3.1524 \(\int \frac{(b+2 c x) (d+e x)^4}{a+b x+c x^2} \, dx\)

Optimal. Leaf size=299 \[ \frac{\left (2 c^2 e^2 \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )-4 b^2 c e^3 (a e+b d)-4 c^3 d^2 e (3 a e+b d)+b^4 e^4+2 c^4 d^4\right ) \log \left (a+b x+c x^2\right )}{2 c^4}+\frac{e^2 x^2 \left (-2 c e (a e+2 b d)+b^2 e^2+12 c^2 d^2\right )}{2 c^2}+\frac{e x \left (-2 c^2 d e (4 a e+3 b d)+b c e^2 (3 a e+4 b d)-b^3 e^3+8 c^3 d^3\right )}{c^3}-\frac{e \sqrt{b^2-4 a c} (2 c d-b e) \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^4}+\frac{e^3 x^3 (8 c d-b e)}{3 c}+\frac{e^4 x^4}{2} \]

[Out]

(e*(8*c^3*d^3 - b^3*e^3 + b*c*e^2*(4*b*d + 3*a*e) - 2*c^2*d*e*(3*b*d + 4*a*e))*x)/c^3 + (e^2*(12*c^2*d^2 + b^2
*e^2 - 2*c*e*(2*b*d + a*e))*x^2)/(2*c^2) + (e^3*(8*c*d - b*e)*x^3)/(3*c) + (e^4*x^4)/2 - (Sqrt[b^2 - 4*a*c]*e*
(2*c*d - b*e)*(2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/c^4 + ((2*c^4*
d^4 + b^4*e^4 - 4*b^2*c*e^3*(b*d + a*e) - 4*c^3*d^2*e*(b*d + 3*a*e) + 2*c^2*e^2*(3*b^2*d^2 + 6*a*b*d*e + a^2*e
^2))*Log[a + b*x + c*x^2])/(2*c^4)

________________________________________________________________________________________

Rubi [A]  time = 0.397274, antiderivative size = 299, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {800, 634, 618, 206, 628} \[ \frac{\left (2 c^2 e^2 \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )-4 b^2 c e^3 (a e+b d)-4 c^3 d^2 e (3 a e+b d)+b^4 e^4+2 c^4 d^4\right ) \log \left (a+b x+c x^2\right )}{2 c^4}+\frac{e^2 x^2 \left (-2 c e (a e+2 b d)+b^2 e^2+12 c^2 d^2\right )}{2 c^2}+\frac{e x \left (-2 c^2 d e (4 a e+3 b d)+b c e^2 (3 a e+4 b d)-b^3 e^3+8 c^3 d^3\right )}{c^3}-\frac{e \sqrt{b^2-4 a c} (2 c d-b e) \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^4}+\frac{e^3 x^3 (8 c d-b e)}{3 c}+\frac{e^4 x^4}{2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e*(8*c^3*d^3 - b^3*e^3 + b*c*e^2*(4*b*d + 3*a*e) - 2*c^2*d*e*(3*b*d + 4*a*e))*x)/c^3 + (e^2*(12*c^2*d^2 + b^2
*e^2 - 2*c*e*(2*b*d + a*e))*x^2)/(2*c^2) + (e^3*(8*c*d - b*e)*x^3)/(3*c) + (e^4*x^4)/2 - (Sqrt[b^2 - 4*a*c]*e*
(2*c*d - b*e)*(2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/c^4 + ((2*c^4*
d^4 + b^4*e^4 - 4*b^2*c*e^3*(b*d + a*e) - 4*c^3*d^2*e*(b*d + 3*a*e) + 2*c^2*e^2*(3*b^2*d^2 + 6*a*b*d*e + a^2*e
^2))*Log[a + b*x + c*x^2])/(2*c^4)

