### 3.2392 $$\int \frac{(d+e x)^4}{(a+b x+c x^2)^{5/2}} \, dx$$

Optimal. Leaf size=295 $-\frac{2 e \sqrt{a+b x+c x^2} (2 c d-b e) \left (-4 c e (2 b d-5 a e)-3 b^2 e^2+8 c^2 d^2\right )}{3 c^2 \left (b^2-4 a c\right )^2}+\frac{4 (d+e x) \left (x (2 c d-b e) \left (-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right )+b^2 \left (-\left (5 c d^2 e-a e^3\right )\right )+4 b c d \left (3 a e^2+c d^2\right )-4 a c e \left (3 a e^2+c d^2\right )\right )}{3 c \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 (d+e x)^3 (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac{e^4 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{c^{5/2}}$

[Out]

(-2*(d + e*x)^3*(b*d - 2*a*e + (2*c*d - b*e)*x))/(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(3/2)) + (4*(d + e*x)*(4*b
*c*d*(c*d^2 + 3*a*e^2) - 4*a*c*e*(c*d^2 + 3*a*e^2) - b^2*(5*c*d^2*e - a*e^3) + (2*c*d - b*e)*(4*c^2*d^2 - b^2*
e^2 - 4*c*e*(b*d - 2*a*e))*x))/(3*c*(b^2 - 4*a*c)^2*Sqrt[a + b*x + c*x^2]) - (2*e*(2*c*d - b*e)*(8*c^2*d^2 - 3
*b^2*e^2 - 4*c*e*(2*b*d - 5*a*e))*Sqrt[a + b*x + c*x^2])/(3*c^2*(b^2 - 4*a*c)^2) + (e^4*ArcTanh[(b + 2*c*x)/(2
*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/c^(5/2)

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Rubi [A]  time = 0.344603, antiderivative size = 295, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.227, Rules used = {738, 818, 640, 621, 206} $-\frac{2 e \sqrt{a+b x+c x^2} (2 c d-b e) \left (-4 c e (2 b d-5 a e)-3 b^2 e^2+8 c^2 d^2\right )}{3 c^2 \left (b^2-4 a c\right )^2}+\frac{4 (d+e x) \left (x (2 c d-b e) \left (-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right )+b^2 \left (-\left (5 c d^2 e-a e^3\right )\right )+4 b c d \left (3 a e^2+c d^2\right )-4 a c e \left (3 a e^2+c d^2\right )\right )}{3 c \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 (d+e x)^3 (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac{e^4 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{c^{5/2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^4/(a + b*x + c*x^2)^(5/2),x]

[Out]

(-2*(d + e*x)^3*(b*d - 2*a*e + (2*c*d - b*e)*x))/(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(3/2)) + (4*(d + e*x)*(4*b
*c*d*(c*d^2 + 3*a*e^2) - 4*a*c*e*(c*d^2 + 3*a*e^2) - b^2*(5*c*d^2*e - a*e^3) + (2*c*d - b*e)*(4*c^2*d^2 - b^2*
e^2 - 4*c*e*(b*d - 2*a*e))*x))/(3*c*(b^2 - 4*a*c)^2*Sqrt[a + b*x + c*x^2]) - (2*e*(2*c*d - b*e)*(8*c^2*d^2 - 3
*b^2*e^2 - 4*c*e*(2*b*d - 5*a*e))*Sqrt[a + b*x + c*x^2])/(3*c^2*(b^2 - 4*a*c)^2) + (e^4*ArcTanh[(b + 2*c*x)/(2
*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/c^(5/2)

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 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 640

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(a + b*x + c*x^2)^(p +
1))/(2*c*(p + 1)), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], 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])

