\(\int \frac {x^3}{(a+\frac {c}{x^2}+\frac {b}{x}) (d+e x)} \, dx\) [61]

Optimal result

Integrand size = 25, antiderivative size = 280 \[ \int \frac {x^3}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) (d+e x)} \, dx=\frac {\left (a^2 d^2+b^2 e^2+a e (b d-c e)\right ) x}{a^3 e^3}-\frac {(a d+b e) x^2}{2 a^2 e^2}+\frac {x^3}{3 a e}+\frac {\left (b^5 d-5 a b^3 c d+5 a^2 b c^2 d-b^4 c e+4 a b^2 c^2 e-2 a^2 c^3 e\right ) \text {arctanh}\left (\frac {b+2 a x}{\sqrt {b^2-4 a c}}\right )}{a^4 \sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )}-\frac {d^5 \log (d+e x)}{e^4 \left (a d^2-e (b d-c e)\right )}+\frac {\left (b^4 d-3 a b^2 c d+a^2 c^2 d-b^3 c e+2 a b c^2 e\right ) \log \left (c+b x+a x^2\right )}{2 a^4 \left (a d^2-e (b d-c e)\right )} \]

[Out]

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

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1583, 1642, 648, 632, 212, 642} \[ \int \frac {x^3}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) (d+e x)} \, dx=-\frac {x^2 (a d+b e)}{2 a^2 e^2}+\frac {\text {arctanh}\left (\frac {2 a x+b}{\sqrt {b^2-4 a c}}\right ) \left (5 a^2 b c^2 d-2 a^2 c^3 e-5 a b^3 c d+4 a b^2 c^2 e+b^5 d-b^4 c e\right )}{a^4 \sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )}+\frac {\left (a^2 c^2 d-3 a b^2 c d+2 a b c^2 e+b^4 d-b^3 c e\right ) \log \left (a x^2+b x+c\right )}{2 a^4 \left (a d^2-e (b d-c e)\right )}+\frac {x \left (a^2 d^2+a e (b d-c e)+b^2 e^2\right )}{a^3 e^3}-\frac {d^5 \log (d+e x)}{e^4 \left (a d^2-e (b d-c e)\right )}+\frac {x^3}{3 a e} \]

[In]

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

[Out]

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

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 1583

Int[(x_)^(m_.)*((a_.) + (b_.)*(x_)^(mn_.) + (c_.)*(x_)^(mn2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_.))^(q_.), x_Symbo
l] :> Int[x^(m - 2*n*p)*(d + e*x^n)^q*(c + b*x^n + a*x^(2*n))^p, x] /; FreeQ[{a, b, c, d, e, m, n, q}, x] && E
qQ[mn, -n] && EqQ[mn2, 2*mn] && IntegerQ[p]

