### 3.2190 $$\int \frac{(d+e x)^5}{(a+b x+c x^2)^2} \, dx$$

Optimal. Leaf size=374 $\frac{(2 c d-b e) \left (-2 c^2 e^2 \left (15 a^2 e^2+10 a b d e+b^2 d^2\right )+4 b^2 c e^3 (5 a e+b d)-4 c^3 d^2 e (b d-5 a e)-3 b^4 e^4+2 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^4 \left (b^2-4 a c\right )^{3/2}}+\frac{e^3 x^2 \left (-2 c e (4 a e+5 b d)+3 b^2 e^2+16 c^2 d^2\right )}{2 c^2 \left (b^2-4 a c\right )}+\frac{e^3 \left (-2 c e (a e+5 b d)+3 b^2 e^2+10 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{2 c^4}+\frac{e^2 x \left (-10 c^2 d e (3 a e+b d)+b c e^2 (11 a e+10 b d)-3 b^3 e^3+12 c^3 d^3\right )}{c^3 \left (b^2-4 a c\right )}+\frac{e^4 x^3 (2 c d-b e)}{c \left (b^2-4 a c\right )}-\frac{(d+e x)^4 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}$

[Out]

(e^2*(12*c^3*d^3 - 3*b^3*e^3 - 10*c^2*d*e*(b*d + 3*a*e) + b*c*e^2*(10*b*d + 11*a*e))*x)/(c^3*(b^2 - 4*a*c)) +
(e^3*(16*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(5*b*d + 4*a*e))*x^2)/(2*c^2*(b^2 - 4*a*c)) + (e^4*(2*c*d - b*e)*x^3)/(c*
(b^2 - 4*a*c)) - ((d + e*x)^4*(b*d - 2*a*e + (2*c*d - b*e)*x))/((b^2 - 4*a*c)*(a + b*x + c*x^2)) + ((2*c*d - b
*e)*(2*c^4*d^4 - 3*b^4*e^4 - 4*c^3*d^2*e*(b*d - 5*a*e) + 4*b^2*c*e^3*(b*d + 5*a*e) - 2*c^2*e^2*(b^2*d^2 + 10*a
*b*d*e + 15*a^2*e^2))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^4*(b^2 - 4*a*c)^(3/2)) + (e^3*(10*c^2*d^2 + 3
*b^2*e^2 - 2*c*e*(5*b*d + a*e))*Log[a + b*x + c*x^2])/(2*c^4)

________________________________________________________________________________________

Rubi [A]  time = 0.72549, antiderivative size = 374, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.3, Rules used = {738, 800, 634, 618, 206, 628} $\frac{(2 c d-b e) \left (-2 c^2 e^2 \left (15 a^2 e^2+10 a b d e+b^2 d^2\right )+4 b^2 c e^3 (5 a e+b d)-4 c^3 d^2 e (b d-5 a e)-3 b^4 e^4+2 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^4 \left (b^2-4 a c\right )^{3/2}}+\frac{e^3 x^2 \left (-2 c e (4 a e+5 b d)+3 b^2 e^2+16 c^2 d^2\right )}{2 c^2 \left (b^2-4 a c\right )}+\frac{e^3 \left (-2 c e (a e+5 b d)+3 b^2 e^2+10 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{2 c^4}+\frac{e^2 x \left (-10 c^2 d e (3 a e+b d)+b c e^2 (11 a e+10 b d)-3 b^3 e^3+12 c^3 d^3\right )}{c^3 \left (b^2-4 a c\right )}+\frac{e^4 x^3 (2 c d-b e)}{c \left (b^2-4 a c\right )}-\frac{(d+e x)^4 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^5/(a + b*x + c*x^2)^2,x]

[Out]

(e^2*(12*c^3*d^3 - 3*b^3*e^3 - 10*c^2*d*e*(b*d + 3*a*e) + b*c*e^2*(10*b*d + 11*a*e))*x)/(c^3*(b^2 - 4*a*c)) +
(e^3*(16*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(5*b*d + 4*a*e))*x^2)/(2*c^2*(b^2 - 4*a*c)) + (e^4*(2*c*d - b*e)*x^3)/(c*
(b^2 - 4*a*c)) - ((d + e*x)^4*(b*d - 2*a*e + (2*c*d - b*e)*x))/((b^2 - 4*a*c)*(a + b*x + c*x^2)) + ((2*c*d - b
*e)*(2*c^4*d^4 - 3*b^4*e^4 - 4*c^3*d^2*e*(b*d - 5*a*e) + 4*b^2*c*e^3*(b*d + 5*a*e) - 2*c^2*e^2*(b^2*d^2 + 10*a
*b*d*e + 15*a^2*e^2))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^4*(b^2 - 4*a*c)^(3/2)) + (e^3*(10*c^2*d^2 + 3
*b^2*e^2 - 2*c*e*(5*b*d + a*e))*Log[a + b*x + c*x^2])/(2*c^4)

