\(\int \frac {x^5}{(c+d x)^3 (a+b x^2)} \, dx\) [158]

Optimal result

Integrand size = 20, antiderivative size = 229 \[ \int \frac {x^5}{(c+d x)^3 \left (a+b x^2\right )} \, dx=\frac {x}{b d^3}+\frac {c^5}{2 d^4 \left (b c^2+a d^2\right ) (c+d x)^2}-\frac {c^4 \left (3 b c^2+5 a d^2\right )}{d^4 \left (b c^2+a d^2\right )^2 (c+d x)}+\frac {a^{5/2} d \left (3 b c^2-a d^2\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{3/2} \left (b c^2+a d^2\right )^3}-\frac {c^3 \left (3 b^2 c^4+9 a b c^2 d^2+10 a^2 d^4\right ) \log (c+d x)}{d^4 \left (b c^2+a d^2\right )^3}+\frac {a^2 c \left (b c^2-3 a d^2\right ) \log \left (a+b x^2\right )}{2 b \left (b c^2+a d^2\right )^3} \] Output:

x/b/d^3+1/2*c^5/d^4/(a*d^2+b*c^2)/(d*x+c)^2-c^4*(5*a*d^2+3*b*c^2)/d^4/(a*d 
^2+b*c^2)^2/(d*x+c)+a^(5/2)*d*(-a*d^2+3*b*c^2)*arctan(b^(1/2)*x/a^(1/2))/b 
^(3/2)/(a*d^2+b*c^2)^3-c^3*(10*a^2*d^4+9*a*b*c^2*d^2+3*b^2*c^4)*ln(d*x+c)/ 
d^4/(a*d^2+b*c^2)^3+1/2*a^2*c*(-3*a*d^2+b*c^2)*ln(b*x^2+a)/b/(a*d^2+b*c^2) 
^3
 

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.00 \[ \int \frac {x^5}{(c+d x)^3 \left (a+b x^2\right )} \, dx=\frac {1}{2} \left (\frac {2 x}{b d^3}+\frac {c^5}{d^4 \left (b c^2+a d^2\right ) (c+d x)^2}-\frac {2 c^4 \left (3 b c^2+5 a d^2\right )}{d^4 \left (b c^2+a d^2\right )^2 (c+d x)}-\frac {2 a^{5/2} d \left (-3 b c^2+a d^2\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{3/2} \left (b c^2+a d^2\right )^3}-\frac {2 \left (3 b^2 c^7+9 a b c^5 d^2+10 a^2 c^3 d^4\right ) \log (c+d x)}{d^4 \left (b c^2+a d^2\right )^3}+\frac {a^2 c \left (b c^2-3 a d^2\right ) \log \left (a+b x^2\right )}{b \left (b c^2+a d^2\right )^3}\right ) \] Input:

Integrate[x^5/((c + d*x)^3*(a + b*x^2)),x]
 

Output:

((2*x)/(b*d^3) + c^5/(d^4*(b*c^2 + a*d^2)*(c + d*x)^2) - (2*c^4*(3*b*c^2 + 
 5*a*d^2))/(d^4*(b*c^2 + a*d^2)^2*(c + d*x)) - (2*a^(5/2)*d*(-3*b*c^2 + a* 
d^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(b^(3/2)*(b*c^2 + a*d^2)^3) - (2*(3*b^2* 
c^7 + 9*a*b*c^5*d^2 + 10*a^2*c^3*d^4)*Log[c + d*x])/(d^4*(b*c^2 + a*d^2)^3 
) + (a^2*c*(b*c^2 - 3*a*d^2)*Log[a + b*x^2])/(b*(b*c^2 + a*d^2)^3))/2
 

Rubi [A] (verified)

Time = 1.06 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.11, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {603, 27, 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.

