\(\int \frac {x^3 (d+e x)}{a+b x+c x^2} \, dx\) [882]

Optimal result

Integrand size = 21, antiderivative size = 169 \[ \int \frac {x^3 (d+e x)}{a+b x+c x^2} \, dx=-\frac {\left (b c d-b^2 e+a c e\right ) x}{c^3}+\frac {(c d-b e) x^2}{2 c^2}+\frac {e x^3}{3 c}+\frac {\left (b^3 c d-3 a b c^2 d-b^4 e+4 a b^2 c e-2 a^2 c^2 e\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^4 \sqrt {b^2-4 a c}}+\frac {\left (b^2 c d-a c^2 d-b^3 e+2 a b c e\right ) \log \left (a+b x+c x^2\right )}{2 c^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {814, 648, 632, 212, 642} \[ \int \frac {x^3 (d+e x)}{a+b x+c x^2} \, dx=\frac {\text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (-2 a^2 c^2 e+4 a b^2 c e-3 a b c^2 d+b^4 (-e)+b^3 c d\right )}{c^4 \sqrt {b^2-4 a c}}-\frac {x \left (a c e+b^2 (-e)+b c d\right )}{c^3}+\frac {\left (2 a b c e-a c^2 d+b^3 (-e)+b^2 c d\right ) \log \left (a+b x+c x^2\right )}{2 c^4}+\frac {x^2 (c d-b e)}{2 c^2}+\frac {e x^3}{3 c} \]

[In]

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

[Out]

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

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 814

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[(d + e*x)^m*((f + g*x)/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.98 \[ \int \frac {x^3 (d+e x)}{a+b x+c x^2} \, dx=\frac {-6 c \left (b c d-b^2 e+a c e\right ) x+3 c^2 (c d-b e) x^2+2 c^3 e x^3+\frac {6 \left (-b^3 c d+3 a b c^2 d+b^4 e-4 a b^2 c e+2 a^2 c^2 e\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}-3 \left (-b^2 c d+a c^2 d+b^3 e-2 a b c e\right ) \log (a+x (b+c x))}{6 c^4} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.08

method	result	size
default	\(-\frac {-\frac {1}{3} e \,x^{3} c^{2}+\frac {1}{2} b c e \,x^{2}-\frac {1}{2} c^{2} d \,x^{2}+x a c e -b^{2} e x +b c d x}{c^{3}}+\frac {\frac {\left (2 a b c e -a \,c^{2} d -b^{3} e +b^{2} c d \right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (a^{2} c e -a \,b^{2} e +a b c d -\frac {\left (2 a b c e -a \,c^{2} d -b^{3} e +b^{2} c d \right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{c^{3}}\)	\(183\)
risch	\(\text {Expression too large to display}\)	\(2940\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 563, normalized size of antiderivative = 3.33 \[ \int \frac {x^3 (d+e x)}{a+b x+c x^2} \, dx=\left [\frac {2 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} e x^{3} + 3 \, {\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d - {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} e\right )} x^{2} - 3 \, \sqrt {b^{2} - 4 \, a c} {\left ({\left (b^{3} c - 3 \, a b c^{2}\right )} d - {\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} e\right )} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c - \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) - 6 \, {\left ({\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d - {\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} e\right )} x + 3 \, {\left ({\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} d - {\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} e\right )} \log \left (c x^{2} + b x + a\right )}{6 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )}}, \frac {2 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} e x^{3} + 3 \, {\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d - {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} e\right )} x^{2} + 6 \, \sqrt {-b^{2} + 4 \, a c} {\left ({\left (b^{3} c - 3 \, a b c^{2}\right )} d - {\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} e\right )} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) - 6 \, {\left ({\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d - {\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} e\right )} x + 3 \, {\left ({\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} d - {\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} e\right )} \log \left (c x^{2} + b x + a\right )}{6 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )}}\right ] \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 840 vs. \(2 (168) = 336\).

