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

Optimal. Leaf size=192 \[ \frac{\left (-12 a^2 c^2 e+12 a b^2 c e-6 a b c^2 d+b^3 c d-2 b^4 e\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{3/2}}-\frac{x \left (6 a c e-2 b^2 e+b c d\right )}{c^2 \left (b^2-4 a c\right )}+\frac{x^2 \left (x \left (2 a c e+b^2 (-e)+b c d\right )+a (2 c d-b e)\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{(c d-2 b e) \log \left (a+b x+c x^2\right )}{2 c^3} \]

[Out]

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

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Rubi [A]  time = 0.312178, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {818, 773, 634, 618, 206, 628} \[ \frac{\left (-12 a^2 c^2 e+12 a b^2 c e-6 a b c^2 d+b^3 c d-2 b^4 e\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{3/2}}-\frac{x \left (6 a c e-2 b^2 e+b c d\right )}{c^2 \left (b^2-4 a c\right )}+\frac{x^2 \left (x \left (2 a c e+b^2 (-e)+b c d\right )+a (2 c d-b e)\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{(c d-2 b e) \log \left (a+b x+c x^2\right )}{2 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 818

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g
- c*(b*e*f + b*d*g + 2*a*e*g))*x))/(c*(p + 1)*(b^2 - 4*a*c)), x] - Dist[1/(c*(p + 1)*(b^2 - 4*a*c)), Int[(d +
e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Simp[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a
*e*(e*f*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*(m + p + 1) + 2*c^2*d*f*(m +
2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2*p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && Ne
Q[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] &&
RationalQ[a, b, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rule 773

Int[(((d_.) + (e_.)*(x_))*((f_) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*g*x)/
c, x] + Dist[1/c, Int[(c*d*f - a*e*g + (c*e*f + c*d*g - b*e*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]