Rule 738

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m - 1)*(
d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/((p + 1)*(b^2 -
4*a*c)), Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2*c*d^2*(2*p + 3) + e*(b*e - 2*d*
c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), 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] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, b, c, d,
e, m, p, x]

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{(d+e x)^5}{\left (a+b x+c x^2\right )^2} \, dx &=-\frac{(d+e x)^4 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{\int \frac{(d+e x)^3 \left (2 c d^2-e (5 b d-8 a e)-3 e (2 c d-b e) x\right )}{a+b x+c x^2} \, dx}{-b^2+4 a c}\\ &=-\frac{(d+e x)^4 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{\int \left (-\frac{e^2 \left (12 c^3 d^3-3 b^3 e^3-10 c^2 d e (b d+3 a e)+b c e^2 (10 b d+11 a e)\right )}{c^3}-\frac{e^3 \left (16 c^2 d^2+3 b^2 e^2-2 c e (5 b d+4 a e)\right ) x}{c^2}-\frac{3 e^4 (2 c d-b e) x^2}{c}+\frac{2 c^4 d^5-3 a b^3 e^5-5 c^3 d^3 e (b d-4 a e)-10 a c^2 d e^3 (b d+3 a e)+a b c e^4 (10 b d+11 a e)-\left (b^2-4 a c\right ) e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right ) x}{c^3 \left (a+b x+c x^2\right )}\right ) \, dx}{-b^2+4 a c}\\ &=\frac{e^2 \left (12 c^3 d^3-3 b^3 e^3-10 c^2 d e (b d+3 a e)+b c e^2 (10 b d+11 a e)\right ) x}{c^3 \left (b^2-4 a c\right )}+\frac{e^3 \left (16 c^2 d^2+3 b^2 e^2-2 c e (5 b d+4 a e)\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}+\frac{e^4 (2 c d-b e) x^3}{c \left (b^2-4 a c\right )}-\frac{(d+e x)^4 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{\int \frac{2 c^4 d^5-3 a b^3 e^5-5 c^3 d^3 e (b d-4 a e)-10 a c^2 d e^3 (b d+3 a e)+a b c e^4 (10 b d+11 a e)-\left (b^2-4 a c\right ) e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right ) x}{a+b x+c x^2} \, dx}{c^3 \left (b^2-4 a c\right )}\\ &=\frac{e^2 \left (12 c^3 d^3-3 b^3 e^3-10 c^2 d e (b d+3 a e)+b c e^2 (10 b d+11 a e)\right ) x}{c^3 \left (b^2-4 a c\right )}+\frac{e^3 \left (16 c^2 d^2+3 b^2 e^2-2 c e (5 b d+4 a e)\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}+\frac{e^4 (2 c d-b e) x^3}{c \left (b^2-4 a c\right )}-\frac{(d+e x)^4 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{\left (e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right )\right ) \int \frac{b+2 c x}{a+b x+c x^2} \, dx}{2 c^4}-\frac{\left ((2 c d-b e) \left (2 c^4 d^4-3 b^4 e^4-4 c^3 d^2 e (b d-5 a e)+4 b^2 c e^3 (b d+5 a e)-2 c^2 e^2 \left (b^2 d^2+10 a b d e+15 a^2 e^2\right )\right )\right ) \int \frac{1}{a+b x+c x^2} \, dx}{2 c^4 \left (b^2-4 a c\right )}\\ &=\frac{e^2 \left (12 c^3 d^3-3 b^3 e^3-10 c^2 d e (b d+3 a e)+b c e^2 (10 b d+11 a e)\right ) x}{c^3 \left (b^2-4 a c\right )}+\frac{e^3 \left (16 c^2 d^2+3 b^2 e^2-2 c e (5 b d+4 a e)\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}+\frac{e^4 (2 c d-b e) x^3}{c \left (b^2-4 a c\right )}-\frac{(d+e x)^4 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right ) \log \left (a+b x+c x^2\right )}{2 c^4}+\frac{\left ((2 c d-b e) \left (2 c^4 d^4-3 b^4 e^4-4 c^3 d^2 e (b d-5 a e)+4 b^2 c e^3 (b d+5 a e)-2 c^2 e^2 \left (b^2 d^2+10 a b d e+15 a^2 e^2\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{c^4 \left (b^2-4 a c\right )}\\ &=\frac{e^2 \left (12 c^3 d^3-3 b^3 e^3-10 c^2 d e (b d+3 a e)+b c e^2 (10 b d+11 a e)\right ) x}{c^3 \left (b^2-4 a c\right )}+\frac{e^3 \left (16 c^2 d^2+3 b^2 e^2-2 c e (5 b d+4 a e)\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}+\frac{e^4 (2 c d-b e) x^3}{c \left (b^2-4 a c\right )}-\frac{(d+e x)^4 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{(2 c d-b e) \left (2 c^4 d^4-3 b^4 e^4-4 c^3 d^2 e (b d-5 a e)+4 b^2 c e^3 (b d+5 a e)-2 c^2 e^2 \left (b^2 d^2+10 a b d e+15 a^2 e^2\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^4 \left (b^2-4 a c\right )^{3/2}}+\frac{e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right ) \log \left (a+b x+c x^2\right )}{2 c^4}\\ \end{align*}

