\(\int \frac {(a+b x+c x^2)^2}{d+e x^3} \, dx\) [73]

Optimal result

Integrand size = 22, antiderivative size = 272 \[ \int \frac {\left (a+b x+c x^2\right )^2}{d+e x^3} \, dx=\frac {2 b c x}{e}+\frac {c^2 x^2}{2 e}+\frac {\left (c^2 d^{4/3}+2 b c d \sqrt [3]{e}-a \left (2 b \sqrt [3]{d}+a \sqrt [3]{e}\right ) e\right ) \arctan \left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{\sqrt {3} d^{2/3} e^{5/3}}-\frac {\left (\sqrt [3]{e} \left (2 b c d-a^2 e\right )-\sqrt [3]{d} \left (c^2 d-2 a b e\right )\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{2/3} e^{5/3}}+\frac {\left (\sqrt [3]{e} \left (2 b c d-a^2 e\right )-\sqrt [3]{d} \left (c^2 d-2 a b e\right )\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{2/3} e^{5/3}}+\frac {\left (b^2+2 a c\right ) \log \left (d+e x^3\right )}{3 e} \]

[Out]

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

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.99, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {1901, 1885, 1874, 31, 648, 631, 210, 642, 266} \[ \int \frac {\left (a+b x+c x^2\right )^2}{d+e x^3} \, dx=\frac {\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (a^2 (-e)-\frac {\sqrt [3]{d} \left (c^2 d-2 a b e\right )}{\sqrt [3]{e}}+2 b c d\right )}{6 d^{2/3} e^{4/3}}-\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (\sqrt [3]{e} \left (2 b c d-a^2 e\right )-\sqrt [3]{d} \left (c^2 d-2 a b e\right )\right )}{3 d^{2/3} e^{5/3}}+\frac {\arctan \left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right ) \left (-a e \left (a \sqrt [3]{e}+2 b \sqrt [3]{d}\right )+2 b c d \sqrt [3]{e}+c^2 d^{4/3}\right )}{\sqrt {3} d^{2/3} e^{5/3}}+\frac {\left (2 a c+b^2\right ) \log \left (d+e x^3\right )}{3 e}+\frac {2 b c x}{e}+\frac {c^2 x^2}{2 e} \]

[In]

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

[Out]

(2*b*c*x)/e + (c^2*x^2)/(2*e) + ((c^2*d^(4/3) + 2*b*c*d*e^(1/3) - a*(2*b*d^(1/3) + a*e^(1/3))*e)*ArcTan[(d^(1/
3) - 2*e^(1/3)*x)/(Sqrt[3]*d^(1/3))])/(Sqrt[3]*d^(2/3)*e^(5/3)) - ((e^(1/3)*(2*b*c*d - a^2*e) - d^(1/3)*(c^2*d
- 2*a*b*e))*Log[d^(1/3) + e^(1/3)*x])/(3*d^(2/3)*e^(5/3)) + ((2*b*c*d - a^2*e - (d^(1/3)*(c^2*d - 2*a*b*e))/e
^(1/3))*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/(6*d^(2/3)*e^(4/3)) + ((b^2 + 2*a*c)*Log[d + e*x^3])/(
3*e)

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 210

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

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
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 1874

Int[((A_) + (B_.)*(x_))/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{r = Numerator[Rt[a/b, 3]], s = Denominator[R
t[a/b, 3]]}, Dist[(-r)*((B*r - A*s)/(3*a*s)), Int[1/(r + s*x), x], x] + Dist[r/(3*a*s), Int[(r*(B*r + 2*A*s) +
s*(B*r - A*s)*x)/(r^2 - r*s*x + s^2*x^2), x], x]] /; FreeQ[{a, b, A, B}, x] && NeQ[a*B^3 - b*A^3, 0] && PosQ[
a/b]

Rule 1885

Int[(P2_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x,
2]}, Int[(A + B*x)/(a + b*x^3), x] + Dist[C, Int[x^2/(a + b*x^3), x], x] /; EqQ[a*B^3 - b*A^3, 0] ||  !Ration
alQ[a/b]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]