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

Mathematica [A]  time = 0.316631, size = 190, normalized size = 0.99 \[ \frac{\frac{2 \left (a^2 c (3 b e-2 c (d+e x))+a b \left (b^2 (-e)+b c (d+4 e x)-3 c^2 d x\right )+b^3 x (c d-b e)\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}-\frac{2 \left (12 a^2 c^2 e-12 a b^2 c e+6 a b c^2 d-b^3 c d+2 b^4 e\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}+(c d-2 b e) \log (a+x (b+c x))+2 c e x}{2 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.011, size = 639, normalized size = 3.3 \begin{align*}{\frac{ex}{{c}^{2}}}+2\,{\frac{x{a}^{2}e}{c \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-4\,{\frac{ax{b}^{2}e}{{c}^{2} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}+3\,{\frac{abdx}{c \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{x{b}^{4}e}{{c}^{3} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-{\frac{x{b}^{3}d}{{c}^{2} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-3\,{\frac{{a}^{2}be}{{c}^{2} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}+2\,{\frac{{a}^{2}d}{c \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{a{b}^{3}e}{{c}^{3} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-{\frac{{b}^{2}ad}{{c}^{2} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-4\,{\frac{\ln \left ( c{x}^{2}+bx+a \right ) abe}{{c}^{2} \left ( 4\,ac-{b}^{2} \right ) }}+2\,{\frac{\ln \left ( c{x}^{2}+bx+a \right ) ad}{c \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ){b}^{3}e}{{c}^{3} \left ( 4\,ac-{b}^{2} \right ) }}-{\frac{\ln \left ( c{x}^{2}+bx+a \right ){b}^{2}d}{2\,{c}^{2} \left ( 4\,ac-{b}^{2} \right ) }}-12\,{\frac{{a}^{2}e}{c \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+12\,{\frac{a{b}^{2}e}{{c}^{2} \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-6\,{\frac{abd}{c \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-2\,{\frac{{b}^{4}e}{{c}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+{\frac{{b}^{3}d}{{c}^{2}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ) \left ( 4\,ac-{b}^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e*x/c^2+2/c/(c*x^2+b*x+a)/(4*a*c-b^2)*x*a^2*e-4/c^2/(c*x^2+b*x+a)/(4*a*c-b^2)*x*a*b^2*e+3/c/(c*x^2+b*x+a)/(4*a
*c-b^2)*x*a*b*d+1/c^3/(c*x^2+b*x+a)/(4*a*c-b^2)*x*b^4*e-1/c^2/(c*x^2+b*x+a)/(4*a*c-b^2)*x*b^3*d-3/c^2/(c*x^2+b
*x+a)*a^2/(4*a*c-b^2)*b*e+2/c/(c*x^2+b*x+a)*a^2/(4*a*c-b^2)*d+1/c^3/(c*x^2+b*x+a)*a/(4*a*c-b^2)*b^3*e-1/c^2/(c
*x^2+b*x+a)*a/(4*a*c-b^2)*b^2*d-4/c^2/(4*a*c-b^2)*ln(c*x^2+b*x+a)*a*b*e+2/c/(4*a*c-b^2)*ln(c*x^2+b*x+a)*a*d+1/
c^3/(4*a*c-b^2)*ln(c*x^2+b*x+a)*b^3*e-1/2/c^2/(4*a*c-b^2)*ln(c*x^2+b*x+a)*b^2*d-12/c/(4*a*c-b^2)^(3/2)*arctan(
(2*c*x+b)/(4*a*c-b^2)^(1/2))*a^2*e+12/c^2/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b^2*e-6/c/(4
*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b*d-2/c^3/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)
^(1/2))*b^4*e+1/c^2/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^3*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^3*(e*x+d)/(c*x^2+b*x+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.42164, size = 2703, normalized size = 14.08 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(2*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*e*x^3 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*e*x^2 + (((b^3*c^2
- 6*a*b*c^3)*d - 2*(b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*e)*x^2 + (a*b^3*c - 6*a^2*b*c^2)*d - 2*(a*b^4 - 6*a^2*b^
2*c + 6*a^3*c^2)*e + ((b^4*c - 6*a*b^2*c^2)*d - 2*(b^5 - 6*a*b^3*c + 6*a^2*b*c^2)*e)*x)*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)) + 2*(a*b^4*c - 6*a^2*b^
2*c^2 + 8*a^3*c^3)*d - 2*(a*b^5 - 7*a^2*b^3*c + 12*a^3*b*c^2)*e + 2*((b^5*c - 7*a*b^3*c^2 + 12*a^2*b*c^3)*d -
(b^6 - 9*a*b^4*c + 26*a^2*b^2*c^2 - 24*a^3*c^3)*e)*x + (((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d - 2*(b^5*c - 8
*a*b^3*c^2 + 16*a^2*b*c^3)*e)*x^2 + (a*b^4*c - 8*a^2*b^2*c^2 + 16*a^3*c^3)*d - 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3
*b*c^2)*e + ((b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d - 2*(b^6 - 8*a*b^4*c + 16*a^2*b^2*c^2)*e)*x)*log(c*x^2 + b
*x + a))/(a*b^4*c^3 - 8*a^2*b^2*c^4 + 16*a^3*c^5 + (b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*x^2 + (b^5*c^3 - 8*a*b
^3*c^4 + 16*a^2*b*c^5)*x), 1/2*(2*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*e*x^3 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2
*b*c^3)*e*x^2 + 2*(((b^3*c^2 - 6*a*b*c^3)*d - 2*(b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*e)*x^2 + (a*b^3*c - 6*a^2*b*
c^2)*d - 2*(a*b^4 - 6*a^2*b^2*c + 6*a^3*c^2)*e + ((b^4*c - 6*a*b^2*c^2)*d - 2*(b^5 - 6*a*b^3*c + 6*a^2*b*c^2)*
e)*x)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + 2*(a*b^4*c - 6*a^2*b^2*c^2 +
8*a^3*c^3)*d - 2*(a*b^5 - 7*a^2*b^3*c + 12*a^3*b*c^2)*e + 2*((b^5*c - 7*a*b^3*c^2 + 12*a^2*b*c^3)*d - (b^6 - 9
*a*b^4*c + 26*a^2*b^2*c^2 - 24*a^3*c^3)*e)*x + (((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d - 2*(b^5*c - 8*a*b^3*c
^2 + 16*a^2*b*c^3)*e)*x^2 + (a*b^4*c - 8*a^2*b^2*c^2 + 16*a^3*c^3)*d - 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*
e + ((b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d - 2*(b^6 - 8*a*b^4*c + 16*a^2*b^2*c^2)*e)*x)*log(c*x^2 + b*x + a))
/(a*b^4*c^3 - 8*a^2*b^2*c^4 + 16*a^3*c^5 + (b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*x^2 + (b^5*c^3 - 8*a*b^3*c^4 +
16*a^2*b*c^5)*x)]

