3.881 \(\int \frac{x^4 (d+e x)}{a+b x+c x^2} \, dx\)

Optimal. Leaf size=229 \[ -\frac{\left (-a^2 c^2 e+3 a b^2 c e-2 a b c^2 d+b^3 c d+b^4 (-e)\right ) \log \left (a+b x+c x^2\right )}{2 c^5}-\frac{\left (-5 a^2 b c^2 e+2 a^2 c^3 d-4 a b^2 c^2 d+5 a b^3 c e+b^4 c d+b^5 (-e)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^5 \sqrt{b^2-4 a c}}-\frac{x^2 \left (a c e+b^2 (-e)+b c d\right )}{2 c^3}+\frac{x \left (2 a b c e-a c^2 d+b^2 c d+b^3 (-e)\right )}{c^4}+\frac{x^3 (c d-b e)}{3 c^2}+\frac{e x^4}{4 c} \]

[Out]

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

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Rubi [A]  time = 0.421258, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {800, 634, 618, 206, 628} \[ -\frac{\left (-a^2 c^2 e+3 a b^2 c e-2 a b c^2 d+b^3 c d+b^4 (-e)\right ) \log \left (a+b x+c x^2\right )}{2 c^5}-\frac{\left (-5 a^2 b c^2 e+2 a^2 c^3 d-4 a b^2 c^2 d+5 a b^3 c e+b^4 c d+b^5 (-e)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^5 \sqrt{b^2-4 a c}}-\frac{x^2 \left (a c e+b^2 (-e)+b c d\right )}{2 c^3}+\frac{x \left (2 a b c e-a c^2 d+b^2 c d+b^3 (-e)\right )}{c^4}+\frac{x^3 (c d-b e)}{3 c^2}+\frac{e x^4}{4 c} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*(d + e*x))/(a + b*x + c*x^2),x]

[Out]

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

Rule 800

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]

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

Mathematica [A]  time = 0.138832, size = 222, normalized size = 0.97 \[ \frac{6 \left (a^2 c^2 e-3 a b^2 c e+2 a b c^2 d-b^3 c d+b^4 e\right ) \log (a+x (b+c x))+\frac{12 \left (-5 a^2 b c^2 e+2 a^2 c^3 d-4 a b^2 c^2 d+5 a b^3 c e+b^4 c d+b^5 (-e)\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}-6 c^2 x^2 \left (a c e+b^2 (-e)+b c d\right )-12 c x \left (-2 a b c e+a c^2 d-b^2 c d+b^3 e\right )+4 c^3 x^3 (c d-b e)+3 c^4 e x^4}{12 c^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^4*(d + e*x))/(a + b*x + c*x^2),x]

[Out]

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

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Maple [B]  time = 0.009, size = 445, normalized size = 1.9 \begin{align*}{\frac{e{x}^{4}}{4\,c}}-{\frac{b{x}^{3}e}{3\,{c}^{2}}}+{\frac{d{x}^{3}}{3\,c}}-{\frac{ae{x}^{2}}{2\,{c}^{2}}}+{\frac{{b}^{2}{x}^{2}e}{2\,{c}^{3}}}-{\frac{b{x}^{2}d}{2\,{c}^{2}}}+2\,{\frac{abex}{{c}^{3}}}-{\frac{adx}{{c}^{2}}}-{\frac{{b}^{3}ex}{{c}^{4}}}+{\frac{{b}^{2}dx}{{c}^{3}}}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ){a}^{2}e}{2\,{c}^{3}}}-{\frac{3\,\ln \left ( c{x}^{2}+bx+a \right ) a{b}^{2}e}{2\,{c}^{4}}}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ) abd}{{c}^{3}}}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ){b}^{4}e}{2\,{c}^{5}}}-{\frac{\ln \left ( c{x}^{2}+bx+a \right ){b}^{3}d}{2\,{c}^{4}}}-5\,{\frac{{a}^{2}be}{{c}^{3}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+2\,{\frac{{a}^{2}d}{{c}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+5\,{\frac{a{b}^{3}e}{{c}^{4}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-4\,{\frac{{b}^{2}ad}{{c}^{3}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{{b}^{5}e}{{c}^{5}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{{b}^{4}d}{{c}^{4}}\arctan \left ({(2\,cx+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^4*(e*x+d)/(c*x^2+b*x+a),x)