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{(b+2 c x) (d+e x)^4}{a+b x+c x^2} \, dx &=\int \left (\frac{e \left (8 c^3 d^3-b^3 e^3+b c e^2 (4 b d+3 a e)-2 c^2 d e (3 b d+4 a e)\right )}{c^3}+\frac{e^2 \left (12 c^2 d^2+b^2 e^2-2 c e (2 b d+a e)\right ) x}{c^2}+\frac{e^3 (8 c d-b e) x^2}{c}+2 e^4 x^3+\frac{-4 a b^2 c d e^3+a b^3 e^4-8 a c^2 d e \left (c d^2-a e^2\right )+b c \left (c^2 d^4+6 a c d^2 e^2-3 a^2 e^4\right )+\left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) x}{c^3 \left (a+b x+c x^2\right )}\right ) \, dx\\ &=\frac{e \left (8 c^3 d^3-b^3 e^3+b c e^2 (4 b d+3 a e)-2 c^2 d e (3 b d+4 a e)\right ) x}{c^3}+\frac{e^2 \left (12 c^2 d^2+b^2 e^2-2 c e (2 b d+a e)\right ) x^2}{2 c^2}+\frac{e^3 (8 c d-b e) x^3}{3 c}+\frac{e^4 x^4}{2}+\frac{\int \frac{-4 a b^2 c d e^3+a b^3 e^4-8 a c^2 d e \left (c d^2-a e^2\right )+b c \left (c^2 d^4+6 a c d^2 e^2-3 a^2 e^4\right )+\left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) x}{a+b x+c x^2} \, dx}{c^3}\\ &=\frac{e \left (8 c^3 d^3-b^3 e^3+b c e^2 (4 b d+3 a e)-2 c^2 d e (3 b d+4 a e)\right ) x}{c^3}+\frac{e^2 \left (12 c^2 d^2+b^2 e^2-2 c e (2 b d+a e)\right ) x^2}{2 c^2}+\frac{e^3 (8 c d-b e) x^3}{3 c}+\frac{e^4 x^4}{2}+\frac{\left (\left (b^2-4 a c\right ) e (2 c d-b e) \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right )\right ) \int \frac{1}{a+b x+c x^2} \, dx}{2 c^4}+\frac{\left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) \int \frac{b+2 c x}{a+b x+c x^2} \, dx}{2 c^4}\\ &=\frac{e \left (8 c^3 d^3-b^3 e^3+b c e^2 (4 b d+3 a e)-2 c^2 d e (3 b d+4 a e)\right ) x}{c^3}+\frac{e^2 \left (12 c^2 d^2+b^2 e^2-2 c e (2 b d+a e)\right ) x^2}{2 c^2}+\frac{e^3 (8 c d-b e) x^3}{3 c}+\frac{e^4 x^4}{2}+\frac{\left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) \log \left (a+b x+c x^2\right )}{2 c^4}-\frac{\left (\left (b^2-4 a c\right ) e (2 c d-b e) \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{c^4}\\ &=\frac{e \left (8 c^3 d^3-b^3 e^3+b c e^2 (4 b d+3 a e)-2 c^2 d e (3 b d+4 a e)\right ) x}{c^3}+\frac{e^2 \left (12 c^2 d^2+b^2 e^2-2 c e (2 b d+a e)\right ) x^2}{2 c^2}+\frac{e^3 (8 c d-b e) x^3}{3 c}+\frac{e^4 x^4}{2}-\frac{\sqrt{b^2-4 a c} e (2 c d-b e) \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^4}+\frac{\left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) \log \left (a+b x+c x^2\right )}{2 c^4}\\ \end{align*}

Mathematica [A]  time = 0.23506, size = 297, normalized size = 0.99 \[ \frac{3 \left (2 c^2 e^2 \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )-4 b^2 c e^3 (a e+b d)-4 c^3 d^2 e (3 a e+b d)+b^4 e^4+2 c^4 d^4\right ) \log (a+x (b+c x))+3 c^2 e^2 x^2 \left (-2 c e (a e+2 b d)+b^2 e^2+12 c^2 d^2\right )+6 c e x \left (-2 c^2 d e (4 a e+3 b d)+b c e^2 (3 a e+4 b d)-b^3 e^3+8 c^3 d^3\right )-6 e \sqrt{4 a c-b^2} (2 c d-b e) \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )+2 c^3 e^3 x^3 (8 c d-b e)+3 c^4 e^4 x^4}{6 c^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*c*e*(8*c^3*d^3 - b^3*e^3 + b*c*e^2*(4*b*d + 3*a*e) - 2*c^2*d*e*(3*b*d + 4*a*e))*x + 3*c^2*e^2*(12*c^2*d^2 +
b^2*e^2 - 2*c*e*(2*b*d + a*e))*x^2 + 2*c^3*e^3*(8*c*d - b*e)*x^3 + 3*c^4*e^4*x^4 - 6*Sqrt[-b^2 + 4*a*c]*e*(2*
c*d - b*e)*(2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]] + 3*(2*c^4*d^4 + b
^4*e^4 - 4*b^2*c*e^3*(b*d + a*e) - 4*c^3*d^2*e*(b*d + 3*a*e) + 2*c^2*e^2*(3*b^2*d^2 + 6*a*b*d*e + a^2*e^2))*Lo
g[a + x*(b + c*x)])/(6*c^4)

