3.1532 \(\int \frac{(b+2 c x) (d+e x)^4}{(a+b x+c x^2)^2} \, dx\)

Optimal. Leaf size=172 \[ \frac{2 e^2 \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{c^3}-\frac{4 e (2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^3 \sqrt{b^2-4 a c}}-\frac{(d+e x)^4}{a+b x+c x^2}+\frac{4 e^3 x (3 c d-b e)}{c^2}+\frac{2 e^4 x^2}{c} \]

[Out]

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

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Rubi [A]  time = 0.21581, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {768, 701, 634, 618, 206, 628} \[ \frac{2 e^2 \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{c^3}-\frac{4 e (2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^3 \sqrt{b^2-4 a c}}-\frac{(d+e x)^4}{a+b x+c x^2}+\frac{4 e^3 x (3 c d-b e)}{c^2}+\frac{2 e^4 x^2}{c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 768

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Sim
p[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(2*c*(p + 1)), x] - Dist[(e*g*m)/(2*c*(p + 1)), Int[(d + e*x)^(m -
1)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[2*c*f - b*g, 0] && LtQ[p, -1]
&& GtQ[m, 0]

Rule 701

Int[((d_.) + (e_.)*(x_))^(m_)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[(d + e*x)
^m, a + b*x + c*x^2, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2,
0] && NeQ[2*c*d - b*e, 0] && IGtQ[m, 1] && (NeQ[d, 0] || GtQ[m, 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{(b+2 c x) (d+e x)^4}{\left (a+b x+c x^2\right )^2} \, dx &=-\frac{(d+e x)^4}{a+b x+c x^2}+(4 e) \int \frac{(d+e x)^3}{a+b x+c x^2} \, dx\\ &=-\frac{(d+e x)^4}{a+b x+c x^2}+(4 e) \int \left (\frac{e^2 (3 c d-b e)}{c^2}+\frac{e^3 x}{c}+\frac{c^2 d^3-3 a c d e^2+a b e^3+e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right ) x}{c^2 \left (a+b x+c x^2\right )}\right ) \, dx\\ &=\frac{4 e^3 (3 c d-b e) x}{c^2}+\frac{2 e^4 x^2}{c}-\frac{(d+e x)^4}{a+b x+c x^2}+\frac{(4 e) \int \frac{c^2 d^3-3 a c d e^2+a b e^3+e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right ) x}{a+b x+c x^2} \, dx}{c^2}\\ &=\frac{4 e^3 (3 c d-b e) x}{c^2}+\frac{2 e^4 x^2}{c}-\frac{(d+e x)^4}{a+b x+c x^2}+\frac{\left (2 e^2 \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right )\right ) \int \frac{b+2 c x}{a+b x+c x^2} \, dx}{c^3}+\frac{\left (2 e (2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )\right ) \int \frac{1}{a+b x+c x^2} \, dx}{c^3}\\ &=\frac{4 e^3 (3 c d-b e) x}{c^2}+\frac{2 e^4 x^2}{c}-\frac{(d+e x)^4}{a+b x+c x^2}+\frac{2 e^2 \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right ) \log \left (a+b x+c x^2\right )}{c^3}-\frac{\left (4 e (2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 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^3}\\ &=\frac{4 e^3 (3 c d-b e) x}{c^2}+\frac{2 e^4 x^2}{c}-\frac{(d+e x)^4}{a+b x+c x^2}-\frac{4 e (2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^3 \sqrt{b^2-4 a c}}+\frac{2 e^2 \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right ) \log \left (a+b x+c x^2\right )}{c^3}\\ \end{align*}