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

\(\Big \downarrow \) 603

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2160

\(\displaystyle \frac {\int \left (\frac {\left (3 b c^2+5 a d^2\right ) c^4}{d^3 \left (b c^2+a d^2\right ) (c+d x)^2}-\frac {\left (3 b^2 c^4+9 a b d^2 c^2+10 a^2 d^4\right ) c^3}{d^3 \left (b c^2+a d^2\right )^2 (c+d x)}+\frac {b c^2+a d^2}{b d^3}+\frac {a^2 \left (a d \left (3 b c^2-a d^2\right )+b c \left (b c^2-3 a d^2\right ) x\right )}{b \left (b c^2+a d^2\right )^2 \left (b x^2+a\right )}\right )dx}{a d^2+b c^2}+\frac {c^5}{2 d^4 (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {a^{5/2} d \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (3 b c^2-a d^2\right )}{b^{3/2} \left (a d^2+b c^2\right )^2}-\frac {c^3 \left (10 a^2 d^4+9 a b c^2 d^2+3 b^2 c^4\right ) \log (c+d x)}{d^4 \left (a d^2+b c^2\right )^2}+\frac {a^2 c \left (b c^2-3 a d^2\right ) \log \left (a+b x^2\right )}{2 b \left (a d^2+b c^2\right )^2}+\frac {x \left (a d^2+b c^2\right )}{b d^3}-\frac {c^4 \left (5 a d^2+3 b c^2\right )}{d^4 (c+d x) \left (a d^2+b c^2\right )}}{a d^2+b c^2}+\frac {c^5}{2 d^4 (c+d x)^2 \left (a d^2+b c^2\right )}\)

Input:

Int[x^5/((c + d*x)^3*(a + b*x^2)),x]
 

Output:

c^5/(2*d^4*(b*c^2 + a*d^2)*(c + d*x)^2) + (((b*c^2 + a*d^2)*x)/(b*d^3) - ( 
c^4*(3*b*c^2 + 5*a*d^2))/(d^4*(b*c^2 + a*d^2)*(c + d*x)) + (a^(5/2)*d*(3*b 
*c^2 - a*d^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(b^(3/2)*(b*c^2 + a*d^2)^2) - ( 
c^3*(3*b^2*c^4 + 9*a*b*c^2*d^2 + 10*a^2*d^4)*Log[c + d*x])/(d^4*(b*c^2 + a 
*d^2)^2) + (a^2*c*(b*c^2 - 3*a*d^2)*Log[a + b*x^2])/(2*b*(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 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 = 216, normalized size of antiderivative = 0.94