________________________________________________________________________________________

Maple [B]  time = 0.009, size = 781, normalized size = 2.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(c*x^2+b*x+a)*d^4+4*e^3/c^2*b^2*d*x-2*e^3/c*x^2*b*d+3*e^4/c^2*a*b*x-8*e^3/c*a*d*x-2/c^3*ln(c*x^2+b*x+a)*a*b^
2*e^4-6*e^2/c*b*d^2*x-2/c*ln(c*x^2+b*x+a)*b*d^3*e-6/c*ln(c*x^2+b*x+a)*a*d^2*e^2+3/c^2*ln(c*x^2+b*x+a)*b^2*d^2*
e^2-2/c^3*ln(c*x^2+b*x+a)*b^3*d*e^3+1/2*e^4*x^4-1/c^4/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^
5*e^4-16/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*d^3*e-20/c^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x+
b)/(4*a*c-b^2)^(1/2))*a*b^2*d*e^3+24/c/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b*d^2*e^2+4/c/(
4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^2*d^3*e+6/c^2*ln(c*x^2+b*x+a)*a*b*d*e^3-8/c^2/(4*a*c-b^
2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*e^4*a^2*b+16/c/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/
2))*a^2*d*e^3+6/c^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b^3*e^4+4/c^3/(4*a*c-b^2)^(1/2)*ar
ctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^4*d*e^3-6/c^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^3*d^
2*e^2-e^4/c^3*b^3*x-e^4/c*x^2*a+1/2*e^4/c^2*x^2*b^2-1/3*e^4/c*x^3*b+1/c^2*ln(c*x^2+b*x+a)*a^2*e^4+1/2/c^4*ln(c
*x^2+b*x+a)*b^4*e^4+8*e*d^3*x+8/3*e^3*x^3*d+6*e^2*x^2*d^2