Mathematica [A]  time = 0.708468, size = 422, normalized size = 1.13 $\frac{\frac{2 \left (-2 b^2 c e^2 \left (2 a^2 e^3-5 a c d e (d+2 e x)+5 c^2 d^3 x\right )+b c^2 \left (5 a^2 e^4 (3 d+e x)-10 a c d^2 e^2 (d+3 e x)-c^2 d^4 (d-5 e x)\right )+2 c^2 \left (-5 a^2 c d e^3 (2 d+e x)+a^3 e^5+5 a c^2 d^3 e (d+2 e x)-c^3 d^5 x\right )-5 b^3 c e^3 \left (a e (d+e x)-2 c d^2 x\right )+b^4 e^4 (a e-5 c d x)+b^5 e^5 x\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}+\frac{2 (b e-2 c d) \left (2 c^2 e^2 \left (15 a^2 e^2+10 a b d e+b^2 d^2\right )-4 b^2 c e^3 (5 a e+b d)+4 c^3 d^2 e (b d-5 a e)+3 b^4 e^4-2 c^4 d^4\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}+e^3 \left (-2 c e (a e+5 b d)+3 b^2 e^2+10 c^2 d^2\right ) \log (a+x (b+c x))+2 c e^4 x (5 c d-2 b e)+c^2 e^5 x^2}{2 c^4}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^5/(a + b*x + c*x^2)^2,x]

[Out]

(2*c*e^4*(5*c*d - 2*b*e)*x + c^2*e^5*x^2 + (2*(b^5*e^5*x + b^4*e^4*(a*e - 5*c*d*x) - 5*b^3*c*e^3*(-2*c*d^2*x +
a*e*(d + e*x)) - 2*b^2*c*e^2*(2*a^2*e^3 + 5*c^2*d^3*x - 5*a*c*d*e*(d + 2*e*x)) + 2*c^2*(a^3*e^5 - c^3*d^5*x -
5*a^2*c*d*e^3*(2*d + e*x) + 5*a*c^2*d^3*e*(d + 2*e*x)) + b*c^2*(-(c^2*d^4*(d - 5*e*x)) + 5*a^2*e^4*(3*d + e*x
) - 10*a*c*d^2*e^2*(d + 3*e*x))))/((b^2 - 4*a*c)*(a + x*(b + c*x))) + (2*(-2*c*d + b*e)*(-2*c^4*d^4 + 3*b^4*e^
4 + 4*c^3*d^2*e*(b*d - 5*a*e) - 4*b^2*c*e^3*(b*d + 5*a*e) + 2*c^2*e^2*(b^2*d^2 + 10*a*b*d*e + 15*a^2*e^2))*Arc
Tan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + e^3*(10*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(5*b*d + a*e))
*Log[a + x*(b + c*x)])/(2*c^4)

________________________________________________________________________________________

Maple [B]  time = 0.165, size = 1542, normalized size = 4.1 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^5/(c*x^2+b*x+a)^2,x)

[Out]