Rubi steps

\begin{align*} \int \frac{(d+e x)^4}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac{2 (d+e x)^3 (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac{2 \int \frac{(d+e x)^2 \left (4 c d^2-e (5 b d-6 a e)-e (2 c d-b e) x\right )}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{3 \left (b^2-4 a c\right )}\\ &=-\frac{2 (d+e x)^3 (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac{4 (d+e x) \left (4 b c d \left (c d^2+3 a e^2\right )-4 a c e \left (c d^2+3 a e^2\right )-b^2 \left (5 c d^2 e-a e^3\right )+(2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x\right )}{3 c \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{4 \int \frac{\frac{1}{2} \left (b^3 d e^3-24 a^2 c e^4+4 b c d e \left (2 c d^2+5 a e^2\right )-2 b^2 \left (6 c d^2 e^2-a e^4\right )\right )+\frac{1}{2} e (2 c d-b e) \left (8 c^2 d^2-3 b^2 e^2-4 c e (2 b d-5 a e)\right ) x}{\sqrt{a+b x+c x^2}} \, dx}{3 c \left (b^2-4 a c\right )^2}\\ &=-\frac{2 (d+e x)^3 (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac{4 (d+e x) \left (4 b c d \left (c d^2+3 a e^2\right )-4 a c e \left (c d^2+3 a e^2\right )-b^2 \left (5 c d^2 e-a e^3\right )+(2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x\right )}{3 c \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 e (2 c d-b e) \left (8 c^2 d^2-3 b^2 e^2-4 c e (2 b d-5 a e)\right ) \sqrt{a+b x+c x^2}}{3 c^2 \left (b^2-4 a c\right )^2}+\frac{e^4 \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{c^2}\\ &=-\frac{2 (d+e x)^3 (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac{4 (d+e x) \left (4 b c d \left (c d^2+3 a e^2\right )-4 a c e \left (c d^2+3 a e^2\right )-b^2 \left (5 c d^2 e-a e^3\right )+(2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x\right )}{3 c \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 e (2 c d-b e) \left (8 c^2 d^2-3 b^2 e^2-4 c e (2 b d-5 a e)\right ) \sqrt{a+b x+c x^2}}{3 c^2 \left (b^2-4 a c\right )^2}+\frac{\left (2 e^4\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x}{\sqrt{a+b x+c x^2}}\right )}{c^2}\\ &=-\frac{2 (d+e x)^3 (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac{4 (d+e x) \left (4 b c d \left (c d^2+3 a e^2\right )-4 a c e \left (c d^2+3 a e^2\right )-b^2 \left (5 c d^2 e-a e^3\right )+(2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x\right )}{3 c \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 e (2 c d-b e) \left (8 c^2 d^2-3 b^2 e^2-4 c e (2 b d-5 a e)\right ) \sqrt{a+b x+c x^2}}{3 c^2 \left (b^2-4 a c\right )^2}+\frac{e^4 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{c^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.862558, size = 397, normalized size = 1.35 $\frac{e^4 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )}{c^{5/2}}-\frac{2 \left (b^3 \left (3 a^2 e^4-18 a c e^4 x^2+c^2 d \left (12 d^2 e x+d^3-18 d e^2 x^2-4 e^3 x^3\right )\right )-2 b^2 c \left (21 a^2 e^4 x+2 a c e \left (18 d^2 e x-2 d^3-6 d e^2 x^2+7 e^3 x^3\right )+3 c^2 d^2 x \left (d^2-8 d e x+2 e^2 x^2\right )\right )-4 b c \left (12 a^2 c d e^2 (d-2 e x)+5 a^3 e^4+3 a c^2 d \left (-4 d^2 e x+d^3+6 d e^2 x^2-4 e^3 x^3\right )+2 c^3 d^3 x^2 (3 d-4 e x)\right )+8 c^2 \left (4 a^2 c e \left (d^3+3 d e^2 x^2+e^3 x^3\right )+a^3 e^3 (8 d+3 e x)-3 a c^2 d^2 x \left (d^2+2 e^2 x^2\right )-2 c^3 d^4 x^3\right )+2 b^4 e^4 x \left (3 a+2 c x^2\right )+3 b^5 e^4 x^2\right )}{3 c^2 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^4/(a + b*x + c*x^2)^(5/2),x]

[Out]