Rule 1642

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^5}{(d+e x) \left (c+b x+a x^2\right )} \, dx \\ & = \int \left (\frac {a^2 d^2+b^2 e^2+a e (b d-c e)}{a^3 e^3}-\frac {(a d+b e) x}{a^2 e^2}+\frac {x^2}{a e}+\frac {d^5}{e^3 \left (-a d^2+e (b d-c e)\right ) (d+e x)}+\frac {c \left (b^3 d-2 a b c d-b^2 c e+a c^2 e\right )+\left (b^4 d-3 a b^2 c d+a^2 c^2 d-b^3 c e+2 a b c^2 e\right ) x}{a^3 \left (a d^2-e (b d-c e)\right ) \left (c+b x+a x^2\right )}\right ) \, dx \\ & = \frac {\left (a^2 d^2+b^2 e^2+a e (b d-c e)\right ) x}{a^3 e^3}-\frac {(a d+b e) x^2}{2 a^2 e^2}+\frac {x^3}{3 a e}-\frac {d^5 \log (d+e x)}{e^4 \left (a d^2-e (b d-c e)\right )}+\frac {\int \frac {c \left (b^3 d-2 a b c d-b^2 c e+a c^2 e\right )+\left (b^4 d-3 a b^2 c d+a^2 c^2 d-b^3 c e+2 a b c^2 e\right ) x}{c+b x+a x^2} \, dx}{a^3 \left (a d^2-e (b d-c e)\right )} \\ & = \frac {\left (a^2 d^2+b^2 e^2+a e (b d-c e)\right ) x}{a^3 e^3}-\frac {(a d+b e) x^2}{2 a^2 e^2}+\frac {x^3}{3 a e}-\frac {d^5 \log (d+e x)}{e^4 \left (a d^2-e (b d-c e)\right )}+\frac {\left (b^4 d-3 a b^2 c d+a^2 c^2 d-b^3 c e+2 a b c^2 e\right ) \int \frac {b+2 a x}{c+b x+a x^2} \, dx}{2 a^4 \left (a d^2-e (b d-c e)\right )}-\frac {\left (b^5 d-5 a b^3 c d+5 a^2 b c^2 d-b^4 c e+4 a b^2 c^2 e-2 a^2 c^3 e\right ) \int \frac {1}{c+b x+a x^2} \, dx}{2 a^4 \left (a d^2-e (b d-c e)\right )} \\ & = \frac {\left (a^2 d^2+b^2 e^2+a e (b d-c e)\right ) x}{a^3 e^3}-\frac {(a d+b e) x^2}{2 a^2 e^2}+\frac {x^3}{3 a e}-\frac {d^5 \log (d+e x)}{e^4 \left (a d^2-e (b d-c e)\right )}+\frac {\left (b^4 d-3 a b^2 c d+a^2 c^2 d-b^3 c e+2 a b c^2 e\right ) \log \left (c+b x+a x^2\right )}{2 a^4 \left (a d^2-e (b d-c e)\right )}+\frac {\left (b^5 d-5 a b^3 c d+5 a^2 b c^2 d-b^4 c e+4 a b^2 c^2 e-2 a^2 c^3 e\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 a x\right )}{a^4 \left (a d^2-e (b d-c e)\right )} \\ & = \frac {\left (a^2 d^2+b^2 e^2+a e (b d-c e)\right ) x}{a^3 e^3}-\frac {(a d+b e) x^2}{2 a^2 e^2}+\frac {x^3}{3 a e}+\frac {\left (b^5 d-5 a b^3 c d+5 a^2 b c^2 d-b^4 c e+4 a b^2 c^2 e-2 a^2 c^3 e\right ) \tanh ^{-1}\left (\frac {b+2 a x}{\sqrt {b^2-4 a c}}\right )}{a^4 \sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )}-\frac {d^5 \log (d+e x)}{e^4 \left (a d^2-e (b d-c e)\right )}+\frac {\left (b^4 d-3 a b^2 c d+a^2 c^2 d-b^3 c e+2 a b c^2 e\right ) \log \left (c+b x+a x^2\right )}{2 a^4 \left (a d^2-e (b d-c e)\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.01 \[ \int \frac {x^3}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) (d+e x)} \, dx=\frac {\left (a^2 d^2+a b d e+b^2 e^2-a c e^2\right ) x}{a^3 e^3}-\frac {(a d+b e) x^2}{2 a^2 e^2}+\frac {x^3}{3 a e}+\frac {\left (b^5 d-5 a b^3 c d+5 a^2 b c^2 d-b^4 c e+4 a b^2 c^2 e-2 a^2 c^3 e\right ) \arctan \left (\frac {b+2 a x}{\sqrt {-b^2+4 a c}}\right )}{a^4 \sqrt {-b^2+4 a c} \left (-a d^2+b d e-c e^2\right )}-\frac {d^5 \log (d+e x)}{e^4 \left (a d^2-b d e+c e^2\right )}+\frac {\left (b^4 d-3 a b^2 c d+a^2 c^2 d-b^3 c e+2 a b c^2 e\right ) \log \left (c+b x+a x^2\right )}{2 a^4 \left (a d^2-b d e+c e^2\right )} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.67 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.02