Rule 1901

Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[Pq/(a + b*x^n), x], x] /; FreeQ[{a, b}, x
] && PolyQ[Pq, x] && IntegerQ[n]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {2 b c}{e}+\frac {c^2 x}{e}-\frac {2 b c d-a^2 e+\left (c^2 d-2 a b e\right ) x-\left (b^2+2 a c\right ) e x^2}{e \left (d+e x^3\right )}\right ) \, dx \\ & = \frac {2 b c x}{e}+\frac {c^2 x^2}{2 e}-\frac {\int \frac {2 b c d-a^2 e+\left (c^2 d-2 a b e\right ) x-\left (b^2+2 a c\right ) e x^2}{d+e x^3} \, dx}{e} \\ & = \frac {2 b c x}{e}+\frac {c^2 x^2}{2 e}-\left (-b^2-2 a c\right ) \int \frac {x^2}{d+e x^3} \, dx-\frac {\int \frac {2 b c d-a^2 e+\left (c^2 d-2 a b e\right ) x}{d+e x^3} \, dx}{e} \\ & = \frac {2 b c x}{e}+\frac {c^2 x^2}{2 e}+\frac {\left (b^2+2 a c\right ) \log \left (d+e x^3\right )}{3 e}-\frac {\int \frac {\sqrt [3]{d} \left (2 \sqrt [3]{e} \left (2 b c d-a^2 e\right )+\sqrt [3]{d} \left (c^2 d-2 a b e\right )\right )+\sqrt [3]{e} \left (-\sqrt [3]{e} \left (2 b c d-a^2 e\right )+\sqrt [3]{d} \left (c^2 d-2 a b e\right )\right ) x}{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx}{3 d^{2/3} e^{4/3}}-\frac {\left (2 b c d-a^2 e-\frac {\sqrt [3]{d} \left (c^2 d-2 a b e\right )}{\sqrt [3]{e}}\right ) \int \frac {1}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{3 d^{2/3} e} \\ & = \frac {2 b c x}{e}+\frac {c^2 x^2}{2 e}-\frac {\left (2 b c d-a^2 e-\frac {\sqrt [3]{d} \left (c^2 d-2 a b e\right )}{\sqrt [3]{e}}\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{2/3} e^{4/3}}+\frac {\left (b^2+2 a c\right ) \log \left (d+e x^3\right )}{3 e}-\frac {\left (c^2 d^{4/3}+2 b c d \sqrt [3]{e}-a \left (2 b \sqrt [3]{d}+a \sqrt [3]{e}\right ) e\right ) \int \frac {1}{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx}{2 \sqrt [3]{d} e^{4/3}}+\frac {\left (2 b c d-a^2 e-\frac {\sqrt [3]{d} \left (c^2 d-2 a b e\right )}{\sqrt [3]{e}}\right ) \int \frac {-\sqrt [3]{d} \sqrt [3]{e}+2 e^{2/3} x}{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx}{6 d^{2/3} e^{4/3}} \\ & = \frac {2 b c x}{e}+\frac {c^2 x^2}{2 e}-\frac {\left (2 b c d-a^2 e-\frac {\sqrt [3]{d} \left (c^2 d-2 a b e\right )}{\sqrt [3]{e}}\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{2/3} e^{4/3}}+\frac {\left (2 b c d-a^2 e-\frac {\sqrt [3]{d} \left (c^2 d-2 a b e\right )}{\sqrt [3]{e}}\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{2/3} e^{4/3}}+\frac {\left (b^2+2 a c\right ) \log \left (d+e x^3\right )}{3 e}-\frac {\left (c^2 d^{4/3}+2 b c d \sqrt [3]{e}-a \left (2 b \sqrt [3]{d}+a \sqrt [3]{e}\right ) e\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{d^{2/3} e^{5/3}} \\ & = \frac {2 b c x}{e}+\frac {c^2 x^2}{2 e}+\frac {\left (c^2 d^{4/3}+2 b c d \sqrt [3]{e}-a \left (2 b \sqrt [3]{d}+a \sqrt [3]{e}\right ) e\right ) \tan ^{-1}\left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{\sqrt {3} d^{2/3} e^{5/3}}-\frac {\left (2 b c d-a^2 e-\frac {\sqrt [3]{d} \left (c^2 d-2 a b e\right )}{\sqrt [3]{e}}\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{2/3} e^{4/3}}+\frac {\left (2 b c d-a^2 e-\frac {\sqrt [3]{d} \left (c^2 d-2 a b e\right )}{\sqrt [3]{e}}\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{2/3} e^{4/3}}+\frac {\left (b^2+2 a c\right ) \log \left (d+e x^3\right )}{3 e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+b x+c x^2\right )^2}{d+e x^3} \, dx=\frac {12 b c e^{2/3} x+3 c^2 e^{2/3} x^2+\frac {2 \sqrt {3} \left (c d^{2/3}-a e^{2/3}\right ) \left (c d^{2/3}+2 b \sqrt [3]{d} \sqrt [3]{e}+a e^{2/3}\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}}{\sqrt {3}}\right )}{d^{2/3}}+\frac {2 \left (c^2 d^{4/3}-2 b c d \sqrt [3]{e}+a \left (-2 b \sqrt [3]{d}+a \sqrt [3]{e}\right ) e\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{d^{2/3}}-\frac {\left (c^2 d^{4/3}-2 b c d \sqrt [3]{e}+a \left (-2 b \sqrt [3]{d}+a \sqrt [3]{e}\right ) e\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{d^{2/3}}+2 \left (b^2+2 a c\right ) e^{2/3} \log \left (d+e x^3\right )}{6 e^{5/3}} \]