Mathematica [A]  time = 0.312665, size = 241, normalized size = 1.4 \[ \frac{\frac{-c e^3 \left (a^2 e+2 a b (2 d+e x)+4 b^2 d x\right )+b^2 e^4 (a+b x)+2 c^2 d e^2 (3 a d+2 a e x+3 b d x)-c^3 d^3 (d+4 e x)}{a+x (b+c x)}+2 e^2 \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \log (a+x (b+c x))+\frac{4 e (b e-2 c d) \left (c e (3 a e+b d)-b^2 e^2-c^2 d^2\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+c e^3 x (8 c d-3 b e)+c^2 e^4 x^2}{c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.013, size = 618, normalized size = 3.6 \begin{align*}{\frac{{e}^{4}{x}^{2}}{c}}-3\,{\frac{b{e}^{4}x}{{c}^{2}}}+8\,{\frac{d{e}^{3}x}{c}}-2\,{\frac{{e}^{4}xab}{{c}^{2} \left ( c{x}^{2}+bx+a \right ) }}+4\,{\frac{{e}^{3}xad}{c \left ( c{x}^{2}+bx+a \right ) }}+{\frac{{e}^{4}x{b}^{3}}{{c}^{3} \left ( c{x}^{2}+bx+a \right ) }}-4\,{\frac{{e}^{3}x{b}^{2}d}{{c}^{2} \left ( c{x}^{2}+bx+a \right ) }}+6\,{\frac{b{e}^{2}x{d}^{2}}{c \left ( c{x}^{2}+bx+a \right ) }}-4\,{\frac{e{d}^{3}x}{c{x}^{2}+bx+a}}-{\frac{{a}^{2}{e}^{4}}{{c}^{2} \left ( c{x}^{2}+bx+a \right ) }}+{\frac{a{b}^{2}{e}^{4}}{{c}^{3} \left ( c{x}^{2}+bx+a \right ) }}-4\,{\frac{abd{e}^{3}}{{c}^{2} \left ( c{x}^{2}+bx+a \right ) }}+6\,{\frac{a{d}^{2}{e}^{2}}{c \left ( c{x}^{2}+bx+a \right ) }}-{\frac{{d}^{4}}{c{x}^{2}+bx+a}}-2\,{\frac{\ln \left ( c{x}^{2}+bx+a \right ){e}^{4}a}{{c}^{2}}}+2\,{\frac{\ln \left ( c{x}^{2}+bx+a \right ){b}^{2}{e}^{4}}{{c}^{3}}}-6\,{\frac{\ln \left ( c{x}^{2}+bx+a \right ) bd{e}^{3}}{{c}^{2}}}+6\,{\frac{\ln \left ( c{x}^{2}+bx+a \right ){e}^{2}{d}^{2}}{c}}+12\,{\frac{ab{e}^{4}}{{c}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-24\,{\frac{ad{e}^{3}}{c\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+8\,{\frac{e{d}^{3}}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-4\,{\frac{{b}^{3}{e}^{4}}{{c}^{3}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+12\,{\frac{{b}^{2}d{e}^{3}}{{c}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-12\,{\frac{b{e}^{2}{d}^{2}}{c\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \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)^2,x)

[Out]

e^4*x^2/c-3*e^4/c^2*b*x+8*e^3/c*d*x-2/c^2/(c*x^2+b*x+a)*e^4*x*a*b+4/c/(c*x^2+b*x+a)*e^3*x*a*d+1/c^3/(c*x^2+b*x
+a)*e^4*x*b^3-4/c^2/(c*x^2+b*x+a)*e^3*x*b^2*d+6/c/(c*x^2+b*x+a)*e^2*x*b*d^2-4/(c*x^2+b*x+a)*e*x*d^3-1/c^2/(c*x
^2+b*x+a)*a^2*e^4+1/c^3/(c*x^2+b*x+a)*a*b^2*e^4-4/c^2/(c*x^2+b*x+a)*a*b*d*e^3+6/c/(c*x^2+b*x+a)*a*d^2*e^2-1/(c
*x^2+b*x+a)*d^4-2/c^2*ln(c*x^2+b*x+a)*e^4*a+2/c^3*ln(c*x^2+b*x+a)*b^2*e^4-6/c^2*ln(c*x^2+b*x+a)*b*d*e^3+6/c*ln
(c*x^2+b*x+a)*e^2*d^2+12/c^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b*e^4-24/c/(4*a*c-b^2)^(1
/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*d*e^3+8*e/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*d^3-
4/c^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^3*e^4+12/c^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/
(4*a*c-b^2)^(1/2))*b^2*d*e^3-12/c/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*e^2*d^2

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.49753, size = 3440, normalized size = 20. \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)^2,x, algorithm="fricas")

[Out]

