### 3.2340 $$\int \frac{\sqrt{a+b x+c x^2}}{(d+e x)^4} \, dx$$

Optimal. Leaf size=215 $-\frac{\left (b^2-4 a c\right ) (2 c d-b e) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{16 \left (a e^2-b d e+c d^2\right )^{5/2}}-\frac{e \left (a+b x+c x^2\right )^{3/2}}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}+\frac{\sqrt{a+b x+c x^2} (2 c d-b e) (-2 a e+x (2 c d-b e)+b d)}{8 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2}$

[Out]

((2*c*d - b*e)*(b*d - 2*a*e + (2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(8*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^2
) - (e*(a + b*x + c*x^2)^(3/2))/(3*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^3) - ((b^2 - 4*a*c)*(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])])/(16*(c*d^2 - b*d*e +
a*e^2)^(5/2))

________________________________________________________________________________________

Rubi [A]  time = 0.147503, antiderivative size = 215, 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 = {730, 720, 724, 206} $-\frac{\left (b^2-4 a c\right ) (2 c d-b e) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{16 \left (a e^2-b d e+c d^2\right )^{5/2}}-\frac{e \left (a+b x+c x^2\right )^{3/2}}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}+\frac{\sqrt{a+b x+c x^2} (2 c d-b e) (-2 a e+x (2 c d-b e)+b d)}{8 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((2*c*d - b*e)*(b*d - 2*a*e + (2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(8*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^2
) - (e*(a + b*x + c*x^2)^(3/2))/(3*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^3) - ((b^2 - 4*a*c)*(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])])/(16*(c*d^2 - b*d*e +
a*e^2)^(5/2))

Rule 730

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m + 1)
*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[(2*c*d - b*e)/(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, m, p}, 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] && EqQ[m + 2*p + 3, 0]

Rule 720

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)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[(p*(b^2 -
4*a*c))/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(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] && EqQ[m +
2*p + 2, 0] && GtQ[p, 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{\sqrt{a+b x+c x^2}}{(d+e x)^4} \, dx &=-\frac{e \left (a+b x+c x^2\right )^{3/2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}+\frac{(2 c d-b e) \int \frac{\sqrt{a+b x+c x^2}}{(d+e x)^3} \, dx}{2 \left (c d^2-b d e+a e^2\right )}\\ &=\frac{(2 c d-b e) (b d-2 a e+(2 c d-b e) x) \sqrt{a+b x+c x^2}}{8 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}-\frac{e \left (a+b x+c x^2\right )^{3/2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}-\frac{\left (\left (b^2-4 a c\right ) (2 c d-b e)\right ) \int \frac{1}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{16 \left (c d^2-b d e+a e^2\right )^2}\\ &=\frac{(2 c d-b e) (b d-2 a e+(2 c d-b e) x) \sqrt{a+b x+c x^2}}{8 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}-\frac{e \left (a+b x+c x^2\right )^{3/2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}+\frac{\left (\left (b^2-4 a c\right ) (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 )}{8 \left (c d^2-b d e+a e^2\right )^2}\\ &=\frac{(2 c d-b e) (b d-2 a e+(2 c d-b e) x) \sqrt{a+b x+c x^2}}{8 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}-\frac{e \left (a+b x+c x^2\right )^{3/2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}-\frac{\left (b^2-4 a c\right ) (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 )}{16 \left (c d^2-b d e+a e^2\right )^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.433355, size = 206, normalized size = 0.96 $\frac{3 (2 c d-b e) \left (\frac{\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right )}{8 \left (e (a e-b d)+c d^2\right )^{3/2}}+\frac{\sqrt{a+x (b+c x)} (-2 a e+b (d-e x)+2 c d x)}{4 (d+e x)^2 \left (e (a e-b d)+c d^2\right )}\right )-\frac{2 e (a+x (b+c x))^{3/2}}{(d+e x)^3}}{6 \left (e (a e-b d)+c d^2\right )}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((-2*e*(a + x*(b + c*x))^(3/2))/(d + e*x)^3 + 3*(2*c*d - b*e)*((Sqrt[a + x*(b + c*x)]*(-2*a*e + 2*c*d*x + b*(d
- e*x)))/(4*(c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^2) + ((b^2 - 4*a*c)*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)])])/(8*(c*d^2 + e*(-(b*d) + a*e))^(3/2))))/(6*(c*d^2
+ e*(-(b*d) + a*e)))

