### 3.2388 $$\int \frac{1}{(d+e x)^2 (a+b x+c x^2)^{3/2}} \, dx$$

Optimal. Leaf size=255 $-\frac{e \sqrt{a+b x+c x^2} \left (-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right )}{\left (b^2-4 a c\right ) (d+e x) \left (a e^2-b d e+c d^2\right )^2}-\frac{2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x) \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}+\frac{3 e^2 (2 c d-b e) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{5/2}}$

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.275376, antiderivative size = 255, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.182, Rules used = {740, 806, 724, 206} $-\frac{e \sqrt{a+b x+c x^2} \left (-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right )}{\left (b^2-4 a c\right ) (d+e x) \left (a e^2-b d e+c d^2\right )^2}-\frac{2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x) \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}+\frac{3 e^2 (2 c d-b e) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{5/2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/((d + e*x)^2*(a + b*x + c*x^2)^(3/2)),x]

[Out]

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

Rule 740

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

Rule 806

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

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 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{1}{(d+e x)^2 \left (a+b x+c x^2\right )^{3/2}} \, dx &=-\frac{2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \sqrt{a+b x+c x^2}}-\frac{2 \int \frac{\frac{1}{2} e \left (2 b c d-3 b^2 e+8 a c e\right )+c e (2 c d-b e) x}{(d+e x)^2 \sqrt{a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \sqrt{a+b x+c x^2}}-\frac{e \left (4 c^2 d^2+3 b^2 e^2-4 c e (b d+2 a e)\right ) \sqrt{a+b x+c x^2}}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac{\left (3 e^2 (2 c d-b e)\right ) \int \frac{1}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{2 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac{2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \sqrt{a+b x+c x^2}}-\frac{e \left (4 c^2 d^2+3 b^2 e^2-4 c e (b d+2 a e)\right ) \sqrt{a+b x+c x^2}}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)}-\frac{\left (3 e^2 (2 c d-b e)\right ) \operatorname{Subst}\left (\int \frac{1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac{-b d+2 a e-(2 c d-b e) x}{\sqrt{a+b x+c x^2}}\right )}{\left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac{2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \sqrt{a+b x+c x^2}}-\frac{e \left (4 c^2 d^2+3 b^2 e^2-4 c e (b d+2 a e)\right ) \sqrt{a+b x+c x^2}}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac{3 e^2 (2 c d-b e) \tanh ^{-1}\left (\frac{b d-2 a e+(2 c d-b e) x}{2 \sqrt{c d^2-b d e+a e^2} \sqrt{a+b x+c x^2}}\right )}{2 \left (c d^2-b d e+a e^2\right )^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.406901, size = 250, normalized size = 0.98 $\frac{2 \left (-\frac{e \sqrt{a+x (b+c x)} \left (-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right )}{2 (d+e x) \left (e (a e-b d)+c d^2\right )}+\frac{3 e^2 \left (b^2-4 a c\right ) (b e-2 c d) \tanh ^{-1}\left (\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right )}{4 \left (e (a e-b d)+c d^2\right )^{3/2}}+\frac{-2 c (a e+c d x)+b^2 e+b c (e x-d)}{(d+e x) \sqrt{a+x (b+c x)}}\right )}{\left (b^2-4 a c\right ) \left (e (a e-b d)+c d^2\right )}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/((d + e*x)^2*(a + b*x + c*x^2)^(3/2)),x]

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.267, size = 1313, normalized size = 5.2 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^2/(c*x^2+b*x+a)^(3/2),x)

[Out]

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

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 21.7937, size = 6063, normalized size = 23.78 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^2/(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")

[Out]