[((b^2*c^3 - 4*a*c^4)*e^4*x^4 - (b^2*c^3 - 4*a*c^4)*d^4 + 6*(a*b^2*c^2 - 4*a^2*c^3)*d^2*e^2 - 4*(a*b^3*c - 4*a
^2*b*c^2)*d*e^3 + (a*b^4 - 5*a^2*b^2*c + 4*a^3*c^2)*e^4 + 2*(4*(b^2*c^3 - 4*a*c^4)*d*e^3 - (b^3*c^2 - 4*a*b*c^
3)*e^4)*x^3 + (8*(b^3*c^2 - 4*a*b*c^3)*d*e^3 - (3*b^4*c - 13*a*b^2*c^2 + 4*a^2*c^3)*e^4)*x^2 + 2*(2*a*c^3*d^3*
e - 3*a*b*c^2*d^2*e^2 + 3*(a*b^2*c - 2*a^2*c^2)*d*e^3 - (a*b^3 - 3*a^2*b*c)*e^4 + (2*c^4*d^3*e - 3*b*c^3*d^2*e
^2 + 3*(b^2*c^2 - 2*a*c^3)*d*e^3 - (b^3*c - 3*a*b*c^2)*e^4)*x^2 + (2*b*c^3*d^3*e - 3*b^2*c^2*d^2*e^2 + 3*(b^3*
c - 2*a*b*c^2)*d*e^3 - (b^4 - 3*a*b^2*c)*e^4)*x)*sqrt(b^2 - 4*a*c)*log((2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c - sq
rt(b^2 - 4*a*c)*(2*c*x + b))/(c*x^2 + b*x + a)) - (4*(b^2*c^3 - 4*a*c^4)*d^3*e - 6*(b^3*c^2 - 4*a*b*c^3)*d^2*e
^2 + 4*(b^4*c - 7*a*b^2*c^2 + 12*a^2*c^3)*d*e^3 - (b^5 - 9*a*b^3*c + 20*a^2*b*c^2)*e^4)*x + 2*(3*(a*b^2*c^2 -
4*a^2*c^3)*d^2*e^2 - 3*(a*b^3*c - 4*a^2*b*c^2)*d*e^3 + (a*b^4 - 5*a^2*b^2*c + 4*a^3*c^2)*e^4 + (3*(b^2*c^3 - 4
*a*c^4)*d^2*e^2 - 3*(b^3*c^2 - 4*a*b*c^3)*d*e^3 + (b^4*c - 5*a*b^2*c^2 + 4*a^2*c^3)*e^4)*x^2 + (3*(b^3*c^2 - 4
*a*b*c^3)*d^2*e^2 - 3*(b^4*c - 4*a*b^2*c^2)*d*e^3 + (b^5 - 5*a*b^3*c + 4*a^2*b*c^2)*e^4)*x)*log(c*x^2 + b*x +
a))/(a*b^2*c^3 - 4*a^2*c^4 + (b^2*c^4 - 4*a*c^5)*x^2 + (b^3*c^3 - 4*a*b*c^4)*x), ((b^2*c^3 - 4*a*c^4)*e^4*x^4
- (b^2*c^3 - 4*a*c^4)*d^4 + 6*(a*b^2*c^2 - 4*a^2*c^3)*d^2*e^2 - 4*(a*b^3*c - 4*a^2*b*c^2)*d*e^3 + (a*b^4 - 5*a
^2*b^2*c + 4*a^3*c^2)*e^4 + 2*(4*(b^2*c^3 - 4*a*c^4)*d*e^3 - (b^3*c^2 - 4*a*b*c^3)*e^4)*x^3 + (8*(b^3*c^2 - 4*
a*b*c^3)*d*e^3 - (3*b^4*c - 13*a*b^2*c^2 + 4*a^2*c^3)*e^4)*x^2 - 4*(2*a*c^3*d^3*e - 3*a*b*c^2*d^2*e^2 + 3*(a*b
^2*c - 2*a^2*c^2)*d*e^3 - (a*b^3 - 3*a^2*b*c)*e^4 + (2*c^4*d^3*e - 3*b*c^3*d^2*e^2 + 3*(b^2*c^2 - 2*a*c^3)*d*e
^3 - (b^3*c - 3*a*b*c^2)*e^4)*x^2 + (2*b*c^3*d^3*e - 3*b^2*c^2*d^2*e^2 + 3*(b^3*c - 2*a*b*c^2)*d*e^3 - (b^4 -
3*a*b^2*c)*e^4)*x)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) - (4*(b^2*c^3 - 4*
a*c^4)*d^3*e - 6*(b^3*c^2 - 4*a*b*c^3)*d^2*e^2 + 4*(b^4*c - 7*a*b^2*c^2 + 12*a^2*c^3)*d*e^3 - (b^5 - 9*a*b^3*c
+ 20*a^2*b*c^2)*e^4)*x + 2*(3*(a*b^2*c^2 - 4*a^2*c^3)*d^2*e^2 - 3*(a*b^3*c - 4*a^2*b*c^2)*d*e^3 + (a*b^4 - 5*
a^2*b^2*c + 4*a^3*c^2)*e^4 + (3*(b^2*c^3 - 4*a*c^4)*d^2*e^2 - 3*(b^3*c^2 - 4*a*b*c^3)*d*e^3 + (b^4*c - 5*a*b^2
*c^2 + 4*a^2*c^3)*e^4)*x^2 + (3*(b^3*c^2 - 4*a*b*c^3)*d^2*e^2 - 3*(b^4*c - 4*a*b^2*c^2)*d*e^3 + (b^5 - 5*a*b^3
*c + 4*a^2*b*c^2)*e^4)*x)*log(c*x^2 + b*x + a))/(a*b^2*c^3 - 4*a^2*c^4 + (b^2*c^4 - 4*a*c^5)*x^2 + (b^3*c^3 -
4*a*b*c^4)*x)]

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Sympy [B]  time = 26.2238, size = 1071, normalized size = 6.23 \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)**2,x)