1/2*e^5*x^2/c^2+30/c/(c*x^2+b*x+a)/(4*a*c-b^2)*x*a*b*d^2*e^3-20/c^2/(c*x^2+b*x+a)/(4*a*c-b^2)*x*a*b^2*d*e^4-3/
2/c^4/(4*a*c-b^2)*ln(c*x^2+b*x+a)*b^4*e^5+3/c^4/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^5*e^5-
10/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*d^4*e+40/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-
b^2)^(1/2))*d^3*a*e^2-10/(c*x^2+b*x+a)/(4*a*c-b^2)*a*d^4*e+1/(c*x^2+b*x+a)/(4*a*c-b^2)*b*d^5-2*e^5/c^3*x*b+5*e
^4/c^2*x*d+4*c/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*d^5-20/c^2/(4*a*c-b^2)*ln(c*x^2+b*x+a)*a*
b*d*e^4+5/c^3/(c*x^2+b*x+a)/(4*a*c-b^2)*x*a*b^3*e^5+5/c^3/(c*x^2+b*x+a)/(4*a*c-b^2)*x*b^4*d*e^4-5/c^2/(c*x^2+b
*x+a)/(4*a*c-b^2)*x*a^2*b*e^5+10/c/(c*x^2+b*x+a)/(4*a*c-b^2)*x*a^2*d*e^4+2*c/(c*x^2+b*x+a)/(4*a*c-b^2)*x*d^5-2
/c^2/(c*x^2+b*x+a)/(4*a*c-b^2)*a^3*e^5-4/c^2/(4*a*c-b^2)*ln(c*x^2+b*x+a)*a^2*e^5+10/c/(c*x^2+b*x+a)/(4*a*c-b^2
)*a*b*d^3*e^2+5/c^3/(c*x^2+b*x+a)/(4*a*c-b^2)*a*b^3*d*e^4-10/c^2/(c*x^2+b*x+a)/(4*a*c-b^2)*x*b^3*d^2*e^3+10/c/
(c*x^2+b*x+a)/(4*a*c-b^2)*x*b^2*d^3*e^2-15/c^2/(c*x^2+b*x+a)/(4*a*c-b^2)*a^2*b*d*e^4-10/c^2/(c*x^2+b*x+a)/(4*a
*c-b^2)*a*b^2*d^2*e^3-10/c^3/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^4*d*e^4+10/c^2/(4*a*c-b^2
)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*d^2*b^3*e^3-1/c^4/(c*x^2+b*x+a)/(4*a*c-b^2)*a*b^4*e^5-1/c^4/(c*x^2
+b*x+a)/(4*a*c-b^2)*x*b^5*e^5+20/c/(c*x^2+b*x+a)/(4*a*c-b^2)*a^2*d^2*e^3+4/c^3/(c*x^2+b*x+a)/(4*a*c-b^2)*a^2*b
^2*e^5+5/c^3/(4*a*c-b^2)*ln(c*x^2+b*x+a)*b^3*d*e^4+7/c^3/(4*a*c-b^2)*ln(c*x^2+b*x+a)*a*b^2*e^5-5/(c*x^2+b*x+a)
/(4*a*c-b^2)*x*b*d^4*e-60/c/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*d*a^2*e^4-5/c^2/(4*a*c-b^2)*
ln(c*x^2+b*x+a)*b^2*d^2*e^3+20/c/(4*a*c-b^2)*ln(c*x^2+b*x+a)*a*d^2*e^3+30/c^2/(4*a*c-b^2)^(3/2)*arctan((2*c*x+
b)/(4*a*c-b^2)^(1/2))*a^2*b*e^5-20/c^3/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b^3*e^5-20/(c*x
^2+b*x+a)/(4*a*c-b^2)*x*a*d^3*e^2-60/c/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b*d^2*e^3+60/c^
2/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b^2*d*e^4

________________________________________________________________________________________

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((e*x+d)^5/(c*x^2+b*x+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 3.33202, size = 5738, normalized size = 15.34 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(c*x^2+b*x+a)^2,x, algorithm="fricas")

[Out]