________________________________________________________________________________________

Maple [B]  time = 0.233, size = 4844, normalized size = 22.5 \begin{align*} \text{output too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^(1/2)/(e*x+d)^4,x)

[Out]

1/2*e/(a*e^2-b*d*e+c*d^2)^3/(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)^(3/2)*c*d*b-3/
4/(a*e^2-b*d*e+c*d^2)^3/((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))*a*
b*c^2*d^2-1/2/e/(a*e^2-b*d*e+c*d^2)^2*c^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))*a*d-1/2*e/(a*e^2-b*d*e+c*d^2)^3*c^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*d-1/2*e/(a*e^2-b*d*e+c*d^2)^3*c^(3/2)*ln((1/2*(b*e-2*c*d)/e+(d/e+x)*c)/c^(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))*a*d*b-5/4/e^2/(a*e^2-b*d*e+c*d^2)^3/((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^3*d^4*b+1/2/e/(a*e^2-b*d*e+c*d^2)^3/((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))*a*c^3*d^3+9/8/e/(a*e^2-b*d*e+c*d^2)^3/((
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^2*d^3*c^2-1/4/e/(a*e^2-b*d
*e+c*d^2)^2*c/((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^2*d+3/4/e^
2/(a*e^2-b*d*e+c*d^2)^2*c^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*d^2-1/2/(a*e^2-b*d*e+c*d^2)^3/(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)^(3/2)*c^
2*d^2+1/2/(a*e^2-b*d*e+c*d^2)^3*c^(5/2)*ln((1/2*(b*e-2*c*d)/e+(d/e+x)*c)/c^(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))*a*d^2-1/2/e/(a*e^2-b*d*e+c*d^2)^3*((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^3*d^3-1/2/e^2/(a*e^2-b*d*e+c*d^2)^2*c^(5/2)*ln((1/2*(b*e-2*c*d)/e+(d/e+x)*c)/c^(
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^2+1/2/e^2/(a*e^2-b*d*e+c*d^2)^3*ln((
1/2*(b*e-2*c*d)/e+(d/e+x)*c)/c^(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))*c^(7/2
)*d^4+1/2/(a*e^2-b*d*e+c*d^2)^3*c^3*((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*d^2+1/
2/e/(a*e^2-b*d*e+c*d^2)^2*c^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-1/8*e^2/(a*e
^2-b*d*e+c*d^2)^3/(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)^(3/2)*b^2+1/(a*e^2-b*d*e
+c*d^2)^3*((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+3/8*e/(a*e^2-b*d*e+c*d^2
)^3/((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))*a*b^2*c*d+5/8/(a*e^2-b
*d*e+c*d^2)^3*ln((1/2*(b*e-2*c*d)/e+(d/e+x)*c)/c^(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))*c^(3/2)*d^2*b^2+1/2/e^3/(a*e^2-b*d*e+c*d^2)^3/((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^4*d^5-1/8*e/(a*e^2-b*d*e+c*d^2)^3*ln((1/2*(b*e-2*c*d)/e+(d/e+x)*c)/c^(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))*c^(1/2)*d*b^3+1/8*e^2/(a*e^2-b*d*e+c*d^2)^3*
((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^3-1/3/e^2/(a*e^2-b*d*e+c*d^2)/(d/e+x)^3*((
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)^(3/2)+1/4/(a*e^2-b*d*e+c*d^2)^2/(d/e+x)^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)^(3/2)*b-1/4/(a*e^2-b*d*e+c*d^2)^2*c*((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-1/8/(a*e^2-b*d*e+c*d^2)^2*c^(1/2)*ln((1/2*(b*e-2*c*d)/e+(d/e+x)*c)
/c^(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))*b^2+1/8*e^2/(a*e^2-b*d*e+c*d^2)^3*
c*((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^2-7/16/(a*e^2-b*d*e+c*d^2)^3/((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*d^2*c+1/4/(a*e^2-b*d*e+c*d^2)^2
*c/((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))*a*b-1/2/e^3/(a*e^2-b*d*
e+c*d^2)^2*c^3/((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))*d^3-5/8*e/(
a*e^2-b*d*e+c*d^2)^3*((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-1/16*e^2/(a*e^2
-b*d*e+c*d^2)^3/((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))*a*b^3-1/2/
e/(a*e^2-b*d*e+c*d^2)^2/(d/e+x)^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)^(3/2)*c*d-1/e/(a
*e^2-b*d*e+c*d^2)^3*ln((1/2*(b*e-2*c*d)/e+(d/e+x)*c)/c^(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))*c^(5/2)*d^3*b+1/8*e^2/(a*e^2-b*d*e+c*d^2)^3*c^(1/2)*ln((1/2*(b*e-2*c*d)/e+(d/e+x)*c)/c^(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))*a*b^2+1/2/e/(a*e^2-b*d*e+c*d^2)^2*c^(3/2)*
ln((1/2*(b*e-2*c*d)/e+(d/e+x)*c)/c^(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*
b+1/16*e/(a*e^2-b*d*e+c*d^2)^3/((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^4*d