[Out]

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

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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^4*(e*x+d)/(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.78964, size = 1519, normalized size = 6.63 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x+d)/(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

[1/12*(3*(b^2*c^4 - 4*a*c^5)*e*x^4 + 4*((b^2*c^4 - 4*a*c^5)*d - (b^3*c^3 - 4*a*b*c^4)*e)*x^3 - 6*((b^3*c^3 - 4
*a*b*c^4)*d - (b^4*c^2 - 5*a*b^2*c^3 + 4*a^2*c^4)*e)*x^2 - 6*sqrt(b^2 - 4*a*c)*((b^4*c - 4*a*b^2*c^2 + 2*a^2*c
^3)*d - (b^5 - 5*a*b^3*c + 5*a^2*b*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)) + 12*((b^4*c^2 - 5*a*b^2*c^3 + 4*a^2*c^4)*d - (b^5*c - 6*a*b^3*c^2 + 8*a^2*b*c^3)*e)*x
- 6*((b^5*c - 6*a*b^3*c^2 + 8*a^2*b*c^3)*d - (b^6 - 7*a*b^4*c + 13*a^2*b^2*c^2 - 4*a^3*c^3)*e)*log(c*x^2 + b*
x + a))/(b^2*c^5 - 4*a*c^6), 1/12*(3*(b^2*c^4 - 4*a*c^5)*e*x^4 + 4*((b^2*c^4 - 4*a*c^5)*d - (b^3*c^3 - 4*a*b*c
^4)*e)*x^3 - 6*((b^3*c^3 - 4*a*b*c^4)*d - (b^4*c^2 - 5*a*b^2*c^3 + 4*a^2*c^4)*e)*x^2 - 12*sqrt(-b^2 + 4*a*c)*(
(b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*d - (b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*e)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b
)/(b^2 - 4*a*c)) + 12*((b^4*c^2 - 5*a*b^2*c^3 + 4*a^2*c^4)*d - (b^5*c - 6*a*b^3*c^2 + 8*a^2*b*c^3)*e)*x - 6*((
b^5*c - 6*a*b^3*c^2 + 8*a^2*b*c^3)*d - (b^6 - 7*a*b^4*c + 13*a^2*b^2*c^2 - 4*a^3*c^3)*e)*log(c*x^2 + b*x + a))
/(b^2*c^5 - 4*a*c^6)]

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Sympy [B]  time = 4.07395, size = 1088, normalized size = 4.75 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(e*x+d)/(c*x**2+b*x+a),x)

[Out]