[1/2*((b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*e^5*x^4 - 2*(b^3*c^4 - 4*a*b*c^5)*d^5 + 20*(a*b^2*c^4 - 4*a^2*c^5)*
d^4*e - 20*(a*b^3*c^3 - 4*a^2*b*c^4)*d^3*e^2 + 20*(a*b^4*c^2 - 6*a^2*b^2*c^3 + 8*a^3*c^4)*d^2*e^3 - 10*(a*b^5*
c - 7*a^2*b^3*c^2 + 12*a^3*b*c^3)*d*e^4 + 2*(a*b^6 - 8*a^2*b^4*c + 18*a^3*b^2*c^2 - 8*a^4*c^3)*e^5 + (10*(b^4*
c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d*e^4 - 3*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*e^5)*x^3 + (10*(b^5*c^2 - 8*a
*b^3*c^3 + 16*a^2*b*c^4)*d*e^4 - (4*b^6*c - 33*a*b^4*c^2 + 72*a^2*b^2*c^3 - 16*a^3*c^4)*e^5)*x^2 - (4*a*c^5*d^
5 - 10*a*b*c^4*d^4*e + 40*a^2*c^4*d^3*e^2 + 10*(a*b^3*c^2 - 6*a^2*b*c^3)*d^2*e^3 - 10*(a*b^4*c - 6*a^2*b^2*c^2
+ 6*a^3*c^3)*d*e^4 + (3*a*b^5 - 20*a^2*b^3*c + 30*a^3*b*c^2)*e^5 + (4*c^6*d^5 - 10*b*c^5*d^4*e + 40*a*c^5*d^3
*e^2 + 10*(b^3*c^3 - 6*a*b*c^4)*d^2*e^3 - 10*(b^4*c^2 - 6*a*b^2*c^3 + 6*a^2*c^4)*d*e^4 + (3*b^5*c - 20*a*b^3*c
^2 + 30*a^2*b*c^3)*e^5)*x^2 + (4*b*c^5*d^5 - 10*b^2*c^4*d^4*e + 40*a*b*c^4*d^3*e^2 + 10*(b^4*c^2 - 6*a*b^2*c^3
)*d^2*e^3 - 10*(b^5*c - 6*a*b^3*c^2 + 6*a^2*b*c^3)*d*e^4 + (3*b^6 - 20*a*b^4*c + 30*a^2*b^2*c^2)*e^5)*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*(2
*(b^2*c^5 - 4*a*c^6)*d^5 - 5*(b^3*c^4 - 4*a*b*c^5)*d^4*e + 10*(b^4*c^3 - 6*a*b^2*c^4 + 8*a^2*c^5)*d^3*e^2 - 10
*(b^5*c^2 - 7*a*b^3*c^3 + 12*a^2*b*c^4)*d^2*e^3 + 5*(b^6*c - 9*a*b^4*c^2 + 26*a^2*b^2*c^3 - 24*a^3*c^4)*d*e^4
- (b^7 - 11*a*b^5*c + 41*a^2*b^3*c^2 - 52*a^3*b*c^3)*e^5)*x + (10*(a*b^4*c^2 - 8*a^2*b^2*c^3 + 16*a^3*c^4)*d^2
*e^3 - 10*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*d*e^4 + (3*a*b^6 - 26*a^2*b^4*c + 64*a^3*b^2*c^2 - 32*a^4*c
^3)*e^5 + (10*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d^2*e^3 - 10*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d*e^4 +
(3*b^6*c - 26*a*b^4*c^2 + 64*a^2*b^2*c^3 - 32*a^3*c^4)*e^5)*x^2 + (10*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*
d^2*e^3 - 10*(b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*d*e^4 + (3*b^7 - 26*a*b^5*c + 64*a^2*b^3*c^2 - 32*a^3*b*c^
3)*e^5)*x)*log(c*x^2 + b*x + a))/(a*b^4*c^4 - 8*a^2*b^2*c^5 + 16*a^3*c^6 + (b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7
)*x^2 + (b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6)*x), 1/2*((b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*e^5*x^4 - 2*(b^3*
c^4 - 4*a*b*c^5)*d^5 + 20*(a*b^2*c^4 - 4*a^2*c^5)*d^4*e - 20*(a*b^3*c^3 - 4*a^2*b*c^4)*d^3*e^2 + 20*(a*b^4*c^2
- 6*a^2*b^2*c^3 + 8*a^3*c^4)*d^2*e^3 - 10*(a*b^5*c - 7*a^2*b^3*c^2 + 12*a^3*b*c^3)*d*e^4 + 2*(a*b^6 - 8*a^2*b
^4*c + 18*a^3*b^2*c^2 - 8*a^4*c^3)*e^5 + (10*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d*e^4 - 3*(b^5*c^2 - 8*a*b^3
*c^3 + 16*a^2*b*c^4)*e^5)*x^3 + (10*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d*e^4 - (4*b^6*c - 33*a*b^4*c^2 + 7
2*a^2*b^2*c^3 - 16*a^3*c^4)*e^5)*x^2 + 2*(4*a*c^5*d^5 - 10*a*b*c^4*d^4*e + 40*a^2*c^4*d^3*e^2 + 10*(a*b^3*c^2
- 6*a^2*b*c^3)*d^2*e^3 - 10*(a*b^4*c - 6*a^2*b^2*c^2 + 6*a^3*c^3)*d*e^4 + (3*a*b^5 - 20*a^2*b^3*c + 30*a^3*b*c
^2)*e^5 + (4*c^6*d^5 - 10*b*c^5*d^4*e + 40*a*c^5*d^3*e^2 + 10*(b^3*c^3 - 6*a*b*c^4)*d^2*e^3 - 10*(b^4*c^2 - 6*
a*b^2*c^3 + 6*a^2*c^4)*d*e^4 + (3*b^5*c - 20*a*b^3*c^2 + 30*a^2*b*c^3)*e^5)*x^2 + (4*b*c^5*d^5 - 10*b^2*c^4*d^
4*e + 40*a*b*c^4*d^3*e^2 + 10*(b^4*c^2 - 6*a*b^2*c^3)*d^2*e^3 - 10*(b^5*c - 6*a*b^3*c^2 + 6*a^2*b*c^3)*d*e^4 +
(3*b^6 - 20*a*b^4*c + 30*a^2*b^2*c^2)*e^5)*x)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2
- 4*a*c)) - 2*(2*(b^2*c^5 - 4*a*c^6)*d^5 - 5*(b^3*c^4 - 4*a*b*c^5)*d^4*e + 10*(b^4*c^3 - 6*a*b^2*c^4 + 8*a^2*c
^5)*d^3*e^2 - 10*(b^5*c^2 - 7*a*b^3*c^3 + 12*a^2*b*c^4)*d^2*e^3 + 5*(b^6*c - 9*a*b^4*c^2 + 26*a^2*b^2*c^3 - 24
*a^3*c^4)*d*e^4 - (b^7 - 11*a*b^5*c + 41*a^2*b^3*c^2 - 52*a^3*b*c^3)*e^5)*x + (10*(a*b^4*c^2 - 8*a^2*b^2*c^3 +
16*a^3*c^4)*d^2*e^3 - 10*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*d*e^4 + (3*a*b^6 - 26*a^2*b^4*c + 64*a^3*b^
2*c^2 - 32*a^4*c^3)*e^5 + (10*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d^2*e^3 - 10*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^
2*b*c^4)*d*e^4 + (3*b^6*c - 26*a*b^4*c^2 + 64*a^2*b^2*c^3 - 32*a^3*c^4)*e^5)*x^2 + (10*(b^5*c^2 - 8*a*b^3*c^3
+ 16*a^2*b*c^4)*d^2*e^3 - 10*(b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*d*e^4 + (3*b^7 - 26*a*b^5*c + 64*a^2*b^3*c
^2 - 32*a^3*b*c^3)*e^5)*x)*log(c*x^2 + b*x + a))/(a*b^4*c^4 - 8*a^2*b^2*c^5 + 16*a^3*c^6 + (b^4*c^5 - 8*a*b^2*
c^6 + 16*a^2*c^7)*x^2 + (b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6)*x)]