[In]

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

[Out]

(12*b*c*e^(2/3)*x + 3*c^2*e^(2/3)*x^2 + (2*Sqrt[3]*(c*d^(2/3) - a*e^(2/3))*(c*d^(2/3) + 2*b*d^(1/3)*e^(1/3) +
a*e^(2/3))*ArcTan[(1 - (2*e^(1/3)*x)/d^(1/3))/Sqrt[3]])/d^(2/3) + (2*(c^2*d^(4/3) - 2*b*c*d*e^(1/3) + a*(-2*b*
d^(1/3) + a*e^(1/3))*e)*Log[d^(1/3) + e^(1/3)*x])/d^(2/3) - ((c^2*d^(4/3) - 2*b*c*d*e^(1/3) + a*(-2*b*d^(1/3)
+ a*e^(1/3))*e)*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/d^(2/3) + 2*(b^2 + 2*a*c)*e^(2/3)*Log[d + e*x^
3])/(6*e^(5/3))

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 1.74 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.31

method	result	size
risch	\(\frac {c^{2} x^{2}}{2 e}+\frac {2 b c x}{e}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} e +d \right )}{\sum }\frac {\left (e \left (2 a c +b^{2}\right ) \textit {\_R}^{2}+\left (2 a e b -c^{2} d \right ) \textit {\_R} +a^{2} e -2 b c d \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{3 e^{2}}\)	\(85\)
default	\(\frac {c \left (\frac {1}{2} c \,x^{2}+2 b x \right )}{e}+\frac {\left (a^{2} e -2 b c d \right ) \left (\frac {\ln \left (x +\left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}{3 e \left (\frac {d}{e}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {d}{e}\right )^{\frac {1}{3}} x +\left (\frac {d}{e}\right )^{\frac {2}{3}}\right )}{6 e \left (\frac {d}{e}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {d}{e}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 e \left (\frac {d}{e}\right )^{\frac {2}{3}}}\right )+\left (2 a e b -c^{2} d \right ) \left (-\frac {\ln \left (x +\left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}{3 e \left (\frac {d}{e}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {d}{e}\right )^{\frac {1}{3}} x +\left (\frac {d}{e}\right )^{\frac {2}{3}}\right )}{6 e \left (\frac {d}{e}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {d}{e}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 e \left (\frac {d}{e}\right )^{\frac {1}{3}}}\right )+\frac {\left (2 a c e +b^{2} e \right ) \ln \left (e \,x^{3}+d \right )}{3 e}}{e}\)	\(252\)

[In]

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

[Out]

1/2*c^2*x^2/e+2*b*c*x/e+1/3/e^2*sum((e*(2*a*c+b^2)*_R^2+(2*a*b*e-c^2*d)*_R+a^2*e-2*b*c*d)/_R^2*ln(x-_R),_R=Roo
tOf(_Z^3*e+d))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.43 (sec) , antiderivative size = 12827, normalized size of antiderivative = 47.16 \[ \int \frac {\left (a+b x+c x^2\right )^2}{d+e x^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

integrate((c*x^2+b*x+a)^2/(e*x^3+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(e>0)', see `assume?` for more
details)Is e