method	result	size
default	\(\frac {\frac {1}{3} a^{2} e^{2} x^{3}-\frac {1}{2} a^{2} d e \,x^{2}-\frac {1}{2} a b \,e^{2} x^{2}+a^{2} d^{2} x +a b d e x -e^{2} a c x +b^{2} e^{2} x}{a^{3} e^{3}}+\frac {\frac {\left (a^{2} c^{2} d -3 a \,b^{2} c d +2 a b \,c^{2} e +d \,b^{4}-b^{3} c e \right ) \ln \left (a \,x^{2}+b x +c \right )}{2 a}+\frac {2 \left (-2 a b \,c^{2} d +a \,c^{3} e +b^{3} c d -b^{2} c^{2} e -\frac {\left (a^{2} c^{2} d -3 a \,b^{2} c d +2 a b \,c^{2} e +d \,b^{4}-b^{3} c e \right ) b}{2 a}\right ) \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{\left (a \,d^{2}-b d e +c \,e^{2}\right ) a^{3}}-\frac {d^{5} \ln \left (e x +d \right )}{e^{4} \left (a \,d^{2}-b d e +c \,e^{2}\right )}\)	\(286\)
risch	\(\frac {x^{3}}{3 a e}-\frac {d \,x^{2}}{2 a \,e^{2}}-\frac {b \,x^{2}}{2 a^{2} e}+\frac {d^{2} x}{a \,e^{3}}+\frac {b d x}{a^{2} e^{2}}-\frac {c x}{a^{2} e}+\frac {b^{2} x}{a^{3} e}-\frac {d^{5} \ln \left (e x +d \right )}{e^{4} \left (a \,d^{2}-b d e +c \,e^{2}\right )}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (4 a^{3} c \,d^{2}-b^{2} d^{2} a^{2}-4 a^{2} b c d e +4 a^{2} c^{2} e^{2}+a \,b^{3} d e -a \,b^{2} c \,e^{2}\right ) \textit {\_Z}^{2}+\left (-4 c^{3} d \,a^{3} e^{3}+13 b^{2} c^{2} d \,e^{3} a^{2}-8 a^{2} b \,c^{3} e^{4}-7 d c \,b^{4} a \,e^{3}+6 a \,b^{3} c^{2} e^{4}+b^{6} d \,e^{3}-b^{5} c \,e^{4}\right ) \textit {\_Z} +a^{3} c^{5} e^{6}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-2 a^{3} d^{2} e +2 a^{2} b d \,e^{2}+6 a^{2} c \,e^{3}-2 a \,b^{2} e^{3}\right ) \textit {\_R}^{2}+\left (-2 a^{5} d^{5}+4 a^{4} c \,d^{3} e^{2}-a^{3} b^{2} d^{3} e^{2}+4 a^{3} b c \,d^{2} e^{3}-3 a^{3} c^{2} d \,e^{4}-a^{2} b^{3} d^{2} e^{3}+2 a^{2} b^{2} c d \,e^{4}-11 a^{2} b \,c^{2} e^{5}+10 a \,b^{3} c \,e^{5}-2 b^{5} e^{5}\right ) \textit {\_R} +a^{5} c^{2} d^{4} e^{3}-3 a^{4} b^{2} c \,d^{4} e^{3}-2 a^{4} b \,c^{2} d^{3} e^{4}-a^{4} c^{3} d^{2} e^{5}+a^{3} b^{4} d^{4} e^{3}+a^{3} b^{3} c \,d^{3} e^{4}+a^{3} b^{2} c^{2} d^{2} e^{5}+a^{3} b \,c^{3} d \,e^{6}+a^{3} c^{4} e^{7}\right ) x +\left (-a^{2} b \,d^{2} e +8 a^{2} c d \,e^{2}-a \,b^{2} d \,e^{2}-a b c \,e^{3}\right ) \textit {\_R}^{2}+\left (-a^{4} b \,d^{5}+4 a^{4} c \,d^{4} e -a^{3} b^{2} d^{4} e +4 a^{3} b c \,d^{3} e^{2}-4 a^{3} c^{2} d^{2} e^{3}-a^{2} b^{3} d^{3} e^{2}+5 a^{2} b^{2} c \,d^{2} e^{3}-10 a^{2} b \,c^{2} d \,e^{4}-a^{2} c^{3} e^{5}-a \,b^{4} d^{2} e^{3}+7 a \,b^{3} c d \,e^{4}+3 a \,b^{2} c^{2} e^{5}-b^{5} d \,e^{4}-b^{4} c \,e^{5}\right ) \textit {\_R} -2 a^{4} b \,c^{2} d^{4} e^{3}-a^{4} c^{3} d^{3} e^{4}+a^{3} b^{3} c \,d^{4} e^{3}+a^{3} b^{2} c^{2} d^{3} e^{4}+a^{3} b \,c^{3} d^{2} e^{5}+a^{3} c^{4} d \,e^{6}\right )}{a^{3} e^{3}}\)	\(861\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 18.16 (sec) , antiderivative size = 1027, normalized size of antiderivative = 3.67 \[ \int \frac {x^3}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) (d+e x)} \, dx=\left [-\frac {6 \, {\left (a^{4} b^{2} - 4 \, a^{5} c\right )} d^{5} \log \left (e x + d\right ) - 2 \, {\left ({\left (a^{4} b^{2} - 4 \, a^{5} c\right )} d^{2} e^{3} - {\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} d e^{4} + {\left (a^{3} b^{2} c - 4 \, a^{4} c^{2}\right )} e^{5}\right )} x^{3} + 3 \, {\left ({\left (a^{4} b^{2} - 4 \, a^{5} c\right )} d^{3} e^{2} - {\left (a^{2} b^{4} - 5 \, a^{3} b^{2} c + 4 \, a^{4} c^{2}\right )} d e^{4} + {\left (a^{2} b^{3} c - 4 \, a^{3} b c^{2}\right )} e^{5}\right )} x^{2} + 3 \, {\left ({\left (b^{5} - 5 \, a b^{3} c + 5 \, a^{2} b c^{2}\right )} d e^{4} - {\left (b^{4} c - 4 \, a b^{2} c^{2} + 2 \, a^{2} c^{3}\right )} e^{5}\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, a^{2} x^{2} + 2 \, a b x + b^{2} - 2 \, a c - \sqrt {b^{2} - 4 \, a c} {\left (2 \, a x + b\right )}}{a x^{2} + b x + c}\right ) - 6 \, {\left ({\left (a^{4} b^{2} - 4 \, a^{5} c\right )} d^{4} e - {\left (a b^{5} - 6 \, a^{2} b^{3} c + 8 \, a^{3} b c^{2}\right )} d e^{4} + {\left (a b^{4} c - 5 \, a^{2} b^{2} c^{2} + 4 \, a^{3} c^{3}\right )} e^{5}\right )} x - 3 \, {\left ({\left (b^{6} - 7 \, a b^{4} c + 13 \, a^{2} b^{2} c^{2} - 4 \, a^{3} c^{3}\right )} d e^{4} - {\left (b^{5} c - 6 \, a b^{3} c^{2} + 8 \, a^{2} b c^{3}\right )} e^{5}\right )} \log \left (a x^{2} + b x + c\right )}{6 \, {\left ({\left (a^{5} b^{2} - 4 \, a^{6} c\right )} d^{2} e^{4} - {\left (a^{4} b^{3} - 4 \, a^{5} b c\right )} d e^{5} + {\left (a^{4} b^{2} c - 4 \, a^{5} c^{2}\right )} e^{6}\right )}}, -\frac {6 \, {\left (a^{4} b^{2} - 4 \, a^{5} c\right )} d^{5} \log \left (e x + d\right ) - 2 \, {\left ({\left (a^{4} b^{2} - 4 \, a^{5} c\right )} d^{2} e^{3} - {\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} d e^{4} + {\left (a^{3} b^{2} c - 4 \, a^{4} c^{2}\right )} e^{5}\right )} x^{3} + 3 \, {\left ({\left (a^{4} b^{2} - 4 \, a^{5} c\right )} d^{3} e^{2} - {\left (a^{2} b^{4} - 5 \, a^{3} b^{2} c + 4 \, a^{4} c^{2}\right )} d e^{4} + {\left (a^{2} b^{3} c - 4 \, a^{3} b c^{2}\right )} e^{5}\right )} x^{2} - 6 \, {\left ({\left (b^{5} - 5 \, a b^{3} c + 5 \, a^{2} b c^{2}\right )} d e^{4} - {\left (b^{4} c - 4 \, a b^{2} c^{2} + 2 \, a^{2} c^{3}\right )} e^{5}\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, a x + b\right )}}{b^{2} - 4 \, a c}\right ) - 6 \, {\left ({\left (a^{4} b^{2} - 4 \, a^{5} c\right )} d^{4} e - {\left (a b^{5} - 6 \, a^{2} b^{3} c + 8 \, a^{3} b c^{2}\right )} d e^{4} + {\left (a b^{4} c - 5 \, a^{2} b^{2} c^{2} + 4 \, a^{3} c^{3}\right )} e^{5}\right )} x - 3 \, {\left ({\left (b^{6} - 7 \, a b^{4} c + 13 \, a^{2} b^{2} c^{2} - 4 \, a^{3} c^{3}\right )} d e^{4} - {\left (b^{5} c - 6 \, a b^{3} c^{2} + 8 \, a^{2} b c^{3}\right )} e^{5}\right )} \log \left (a x^{2} + b x + c\right )}{6 \, {\left ({\left (a^{5} b^{2} - 4 \, a^{6} c\right )} d^{2} e^{4} - {\left (a^{4} b^{3} - 4 \, a^{5} b c\right )} d e^{5} + {\left (a^{4} b^{2} c - 4 \, a^{5} c^{2}\right )} e^{6}\right )}}\right ] \]