method	result	size
default	\(\frac {x}{b \,d^{3}}-\frac {a^{2} \left (\frac {\left (3 a b c \,d^{2}-c^{3} b^{2}\right ) \ln \left (b \,x^{2}+a \right )}{2 b}+\frac {\left (a^{2} d^{3}-3 a b \,c^{2} d \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}}\right )}{\left (a \,d^{2}+b \,c^{2}\right )^{3} b}-\frac {c^{4} \left (5 a \,d^{2}+3 b \,c^{2}\right )}{d^{4} \left (a \,d^{2}+b \,c^{2}\right )^{2} \left (d x +c \right )}+\frac {c^{5}}{2 d^{4} \left (a \,d^{2}+b \,c^{2}\right ) \left (d x +c \right )^{2}}-\frac {c^{3} \left (10 a^{2} d^{4}+9 b \,c^{2} d^{2} a +3 b^{2} c^{4}\right ) \ln \left (d x +c \right )}{d^{4} \left (a \,d^{2}+b \,c^{2}\right )^{3}}\)	\(216\)
risch	\(\frac {x}{b \,d^{3}}+\frac {-\frac {b \,c^{4} \left (5 a \,d^{2}+3 b \,c^{2}\right ) x}{a^{2} d^{4}+2 b \,c^{2} d^{2} a +b^{2} c^{4}}-\frac {b \,c^{5} \left (9 a \,d^{2}+5 b \,c^{2}\right )}{2 d \left (a^{2} d^{4}+2 b \,c^{2} d^{2} a +b^{2} c^{4}\right )}}{b \,d^{3} \left (d x +c \right )^{2}}-\frac {10 c^{3} \ln \left (d x +c \right ) a^{2}}{a^{3} d^{6}+3 a^{2} b \,c^{2} d^{4}+3 a \,b^{2} c^{4} d^{2}+b^{3} c^{6}}-\frac {9 c^{5} \ln \left (d x +c \right ) b a}{d^{2} \left (a^{3} d^{6}+3 a^{2} b \,c^{2} d^{4}+3 a \,b^{2} c^{4} d^{2}+b^{3} c^{6}\right )}-\frac {3 c^{7} \ln \left (d x +c \right ) b^{2}}{d^{4} \left (a^{3} d^{6}+3 a^{2} b \,c^{2} d^{4}+3 a \,b^{2} c^{4} d^{2}+b^{3} c^{6}\right )}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a^{3} b \,d^{6}+3 a^{2} b^{2} c^{2} d^{4}+3 a \,b^{3} c^{4} d^{2}+b^{4} c^{6}\right ) \textit {\_Z}^{2}+\left (6 a^{3} b c \,d^{5}-2 a^{2} b^{2} c^{3} d^{3}\right ) \textit {\_Z} +d^{6} a^{4}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (3 a^{5} b \,d^{11}+11 a^{4} b^{2} c^{2} d^{9}+14 a^{3} b^{3} c^{4} d^{7}+6 a^{2} b^{4} c^{6} d^{5}-a \,b^{5} c^{8} d^{3}-b^{6} c^{10} d \right ) \textit {\_R}^{2}+\left (13 a^{5} b c \,d^{10}+31 a^{4} b^{2} c^{3} d^{8}+17 a^{3} b^{3} c^{5} d^{6}-13 a^{2} b^{4} c^{7} d^{4}-18 a \,b^{5} c^{9} d^{2}-6 b^{6} c^{11}\right ) \textit {\_R} +2 a^{6} d^{11}+2 a^{5} b \,c^{2} d^{9}+20 a^{4} b^{2} c^{4} d^{7}+18 a^{3} b^{3} c^{6} d^{5}+6 a^{2} b^{4} c^{8} d^{3}\right ) x +\left (4 a^{5} b c \,d^{10}+16 a^{4} b^{2} c^{3} d^{8}+24 a^{3} b^{3} c^{5} d^{6}+16 a^{2} b^{4} c^{7} d^{4}+4 a \,b^{5} c^{9} d^{2}\right ) \textit {\_R}^{2}+\left (a^{6} d^{11}+15 a^{5} b \,c^{2} d^{9}+51 a^{4} b^{2} c^{4} d^{7}+73 a^{3} b^{3} c^{6} d^{5}+48 a^{2} b^{4} c^{8} d^{3}+12 a \,b^{5} c^{10} d \right ) \textit {\_R} +2 a^{6} c \,d^{10}+22 a^{5} b \,c^{3} d^{8}+18 a^{4} b^{2} c^{5} d^{6}+6 a^{3} b^{3} c^{7} d^{4}\right )}{2 b \,d^{3}}\)	\(797\)

Input:

int(x^5/(d*x+c)^3/(b*x^2+a),x,method=_RETURNVERBOSE)
 

Output:

x/b/d^3-a^2/(a*d^2+b*c^2)^3/b*(1/2*(3*a*b*c*d^2-b^2*c^3)/b*ln(b*x^2+a)+(a^ 
2*d^3-3*a*b*c^2*d)/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2)))-c^4*(5*a*d^2+3*b*c 
^2)/d^4/(a*d^2+b*c^2)^2/(d*x+c)+1/2*c^5/d^4/(a*d^2+b*c^2)/(d*x+c)^2-c^3*(1 
0*a^2*d^4+9*a*b*c^2*d^2+3*b^2*c^4)*ln(d*x+c)/d^4/(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. 625 vs. \(2 (217) = 434\).

