\(\int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^4} \, dx\) [2340]

Optimal result

Integrand size = 22, antiderivative size = 215 \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^4} \, dx=\frac {(2 c d-b e) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{8 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}-\frac {e \left (a+b x+c x^2\right )^{3/2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}-\frac {\left (b^2-4 a c\right ) (2 c d-b e) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{16 \left (c d^2-b d e+a e^2\right )^{5/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {744, 734, 738, 212} \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^4} \, dx=-\frac {\left (b^2-4 a c\right ) (2 c d-b e) \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{16 \left (a e^2-b d e+c d^2\right )^{5/2}}-\frac {e \left (a+b x+c x^2\right )^{3/2}}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} (2 c d-b e) (-2 a e+x (2 c d-b e)+b d)}{8 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2} \]

[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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 734

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] /; Free
Q[{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 738

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 744

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]

Rubi steps \begin{align*} \text {integral}& = -\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 ) \text {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] (verified)

Time = 10.26 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^4} \, dx=\frac {-\frac {2 e (a+x (b+c x))^{3/2}}{(d+e x)^3}+3 (2 c d-b e) \left (\frac {\sqrt {a+x (b+c x)} (-2 a e+2 c d x+b (d-e x))}{4 \left (c d^2+e (-b d+a e)\right ) (d+e x)^2}+\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{8 \left (c d^2+e (-b d+a e)\right )^{3/2}}\right )}{6 \left (c d^2+e (-b d+a e)\right )} \]

[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] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1252\) vs. \(2(197)=394\).

Time = 0.51 (sec) , antiderivative size = 1253, normalized size of antiderivative = 5.83

method	result	size
default	\(\text {Expression too large to display}\)	\(1253\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 954 vs. \(2 (197) = 394\).

Time = 2.14 (sec) , antiderivative size = 1950, normalized size of antiderivative = 9.07 \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^4} \, dx=\text {Too large to display} \]

[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]

\[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^4} \, dx=\int \frac {\sqrt {a + b x + c x^{2}}}{\left (d + e x\right )^{4}}\, dx \]

[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)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^4} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*e^2-b*d*e>0)', see `assume?`
for more de

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1992 vs. \(2 (197) = 394\).

Time = 0.36 (sec) , antiderivative size = 1992, normalized size of antiderivative = 9.27 \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^4} \, dx=\text {Too large to display} \]

[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*(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 - 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 + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*c^(7/2)*d^4*e - 96*(sqrt(c)*x - sqrt(c*x^
2 + b*x + a))^4*b*c^(5/2)*d^3*e^2 + 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 - 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 + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*c^(
3/2)*e^5 + 32*(sqrt(c)*x - sqrt(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 - 84*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^2*c^2*d^3*e^2 - 112*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*
c^3*d^3*e^2 + 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 - 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 + 8*(sqrt(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 + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^2*b*c^(7/2)*d^5 - 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 - 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 + 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 - 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 + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b^2*sqrt(c)*e^5 + 24*(sqrt(c)*
x - sqrt(c*x^2 + b*x + a))*b^2*c^3*d^5 - 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 + 12*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^4*c*d^3*e^2 - 12*(sqrt(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 + 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 - 6*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^4*d*e^4 - 30*(sqr
t(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 + 3*(sq
rt(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 + 4*b^3*c^(5
/2)*d^5 - 4*b^4*c^(3/2)*d^4*e - 12*a*b^2*c^(5/2)*d^4*e + 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 - 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 + 3*a^2*b
^3*sqrt(c)*d*e^4 - 28*a^3*b*c^(3/2)*d*e^4 + 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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^4} \, dx=\int \frac {\sqrt {c\,x^2+b\,x+a}}{{\left (d+e\,x\right )}^4} \,d x \]

[In]

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

[Out]

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