________________________________________________________________________________________

Sympy [B]  time = 26.3127, size = 2669, normalized size = 7.14 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**5/(c*x**2+b*x+a)**2,x)

[Out]

(-e**3*(2*a*c*e**2 - 3*b**2*e**2 + 10*b*c*d*e - 10*c**2*d**2)/(2*c**4) - sqrt(-(4*a*c - b**2)**3)*(b*e - 2*c*d
)*(30*a**2*c**2*e**4 - 20*a*b**2*c*e**4 + 20*a*b*c**2*d*e**3 - 20*a*c**3*d**2*e**2 + 3*b**4*e**4 - 4*b**3*c*d*
e**3 + 2*b**2*c**2*d**2*e**2 + 4*b*c**3*d**3*e - 2*c**4*d**4)/(2*c**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a
*b**4*c - b**6)))*log(x + (16*a**3*c**2*e**5 - 17*a**2*b**2*c*e**5 + 50*a**2*b*c**2*d*e**4 + 16*a**2*c**5*(-e*
*3*(2*a*c*e**2 - 3*b**2*e**2 + 10*b*c*d*e - 10*c**2*d**2)/(2*c**4) - sqrt(-(4*a*c - b**2)**3)*(b*e - 2*c*d)*(3
0*a**2*c**2*e**4 - 20*a*b**2*c*e**4 + 20*a*b*c**2*d*e**3 - 20*a*c**3*d**2*e**2 + 3*b**4*e**4 - 4*b**3*c*d*e**3
+ 2*b**2*c**2*d**2*e**2 + 4*b*c**3*d**3*e - 2*c**4*d**4)/(2*c**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**
4*c - b**6))) - 80*a**2*c**3*d**2*e**3 + 3*a*b**4*e**5 - 10*a*b**3*c*d*e**4 - 8*a*b**2*c**4*(-e**3*(2*a*c*e**2
- 3*b**2*e**2 + 10*b*c*d*e - 10*c**2*d**2)/(2*c**4) - sqrt(-(4*a*c - b**2)**3)*(b*e - 2*c*d)*(30*a**2*c**2*e*
*4 - 20*a*b**2*c*e**4 + 20*a*b*c**2*d*e**3 - 20*a*c**3*d**2*e**2 + 3*b**4*e**4 - 4*b**3*c*d*e**3 + 2*b**2*c**2
*d**2*e**2 + 4*b*c**3*d**3*e - 2*c**4*d**4)/(2*c**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))
+ 10*a*b**2*c**2*d**2*e**3 + 20*a*b*c**3*d**3*e**2 + b**4*c**3*(-e**3*(2*a*c*e**2 - 3*b**2*e**2 + 10*b*c*d*e -
10*c**2*d**2)/(2*c**4) - sqrt(-(4*a*c - b**2)**3)*(b*e - 2*c*d)*(30*a**2*c**2*e**4 - 20*a*b**2*c*e**4 + 20*a*
b*c**2*d*e**3 - 20*a*c**3*d**2*e**2 + 3*b**4*e**4 - 4*b**3*c*d*e**3 + 2*b**2*c**2*d**2*e**2 + 4*b*c**3*d**3*e
- 2*c**4*d**4)/(2*c**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) - 5*b**2*c**3*d**4*e + 2*b*c*
*4*d**5)/(30*a**2*b*c**2*e**5 - 60*a**2*c**3*d*e**4 - 20*a*b**3*c*e**5 + 60*a*b**2*c**2*d*e**4 - 60*a*b*c**3*d
**2*e**3 + 40*a*c**4*d**3*e**2 + 3*b**5*e**5 - 10*b**4*c*d*e**4 + 10*b**3*c**2*d**2*e**3 - 10*b*c**4*d**4*e +
4*c**5*d**5)) + (-e**3*(2*a*c*e**2 - 3*b**2*e**2 + 10*b*c*d*e - 10*c**2*d**2)/(2*c**4) + sqrt(-(4*a*c - b**2)*
*3)*(b*e - 2*c*d)*(30*a**2*c**2*e**4 - 20*a*b**2*c*e**4 + 20*a*b*c**2*d*e**3 - 20*a*c**3*d**2*e**2 + 3*b**4*e*