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.59543, size = 1480, normalized size = 4.95 \begin{align*} \left [\frac{3 \, c^{4} e^{4} x^{4} + 2 \,{\left (8 \, c^{4} d e^{3} - b c^{3} e^{4}\right )} x^{3} + 3 \,{\left (12 \, c^{4} d^{2} e^{2} - 4 \, b c^{3} d e^{3} +{\left (b^{2} c^{2} - 2 \, a c^{3}\right )} e^{4}\right )} x^{2} + 3 \,{\left (4 \, c^{3} d^{3} e - 6 \, b c^{2} d^{2} e^{2} + 4 \,{\left (b^{2} c - a c^{2}\right )} d e^{3} -{\left (b^{3} - 2 \, a b c\right )} e^{4}\right )} \sqrt{b^{2} - 4 \, a c} \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 (8 \, c^{4} d^{3} e - 6 \, b c^{3} d^{2} e^{2} + 4 \,{\left (b^{2} c^{2} - 2 \, a c^{3}\right )} d e^{3} -{\left (b^{3} c - 3 \, a b c^{2}\right )} e^{4}\right )} x + 3 \,{\left (2 \, c^{4} d^{4} - 4 \, b c^{3} d^{3} e + 6 \,{\left (b^{2} c^{2} - 2 \, a c^{3}\right )} d^{2} e^{2} - 4 \,{\left (b^{3} c - 3 \, a b c^{2}\right )} d e^{3} +{\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} e^{4}\right )} \log \left (c x^{2} + b x + a\right )}{6 \, c^{4}}, \frac{3 \, c^{4} e^{4} x^{4} + 2 \,{\left (8 \, c^{4} d e^{3} - b c^{3} e^{4}\right )} x^{3} + 3 \,{\left (12 \, c^{4} d^{2} e^{2} - 4 \, b c^{3} d e^{3} +{\left (b^{2} c^{2} - 2 \, a c^{3}\right )} e^{4}\right )} x^{2} - 6 \,{\left (4 \, c^{3} d^{3} e - 6 \, b c^{2} d^{2} e^{2} + 4 \,{\left (b^{2} c - a c^{2}\right )} d e^{3} -{\left (b^{3} - 2 \, a b c\right )} e^{4}\right )} \sqrt{-b^{2} + 4 \, a c} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + 6 \,{\left (8 \, c^{4} d^{3} e - 6 \, b c^{3} d^{2} e^{2} + 4 \,{\left (b^{2} c^{2} - 2 \, a c^{3}\right )} d e^{3} -{\left (b^{3} c - 3 \, a b c^{2}\right )} e^{4}\right )} x + 3 \,{\left (2 \, c^{4} d^{4} - 4 \, b c^{3} d^{3} e + 6 \,{\left (b^{2} c^{2} - 2 \, a c^{3}\right )} d^{2} e^{2} - 4 \,{\left (b^{3} c - 3 \, a b c^{2}\right )} d e^{3} +{\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} e^{4}\right )} \log \left (c x^{2} + b x + a\right )}{6 \, c^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [B]  time = 7.87196, size = 1064, normalized size = 3.56 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e**4*x**4/2 + (-e*sqrt(-4*a*c + b**2)*(b*e - 2*c*d)*(2*a*c*e**2 - b**2*e**2 + 2*b*c*d*e - 2*c**2*d**2)/(2*c**4
) + (2*a**2*c**2*e**4 - 4*a*b**2*c*e**4 + 12*a*b*c**2*d*e**3 - 12*a*c**3*d**2*e**2 + b**4*e**4 - 4*b**3*c*d*e*
*3 + 6*b**2*c**2*d**2*e**2 - 4*b*c**3*d**3*e + 2*c**4*d**4)/(2*c**4))*log(x + (a**2*c*e**4 - a*b**2*e**4 + 4*a
*b*c*d*e**3 - 6*a*c**2*d**2*e**2 + c**3*d**4 - c**3*(-e*sqrt(-4*a*c + b**2)*(b*e - 2*c*d)*(2*a*c*e**2 - b**2*e
**2 + 2*b*c*d*e - 2*c**2*d**2)/(2*c**4) + (2*a**2*c**2*e**4 - 4*a*b**2*c*e**4 + 12*a*b*c**2*d*e**3 - 12*a*c**3
*d**2*e**2 + b**4*e**4 - 4*b**3*c*d*e**3 + 6*b**2*c**2*d**2*e**2 - 4*b*c**3*d**3*e + 2*c**4*d**4)/(2*c**4)))/(
2*a*b*c*e**4 - 4*a*c**2*d*e**3 - b**3*e**4 + 4*b**2*c*d*e**3 - 6*b*c**2*d**2*e**2 + 4*c**3*d**3*e)) + (e*sqrt(
-4*a*c + b**2)*(b*e - 2*c*d)*(2*a*c*e**2 - b**2*e**2 + 2*b*c*d*e - 2*c**2*d**2)/(2*c**4) + (2*a**2*c**2*e**4 -
4*a*b**2*c*e**4 + 12*a*b*c**2*d*e**3 - 12*a*c**3*d**2*e**2 + b**4*e**4 - 4*b**3*c*d*e**3 + 6*b**2*c**2*d**2*e
**2 - 4*b*c**3*d**3*e + 2*c**4*d**4)/(2*c**4))*log(x + (a**2*c*e**4 - a*b**2*e**4 + 4*a*b*c*d*e**3 - 6*a*c**2*
d**2*e**2 + c**3*d**4 - c**3*(e*sqrt(-4*a*c + b**2)*(b*e - 2*c*d)*(2*a*c*e**2 - b**2*e**2 + 2*b*c*d*e - 2*c**2
*d**2)/(2*c**4) + (2*a**2*c**2*e**4 - 4*a*b**2*c*e**4 + 12*a*b*c**2*d*e**3 - 12*a*c**3*d**2*e**2 + b**4*e**4 -
4*b**3*c*d*e**3 + 6*b**2*c**2*d**2*e**2 - 4*b*c**3*d**3*e + 2*c**4*d**4)/(2*c**4)))/(2*a*b*c*e**4 - 4*a*c**2*
d*e**3 - b**3*e**4 + 4*b**2*c*d*e**3 - 6*b*c**2*d**2*e**2 + 4*c**3*d**3*e)) - x**3*(b*e**4 - 8*c*d*e**3)/(3*c)
- x**2*(2*a*c*e**4 - b**2*e**4 + 4*b*c*d*e**3 - 12*c**2*d**2*e**2)/(2*c**2) + x*(3*a*b*c*e**4 - 8*a*c**2*d*e*
*3 - b**3*e**4 + 4*b**2*c*d*e**3 - 6*b*c**2*d**2*e**2 + 8*c**3*d**3*e)/c**3