(-2*(3*b^5*e^4*x^2 + 2*b^4*e^4*x*(3*a + 2*c*x^2) + b^3*(3*a^2*e^4 - 18*a*c*e^4*x^2 + c^2*d*(d^3 + 12*d^2*e*x -
18*d*e^2*x^2 - 4*e^3*x^3)) - 4*b*c*(5*a^3*e^4 + 2*c^3*d^3*x^2*(3*d - 4*e*x) + 12*a^2*c*d*e^2*(d - 2*e*x) + 3*
a*c^2*d*(d^3 - 4*d^2*e*x + 6*d*e^2*x^2 - 4*e^3*x^3)) + 8*c^2*(-2*c^3*d^4*x^3 + a^3*e^3*(8*d + 3*e*x) - 3*a*c^2
*d^2*x*(d^2 + 2*e^2*x^2) + 4*a^2*c*e*(d^3 + 3*d*e^2*x^2 + e^3*x^3)) - 2*b^2*c*(21*a^2*e^4*x + 3*c^2*d^2*x*(d^2
- 8*d*e*x + 2*e^2*x^2) + 2*a*c*e*(-2*d^3 + 18*d^2*e*x - 6*d*e^2*x^2 + 7*e^3*x^3))))/(3*c^2*(b^2 - 4*a*c)^2*(a
+ x*(b + c*x))^(3/2)) + (e^4*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/c^(5/2)

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Maple [B]  time = 0.053, size = 1550, normalized size = 5.3 \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)^4/(c*x^2+b*x+a)^(5/2),x)

[Out]

8*d^2*e^2*b^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x+4*d^2*e^2*b^3/c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)-1/24*e^4*b
^4/c^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x-1/3*e^4*b^4/c^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x-4*d*e^3*b/c*a/(4*
a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x+e^4/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-1/3*e^4*x^3/c/(c*x^2+b*
x+a)^(3/2)-1/48*e^4*b^3/c^4/(c*x^2+b*x+a)^(3/2)-e^4/c^2*x/(c*x^2+b*x+a)^(1/2)+1/2*e^4/c^3*b/(c*x^2+b*x+a)^(1/2
)-4/3*d^3*e/c/(c*x^2+b*x+a)^(3/2)+2/3*d^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*b-4*d*e^3*x^2/c/(c*x^2+b*x+a)^(3/2)+
1/6*d*e^3*b^2/c^3/(c*x^2+b*x+a)^(3/2)-32*d*e^3*b*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x+1/2*e^4*b^2/c^2*a/(4*a*
c-b^2)/(c*x^2+b*x+a)^(3/2)*x+4*e^4*b^2/c*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x-16*d*e^3*b^2/c*a/(4*a*c-b^2)^2/
(c*x^2+b*x+a)^(1/2)+d^2*e^2*b^2/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x-2*d*e^3*b^2/c^2*a/(4*a*c-b^2)/(c*x^2+b*x+a
)^(3/2)-32/3*d^3*e*b^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)+4/3*d^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*c-3*d^2*e^2
*x/c/(c*x^2+b*x+a)^(3/2)+1/2*d^2*e^2*b/c^2/(c*x^2+b*x+a)^(3/2)-8/3*d*e^3*a/c^2/(c*x^2+b*x+a)^(3/2)+1/2*e^4*b/c
^2*x^2/(c*x^2+b*x+a)^(3/2)+1/8*e^4*b^2/c^3*x/(c*x^2+b*x+a)^(3/2)-1/48*e^4*b^5/c^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(3
/2)-1/6*e^4*b^5/c^3/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)+1/3*e^4*b/c^3*a/(c*x^2+b*x+a)^(3/2)+1/2*e^4/c^3*b^3/(4*a
*c-b^2)/(c*x^2+b*x+a)^(1/2)+32/3*d^4*c^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x+16/3*d^4*c/(4*a*c-b^2)^2/(c*x^2+b
*x+a)^(1/2)*b+2*d^2*e^2*a/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*b+32*d^2*e^2*a*c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)
*x-64/3*d^3*e*b*c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x+1/3*d*e^3*b^3/c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x+8/3*
d*e^3*b^3/c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x+1/2*d^2*e^2*b^3/c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+1/4*e^4*b^
3/c^3*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+2*e^4*b^3/c^2*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)+4*d^2*e^2*a/(4*a*c-b
^2)/(c*x^2+b*x+a)^(3/2)*x+16*d^2*e^2*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*b-8/3*d^3*e*b/(4*a*c-b^2)/(c*x^2+b*x+
a)^(3/2)*x+4/3*d*e^3*b^4/c^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)-d*e^3*b/c^2*x/(c*x^2+b*x+a)^(3/2)+1/6*d*e^3*b^4
/c^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+e^4/c^2*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x-4/3*d^3*e*b^2/c/(4*a*c-b^2)
/(c*x^2+b*x+a)^(3/2)