[-1/4*(3*(2*(a*b^2*c - 4*a^2*c^2)*d^2*e^2 - (a*b^3 - 4*a^2*b*c)*d*e^3 + (2*(b^2*c^2 - 4*a*c^3)*d*e^3 - (b^3*c
- 4*a*b*c^2)*e^4)*x^3 + (2*(b^2*c^2 - 4*a*c^3)*d^2*e^2 + (b^3*c - 4*a*b*c^2)*d*e^3 - (b^4 - 4*a*b^2*c)*e^4)*x^
2 + (2*(b^3*c - 4*a*b*c^2)*d^2*e^2 - (b^4 - 6*a*b^2*c + 8*a^2*c^2)*d*e^3 - (a*b^3 - 4*a^2*b*c)*e^4)*x)*sqrt(c*
d^2 - b*d*e + a*e^2)*log((8*a*b*d*e - 8*a^2*e^2 - (b^2 + 4*a*c)*d^2 - (8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e
^2)*x^2 + 4*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d - b*e)*x) - 2*(4*b*c*d^2 +
4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x)/(e^2*x^2 + 2*d*e*x + d^2)) + 4*(2*b*c^3*d^5 - 2*(3*b^2*c^2 - 4*a*c^3)*d^4
*e + 6*(b^3*c - 2*a*b*c^2)*d^3*e^2 - (2*b^4 - 3*a*b^2*c - 4*a^2*c^2)*d^2*e^3 + (a*b^3 - 2*a^2*b*c)*d*e^4 + (a^
2*b^2 - 4*a^3*c)*e^5 + (4*c^4*d^4*e - 8*b*c^3*d^3*e^2 + (7*b^2*c^2 - 4*a*c^3)*d^2*e^3 - (3*b^3*c - 4*a*b*c^2)*
d*e^4 + (3*a*b^2*c - 8*a^2*c^2)*e^5)*x^2 + (4*c^4*d^5 - 6*b*c^3*d^4*e + 8*a*c^3*d^3*e^2 + (5*b^3*c - 16*a*b*c^
2)*d^2*e^3 - (3*b^4 - 8*a*b^2*c - 4*a^2*c^2)*d*e^4 + (3*a*b^3 - 10*a^2*b*c)*e^5)*x)*sqrt(c*x^2 + b*x + a))/((a
*b^2*c^3 - 4*a^2*c^4)*d^7 - 3*(a*b^3*c^2 - 4*a^2*b*c^3)*d^6*e + 3*(a*b^4*c - 3*a^2*b^2*c^2 - 4*a^3*c^3)*d^5*e^
2 - (a*b^5 + 2*a^2*b^3*c - 24*a^3*b*c^2)*d^4*e^3 + 3*(a^2*b^4 - 3*a^3*b^2*c - 4*a^4*c^2)*d^3*e^4 - 3*(a^3*b^3
- 4*a^4*b*c)*d^2*e^5 + (a^4*b^2 - 4*a^5*c)*d*e^6 + ((b^2*c^4 - 4*a*c^5)*d^6*e - 3*(b^3*c^3 - 4*a*b*c^4)*d^5*e^
2 + 3*(b^4*c^2 - 3*a*b^2*c^3 - 4*a^2*c^4)*d^4*e^3 - (b^5*c + 2*a*b^3*c^2 - 24*a^2*b*c^3)*d^3*e^4 + 3*(a*b^4*c
- 3*a^2*b^2*c^2 - 4*a^3*c^3)*d^2*e^5 - 3*(a^2*b^3*c - 4*a^3*b*c^2)*d*e^6 + (a^3*b^2*c - 4*a^4*c^2)*e^7)*x^3 +
((b^2*c^4 - 4*a*c^5)*d^7 - 2*(b^3*c^3 - 4*a*b*c^4)*d^6*e + 3*(a*b^2*c^3 - 4*a^2*c^4)*d^5*e^2 + (2*b^5*c - 11*a
*b^3*c^2 + 12*a^2*b*c^3)*d^4*e^3 - (b^6 - a*b^4*c - 15*a^2*b^2*c^2 + 12*a^3*c^3)*d^3*e^4 + 3*(a*b^5 - 4*a^2*b^
3*c)*d^2*e^5 - (3*a^2*b^4 - 13*a^3*b^2*c + 4*a^4*c^2)*d*e^6 + (a^3*b^3 - 4*a^4*b*c)*e^7)*x^2 + ((b^3*c^3 - 4*a
*b*c^4)*d^7 - (3*b^4*c^2 - 13*a*b^2*c^3 + 4*a^2*c^4)*d^6*e + 3*(b^5*c - 4*a*b^3*c^2)*d^5*e^2 - (b^6 - a*b^4*c
- 15*a^2*b^2*c^2 + 12*a^3*c^3)*d^4*e^3 + (2*a*b^5 - 11*a^2*b^3*c + 12*a^3*b*c^2)*d^3*e^4 + 3*(a^3*b^2*c - 4*a^
4*c^2)*d^2*e^5 - 2*(a^3*b^3 - 4*a^4*b*c)*d*e^6 + (a^4*b^2 - 4*a^5*c)*e^7)*x), 1/2*(3*(2*(a*b^2*c - 4*a^2*c^2)*