(-sqrt(-4*a*c + b**2)*(5*a**2*b*c**2*e - 2*a**2*c**3*d - 5*a*b**3*c*e + 4*a*b**2*c**2*d + b**5*e - b**4*c*d)/(
2*c**5*(4*a*c - b**2)) + (a**2*c**2*e - 3*a*b**2*c*e + 2*a*b*c**2*d + b**4*e - b**3*c*d)/(2*c**5))*log(x + (2*
a**3*c**2*e - 4*a**2*b**2*c*e + 3*a**2*b*c**2*d + a*b**4*e - a*b**3*c*d - 4*a*c**5*(-sqrt(-4*a*c + b**2)*(5*a*
*2*b*c**2*e - 2*a**2*c**3*d - 5*a*b**3*c*e + 4*a*b**2*c**2*d + b**5*e - b**4*c*d)/(2*c**5*(4*a*c - b**2)) + (a
**2*c**2*e - 3*a*b**2*c*e + 2*a*b*c**2*d + b**4*e - b**3*c*d)/(2*c**5)) + b**2*c**4*(-sqrt(-4*a*c + b**2)*(5*a
**2*b*c**2*e - 2*a**2*c**3*d - 5*a*b**3*c*e + 4*a*b**2*c**2*d + b**5*e - b**4*c*d)/(2*c**5*(4*a*c - b**2)) + (
a**2*c**2*e - 3*a*b**2*c*e + 2*a*b*c**2*d + b**4*e - b**3*c*d)/(2*c**5)))/(5*a**2*b*c**2*e - 2*a**2*c**3*d - 5
*a*b**3*c*e + 4*a*b**2*c**2*d + b**5*e - b**4*c*d)) + (sqrt(-4*a*c + b**2)*(5*a**2*b*c**2*e - 2*a**2*c**3*d -
5*a*b**3*c*e + 4*a*b**2*c**2*d + b**5*e - b**4*c*d)/(2*c**5*(4*a*c - b**2)) + (a**2*c**2*e - 3*a*b**2*c*e + 2*
a*b*c**2*d + b**4*e - b**3*c*d)/(2*c**5))*log(x + (2*a**3*c**2*e - 4*a**2*b**2*c*e + 3*a**2*b*c**2*d + a*b**4*
e - a*b**3*c*d - 4*a*c**5*(sqrt(-4*a*c + b**2)*(5*a**2*b*c**2*e - 2*a**2*c**3*d - 5*a*b**3*c*e + 4*a*b**2*c**2
*d + b**5*e - b**4*c*d)/(2*c**5*(4*a*c - b**2)) + (a**2*c**2*e - 3*a*b**2*c*e + 2*a*b*c**2*d + b**4*e - b**3*c
*d)/(2*c**5)) + b**2*c**4*(sqrt(-4*a*c + b**2)*(5*a**2*b*c**2*e - 2*a**2*c**3*d - 5*a*b**3*c*e + 4*a*b**2*c**2
*d + b**5*e - b**4*c*d)/(2*c**5*(4*a*c - b**2)) + (a**2*c**2*e - 3*a*b**2*c*e + 2*a*b*c**2*d + b**4*e - b**3*c
*d)/(2*c**5)))/(5*a**2*b*c**2*e - 2*a**2*c**3*d - 5*a*b**3*c*e + 4*a*b**2*c**2*d + b**5*e - b**4*c*d)) + e*x**
4/(4*c) - x**3*(b*e - c*d)/(3*c**2) - x**2*(a*c*e - b**2*e + b*c*d)/(2*c**3) + x*(2*a*b*c*e - a*c**2*d - b**3*
e + b**2*c*d)/c**4

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Giac [A]  time = 1.30405, size = 333, normalized size = 1.45 \begin{align*} \frac{3 \, c^{3} x^{4} e + 4 \, c^{3} d x^{3} - 4 \, b c^{2} x^{3} e - 6 \, b c^{2} d x^{2} + 6 \, b^{2} c x^{2} e - 6 \, a c^{2} x^{2} e + 12 \, b^{2} c d x - 12 \, a c^{2} d x - 12 \, b^{3} x e + 24 \, a b c x e}{12 \, c^{4}} - \frac{{\left (b^{3} c d - 2 \, a b c^{2} d - b^{4} e + 3 \, a b^{2} c e - a^{2} c^{2} e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{5}} + \frac{{\left (b^{4} c d - 4 \, a b^{2} c^{2} d + 2 \, a^{2} c^{3} d - b^{5} e + 5 \, a b^{3} c e - 5 \, a^{2} b 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^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x+d)/(c*x^2+b*x+a),x, algorithm="giac")

[Out]

1/12*(3*c^3*x^4*e + 4*c^3*d*x^3 - 4*b*c^2*x^3*e - 6*b*c^2*d*x^2 + 6*b^2*c*x^2*e - 6*a*c^2*x^2*e + 12*b^2*c*d*x
- 12*a*c^2*d*x - 12*b^3*x*e + 24*a*b*c*x*e)/c^4 - 1/2*(b^3*c*d - 2*a*b*c^2*d - b^4*e + 3*a*b^2*c*e - a^2*c^2*
e)*log(c*x^2 + b*x + a)/c^5 + (b^4*c*d - 4*a*b^2*c^2*d + 2*a^2*c^3*d - b^5*e + 5*a*b^3*c*e - 5*a^2*b*c^2*e)*ar
ctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*c^5)