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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((e*x+d)^4/(c*x^2+b*x+a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 11.0915, size = 3106, normalized size = 10.53 \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)^4/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")

[Out]

[1/6*(3*((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*e^4*x^4 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*e^4*x^3 + (b^6
- 6*a*b^4*c + 32*a^3*c^3)*e^4*x^2 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*e^4*x + (a^2*b^4 - 8*a^3*b^2*c + 16
*a^4*c^2)*e^4)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 - 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) +
4*(48*a^2*b*c^3*d^2*e^2 - 64*a^3*c^3*d*e^3 - (b^3*c^3 - 12*a*b*c^4)*d^4 - 8*(a*b^2*c^3 + 4*a^2*c^4)*d^3*e - (
3*a^2*b^3*c - 20*a^3*b*c^2)*e^4 + 4*(4*c^6*d^4 - 8*b*c^5*d^3*e + 3*(b^2*c^4 + 4*a*c^5)*d^2*e^2 + (b^3*c^3 - 12
*a*b*c^4)*d*e^3 - (b^4*c^2 - 7*a*b^2*c^3 + 8*a^2*c^4)*e^4)*x^3 + 3*(8*b*c^5*d^4 - 16*b^2*c^4*d^3*e + 6*(b^3*c^
3 + 4*a*b*c^4)*d^2*e^2 - 8*(a*b^2*c^3 + 4*a^2*c^4)*d*e^3 - (b^5*c - 6*a*b^3*c^2)*e^4)*x^2 + 6*(12*a*b^2*c^3*d^
2*e^2 - 16*a^2*b*c^3*d*e^3 + (b^2*c^4 + 4*a*c^5)*d^4 - 2*(b^3*c^3 + 4*a*b*c^4)*d^3*e - (a*b^4*c - 7*a^2*b^2*c^
2 + 4*a^3*c^3)*e^4)*x)*sqrt(c*x^2 + b*x + a))/(a^2*b^4*c^3 - 8*a^3*b^2*c^4 + 16*a^4*c^5 + (b^4*c^5 - 8*a*b^2*c
^6 + 16*a^2*c^7)*x^4 + 2*(b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6)*x^3 + (b^6*c^3 - 6*a*b^4*c^4 + 32*a^3*c^6)*x^2
+ 2*(a*b^5*c^3 - 8*a^2*b^3*c^4 + 16*a^3*b*c^5)*x), -1/3*(3*((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*e^4*x^4 + 2*
(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*e^4*x^3 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*e^4*x^2 + 2*(a*b^5 - 8*a^2*b^3*c
+ 16*a^3*b*c^2)*e^4*x + (a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2)*e^4)*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(
2*c*x + b)*sqrt(-c)/(c^2*x^2 + b*c*x + a*c)) - 2*(48*a^2*b*c^3*d^2*e^2 - 64*a^3*c^3*d*e^3 - (b^3*c^3 - 12*a*b*
c^4)*d^4 - 8*(a*b^2*c^3 + 4*a^2*c^4)*d^3*e - (3*a^2*b^3*c - 20*a^3*b*c^2)*e^4 + 4*(4*c^6*d^4 - 8*b*c^5*d^3*e +
3*(b^2*c^4 + 4*a*c^5)*d^2*e^2 + (b^3*c^3 - 12*a*b*c^4)*d*e^3 - (b^4*c^2 - 7*a*b^2*c^3 + 8*a^2*c^4)*e^4)*x^3 +
3*(8*b*c^5*d^4 - 16*b^2*c^4*d^3*e + 6*(b^3*c^3 + 4*a*b*c^4)*d^2*e^2 - 8*(a*b^2*c^3 + 4*a^2*c^4)*d*e^3 - (b^5*
c - 6*a*b^3*c^2)*e^4)*x^2 + 6*(12*a*b^2*c^3*d^2*e^2 - 16*a^2*b*c^3*d*e^3 + (b^2*c^4 + 4*a*c^5)*d^4 - 2*(b^3*c^
3 + 4*a*b*c^4)*d^3*e - (a*b^4*c - 7*a^2*b^2*c^2 + 4*a^3*c^3)*e^4)*x)*sqrt(c*x^2 + b*x + a))/(a^2*b^4*c^3 - 8*a
^3*b^2*c^4 + 16*a^4*c^5 + (b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*x^4 + 2*(b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6)*
x^3 + (b^6*c^3 - 6*a*b^4*c^4 + 32*a^3*c^6)*x^2 + 2*(a*b^5*c^3 - 8*a^2*b^3*c^4 + 16*a^3*b*c^5)*x)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**4/(c*x**2+b*x+a)**(5/2),x)