Time = 2.80 (sec) , antiderivative size = 1269, normalized size of antiderivative = 5.54 \[ \int \frac {x^5}{(c+d x)^3 \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:

integrate(x^5/(d*x+c)^3/(b*x^2+a),x, algorithm="fricas")
 

Output:

[-1/2*(5*b^3*c^9 + 14*a*b^2*c^7*d^2 + 9*a^2*b*c^5*d^4 - 2*(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^3 - 4*(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^2 + (3*a^2*b*c^4*d^5 - a^3*c^2*d 
^7 + (3*a^2*b*c^2*d^7 - a^3*d^9)*x^2 + 2*(3*a^2*b*c^3*d^6 - a^3*c*d^8)*x)* 
sqrt(-a/b)*log((b*x^2 - 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) + 2*(2*b^3*c^8* 
d + 5*a*b^2*c^6*d^3 + 2*a^2*b*c^4*d^5 - a^3*c^2*d^7)*x - (a^2*b*c^5*d^4 - 
3*a^3*c^3*d^6 + (a^2*b*c^3*d^6 - 3*a^3*c*d^8)*x^2 + 2*(a^2*b*c^4*d^5 - 3*a 
^3*c^2*d^7)*x)*log(b*x^2 + a) + 2*(3*b^3*c^9 + 9*a*b^2*c^7*d^2 + 10*a^2*b* 
c^5*d^4 + (3*b^3*c^7*d^2 + 9*a*b^2*c^5*d^4 + 10*a^2*b*c^3*d^6)*x^2 + 2*(3* 
b^3*c^8*d + 9*a*b^2*c^6*d^3 + 10*a^2*b*c^4*d^5)*x)*log(d*x + c))/(b^4*c^8* 
d^4 + 3*a*b^3*c^6*d^6 + 3*a^2*b^2*c^4*d^8 + a^3*b*c^2*d^10 + (b^4*c^6*d^6 
+ 3*a*b^3*c^4*d^8 + 3*a^2*b^2*c^2*d^10 + a^3*b*d^12)*x^2 + 2*(b^4*c^7*d^5 
+ 3*a*b^3*c^5*d^7 + 3*a^2*b^2*c^3*d^9 + a^3*b*c*d^11)*x), -1/2*(5*b^3*c^9 
+ 14*a*b^2*c^7*d^2 + 9*a^2*b*c^5*d^4 - 2*(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^3 - 4*(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^2 - 2*(3*a^2*b*c^4*d^5 - a^3*c^2*d^7 + (3*a^2*b*c 
^2*d^7 - a^3*d^9)*x^2 + 2*(3*a^2*b*c^3*d^6 - a^3*c*d^8)*x)*sqrt(a/b)*arcta 
n(b*x*sqrt(a/b)/a) + 2*(2*b^3*c^8*d + 5*a*b^2*c^6*d^3 + 2*a^2*b*c^4*d^5 - 
a^3*c^2*d^7)*x - (a^2*b*c^5*d^4 - 3*a^3*c^3*d^6 + (a^2*b*c^3*d^6 - 3*a^3*c 
*d^8)*x^2 + 2*(a^2*b*c^4*d^5 - 3*a^3*c^2*d^7)*x)*log(b*x^2 + a) + 2*(3*...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^5}{(c+d x)^3 \left (a+b x^2\right )} \, dx=\text {Timed out} \] Input:

integrate(x**5/(d*x+c)**3/(b*x**2+a),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.64 \[ \int \frac {x^5}{(c+d x)^3 \left (a+b x^2\right )} \, dx=\frac {{\left (a^{2} b c^{3} - 3 \, a^{3} c d^{2}\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (b^{4} c^{6} + 3 \, a b^{3} c^{4} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{4} + a^{3} b d^{6}\right )}} - \frac {{\left (3 \, b^{2} c^{7} + 9 \, a b c^{5} d^{2} + 10 \, a^{2} c^{3} d^{4}\right )} \log \left (d x + c\right )}{b^{3} c^{6} d^{4} + 3 \, a b^{2} c^{4} d^{6} + 3 \, a^{2} b c^{2} d^{8} + a^{3} d^{10}} + \frac {{\left (3 \, a^{3} b c^{2} d - a^{4} d^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{{\left (b^{4} c^{6} + 3 \, a b^{3} c^{4} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{4} + a^{3} b d^{6}\right )} \sqrt {a b}} - \frac {5 \, b c^{7} + 9 \, a c^{5} d^{2} + 2 \, {\left (3 \, b c^{6} d + 5 \, a c^{4} d^{3}\right )} x}{2 \, {\left (b^{2} c^{6} d^{4} + 2 \, a b c^{4} d^{6} + a^{2} c^{2} d^{8} + {\left (b^{2} c^{4} d^{6} + 2 \, a b c^{2} d^{8} + a^{2} d^{10}\right )} x^{2} + 2 \, {\left (b^{2} c^{5} d^{5} + 2 \, a b c^{3} d^{7} + a^{2} c d^{9}\right )} x\right )}} + \frac {x}{b d^{3}} \] Input:

integrate(x^5/(d*x+c)^3/(b*x^2+a),x, algorithm="maxima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.45 \[ \int \frac {x^5}{(c+d x)^3 \left (a+b x^2\right )} \, dx=\frac {{\left (a^{2} b c^{3} - 3 \, a^{3} c d^{2}\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (b^{4} c^{6} + 3 \, a b^{3} c^{4} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{4} + a^{3} b d^{6}\right )}} - \frac {{\left (3 \, b^{2} c^{7} + 9 \, a b c^{5} d^{2} + 10 \, a^{2} c^{3} d^{4}\right )} \log \left ({\left | d x + c \right |}\right )}{b^{3} c^{6} d^{4} + 3 \, a b^{2} c^{4} d^{6} + 3 \, a^{2} b c^{2} d^{8} + a^{3} d^{10}} + \frac {{\left (3 \, a^{3} b c^{2} d - a^{4} d^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{{\left (b^{4} c^{6} + 3 \, a b^{3} c^{4} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{4} + a^{3} b d^{6}\right )} \sqrt {a b}} + \frac {x}{b d^{3}} - \frac {5 \, b^{2} c^{9} + 14 \, a b c^{7} d^{2} + 9 \, a^{2} c^{5} d^{4} + 2 \, {\left (3 \, b^{2} c^{8} d + 8 \, a b c^{6} d^{3} + 5 \, a^{2} c^{4} d^{5}\right )} x}{2 \, {\left (b c^{2} + a d^{2}\right )}^{3} {\left (d x + c\right )}^{2} d^{4}} \] Input:

integrate(x^5/(d*x+c)^3/(b*x^2+a),x, algorithm="giac")
 

Output:

1/2*(a^2*b*c^3 - 3*a^3*c*d^2)*log(b*x^2 + a)/(b^4*c^6 + 3*a*b^3*c^4*d^2 + 
3*a^2*b^2*c^2*d^4 + a^3*b*d^6) - (3*b^2*c^7 + 9*a*b*c^5*d^2 + 10*a^2*c^3*d 
^4)*log(abs(d*x + c))/(b^3*c^6*d^4 + 3*a*b^2*c^4*d^6 + 3*a^2*b*c^2*d^8 + a 
^3*d^10) + (3*a^3*b*c^2*d - a^4*d^3)*arctan(b*x/sqrt(a*b))/((b^4*c^6 + 3*a 
*b^3*c^4*d^2 + 3*a^2*b^2*c^2*d^4 + a^3*b*d^6)*sqrt(a*b)) + x/(b*d^3) - 1/2 
*(5*b^2*c^9 + 14*a*b*c^7*d^2 + 9*a^2*c^5*d^4 + 2*(3*b^2*c^8*d + 8*a*b*c^6* 
d^3 + 5*a^2*c^4*d^5)*x)/((b*c^2 + a*d^2)^3*(d*x + c)^2*d^4)
 

Mupad [B] (verification not implemented)