*4 - 4*b**3*c*d*e**3 + 2*b**2*c**2*d**2*e**2 + 4*b*c**3*d**3*e - 2*c**4*d**4)/(2*c**4*(64*a**3*c**3 - 48*a**2*
b**2*c**2 + 12*a*b**4*c - b**6)))*log(x + (16*a**3*c**2*e**5 - 17*a**2*b**2*c*e**5 + 50*a**2*b*c**2*d*e**4 + 1
6*a**2*c**5*(-e**3*(2*a*c*e**2 - 3*b**2*e**2 + 10*b*c*d*e - 10*c**2*d**2)/(2*c**4) + sqrt(-(4*a*c - b**2)**3)*
(b*e - 2*c*d)*(30*a**2*c**2*e**4 - 20*a*b**2*c*e**4 + 20*a*b*c**2*d*e**3 - 20*a*c**3*d**2*e**2 + 3*b**4*e**4 -
4*b**3*c*d*e**3 + 2*b**2*c**2*d**2*e**2 + 4*b*c**3*d**3*e - 2*c**4*d**4)/(2*c**4*(64*a**3*c**3 - 48*a**2*b**2
*c**2 + 12*a*b**4*c - b**6))) - 80*a**2*c**3*d**2*e**3 + 3*a*b**4*e**5 - 10*a*b**3*c*d*e**4 - 8*a*b**2*c**4*(-
e**3*(2*a*c*e**2 - 3*b**2*e**2 + 10*b*c*d*e - 10*c**2*d**2)/(2*c**4) + sqrt(-(4*a*c - b**2)**3)*(b*e - 2*c*d)*
(30*a**2*c**2*e**4 - 20*a*b**2*c*e**4 + 20*a*b*c**2*d*e**3 - 20*a*c**3*d**2*e**2 + 3*b**4*e**4 - 4*b**3*c*d*e*
*3 + 2*b**2*c**2*d**2*e**2 + 4*b*c**3*d**3*e - 2*c**4*d**4)/(2*c**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b
**4*c - b**6))) + 10*a*b**2*c**2*d**2*e**3 + 20*a*b*c**3*d**3*e**2 + b**4*c**3*(-e**3*(2*a*c*e**2 - 3*b**2*e**
2 + 10*b*c*d*e - 10*c**2*d**2)/(2*c**4) + sqrt(-(4*a*c - b**2)**3)*(b*e - 2*c*d)*(30*a**2*c**2*e**4 - 20*a*b**
2*c*e**4 + 20*a*b*c**2*d*e**3 - 20*a*c**3*d**2*e**2 + 3*b**4*e**4 - 4*b**3*c*d*e**3 + 2*b**2*c**2*d**2*e**2 +
4*b*c**3*d**3*e - 2*c**4*d**4)/(2*c**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) - 5*b**2*c**3
*d**4*e + 2*b*c**4*d**5)/(30*a**2*b*c**2*e**5 - 60*a**2*c**3*d*e**4 - 20*a*b**3*c*e**5 + 60*a*b**2*c**2*d*e**4
- 60*a*b*c**3*d**2*e**3 + 40*a*c**4*d**3*e**2 + 3*b**5*e**5 - 10*b**4*c*d*e**4 + 10*b**3*c**2*d**2*e**3 - 10*
b*c**4*d**4*e + 4*c**5*d**5)) - (2*a**3*c**2*e**5 - 4*a**2*b**2*c*e**5 + 15*a**2*b*c**2*d*e**4 - 20*a**2*c**3*
d**2*e**3 + a*b**4*e**5 - 5*a*b**3*c*d*e**4 + 10*a*b**2*c**2*d**2*e**3 - 10*a*b*c**3*d**3*e**2 + 10*a*c**4*d**
4*e - b*c**4*d**5 + x*(5*a**2*b*c**2*e**5 - 10*a**2*c**3*d*e**4 - 5*a*b**3*c*e**5 + 20*a*b**2*c**2*d*e**4 - 30
*a*b*c**3*d**2*e**3 + 20*a*c**4*d**3*e**2 + b**5*e**5 - 5*b**4*c*d*e**4 + 10*b**3*c**2*d**2*e**3 - 10*b**2*c**
3*d**3*e**2 + 5*b*c**4*d**4*e - 2*c**5*d**5))/(4*a**2*c**5 - a*b**2*c**4 + x**2*(4*a*c**6 - b**2*c**5) + x*(4*
a*b*c**5 - b**3*c**4)) + e**5*x**2/(2*c**2) - x*(2*b*e**5 - 5*c*d*e**4)/c**3