[Out]

Timed out

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Giac [A]  time = 1.15889, size = 741, normalized size = 2.51 \begin{align*} \frac{2 \,{\left ({\left ({\left (\frac{4 \,{\left (4 \, c^{5} d^{4} - 8 \, b c^{4} d^{3} e + 3 \, b^{2} c^{3} d^{2} e^{2} + 12 \, a c^{4} d^{2} e^{2} + b^{3} c^{2} d e^{3} - 12 \, a b c^{3} d e^{3} - b^{4} c e^{4} + 7 \, a b^{2} c^{2} e^{4} - 8 \, a^{2} c^{3} e^{4}\right )} x}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}} + \frac{3 \,{\left (8 \, b c^{4} d^{4} - 16 \, b^{2} c^{3} d^{3} e + 6 \, b^{3} c^{2} d^{2} e^{2} + 24 \, a b c^{3} d^{2} e^{2} - 8 \, a b^{2} c^{2} d e^{3} - 32 \, a^{2} c^{3} d e^{3} - b^{5} e^{4} + 6 \, a b^{3} c e^{4}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac{6 \,{\left (b^{2} c^{3} d^{4} + 4 \, a c^{4} d^{4} - 2 \, b^{3} c^{2} d^{3} e - 8 \, a b c^{3} d^{3} e + 12 \, a b^{2} c^{2} d^{2} e^{2} - 16 \, a^{2} b c^{2} d e^{3} - a b^{4} e^{4} + 7 \, a^{2} b^{2} c e^{4} - 4 \, a^{3} c^{2} e^{4}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x - \frac{b^{3} c^{2} d^{4} - 12 \, a b c^{3} d^{4} + 8 \, a b^{2} c^{2} d^{3} e + 32 \, a^{2} c^{3} d^{3} e - 48 \, a^{2} b c^{2} d^{2} e^{2} + 64 \, a^{3} c^{2} d e^{3} + 3 \, a^{2} b^{3} e^{4} - 20 \, a^{3} b c e^{4}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )}}{3 \,{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}} - \frac{e^{4} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{c^{\frac{5}{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")

[Out]

2/3*(((4*(4*c^5*d^4 - 8*b*c^4*d^3*e + 3*b^2*c^3*d^2*e^2 + 12*a*c^4*d^2*e^2 + b^3*c^2*d*e^3 - 12*a*b*c^3*d*e^3
- b^4*c*e^4 + 7*a*b^2*c^2*e^4 - 8*a^2*c^3*e^4)*x/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4) + 3*(8*b*c^4*d^4 - 16*b^
2*c^3*d^3*e + 6*b^3*c^2*d^2*e^2 + 24*a*b*c^3*d^2*e^2 - 8*a*b^2*c^2*d*e^3 - 32*a^2*c^3*d*e^3 - b^5*e^4 + 6*a*b^
3*c*e^4)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4))*x + 6*(b^2*c^3*d^4 + 4*a*c^4*d^4 - 2*b^3*c^2*d^3*e - 8*a*b*c^3*
d^3*e + 12*a*b^2*c^2*d^2*e^2 - 16*a^2*b*c^2*d*e^3 - a*b^4*e^4 + 7*a^2*b^2*c*e^4 - 4*a^3*c^2*e^4)/(b^4*c^2 - 8*
a*b^2*c^3 + 16*a^2*c^4))*x - (b^3*c^2*d^4 - 12*a*b*c^3*d^4 + 8*a*b^2*c^2*d^3*e + 32*a^2*c^3*d^3*e - 48*a^2*b*c
^2*d^2*e^2 + 64*a^3*c^2*d*e^3 + 3*a^2*b^3*e^4 - 20*a^3*b*c*e^4)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4))/(c*x^2 +
b*x + a)^(3/2) - e^4*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(5/2)