Time = 8.39 (sec) , antiderivative size = 470, normalized size of antiderivative = 2.05 \[ \int \frac {x^5}{(c+d x)^3 \left (a+b x^2\right )} \, dx=\frac {x}{b\,d^3}-\frac {\frac {5\,b^2\,c^7+9\,a\,b\,c^5\,d^2}{2\,d\,\left (a^2\,d^4+2\,a\,b\,c^2\,d^2+b^2\,c^4\right )}+\frac {x\,\left (3\,b^2\,c^6+5\,a\,b\,c^4\,d^2\right )}{a^2\,d^4+2\,a\,b\,c^2\,d^2+b^2\,c^4}}{b\,c^2\,d^3+2\,b\,c\,d^4\,x+b\,d^5\,x^2}+\frac {\ln \left (\sqrt {-a^5\,b^3}+a^2\,b^2\,x\right )\,\left (a^2\,b^3\,c^3-a\,d^3\,\sqrt {-a^5\,b^3}-3\,a^3\,b^2\,c\,d^2+3\,b\,c^2\,d\,\sqrt {-a^5\,b^3}\right )}{2\,\left (a^3\,b^3\,d^6+3\,a^2\,b^4\,c^2\,d^4+3\,a\,b^5\,c^4\,d^2+b^6\,c^6\right )}+\frac {\ln \left (\sqrt {-a^5\,b^3}-a^2\,b^2\,x\right )\,\left (a^2\,b^3\,c^3+a\,d^3\,\sqrt {-a^5\,b^3}-3\,a^3\,b^2\,c\,d^2-3\,b\,c^2\,d\,\sqrt {-a^5\,b^3}\right )}{2\,\left (a^3\,b^3\,d^6+3\,a^2\,b^4\,c^2\,d^4+3\,a\,b^5\,c^4\,d^2+b^6\,c^6\right )}-\frac {\ln \left (c+d\,x\right )\,\left (10\,a^2\,c^3\,d^4+9\,a\,b\,c^5\,d^2+3\,b^2\,c^7\right )}{a^3\,d^{10}+3\,a^2\,b\,c^2\,d^8+3\,a\,b^2\,c^4\,d^6+b^3\,c^6\,d^4} \] Input:

int(x^5/((a + b*x^2)*(c + d*x)^3),x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 785, normalized size of antiderivative = 3.43 \[ \int \frac {x^5}{(c+d x)^3 \left (a+b x^2\right )} \, dx =\text {Too large to display} \] Input:

int(x^5/(d*x+c)^3/(b*x^2+a),x)
 

Output:

( - 2*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3*c**2*d**7 - 4*sqr 
t(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3*c*d**8*x - 2*sqrt(b)*sqrt( 
a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3*d**9*x**2 + 6*sqrt(b)*sqrt(a)*atan(( 
b*x)/(sqrt(b)*sqrt(a)))*a**2*b*c**4*d**5 + 12*sqrt(b)*sqrt(a)*atan((b*x)/( 
sqrt(b)*sqrt(a)))*a**2*b*c**3*d**6*x + 6*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt( 
b)*sqrt(a)))*a**2*b*c**2*d**7*x**2 - 3*log(a + b*x**2)*a**3*b*c**3*d**6 - 
6*log(a + b*x**2)*a**3*b*c**2*d**7*x - 3*log(a + b*x**2)*a**3*b*c*d**8*x** 
2 + log(a + b*x**2)*a**2*b**2*c**5*d**4 + 2*log(a + b*x**2)*a**2*b**2*c**4 
*d**5*x + log(a + b*x**2)*a**2*b**2*c**3*d**6*x**2 - 20*log(c + d*x)*a**2* 
b**2*c**5*d**4 - 40*log(c + d*x)*a**2*b**2*c**4*d**5*x - 20*log(c + d*x)*a 
**2*b**2*c**3*d**6*x**2 - 18*log(c + d*x)*a*b**3*c**7*d**2 - 36*log(c + d* 
x)*a*b**3*c**6*d**3*x - 18*log(c + d*x)*a*b**3*c**5*d**4*x**2 - 6*log(c + 
d*x)*b**4*c**9 - 12*log(c + d*x)*b**4*c**8*d*x - 6*log(c + d*x)*b**4*c**7* 
d**2*x**2 - a**3*b*c**3*d**6 + 3*a**3*b*c*d**8*x**2 + 2*a**3*b*d**9*x**3 - 
 7*a**2*b**2*c**5*d**4 + 14*a**2*b**2*c**3*d**6*x**2 + 6*a**2*b**2*c**2*d* 
*7*x**3 - 9*a*b**3*c**7*d**2 + 17*a*b**3*c**5*d**4*x**2 + 6*a*b**3*c**4*d* 
*5*x**3 - 3*b**4*c**9 + 6*b**4*c**7*d**2*x**2 + 2*b**4*c**6*d**3*x**3)/(2* 
b**2*d**4*(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*...