________________________________________________________________________________________

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((c*x^2+b*x+a)^(1/2)/(e*x+d)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 49.9147, size = 4016, normalized size = 18.68 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(1/2)/(e*x+d)^4,x, algorithm="fricas")

[Out]

[1/96*(3*(2*(b^2*c - 4*a*c^2)*d^4 - (b^3 - 4*a*b*c)*d^3*e + (2*(b^2*c - 4*a*c^2)*d*e^3 - (b^3 - 4*a*b*c)*e^4)*
x^3 + 3*(2*(b^2*c - 4*a*c^2)*d^2*e^2 - (b^3 - 4*a*b*c)*d*e^3)*x^2 + 3*(2*(b^2*c - 4*a*c^2)*d^3*e - (b^3 - 4*a*
b*c)*d^2*e^2)*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*(6*b*c^2*d^5 + 22*
a^2*b*d*e^4 - 8*a^3*e^5 - (9*b^2*c + 20*a*c^2)*d^4*e + (3*b^3 + 40*a*b*c)*d^3*e^2 - (17*a*b^2 + 28*a^2*c)*d^2*
e^3 + (4*c^3*d^4*e - 8*b*c^2*d^3*e^2 + (7*b^2*c - 4*a*c^2)*d^2*e^3 - (3*b^3 - 4*a*b*c)*d*e^4 + (3*a*b^2 - 8*a^
2*c)*e^5)*x^2 + 2*(6*c^3*d^5 - 13*b*c^2*d^4*e + 11*b^2*c*d^3*e^2 - a^2*b*e^5 - 2*(2*b^3 + a*b*c)*d^2*e^3 + (5*
a*b^2 - 6*a^2*c)*d*e^4)*x)*sqrt(c*x^2 + b*x + a))/(c^3*d^9 - 3*b*c^2*d^8*e - 3*a^2*b*d^4*e^5 + a^3*d^3*e^6 + 3
*(b^2*c + a*c^2)*d^7*e^2 - (b^3 + 6*a*b*c)*d^6*e^3 + 3*(a*b^2 + a^2*c)*d^5*e^4 + (c^3*d^6*e^3 - 3*b*c^2*d^5*e^
4 - 3*a^2*b*d*e^8 + a^3*e^9 + 3*(b^2*c + a*c^2)*d^4*e^5 - (b^3 + 6*a*b*c)*d^3*e^6 + 3*(a*b^2 + a^2*c)*d^2*e^7)
*x^3 + 3*(c^3*d^7*e^2 - 3*b*c^2*d^6*e^3 - 3*a^2*b*d^2*e^7 + a^3*d*e^8 + 3*(b^2*c + a*c^2)*d^5*e^4 - (b^3 + 6*a
*b*c)*d^4*e^5 + 3*(a*b^2 + a^2*c)*d^3*e^6)*x^2 + 3*(c^3*d^8*e - 3*b*c^2*d^7*e^2 - 3*a^2*b*d^3*e^6 + a^3*d^2*e^
7 + 3*(b^2*c + a*c^2)*d^6*e^3 - (b^3 + 6*a*b*c)*d^5*e^4 + 3*(a*b^2 + a^2*c)*d^4*e^5)*x), -1/48*(3*(2*(b^2*c -