[In]

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

[Out]

[-1/6*(6*(a^4*b^2 - 4*a^5*c)*d^5*log(e*x + d) - 2*((a^4*b^2 - 4*a^5*c)*d^2*e^3 - (a^3*b^3 - 4*a^4*b*c)*d*e^4 +
(a^3*b^2*c - 4*a^4*c^2)*e^5)*x^3 + 3*((a^4*b^2 - 4*a^5*c)*d^3*e^2 - (a^2*b^4 - 5*a^3*b^2*c + 4*a^4*c^2)*d*e^4
+ (a^2*b^3*c - 4*a^3*b*c^2)*e^5)*x^2 + 3*((b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*d*e^4 - (b^4*c - 4*a*b^2*c^2 + 2*a^
2*c^3)*e^5)*sqrt(b^2 - 4*a*c)*log((2*a^2*x^2 + 2*a*b*x + b^2 - 2*a*c - sqrt(b^2 - 4*a*c)*(2*a*x + b))/(a*x^2 +
b*x + c)) - 6*((a^4*b^2 - 4*a^5*c)*d^4*e - (a*b^5 - 6*a^2*b^3*c + 8*a^3*b*c^2)*d*e^4 + (a*b^4*c - 5*a^2*b^2*c
^2 + 4*a^3*c^3)*e^5)*x - 3*((b^6 - 7*a*b^4*c + 13*a^2*b^2*c^2 - 4*a^3*c^3)*d*e^4 - (b^5*c - 6*a*b^3*c^2 + 8*a^
2*b*c^3)*e^5)*log(a*x^2 + b*x + c))/((a^5*b^2 - 4*a^6*c)*d^2*e^4 - (a^4*b^3 - 4*a^5*b*c)*d*e^5 + (a^4*b^2*c -
4*a^5*c^2)*e^6), -1/6*(6*(a^4*b^2 - 4*a^5*c)*d^5*log(e*x + d) - 2*((a^4*b^2 - 4*a^5*c)*d^2*e^3 - (a^3*b^3 - 4*
a^4*b*c)*d*e^4 + (a^3*b^2*c - 4*a^4*c^2)*e^5)*x^3 + 3*((a^4*b^2 - 4*a^5*c)*d^3*e^2 - (a^2*b^4 - 5*a^3*b^2*c +
4*a^4*c^2)*d*e^4 + (a^2*b^3*c - 4*a^3*b*c^2)*e^5)*x^2 - 6*((b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*d*e^4 - (b^4*c - 4*
a*b^2*c^2 + 2*a^2*c^3)*e^5)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*a*x + b)/(b^2 - 4*a*c)) - 6*((a^4
*b^2 - 4*a^5*c)*d^4*e - (a*b^5 - 6*a^2*b^3*c + 8*a^3*b*c^2)*d*e^4 + (a*b^4*c - 5*a^2*b^2*c^2 + 4*a^3*c^3)*e^5)
*x - 3*((b^6 - 7*a*b^4*c + 13*a^2*b^2*c^2 - 4*a^3*c^3)*d*e^4 - (b^5*c - 6*a*b^3*c^2 + 8*a^2*b*c^3)*e^5)*log(a*
x^2 + b*x + c))/((a^5*b^2 - 4*a^6*c)*d^2*e^4 - (a^4*b^3 - 4*a^5*b*c)*d*e^5 + (a^4*b^2*c - 4*a^5*c^2)*e^6)]