d^2*e^2 - (a*b^3 - 4*a^2*b*c)*d*e^3 + (2*(b^2*c^2 - 4*a*c^3)*d*e^3 - (b^3*c - 4*a*b*c^2)*e^4)*x^3 + (2*(b^2*c^
2 - 4*a*c^3)*d^2*e^2 + (b^3*c - 4*a*b*c^2)*d*e^3 - (b^4 - 4*a*b^2*c)*e^4)*x^2 + (2*(b^3*c - 4*a*b*c^2)*d^2*e^2
- (b^4 - 6*a*b^2*c + 8*a^2*c^2)*d*e^3 - (a*b^3 - 4*a^2*b*c)*e^4)*x)*sqrt(-c*d^2 + b*d*e - a*e^2)*arctan(-1/2*
sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d - b*e)*x)/(a*c*d^2 - a*b*d*e + a^2*e^
2 + (c^2*d^2 - b*c*d*e + a*c*e^2)*x^2 + (b*c*d^2 - b^2*d*e + a*b*e^2)*x)) - 2*(2*b*c^3*d^5 - 2*(3*b^2*c^2 - 4*
a*c^3)*d^4*e + 6*(b^3*c - 2*a*b*c^2)*d^3*e^2 - (2*b^4 - 3*a*b^2*c - 4*a^2*c^2)*d^2*e^3 + (a*b^3 - 2*a^2*b*c)*d
*e^4 + (a^2*b^2 - 4*a^3*c)*e^5 + (4*c^4*d^4*e - 8*b*c^3*d^3*e^2 + (7*b^2*c^2 - 4*a*c^3)*d^2*e^3 - (3*b^3*c - 4
*a*b*c^2)*d*e^4 + (3*a*b^2*c - 8*a^2*c^2)*e^5)*x^2 + (4*c^4*d^5 - 6*b*c^3*d^4*e + 8*a*c^3*d^3*e^2 + (5*b^3*c -
16*a*b*c^2)*d^2*e^3 - (3*b^4 - 8*a*b^2*c - 4*a^2*c^2)*d*e^4 + (3*a*b^3 - 10*a^2*b*c)*e^5)*x)*sqrt(c*x^2 + b*x
+ a))/((a*b^2*c^3 - 4*a^2*c^4)*d^7 - 3*(a*b^3*c^2 - 4*a^2*b*c^3)*d^6*e + 3*(a*b^4*c - 3*a^2*b^2*c^2 - 4*a^3*c
^3)*d^5*e^2 - (a*b^5 + 2*a^2*b^3*c - 24*a^3*b*c^2)*d^4*e^3 + 3*(a^2*b^4 - 3*a^3*b^2*c - 4*a^4*c^2)*d^3*e^4 - 3
*(a^3*b^3 - 4*a^4*b*c)*d^2*e^5 + (a^4*b^2 - 4*a^5*c)*d*e^6 + ((b^2*c^4 - 4*a*c^5)*d^6*e - 3*(b^3*c^3 - 4*a*b*c
^4)*d^5*e^2 + 3*(b^4*c^2 - 3*a*b^2*c^3 - 4*a^2*c^4)*d^4*e^3 - (b^5*c + 2*a*b^3*c^2 - 24*a^2*b*c^3)*d^3*e^4 + 3
*(a*b^4*c - 3*a^2*b^2*c^2 - 4*a^3*c^3)*d^2*e^5 - 3*(a^2*b^3*c - 4*a^3*b*c^2)*d*e^6 + (a^3*b^2*c - 4*a^4*c^2)*e
^7)*x^3 + ((b^2*c^4 - 4*a*c^5)*d^7 - 2*(b^3*c^3 - 4*a*b*c^4)*d^6*e + 3*(a*b^2*c^3 - 4*a^2*c^4)*d^5*e^2 + (2*b^
5*c - 11*a*b^3*c^2 + 12*a^2*b*c^3)*d^4*e^3 - (b^6 - a*b^4*c - 15*a^2*b^2*c^2 + 12*a^3*c^3)*d^3*e^4 + 3*(a*b^5
- 4*a^2*b^3*c)*d^2*e^5 - (3*a^2*b^4 - 13*a^3*b^2*c + 4*a^4*c^2)*d*e^6 + (a^3*b^3 - 4*a^4*b*c)*e^7)*x^2 + ((b^3
*c^3 - 4*a*b*c^4)*d^7 - (3*b^4*c^2 - 13*a*b^2*c^3 + 4*a^2*c^4)*d^6*e + 3*(b^5*c - 4*a*b^3*c^2)*d^5*e^2 - (b^6
- a*b^4*c - 15*a^2*b^2*c^2 + 12*a^3*c^3)*d^4*e^3 + (2*a*b^5 - 11*a^2*b^3*c + 12*a^3*b*c^2)*d^3*e^4 + 3*(a^3*b^
2*c - 4*a^4*c^2)*d^2*e^5 - 2*(a^3*b^3 - 4*a^4*b*c)*d*e^6 + (a^4*b^2 - 4*a^5*c)*e^7)*x)]

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^2/(c*x^2+b*x+a)^(3/2),x, algorithm="giac")

[Out]

Exception raised: RuntimeError