4*a*c^2)*d^4 - (b^3 - 4*a*b*c)*d^3*e + (2*(b^2*c - 4*a*c^2)*d*e^3 - (b^3 - 4*a*b*c)*e^4)*x^3 + 3*(2*(b^2*c - 4
*a*c^2)*d^2*e^2 - (b^3 - 4*a*b*c)*d*e^3)*x^2 + 3*(2*(b^2*c - 4*a*c^2)*d^3*e - (b^3 - 4*a*b*c)*d^2*e^2)*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*(6*b*c^2*d^5 + 22*a^2*b*d*e^4 - 8*a^3*e^5 - (9*b^2*c + 20*a*c^2)*d^4*e + (3*b^3 + 40*a*b*c)*d^3*e^2 - (17
*a*b^2 + 28*a^2*c)*d^2*e^3 + (4*c^3*d^4*e - 8*b*c^2*d^3*e^2 + (7*b^2*c - 4*a*c^2)*d^2*e^3 - (3*b^3 - 4*a*b*c)*
d*e^4 + (3*a*b^2 - 8*a^2*c)*e^5)*x^2 + 2*(6*c^3*d^5 - 13*b*c^2*d^4*e + 11*b^2*c*d^3*e^2 - a^2*b*e^5 - 2*(2*b^3
+ a*b*c)*d^2*e^3 + (5*a*b^2 - 6*a^2*c)*d*e^4)*x)*sqrt(c*x^2 + b*x + a))/(c^3*d^9 - 3*b*c^2*d^8*e - 3*a^2*b*d^
4*e^5 + a^3*d^3*e^6 + 3*(b^2*c + a*c^2)*d^7*e^2 - (b^3 + 6*a*b*c)*d^6*e^3 + 3*(a*b^2 + a^2*c)*d^5*e^4 + (c^3*d
^6*e^3 - 3*b*c^2*d^5*e^4 - 3*a^2*b*d*e^8 + a^3*e^9 + 3*(b^2*c + a*c^2)*d^4*e^5 - (b^3 + 6*a*b*c)*d^3*e^6 + 3*(
a*b^2 + a^2*c)*d^2*e^7)*x^3 + 3*(c^3*d^7*e^2 - 3*b*c^2*d^6*e^3 - 3*a^2*b*d^2*e^7 + a^3*d*e^8 + 3*(b^2*c + a*c^
2)*d^5*e^4 - (b^3 + 6*a*b*c)*d^4*e^5 + 3*(a*b^2 + a^2*c)*d^3*e^6)*x^2 + 3*(c^3*d^8*e - 3*b*c^2*d^7*e^2 - 3*a^2
*b*d^3*e^6 + a^3*d^2*e^7 + 3*(b^2*c + a*c^2)*d^6*e^3 - (b^3 + 6*a*b*c)*d^5*e^4 + 3*(a*b^2 + a^2*c)*d^4*e^5)*x)
]

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a + b x + c x^{2}}}{\left (d + e x\right )^{4}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**(1/2)/(e*x+d)**4,x)

[Out]

Integral(sqrt(a + b*x + c*x**2)/(d + e*x)**4, x)

________________________________________________________________________________________

Giac [B]  time = 1.58178, size = 2631, normalized size = 12.24 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(1/2)/(e*x+d)^4,x, algorithm="giac")

[Out]