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^3}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) (d+e x)} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more deta

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.05 \[ \int \frac {x^3}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) (d+e x)} \, dx=-\frac {d^{5} \log \left ({\left | e x + d \right |}\right )}{a d^{2} e^{4} - b d e^{5} + c e^{6}} + \frac {{\left (b^{4} d - 3 \, a b^{2} c d + a^{2} c^{2} d - b^{3} c e + 2 \, a b c^{2} e\right )} \log \left (a x^{2} + b x + c\right )}{2 \, {\left (a^{5} d^{2} - a^{4} b d e + a^{4} c e^{2}\right )}} - \frac {{\left (b^{5} d - 5 \, a b^{3} c d + 5 \, a^{2} b c^{2} d - b^{4} c e + 4 \, a b^{2} c^{2} e - 2 \, a^{2} c^{3} e\right )} \arctan \left (\frac {2 \, a x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (a^{5} d^{2} - a^{4} b d e + a^{4} c e^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {2 \, a^{2} e^{2} x^{3} - 3 \, a^{2} d e x^{2} - 3 \, a b e^{2} x^{2} + 6 \, a^{2} d^{2} x + 6 \, a b d e x + 6 \, b^{2} e^{2} x - 6 \, a c e^{2} x}{6 \, a^{3} e^{3}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 11.49 (sec) , antiderivative size = 2490, normalized size of antiderivative = 8.89 \[ \int \frac {x^3}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) (d+e x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