Time = 1.48 (sec) , antiderivative size = 840, normalized size of antiderivative = 4.97 \[ \int \frac {x^3 (d+e x)}{a+b x+c x^2} \, dx=x^{2} \left (- \frac {b e}{2 c^{2}} + \frac {d}{2 c}\right ) + x \left (- \frac {a e}{c^{2}} + \frac {b^{2} e}{c^{3}} - \frac {b d}{c^{2}}\right ) + \left (- \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a^{2} c^{2} e - 4 a b^{2} c e + 3 a b c^{2} d + b^{4} e - b^{3} c d\right )}{2 c^{4} \cdot \left (4 a c - b^{2}\right )} + \frac {2 a b c e - a c^{2} d - b^{3} e + b^{2} c d}{2 c^{4}}\right ) \log {\left (x + \frac {- 3 a^{2} b c e + 2 a^{2} c^{2} d + a b^{3} e - a b^{2} c d + 4 a c^{4} \left (- \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a^{2} c^{2} e - 4 a b^{2} c e + 3 a b c^{2} d + b^{4} e - b^{3} c d\right )}{2 c^{4} \cdot \left (4 a c - b^{2}\right )} + \frac {2 a b c e - a c^{2} d - b^{3} e + b^{2} c d}{2 c^{4}}\right ) - b^{2} c^{3} \left (- \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a^{2} c^{2} e - 4 a b^{2} c e + 3 a b c^{2} d + b^{4} e - b^{3} c d\right )}{2 c^{4} \cdot \left (4 a c - b^{2}\right )} + \frac {2 a b c e - a c^{2} d - b^{3} e + b^{2} c d}{2 c^{4}}\right )}{2 a^{2} c^{2} e - 4 a b^{2} c e + 3 a b c^{2} d + b^{4} e - b^{3} c d} \right )} + \left (\frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a^{2} c^{2} e - 4 a b^{2} c e + 3 a b c^{2} d + b^{4} e - b^{3} c d\right )}{2 c^{4} \cdot \left (4 a c - b^{2}\right )} + \frac {2 a b c e - a c^{2} d - b^{3} e + b^{2} c d}{2 c^{4}}\right ) \log {\left (x + \frac {- 3 a^{2} b c e + 2 a^{2} c^{2} d + a b^{3} e - a b^{2} c d + 4 a c^{4} \left (\frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a^{2} c^{2} e - 4 a b^{2} c e + 3 a b c^{2} d + b^{4} e - b^{3} c d\right )}{2 c^{4} \cdot \left (4 a c - b^{2}\right )} + \frac {2 a b c e - a c^{2} d - b^{3} e + b^{2} c d}{2 c^{4}}\right ) - b^{2} c^{3} \left (\frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a^{2} c^{2} e - 4 a b^{2} c e + 3 a b c^{2} d + b^{4} e - b^{3} c d\right )}{2 c^{4} \cdot \left (4 a c - b^{2}\right )} + \frac {2 a b c e - a c^{2} d - b^{3} e + b^{2} c d}{2 c^{4}}\right )}{2 a^{2} c^{2} e - 4 a b^{2} c e + 3 a b c^{2} d + b^{4} e - b^{3} c d} \right )} + \frac {e x^{3}}{3 c} \]

[In]

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

[Out]

