3.61 $$\int \frac{x^3}{(a+\frac{c}{x^2}+\frac{b}{x}) (d+e x)} \, dx$$

Optimal. Leaf size=280 $\frac{\left (a^2 c^2 d-3 a b^2 c d+2 a b c^2 e-b^3 c e+b^4 d\right ) \log \left (a x^2+b x+c\right )}{2 a^4 \left (a d^2-e (b d-c e)\right )}+\frac{\left (5 a^2 b c^2 d-2 a^2 c^3 e+4 a b^2 c^2 e-5 a b^3 c d-b^4 c e+b^5 d\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{a^4 \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )}+\frac{x \left (a^2 d^2+a e (b d-c e)+b^2 e^2\right )}{a^3 e^3}-\frac{x^2 (a d+b e)}{2 a^2 e^2}-\frac{d^5 \log (d+e x)}{e^4 \left (a d^2-e (b d-c e)\right )}+\frac{x^3}{3 a e}$

[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)))

________________________________________________________________________________________

Rubi [A]  time = 0.59737, antiderivative size = 280, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.24, Rules used = {1569, 1628, 634, 618, 206, 628} $\frac{\left (a^2 c^2 d-3 a b^2 c d+2 a b c^2 e-b^3 c e+b^4 d\right ) \log \left (a x^2+b x+c\right )}{2 a^4 \left (a d^2-e (b d-c e)\right )}+\frac{\left (5 a^2 b c^2 d-2 a^2 c^3 e+4 a b^2 c^2 e-5 a b^3 c d-b^4 c e+b^5 d\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{a^4 \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )}+\frac{x \left (a^2 d^2+a e (b d-c e)+b^2 e^2\right )}{a^3 e^3}-\frac{x^2 (a d+b e)}{2 a^2 e^2}-\frac{d^5 \log (d+e x)}{e^4 \left (a d^2-e (b d-c e)\right )}+\frac{x^3}{3 a e}$

Antiderivative was successfully veriﬁed.

[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 1569

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 1628

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]

Rule 634

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 618

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 206

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

Rule 628

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]

Rubi steps

\begin{align*} \int \frac{x^3}{\left (a+\frac{c}{x^2}+\frac{b}{x}\right ) (d+e x)} \, dx &=\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 ) \operatorname{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]  time = 0.241123, size = 283, normalized size = 1.01 $\frac{\left (a^2 c^2 d-3 a b^2 c d+2 a b c^2 e-b^3 c e+b^4 d\right ) \log \left (a x^2+b x+c\right )}{2 a^4 \left (a d^2-b d e+c e^2\right )}+\frac{\left (5 a^2 b c^2 d-2 a^2 c^3 e+4 a b^2 c^2 e-5 a b^3 c d-b^4 c e+b^5 d\right ) \tan ^{-1}\left (\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right )}{a^4 \sqrt{4 a c-b^2} \left (-a d^2+b d e-c e^2\right )}+\frac{x \left (a^2 d^2+a b d e-a c e^2+b^2 e^2\right )}{a^3 e^3}-\frac{x^2 (a d+b e)}{2 a^2 e^2}-\frac{d^5 \log (d+e x)}{e^4 \left (a d^2-b d e+c e^2\right )}+\frac{x^3}{3 a e}$

Antiderivative was successfully veriﬁed.

[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 [B]  time = 0.014, size = 662, normalized size = 2.4 \begin{align*}{\frac{{x}^{3}}{3\,ae}}-{\frac{{x}^{2}d}{2\,a{e}^{2}}}-{\frac{{x}^{2}b}{2\,e{a}^{2}}}+{\frac{{d}^{2}x}{{e}^{3}a}}+{\frac{bdx}{{a}^{2}{e}^{2}}}-{\frac{cx}{e{a}^{2}}}+{\frac{{b}^{2}x}{e{a}^{3}}}-{\frac{{d}^{5}\ln \left ( ex+d \right ) }{{e}^{4} \left ( a{d}^{2}-bde+{e}^{2}c \right ) }}+{\frac{\ln \left ( a{x}^{2}+bx+c \right ){c}^{2}d}{ \left ( 2\,a{d}^{2}-2\,bde+2\,{e}^{2}c \right ){a}^{2}}}-{\frac{3\,\ln \left ( a{x}^{2}+bx+c \right ){b}^{2}cd}{ \left ( 2\,a{d}^{2}-2\,bde+2\,{e}^{2}c \right ){a}^{3}}}+{\frac{\ln \left ( a{x}^{2}+bx+c \right ) b{c}^{2}e}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ){a}^{3}}}+{\frac{\ln \left ( a{x}^{2}+bx+c \right ){b}^{4}d}{ \left ( 2\,a{d}^{2}-2\,bde+2\,{e}^{2}c \right ){a}^{4}}}-{\frac{\ln \left ( a{x}^{2}+bx+c \right ){b}^{3}ce}{ \left ( 2\,a{d}^{2}-2\,bde+2\,{e}^{2}c \right ){a}^{4}}}-5\,{\frac{d{c}^{2}b}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ){a}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+2\,{\frac{e{c}^{3}}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ){a}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+5\,{\frac{{b}^{3}cd}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ){a}^{3}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-4\,{\frac{{b}^{2}{c}^{2}e}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ){a}^{3}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{{b}^{5}d}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ){a}^{4}}\arctan \left ({(2\,ax+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{{b}^{4}ce}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ){a}^{4}}\arctan \left ({(2\,ax+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 137.062, size = 2079, normalized size = 7.42 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.10794, size = 398, normalized size = 1.42 \begin{align*} -\frac{d^{5} \log \left ({\left | x e + 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{{\left (2 \, a^{2} x^{3} e^{2} - 3 \, a^{2} d x^{2} e + 6 \, a^{2} d^{2} x - 3 \, a b x^{2} e^{2} + 6 \, a b d x e + 6 \, b^{2} x e^{2} - 6 \, a c x e^{2}\right )} e^{\left (-3\right )}}{6 \, a^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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