-1/8*(2*b^2*c*d - 8*a*c^2*d - b^3*e + 4*a*b*c*e)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x + a))*e + sqrt(c)*d)/s
qrt(-c*d^2 + b*d*e - a*e^2))/((c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 + 2*a*c*d^2*e^2 - 2*a*b*d*e^3 + a^2*e^4)*sq
rt(-c*d^2 + b*d*e - a*e^2)) + 1/24*(48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*c^(7/2)*d^4*e + 32*(sqrt(c)*x - s
qrt(c*x^2 + b*x + a))^3*c^4*d^5 + 16*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b*c^3*d^4*e + 48*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))^2*b*c^(7/2)*d^5 - 96*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b*c^(5/2)*d^3*e^2 - 36*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^2*b^2*c^(5/2)*d^4*e - 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*c^(7/2)*d^4*e + 24*
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^2*c^3*d^5 - 84*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^2*c^2*d^3*e^2 - 1
12*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*c^3*d^3*e^2 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^3*c^2*d^4*e
- 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b*c^3*d^4*e + 4*b^3*c^(5/2)*d^5 + 78*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^4*b^2*c^(3/2)*d^2*e^3 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*c^(5/2)*d^2*e^3 - 6*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))^2*b^3*c^(3/2)*d^3*e^2 - 72*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b*c^(5/2)*d^3*e^2 - 4*b^
4*c^(3/2)*d^4*e - 12*a*b^2*c^(5/2)*d^4*e + 6*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^2*c*d*e^4 - 24*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^5*a*c^2*d*e^4 + 74*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^3*c*d^2*e^3 + 120*(sqrt(c
)*x - sqrt(c*x^2 + b*x + a))^3*a*b*c^2*d^2*e^3 + 12*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^4*c*d^3*e^2 - 12*(sq
rt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^2*c^2*d^3*e^2 + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*c^3*d^3*e^2 -
15*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^3*sqrt(c)*d*e^4 - 36*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b*c^(3
/2)*d*e^4 + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^4*sqrt(c)*d^2*e^3 + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^2*a*b^2*c^(3/2)*d^2*e^3 + 192*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*c^(5/2)*d^2*e^3 + 3*b^5*sqrt(c)*d^
3*e^2 - 2*a*b^3*c^(3/2)*d^3*e^2 + 24*a^2*b*c^(5/2)*d^3*e^2 - 3*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^3*e^5 +
12*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b*c*e^5 - 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^4*d*e^4 - 144*
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^2*c*d*e^4 + 96*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*c^2*d*e^4 +
3*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^5*d^2*e^3 - 18*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^3*c*d^2*e^3 +
120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b*c^2*d^2*e^3 + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*c^(3/
2)*e^5 - 72*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^3*sqrt(c)*d*e^4 - 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
^2*a^2*b*c^(3/2)*d*e^4 - 6*a*b^4*sqrt(c)*d^2*e^3 + 18*a^2*b^2*c^(3/2)*d^2*e^3 - 8*a^3*c^(5/2)*d^2*e^3 + 8*(sqr
t(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^3*e^5 + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b*c*e^5 - 6*(sqrt(c
)*x - sqrt(c*x^2 + b*x + a))*a*b^4*d*e^4 - 30*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^2*c*d*e^4 - 72*(sqrt(c
)*x - sqrt(c*x^2 + b*x + a))*a^3*c^2*d*e^4 + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b^2*sqrt(c)*e^5 + 3*
a^2*b^3*sqrt(c)*d*e^4 - 28*a^3*b*c^(3/2)*d*e^4 + 3*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^3*e^5 + 36*(sqrt(
c)*x - sqrt(c*x^2 + b*x + a))*a^3*b*c*e^5 + 16*a^4*c^(3/2)*e^5)/((c^2*d^4*e^2 - 2*b*c*d^3*e^3 + b^2*d^2*e^4 +
2*a*c*d^2*e^4 - 2*a*b*d*e^5 + a^2*e^6)*((sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))*sqrt(c)*d + b*d - a*e)^3)