(log(4*a^5*c*d^7 - a^4*b^2*d^7 + b^3*c^3*e^7 - b^6*d^3*e^4 - 6*a^2*c^4*d*e^6 - 3*b^4*c^2*d*e^6 + 3*b^5*c*d^2*e
^5 - 2*a^2*c^4*e^7*x - b^2*c^3*e^7*(b^2 - 4*a*c)^(1/2) + b^5*d^3*e^4*(b^2 - 4*a*c)^(1/2) + 2*a^3*c^3*d^3*e^4 -
4*a^4*c^2*d^5*e^2 - 3*a*b*c^4*e^7 + a^4*b*d^7*(b^2 - 4*a*c)^(1/2) + a*c^4*e^7*(b^2 - 4*a*c)^(1/2) + 2*a^5*d^7
*x*(b^2 - 4*a*c)^(1/2) - 3*a^2*c^3*d^2*e^5*(b^2 - 4*a*c)^(1/2) + 8*a^5*c*d^6*e*x - 9*a^2*b^2*c^2*d^3*e^4 - 4*a
^4*c*d^6*e*(b^2 - 4*a*c)^(1/2) + 12*a*b^2*c^3*d*e^6 + 6*a*b^4*c*d^3*e^4 + a*b^2*c^3*e^7*x - a*b^5*d^3*e^4*x -
2*a^4*b^2*d^6*e*x + 3*b^3*c^2*d*e^6*(b^2 - 4*a*c)^(1/2) - 3*b^4*c*d^2*e^5*(b^2 - 4*a*c)^(1/2) - 15*a*b^3*c^2*d
^2*e^5 + 15*a^2*b*c^3*d^2*e^5 + a^3*b^2*c*d^5*e^2 + a^3*b^3*d^5*e^2*x + 6*a^3*c^3*d^2*e^5*x - 4*a*b^3*c*d^3*e^
4*(b^2 - 4*a*c)^(1/2) + a^3*b*c*d^5*e^2*(b^2 - 4*a*c)^(1/2) + a*b^4*d^3*e^4*x*(b^2 - 4*a*c)^(1/2) - 3*a^2*c^3*
d*e^6*x*(b^2 - 4*a*c)^(1/2) - 2*a^4*c*d^5*e^2*x*(b^2 - 4*a*c)^(1/2) + 5*a^2*b^3*c*d^3*e^4*x - 5*a^3*b*c^2*d^3*
e^4*x + 9*a*b^2*c^2*d^2*e^5*(b^2 - 4*a*c)^(1/2) + 3*a^2*b*c^2*d^3*e^4*(b^2 - 4*a*c)^(1/2) + a^3*b^2*d^5*e^2*x*
(b^2 - 4*a*c)^(1/2) + a^3*c^2*d^3*e^4*x*(b^2 - 4*a*c)^(1/2) - 12*a^2*b^2*c^2*d^2*e^5*x - 6*a*b*c^3*d*e^6*(b^2
- 4*a*c)^(1/2) - a*b*c^3*e^7*x*(b^2 - 4*a*c)^(1/2) - 2*a^4*b*d^6*e*x*(b^2 - 4*a*c)^(1/2) - 3*a*b^3*c^2*d*e^6*x
+ 3*a*b^4*c*d^2*e^5*x + 9*a^2*b*c^3*d*e^6*x - 4*a^4*b*c*d^5*e^2*x + 3*a*b^2*c^2*d*e^6*x*(b^2 - 4*a*c)^(1/2) -
3*a*b^3*c*d^2*e^5*x*(b^2 - 4*a*c)^(1/2) + 6*a^2*b*c^2*d^2*e^5*x*(b^2 - 4*a*c)^(1/2) - 3*a^2*b^2*c*d^3*e^4*x*(
b^2 - 4*a*c)^(1/2))*(b^5*d*(b^2 - 4*a*c)^(1/2) - b^6*d + 4*a^3*c^3*d + b^5*c*e - 13*a^2*b^2*c^2*d + 7*a*b^4*c*
d - b^4*c*e*(b^2 - 4*a*c)^(1/2) - 6*a*b^3*c^2*e + 8*a^2*b*c^3*e - 2*a^2*c^3*e*(b^2 - 4*a*c)^(1/2) + 5*a^2*b*c^
2*d*(b^2 - 4*a*c)^(1/2) + 4*a*b^2*c^2*e*(b^2 - 4*a*c)^(1/2) - 5*a*b^3*c*d*(b^2 - 4*a*c)^(1/2)))/(2*(4*a^6*c*d^
2 - a^5*b^2*d^2 + 4*a^5*c^2*e^2 - a^4*b^2*c*e^2 + a^4*b^3*d*e - 4*a^5*b*c*d*e)) - (d^5*log(d + e*x))/(c*e^6 +