[Out]

(-2*e**2*(a*c*e**2 - b**2*e**2 + 3*b*c*d*e - 3*c**2*d**2)/c**3 - 2*e*sqrt(-4*a*c + b**2)*(b*e - 2*c*d)*(3*a*c*
e**2 - b**2*e**2 + b*c*d*e - c**2*d**2)/(c**3*(4*a*c - b**2)))*log(x + (8*a**2*c*e**4 - 4*a*b**2*e**4 + 12*a*b
*c*d*e**3 + 4*a*c**3*(-2*e**2*(a*c*e**2 - b**2*e**2 + 3*b*c*d*e - 3*c**2*d**2)/c**3 - 2*e*sqrt(-4*a*c + b**2)*
(b*e - 2*c*d)*(3*a*c*e**2 - b**2*e**2 + b*c*d*e - c**2*d**2)/(c**3*(4*a*c - b**2))) - 24*a*c**2*d**2*e**2 - b*
*2*c**2*(-2*e**2*(a*c*e**2 - b**2*e**2 + 3*b*c*d*e - 3*c**2*d**2)/c**3 - 2*e*sqrt(-4*a*c + b**2)*(b*e - 2*c*d)
*(3*a*c*e**2 - b**2*e**2 + b*c*d*e - c**2*d**2)/(c**3*(4*a*c - b**2))) + 4*b*c**2*d**3*e)/(12*a*b*c*e**4 - 24*
a*c**2*d*e**3 - 4*b**3*e**4 + 12*b**2*c*d*e**3 - 12*b*c**2*d**2*e**2 + 8*c**3*d**3*e)) + (-2*e**2*(a*c*e**2 -
b**2*e**2 + 3*b*c*d*e - 3*c**2*d**2)/c**3 + 2*e*sqrt(-4*a*c + b**2)*(b*e - 2*c*d)*(3*a*c*e**2 - b**2*e**2 + b*
c*d*e - c**2*d**2)/(c**3*(4*a*c - b**2)))*log(x + (8*a**2*c*e**4 - 4*a*b**2*e**4 + 12*a*b*c*d*e**3 + 4*a*c**3*
(-2*e**2*(a*c*e**2 - b**2*e**2 + 3*b*c*d*e - 3*c**2*d**2)/c**3 + 2*e*sqrt(-4*a*c + b**2)*(b*e - 2*c*d)*(3*a*c*
e**2 - b**2*e**2 + b*c*d*e - c**2*d**2)/(c**3*(4*a*c - b**2))) - 24*a*c**2*d**2*e**2 - b**2*c**2*(-2*e**2*(a*c
*e**2 - b**2*e**2 + 3*b*c*d*e - 3*c**2*d**2)/c**3 + 2*e*sqrt(-4*a*c + b**2)*(b*e - 2*c*d)*(3*a*c*e**2 - b**2*e
**2 + b*c*d*e - c**2*d**2)/(c**3*(4*a*c - b**2))) + 4*b*c**2*d**3*e)/(12*a*b*c*e**4 - 24*a*c**2*d*e**3 - 4*b**
3*e**4 + 12*b**2*c*d*e**3 - 12*b*c**2*d**2*e**2 + 8*c**3*d**3*e)) - (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 + x*(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))/(a*c**3 + b*c**3*x + c**4*x**2) + e**4*x**2/c - x*(3*b*e**4 - 8*c*d*e**3)/c**2

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Giac [A]  time = 1.17184, size = 385, normalized size = 2.24 \begin{align*} \frac{2 \,{\left (3 \, c^{2} d^{2} e^{2} - 3 \, b c d e^{3} + b^{2} e^{4} - a c e^{4}\right )} \log \left (c x^{2} + b x + a\right )}{c^{3}} + \frac{4 \,{\left (2 \, c^{3} d^{3} e - 3 \, b c^{2} d^{2} e^{2} + 3 \, b^{2} c d e^{3} - 6 \, a c^{2} d e^{3} - b^{3} e^{4} + 3 \, a b c e^{4}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} c^{3}} + \frac{c^{3} x^{2} e^{4} + 8 \, c^{3} d x e^{3} - 3 \, b c^{2} x e^{4}}{c^{4}} - \frac{c^{3} d^{4} - 6 \, a c^{2} d^{2} e^{2} + 4 \, a b c d e^{3} - a b^{2} e^{4} + a^{2} c e^{4} +{\left (4 \, c^{3} d^{3} e - 6 \, b c^{2} d^{2} e^{2} + 4 \, b^{2} c d e^{3} - 4 \, a c^{2} d e^{3} - b^{3} e^{4} + 2 \, a b c e^{4}\right )} x}{{\left (c x^{2} + b x + a\right )} c^{3}} \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)^2,x, algorithm="giac")

[Out]

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