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Sympy [B]  time = 4.93313, size = 1248, normalized size = 6.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-sqrt(-(4*a*c - b**2)**3)*(12*a**2*c**2*e - 12*a*b**2*c*e + 6*a*b*c**2*d + 2*b**4*e - b**3*c*d)/(2*c**3*(64*a
**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (2*b*e - c*d)/(2*c**3))*log(x + (-10*a**2*b*c*e - 16*a**
2*c**4*(-sqrt(-(4*a*c - b**2)**3)*(12*a**2*c**2*e - 12*a*b**2*c*e + 6*a*b*c**2*d + 2*b**4*e - b**3*c*d)/(2*c**
3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (2*b*e - c*d)/(2*c**3)) + 8*a**2*c**2*d + 2*a*b**
3*e + 8*a*b**2*c**3*(-sqrt(-(4*a*c - b**2)**3)*(12*a**2*c**2*e - 12*a*b**2*c*e + 6*a*b*c**2*d + 2*b**4*e - b**
3*c*d)/(2*c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (2*b*e - c*d)/(2*c**3)) - a*b**2*c*d
- b**4*c**2*(-sqrt(-(4*a*c - b**2)**3)*(12*a**2*c**2*e - 12*a*b**2*c*e + 6*a*b*c**2*d + 2*b**4*e - b**3*c*d)/
(2*c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (2*b*e - c*d)/(2*c**3)))/(12*a**2*c**2*e -
12*a*b**2*c*e + 6*a*b*c**2*d + 2*b**4*e - b**3*c*d)) + (sqrt(-(4*a*c - b**2)**3)*(12*a**2*c**2*e - 12*a*b**2*c
*e + 6*a*b*c**2*d + 2*b**4*e - b**3*c*d)/(2*c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (2
*b*e - c*d)/(2*c**3))*log(x + (-10*a**2*b*c*e - 16*a**2*c**4*(sqrt(-(4*a*c - b**2)**3)*(12*a**2*c**2*e - 12*a*
b**2*c*e + 6*a*b*c**2*d + 2*b**4*e - b**3*c*d)/(2*c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)
) - (2*b*e - c*d)/(2*c**3)) + 8*a**2*c**2*d + 2*a*b**3*e + 8*a*b**2*c**3*(sqrt(-(4*a*c - b**2)**3)*(12*a**2*c*
*2*e - 12*a*b**2*c*e + 6*a*b*c**2*d + 2*b**4*e - b**3*c*d)/(2*c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b*
*4*c - b**6)) - (2*b*e - c*d)/(2*c**3)) - a*b**2*c*d - b**4*c**2*(sqrt(-(4*a*c - b**2)**3)*(12*a**2*c**2*e - 1
2*a*b**2*c*e + 6*a*b*c**2*d + 2*b**4*e - b**3*c*d)/(2*c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b
**6)) - (2*b*e - c*d)/(2*c**3)))/(12*a**2*c**2*e - 12*a*b**2*c*e + 6*a*b*c**2*d + 2*b**4*e - b**3*c*d)) + (-3*
a**2*b*c*e + 2*a**2*c**2*d + a*b**3*e - a*b**2*c*d + x*(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))/(4*a**2*c**4 - a*b**2*c**3 + x**2*(4*a*c**5 - b**2*c**4) + x*(4*a*b*c**4 - b**3*c**3)) + e*x/c**2

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Giac [A]  time = 1.24602, size = 317, normalized size = 1.65 \begin{align*} -\frac{{\left (b^{3} c d - 6 \, a b c^{2} d - 2 \, b^{4} e + 12 \, a b^{2} c e - 12 \, a^{2} c^{2} e\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{x e}{c^{2}} + \frac{{\left (c d - 2 \, b e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{3}} + \frac{\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 )} x}{c} + \frac{a b^{2} c d - 2 \, a^{2} c^{2} d - a b^{3} e + 3 \, a^{2} b c e}{c}}{{\left (c x^{2} + b x + a\right )}{\left (b^{2} - 4 \, a c\right )} c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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