^{2} c d^{2} + {\left (b^{2} c^{2} d - 3 \, a b d^{3}\right )} x^{3} - {\left (a b c^{2} d + 5 \, a^{2} d^{3}\right )} x}{8 \, {\left (a^{3} b^{2} c^{4} + 2 \, a^{4} b c^{2} d^{2} + a^{5} d^{4} + {\left (a b^{4} c^{4} + 2 \, a^{2} b^{3} c^{2} d^{2} + a^{3} b^{2} d^{4}\right )} x^{4} + 2 \, {\left (a^{2} b^{3} c^{4} + 2 \, a^{3} b^{2} c^{2} d^{2} + a^{4} b d^{4}\right )} x^{2}\right )}} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.71 \[ \int \frac {x}{(c+d x) \left (a+b x^2\right )^3} \, dx=-\frac {c d^{5} \log \left ({\left | d x + c \right |}\right )}{b^{3} c^{6} d + 3 \, a b^{2} c^{4} d^{3} + 3 \, a^{2} b c^{2} d^{5} + a^{3} d^{7}} + \frac {c d^{4} \log \left (b x^{2} + a\right )}{2 \, {\left (b^{3} c^{6} + 3 \, a b^{2} c^{4} d^{2} + 3 \, a^{2} b c^{2} d^{4} + a^{3} d^{6}\right )}} - \frac {{\left (b^{2} c^{4} d + 6 \, a b c^{2} d^{3} - 3 \, a^{2} d^{5}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, {\left (a b^{3} c^{6} + 3 \, a^{2} b^{2} c^{4} d^{2} + 3 \, a^{3} b c^{2} d^{4} + a^{4} d^{6}\right )} \sqrt {a b}} - \frac {2 \, a b^{2} c^{5} + 8 \, a^{2} b c^{3} d^{2} + 6 \, a^{3} c d^{4} + {\left (b^{3} c^{4} d - 2 \, a b^{2} c^{2} d^{3} - 3 \, a^{2} b d^{5}\right )} x^{3} + 4 \, {\left (a b^{2} c^{3} d^{2} + a^{2} b c d^{4}\right )} x^{2} - {\left (a b^{2} c^{4} d + 6 \, a^{2} b c^{2} d^{3} + 5 \, a^{3} d^{5}\right )} x}{8 \, {\left (b c^{2} + a d^{2}\right )}^{3} {\left (b x^{2} + a\right )}^{2} a} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 8.17 (sec) , antiderivative size = 901, normalized size of antiderivative = 4.42 \[ \int \frac {x}{(c+d x) \left (a+b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(log(b^5*c^12*(-a^3*b)^(3/2) - 9*a^9*d^12*(-a^3*b)^(1/2) + a^4*b^7*c^12*x 
- 1119*a*c^4*d^8*(-a^3*b)^(5/2) - 612*b*c^6*d^6*(-a^3*b)^(5/2) + 558*a^5*c 
^2*d^10*(-a^3*b)^(3/2) + 9*a^10*b*d^12*x + 55*a^2*b^3*c^8*d^4*(-a^3*b)^(3/ 
2) + 14*a^5*b^6*c^10*d^2*x + 55*a^6*b^5*c^8*d^4*x + 612*a^7*b^4*c^6*d^6*x 
+ 1119*a^8*b^3*c^4*d^8*x + 558*a^9*b^2*c^2*d^10*x + 14*a*b^4*c^10*d^2*(-a^ 
3*b)^(3/2))*(b*((a^3*c*d^4)/2 + (3*a*c^2*d^3*(-a^3*b)^(1/2))/8) - (3*a^2*d 
^5*(-a^3*b)^(1/2))/16 + (b^2*c^4*d*(-a^3*b)^(1/2))/16))/(a^6*b*d^6 + a^3*b 
^4*c^6 + 3*a^4*b^3*c^4*d^2 + 3*a^5*b^2*c^2*d^4) - ((b*c^3 + 3*a*c*d^2)/(4* 
(a^2*d^4 + b^2*c^4 + 2*a*b*c^2*d^2)) + (x^3*(b^2*c^2*d - 3*a*b*d^3))/(8*a* 
(a^2*d^4 + b^2*c^4 + 2*a*b*c^2*d^2)) - (d*x*(5*a*d^2 + b*c^2))/(8*(a^2*d^4 
 + b^2*c^4 + 2*a*b*c^2*d^2)) + (b*c*d^2*x^2)/(2*(a^2*d^4 + b^2*c^4 + 2*a*b 
*c^2*d^2)))/(a^2 + b^2*x^4 + 2*a*b*x^2) + (log(9*a^9*d^12*(-a^3*b)^(1/2) - 
 b^5*c^12*(-a^3*b)^(3/2) + a^4*b^7*c^12*x + 1119*a*c^4*d^8*(-a^3*b)^(5/2) 
+ 612*b*c^6*d^6*(-a^3*b)^(5/2) - 558*a^5*c^2*d^10*(-a^3*b)^(3/2) + 9*a^10* 
b*d^12*x - 55*a^2*b^3*c^8*d^4*(-a^3*b)^(3/2) + 14*a^5*b^6*c^10*d^2*x + 55* 
a^6*b^5*c^8*d^4*x + 612*a^7*b^4*c^6*d^6*x + 1119*a^8*b^3*c^4*d^8*x + 558*a 
^9*b^2*c^2*d^10*x - 14*a*b^4*c^10*d^2*(-a^3*b)^(3/2))*(b*((a^3*c*d^4)/2 - 
(3*a*c^2*d^3*(-a^3*b)^(1/2))/8) + (3*a^2*d^5*(-a^3*b)^(1/2))/16 - (b^2*c^4 
*d*(-a^3*b)^(1/2))/16))/(a^6*b*d^6 + a^3*b^4*c^6 + 3*a^4*b^3*c^4*d^2 + 3*a 
^5*b^2*c^2*d^4) - (c*d^4*log(c + d*x))/(a*d^2 + b*c^2)^3
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 695, normalized size of antiderivative = 3.41 \[ \int \frac {x}{(c+d x) \left (a+b x^2\right )^3} \, dx=\frac {-2 a^{2} b^{3} c^{5}-4 a^{4} b c \,d^{4}-12 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b^{2} c^{2} d^{3} x^{2}-2 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{3} c^{4} d \,x^{2}-6 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{3} c^{2} d^{3} x^{4}-6 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{3} b \,c^{2} d^{3}+6 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{3} b \,d^{5} x^{2}-\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b^{2} c^{4} d +3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b^{2} d^{5} x^{4}-\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b^{4} c^{4} d \,x^{4}+3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{4} d^{5}+6 a^{3} b^{2} c^{2} d^{3} x +a^{2} b^{3} c^{4} d x +2 a^{2} b^{3} c^{2} d^{3} x^{3}+2 a^{2} b^{3} c \,d^{4} x^{4}-a \,b^{4} c^{4} d \,x^{3}+2 a \,b^{4} c^{3} d^{2} x^{4}+4 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{4} b c \,d^{4}-8 \,\mathrm {log}\left (d x +c \right ) a^{4} b c \,d^{4}+5 a^{4} b \,d^{5} x -6 a^{3} b^{2} c^{3} d^{2}+3 a^{3} b^{2} d^{5} x^{3}+8 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{3} b^{2} c \,d^{4} x^{2}+4 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{2} b^{3} c \,d^{4} x^{4}-16 \,\mathrm {log}\left (d x +c \right ) a^{3} b^{2} c \,d^{4} x^{2}-8 \,\mathrm {log}\left (d x +c \right ) a^{2} b^{3} c \,d^{4} x^{4}}{8 a^{2} b \left (a^{3} b^{2} d^{6} x^{4}+3 a^{2} b^{3} c^{2} d^{4} x^{4}+3 a \,b^{4} c^{4} d^{2} x^{4}+b^{5} c^{6} x^{4}+2 a^{4} b \,d^{6} x^{2}+6 a^{3} b^{2} c^{2} d^{4} x^{2}+6 a^{2} b^{3} c^{4} d^{2} x^{2}+2 a \,b^{4} c^{6} x^{2}+a^{5} d^{6}+3 a^{4} b \,c^{2} d^{4}+3 a^{3} b^{2} c^{4} d^{2}+a^{2} b^{3} c^{6}\right )} \] Input:

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

Output:

(3*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**4*d**5 - 6*sqrt(b)*sqr 
t(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3*b*c**2*d**3 + 6*sqrt(b)*sqrt(a)*at 
an((b*x)/(sqrt(b)*sqrt(a)))*a**3*b*d**5*x**2 - sqrt(b)*sqrt(a)*atan((b*x)/ 
(sqrt(b)*sqrt(a)))*a**2*b**2*c**4*d - 12*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt( 
b)*sqrt(a)))*a**2*b**2*c**2*d**3*x**2 + 3*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt 
(b)*sqrt(a)))*a**2*b**2*d**5*x**4 - 2*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)* 
sqrt(a)))*a*b**3*c**4*d*x**2 - 6*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt( 
a)))*a*b**3*c**2*d**3*x**4 - sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a))) 
*b**4*c**4*d*x**4 + 4*log(a + b*x**2)*a**4*b*c*d**4 + 8*log(a + b*x**2)*a* 
*3*b**2*c*d**4*x**2 + 4*log(a + b*x**2)*a**2*b**3*c*d**4*x**4 - 8*log(c + 
d*x)*a**4*b*c*d**4 - 16*log(c + d*x)*a**3*b**2*c*d**4*x**2 - 8*log(c + d*x 
)*a**2*b**3*c*d**4*x**4 - 4*a**4*b*c*d**4 + 5*a**4*b*d**5*x - 6*a**3*b**2* 
c**3*d**2 + 6*a**3*b**2*c**2*d**3*x + 3*a**3*b**2*d**5*x**3 - 2*a**2*b**3* 
c**5 + a**2*b**3*c**4*d*x + 2*a**2*b**3*c**2*d**3*x**3 + 2*a**2*b**3*c*d** 
4*x**4 - a*b**4*c**4*d*x**3 + 2*a*b**4*c**3*d**2*x**4)/(8*a**2*b*(a**5*d** 
6 + 3*a**4*b*c**2*d**4 + 2*a**4*b*d**6*x**2 + 3*a**3*b**2*c**4*d**2 + 6*a* 
*3*b**2*c**2*d**4*x**2 + a**3*b**2*d**6*x**4 + a**2*b**3*c**6 + 6*a**2*b** 
3*c**4*d**2*x**2 + 3*a**2*b**3*c**2*d**4*x**4 + 2*a*b**4*c**6*x**2 + 3*a*b 
**4*c**4*d**2*x**4 + b**5*c**6*x**4))