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.04 \[ \int \frac {\left (a+b x+c x^2\right )^2}{d+e x^3} \, dx=\frac {{\left (b^{2} + 2 \, a c\right )} \log \left ({\left | e x^{3} + d \right |}\right )}{3 \, e} + \frac {\sqrt {3} {\left (2 \, b c d e - a^{2} e^{2} - \left (-d e^{2}\right )^{\frac {1}{3}} c^{2} d + 2 \, \left (-d e^{2}\right )^{\frac {1}{3}} a b e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {d}{e}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {d}{e}\right )^{\frac {1}{3}}}\right )}{3 \, \left (-d e^{2}\right )^{\frac {2}{3}} e} + \frac {{\left (2 \, b c d e - a^{2} e^{2} + \left (-d e^{2}\right )^{\frac {1}{3}} c^{2} d - 2 \, \left (-d e^{2}\right )^{\frac {1}{3}} a b e\right )} \log \left (x^{2} + x \left (-\frac {d}{e}\right )^{\frac {1}{3}} + \left (-\frac {d}{e}\right )^{\frac {2}{3}}\right )}{6 \, \left (-d e^{2}\right )^{\frac {2}{3}} e} + \frac {c^{2} e x^{2} + 4 \, b c e x}{2 \, e^{2}} + \frac {{\left (c^{2} d e^{4} \left (-\frac {d}{e}\right )^{\frac {1}{3}} - 2 \, a b e^{5} \left (-\frac {d}{e}\right )^{\frac {1}{3}} + 2 \, b c d e^{4} - a^{2} e^{5}\right )} \left (-\frac {d}{e}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {d}{e}\right )^{\frac {1}{3}} \right |}\right )}{3 \, d e^{5}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 10.38 (sec) , antiderivative size = 769, normalized size of antiderivative = 2.83 \[ \int \frac {\left (a+b x+c x^2\right )^2}{d+e x^3} \, dx=\left (\sum _{k=1}^3\ln \left (\frac {2\,a^3\,b\,e^2+3\,a^2\,c^2\,d\,e+b^4\,d\,e+2\,b\,c^3\,d^2}{e}+\frac {x\,\left (-2\,a^3\,c\,e^2+3\,a^2\,b^2\,e^2+2\,b^3\,c\,d\,e+c^4\,d^2\right )}{e}-\mathrm {root}\left (27\,d^2\,e^5\,z^3-54\,a\,c\,d^2\,e^4\,z^2-27\,b^2\,d^2\,e^4\,z^2+27\,a^2\,c^2\,d^2\,e^3\,z+18\,b\,c^3\,d^3\,e^2\,z+18\,a^3\,b\,d\,e^4\,z+9\,b^4\,d^2\,e^3\,z+6\,a\,b^4\,c\,d^2\,e^2-9\,a^2\,b^2\,c^2\,d^2\,e^2-6\,a^4\,b\,c\,d\,e^3-6\,a\,b\,c^4\,d^3\,e-2\,a^3\,c^3\,d^2\,e^2+2\,b^3\,c^3\,d^3\,e+2\,a^3\,b^3\,d\,e^3-b^6\,d^2\,e^2-c^6\,d^4-a^6\,e^4,z,k\right )\,e\,\left (2\,b^2\,d-\mathrm {root}\left (27\,d^2\,e^5\,z^3-54\,a\,c\,d^2\,e^4\,z^2-27\,b^2\,d^2\,e^4\,z^2+27\,a^2\,c^2\,d^2\,e^3\,z+18\,b\,c^3\,d^3\,e^2\,z+18\,a^3\,b\,d\,e^4\,z+9\,b^4\,d^2\,e^3\,z+6\,a\,b^4\,c\,d^2\,e^2-9\,a^2\,b^2\,c^2\,d^2\,e^2-6\,a^4\,b\,c\,d\,e^3-6\,a\,b\,c^4\,d^3\,e-2\,a^3\,c^3\,d^2\,e^2+2\,b^3\,c^3\,d^3\,e+2\,a^3\,b^3\,d\,e^3-b^6\,d^2\,e^2-c^6\,d^4-a^6\,e^4,z,k\right )\,d\,e\,3+4\,a\,c\,d-a^2\,e\,x+2\,b\,c\,d\,x\right )\,3\right )\,\mathrm {root}\left (27\,d^2\,e^5\,z^3-54\,a\,c\,d^2\,e^4\,z^2-27\,b^2\,d^2\,e^4\,z^2+27\,a^2\,c^2\,d^2\,e^3\,z+18\,b\,c^3\,d^3\,e^2\,z+18\,a^3\,b\,d\,e^4\,z+9\,b^4\,d^2\,e^3\,z+6\,a\,b^4\,c\,d^2\,e^2-9\,a^2\,b^2\,c^2\,d^2\,e^2-6\,a^4\,b\,c\,d\,e^3-6\,a\,b\,c^4\,d^3\,e-2\,a^3\,c^3\,d^2\,e^2+2\,b^3\,c^3\,d^3\,e+2\,a^3\,b^3\,d\,e^3-b^6\,d^2\,e^2-c^6\,d^4-a^6\,e^4,z,k\right )\right )+\frac {c^2\,x^2}{2\,e}+\frac {2\,b\,c\,x}{e} \]