________________________________________________________________________________________

Giac [A]  time = 1.18247, size = 541, normalized size = 1.81 \begin{align*} \frac{{\left (2 \, c^{4} d^{4} - 4 \, b c^{3} d^{3} e + 6 \, b^{2} c^{2} d^{2} e^{2} - 12 \, a c^{3} d^{2} e^{2} - 4 \, b^{3} c d e^{3} + 12 \, a b c^{2} d e^{3} + b^{4} e^{4} - 4 \, a b^{2} c e^{4} + 2 \, a^{2} c^{2} e^{4}\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{4}} + \frac{{\left (4 \, b^{2} c^{3} d^{3} e - 16 \, a c^{4} d^{3} e - 6 \, b^{3} c^{2} d^{2} e^{2} + 24 \, a b c^{3} d^{2} e^{2} + 4 \, b^{4} c d e^{3} - 20 \, a b^{2} c^{2} d e^{3} + 16 \, a^{2} c^{3} d e^{3} - b^{5} e^{4} + 6 \, a b^{3} c e^{4} - 8 \, a^{2} b c^{2} e^{4}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} c^{4}} + \frac{3 \, c^{4} x^{4} e^{4} + 16 \, c^{4} d x^{3} e^{3} + 36 \, c^{4} d^{2} x^{2} e^{2} + 48 \, c^{4} d^{3} x e - 2 \, b c^{3} x^{3} e^{4} - 12 \, b c^{3} d x^{2} e^{3} - 36 \, b c^{3} d^{2} x e^{2} + 3 \, b^{2} c^{2} x^{2} e^{4} - 6 \, a c^{3} x^{2} e^{4} + 24 \, b^{2} c^{2} d x e^{3} - 48 \, a c^{3} d x e^{3} - 6 \, b^{3} c x e^{4} + 18 \, a b c^{2} x e^{4}}{6 \, c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*c^4*d^4 - 4*b*c^3*d^3*e + 6*b^2*c^2*d^2*e^2 - 12*a*c^3*d^2*e^2 - 4*b^3*c*d*e^3 + 12*a*b*c^2*d*e^3 + b^4
*e^4 - 4*a*b^2*c*e^4 + 2*a^2*c^2*e^4)*log(c*x^2 + b*x + a)/c^4 + (4*b^2*c^3*d^3*e - 16*a*c^4*d^3*e - 6*b^3*c^2
*d^2*e^2 + 24*a*b*c^3*d^2*e^2 + 4*b^4*c*d*e^3 - 20*a*b^2*c^2*d*e^3 + 16*a^2*c^3*d*e^3 - b^5*e^4 + 6*a*b^3*c*e^
4 - 8*a^2*b*c^2*e^4)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*c^4) + 1/6*(3*c^4*x^4*e^4 + 16
*c^4*d*x^3*e^3 + 36*c^4*d^2*x^2*e^2 + 48*c^4*d^3*x*e - 2*b*c^3*x^3*e^4 - 12*b*c^3*d*x^2*e^3 - 36*b*c^3*d^2*x*e
^2 + 3*b^2*c^2*x^2*e^4 - 6*a*c^3*x^2*e^4 + 24*b^2*c^2*d*x*e^3 - 48*a*c^3*d*x*e^3 - 6*b^3*c*x*e^4 + 18*a*b*c^2*
x*e^4)/c^4