a*d^2*e^4 - b*d*e^5) - x*((b*d + c*e)/(a^2*e^2) - (a*d + b*e)^2/(a^3*e^3)) + (log(a^4*b^2*d^7 - 4*a^5*c*d^7 -
b^3*c^3*e^7 + b^6*d^3*e^4 + 6*a^2*c^4*d*e^6 + 3*b^4*c^2*d*e^6 - 3*b^5*c*d^2*e^5 + 2*a^2*c^4*e^7*x - b^2*c^3*e^
7*(b^2 - 4*a*c)^(1/2) + b^5*d^3*e^4*(b^2 - 4*a*c)^(1/2) - 2*a^3*c^3*d^3*e^4 + 4*a^4*c^2*d^5*e^2 + 3*a*b*c^4*e^
7 + a^4*b*d^7*(b^2 - 4*a*c)^(1/2) + a*c^4*e^7*(b^2 - 4*a*c)^(1/2) + 2*a^5*d^7*x*(b^2 - 4*a*c)^(1/2) - 3*a^2*c^
3*d^2*e^5*(b^2 - 4*a*c)^(1/2) - 8*a^5*c*d^6*e*x + 9*a^2*b^2*c^2*d^3*e^4 - 4*a^4*c*d^6*e*(b^2 - 4*a*c)^(1/2) -
12*a*b^2*c^3*d*e^6 - 6*a*b^4*c*d^3*e^4 - a*b^2*c^3*e^7*x + a*b^5*d^3*e^4*x + 2*a^4*b^2*d^6*e*x + 3*b^3*c^2*d*e
^6*(b^2 - 4*a*c)^(1/2) - 3*b^4*c*d^2*e^5*(b^2 - 4*a*c)^(1/2) + 15*a*b^3*c^2*d^2*e^5 - 15*a^2*b*c^3*d^2*e^5 - a
^3*b^2*c*d^5*e^2 - a^3*b^3*d^5*e^2*x - 6*a^3*c^3*d^2*e^5*x - 4*a*b^3*c*d^3*e^4*(b^2 - 4*a*c)^(1/2) + a^3*b*c*d
^5*e^2*(b^2 - 4*a*c)^(1/2) + a*b^4*d^3*e^4*x*(b^2 - 4*a*c)^(1/2) - 3*a^2*c^3*d*e^6*x*(b^2 - 4*a*c)^(1/2) - 2*a
^4*c*d^5*e^2*x*(b^2 - 4*a*c)^(1/2) - 5*a^2*b^3*c*d^3*e^4*x + 5*a^3*b*c^2*d^3*e^4*x + 9*a*b^2*c^2*d^2*e^5*(b^2
- 4*a*c)^(1/2) + 3*a^2*b*c^2*d^3*e^4*(b^2 - 4*a*c)^(1/2) + a^3*b^2*d^5*e^2*x*(b^2 - 4*a*c)^(1/2) + a^3*c^2*d^3
*e^4*x*(b^2 - 4*a*c)^(1/2) + 12*a^2*b^2*c^2*d^2*e^5*x - 6*a*b*c^3*d*e^6*(b^2 - 4*a*c)^(1/2) - a*b*c^3*e^7*x*(b
^2 - 4*a*c)^(1/2) - 2*a^4*b*d^6*e*x*(b^2 - 4*a*c)^(1/2) + 3*a*b^3*c^2*d*e^6*x - 3*a*b^4*c*d^2*e^5*x - 9*a^2*b*
c^3*d*e^6*x + 4*a^4*b*c*d^5*e^2*x + 3*a*b^2*c^2*d*e^6*x*(b^2 - 4*a*c)^(1/2) - 3*a*b^3*c*d^2*e^5*x*(b^2 - 4*a*c
)^(1/2) + 6*a^2*b*c^2*d^2*e^5*x*(b^2 - 4*a*c)^(1/2) - 3*a^2*b^2*c*d^3*e^4*x*(b^2 - 4*a*c)^(1/2))*(4*a^3*c^3*d
- b^5*d*(b^2 - 4*a*c)^(1/2) - b^6*d + b^5*c*e - 13*a^2*b^2*c^2*d + 7*a*b^4*c*d + b^4*c*e*(b^2 - 4*a*c)^(1/2) -
6*a*b^3*c^2*e + 8*a^2*b*c^3*e + 2*a^2*c^3*e*(b^2 - 4*a*c)^(1/2) - 5*a^2*b*c^2*d*(b^2 - 4*a*c)^(1/2) - 4*a*b^2
*c^2*e*(b^2 - 4*a*c)^(1/2) + 5*a*b^3*c*d*(b^2 - 4*a*c)^(1/2)))/(2*(4*a^6*c*d^2 - a^5*b^2*d^2 + 4*a^5*c^2*e^2 -
a^4*b^2*c*e^2 + a^4*b^3*d*e - 4*a^5*b*c*d*e)) + x^3/(3*a*e) - (x^2*(a*d + b*e))/(2*a^2*e^2)