________________________________________________________________________________________

Giac [A]  time = 1.128, size = 690, normalized size = 1.84 \begin{align*} -\frac{{\left (4 \, c^{5} d^{5} - 10 \, b c^{4} d^{4} e + 40 \, a c^{4} d^{3} e^{2} + 10 \, b^{3} c^{2} d^{2} e^{3} - 60 \, a b c^{3} d^{2} e^{3} - 10 \, b^{4} c d e^{4} + 60 \, a b^{2} c^{2} d e^{4} - 60 \, a^{2} c^{3} d e^{4} + 3 \, b^{5} e^{5} - 20 \, a b^{3} c e^{5} + 30 \, a^{2} b c^{2} e^{5}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{4} - 4 \, a c^{5}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{{\left (10 \, c^{2} d^{2} e^{3} - 10 \, b c d e^{4} + 3 \, b^{2} e^{5} - 2 \, a c e^{5}\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{4}} + \frac{c^{2} x^{2} e^{5} + 10 \, c^{2} d x e^{4} - 4 \, b c x e^{5}}{2 \, c^{4}} - \frac{b c^{4} d^{5} - 10 \, a c^{4} d^{4} e + 10 \, a b c^{3} d^{3} e^{2} - 10 \, a b^{2} c^{2} d^{2} e^{3} + 20 \, a^{2} c^{3} d^{2} e^{3} + 5 \, a b^{3} c d e^{4} - 15 \, a^{2} b c^{2} d e^{4} - a b^{4} e^{5} + 4 \, a^{2} b^{2} c e^{5} - 2 \, a^{3} c^{2} e^{5} +{\left (2 \, c^{5} d^{5} - 5 \, b c^{4} d^{4} e + 10 \, b^{2} c^{3} d^{3} e^{2} - 20 \, a c^{4} d^{3} e^{2} - 10 \, b^{3} c^{2} d^{2} e^{3} + 30 \, a b c^{3} d^{2} e^{3} + 5 \, b^{4} c d e^{4} - 20 \, a b^{2} c^{2} d e^{4} + 10 \, a^{2} c^{3} d e^{4} - b^{5} e^{5} + 5 \, a b^{3} c e^{5} - 5 \, a^{2} b c^{2} e^{5}\right )} x}{{\left (c x^{2} + b x + a\right )}{\left (b^{2} - 4 \, a c\right )} c^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(c*x^2+b*x+a)^2,x, algorithm="giac")

[Out]

-(4*c^5*d^5 - 10*b*c^4*d^4*e + 40*a*c^4*d^3*e^2 + 10*b^3*c^2*d^2*e^3 - 60*a*b*c^3*d^2*e^3 - 10*b^4*c*d*e^4 + 6
0*a*b^2*c^2*d*e^4 - 60*a^2*c^3*d*e^4 + 3*b^5*e^5 - 20*a*b^3*c*e^5 + 30*a^2*b*c^2*e^5)*arctan((2*c*x + b)/sqrt(
-b^2 + 4*a*c))/((b^2*c^4 - 4*a*c^5)*sqrt(-b^2 + 4*a*c)) + 1/2*(10*c^2*d^2*e^3 - 10*b*c*d*e^4 + 3*b^2*e^5 - 2*a
*c*e^5)*log(c*x^2 + b*x + a)/c^4 + 1/2*(c^2*x^2*e^5 + 10*c^2*d*x*e^4 - 4*b*c*x*e^5)/c^4 - (b*c^4*d^5 - 10*a*c^
4*d^4*e + 10*a*b*c^3*d^3*e^2 - 10*a*b^2*c^2*d^2*e^3 + 20*a^2*c^3*d^2*e^3 + 5*a*b^3*c*d*e^4 - 15*a^2*b*c^2*d*e^
4 - a*b^4*e^5 + 4*a^2*b^2*c*e^5 - 2*a^3*c^2*e^5 + (2*c^5*d^5 - 5*b*c^4*d^4*e + 10*b^2*c^3*d^3*e^2 - 20*a*c^4*d
^3*e^2 - 10*b^3*c^2*d^2*e^3 + 30*a*b*c^3*d^2*e^3 + 5*b^4*c*d*e^4 - 20*a*b^2*c^2*d*e^4 + 10*a^2*c^3*d*e^4 - b^5
*e^5 + 5*a*b^3*c*e^5 - 5*a^2*b*c^2*e^5)*x)/((c*x^2 + b*x + a)*(b^2 - 4*a*c)*c^4)