x**2*(-b*e/(2*c**2) + d/(2*c)) + x*(-a*e/c**2 + b**2*e/c**3 - b*d/c**2) + (-sqrt(-4*a*c + b**2)*(2*a**2*c**2*e
- 4*a*b**2*c*e + 3*a*b*c**2*d + b**4*e - b**3*c*d)/(2*c**4*(4*a*c - b**2)) + (2*a*b*c*e - a*c**2*d - b**3*e +
b**2*c*d)/(2*c**4))*log(x + (-3*a**2*b*c*e + 2*a**2*c**2*d + a*b**3*e - a*b**2*c*d + 4*a*c**4*(-sqrt(-4*a*c +
b**2)*(2*a**2*c**2*e - 4*a*b**2*c*e + 3*a*b*c**2*d + b**4*e - b**3*c*d)/(2*c**4*(4*a*c - b**2)) + (2*a*b*c*e
- a*c**2*d - b**3*e + b**2*c*d)/(2*c**4)) - b**2*c**3*(-sqrt(-4*a*c + b**2)*(2*a**2*c**2*e - 4*a*b**2*c*e + 3*
a*b*c**2*d + b**4*e - b**3*c*d)/(2*c**4*(4*a*c - b**2)) + (2*a*b*c*e - a*c**2*d - b**3*e + b**2*c*d)/(2*c**4))
)/(2*a**2*c**2*e - 4*a*b**2*c*e + 3*a*b*c**2*d + b**4*e - b**3*c*d)) + (sqrt(-4*a*c + b**2)*(2*a**2*c**2*e - 4
*a*b**2*c*e + 3*a*b*c**2*d + b**4*e - b**3*c*d)/(2*c**4*(4*a*c - b**2)) + (2*a*b*c*e - a*c**2*d - b**3*e + b**
2*c*d)/(2*c**4))*log(x + (-3*a**2*b*c*e + 2*a**2*c**2*d + a*b**3*e - a*b**2*c*d + 4*a*c**4*(sqrt(-4*a*c + b**2
)*(2*a**2*c**2*e - 4*a*b**2*c*e + 3*a*b*c**2*d + b**4*e - b**3*c*d)/(2*c**4*(4*a*c - b**2)) + (2*a*b*c*e - a*c
**2*d - b**3*e + b**2*c*d)/(2*c**4)) - b**2*c**3*(sqrt(-4*a*c + b**2)*(2*a**2*c**2*e - 4*a*b**2*c*e + 3*a*b*c*
*2*d + b**4*e - b**3*c*d)/(2*c**4*(4*a*c - b**2)) + (2*a*b*c*e - a*c**2*d - b**3*e + b**2*c*d)/(2*c**4)))/(2*a
**2*c**2*e - 4*a*b**2*c*e + 3*a*b*c**2*d + b**4*e - b**3*c*d)) + e*x**3/(3*c)

Maxima [F(-2)]

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

[In]

integrate(x^3*(e*x+d)/(c*x^2+b*x+a),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 = 0.26 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.00 \[ \int \frac {x^3 (d+e x)}{a+b x+c x^2} \, dx=\frac {2 \, c^{2} e x^{3} + 3 \, c^{2} d x^{2} - 3 \, b c e x^{2} - 6 \, b c d x + 6 \, b^{2} e x - 6 \, a c e x}{6 \, c^{3}} + \frac {{\left (b^{2} c d - a c^{2} d - b^{3} e + 2 \, a b c e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{4}} - \frac {{\left (b^{3} c d - 3 \, a b c^{2} d - b^{4} e + 4 \, a b^{2} c e - 2 \, a^{2} c^{2} e\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c} c^{4}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 10.15 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.31 \[ \int \frac {x^3 (d+e x)}{a+b x+c x^2} \, dx=x^2\,\left (\frac {d}{2\,c}-\frac {b\,e}{2\,c^2}\right )-x\,\left (\frac {b\,\left (\frac {d}{c}-\frac {b\,e}{c^2}\right )}{c}+\frac {a\,e}{c^2}\right )+\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (8\,e\,a^2\,b\,c^2-4\,d\,a^2\,c^3-6\,e\,a\,b^3\,c+5\,d\,a\,b^2\,c^2+e\,b^5-d\,b^4\,c\right )}{2\,\left (4\,a\,c^5-b^2\,c^4\right )}+\frac {e\,x^3}{3\,c}+\frac {\mathrm {atan}\left (\frac {b}{\sqrt {4\,a\,c-b^2}}+\frac {2\,c\,x}{\sqrt {4\,a\,c-b^2}}\right )\,\left (2\,e\,a^2\,c^2-4\,e\,a\,b^2\,c+3\,d\,a\,b\,c^2+e\,b^4-d\,b^3\,c\right )}{c^4\,\sqrt {4\,a\,c-b^2}} \]

[In]

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

[Out]

x^2*(d/(2*c) - (b*e)/(2*c^2)) - x*((b*(d/c - (b*e)/c^2))/c + (a*e)/c^2) + (log(a + b*x + c*x^2)*(b^5*e - 4*a^2
*c^3*d - b^4*c*d - 6*a*b^3*c*e + 5*a*b^2*c^2*d + 8*a^2*b*c^2*e))/(2*(4*a*c^5 - b^2*c^4)) + (e*x^3)/(3*c) + (at
an(b/(4*a*c - b^2)^(1/2) + (2*c*x)/(4*a*c - b^2)^(1/2))*(b^4*e + 2*a^2*c^2*e - b^3*c*d + 3*a*b*c^2*d - 4*a*b^2
*c*e))/(c^4*(4*a*c - b^2)^(1/2))