[In]

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

[Out]

symsum(log((2*a^3*b*e^2 + 2*b*c^3*d^2 + b^4*d*e + 3*a^2*c^2*d*e)/e + (x*(c^4*d^2 - 2*a^3*c*e^2 + 3*a^2*b^2*e^2
+ 2*b^3*c*d*e))/e - 3*root(27*d^2*e^5*z^3 - 54*a*c*d^2*e^4*z^2 - 27*b^2*d^2*e^4*z^2 + 27*a^2*c^2*d^2*e^3*z +
18*b*c^3*d^3*e^2*z + 18*a^3*b*d*e^4*z + 9*b^4*d^2*e^3*z + 6*a*b^4*c*d^2*e^2 - 9*a^2*b^2*c^2*d^2*e^2 - 6*a^4*b*
c*d*e^3 - 6*a*b*c^4*d^3*e - 2*a^3*c^3*d^2*e^2 + 2*b^3*c^3*d^3*e + 2*a^3*b^3*d*e^3 - b^6*d^2*e^2 - c^6*d^4 - a^
6*e^4, z, k)*e*(2*b^2*d - 3*root(27*d^2*e^5*z^3 - 54*a*c*d^2*e^4*z^2 - 27*b^2*d^2*e^4*z^2 + 27*a^2*c^2*d^2*e^3
*z + 18*b*c^3*d^3*e^2*z + 18*a^3*b*d*e^4*z + 9*b^4*d^2*e^3*z + 6*a*b^4*c*d^2*e^2 - 9*a^2*b^2*c^2*d^2*e^2 - 6*a
^4*b*c*d*e^3 - 6*a*b*c^4*d^3*e - 2*a^3*c^3*d^2*e^2 + 2*b^3*c^3*d^3*e + 2*a^3*b^3*d*e^3 - b^6*d^2*e^2 - c^6*d^4
- a^6*e^4, z, k)*d*e + 4*a*c*d - a^2*e*x + 2*b*c*d*x))*root(27*d^2*e^5*z^3 - 54*a*c*d^2*e^4*z^2 - 27*b^2*d^2*
e^4*z^2 + 27*a^2*c^2*d^2*e^3*z + 18*b*c^3*d^3*e^2*z + 18*a^3*b*d*e^4*z + 9*b^4*d^2*e^3*z + 6*a*b^4*c*d^2*e^2 -
9*a^2*b^2*c^2*d^2*e^2 - 6*a^4*b*c*d*e^3 - 6*a*b*c^4*d^3*e - 2*a^3*c^3*d^2*e^2 + 2*b^3*c^3*d^3*e + 2*a^3*b^3*d
*e^3 - b^6*d^2*e^2 - c^6*d^4 - a^6*e^4, z, k), k, 1, 3) + (c^2*x^2)/(2*e) + (2*b*c*x)/e