\(\int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^6} \, dx\) [2342]

Optimal result

Integrand size = 22, antiderivative size = 402 \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^6} \, dx=\frac {(2 c d-b e) \left (16 c^2 d^2+7 b^2 e^2-4 c e (4 b d+3 a e)\right ) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{128 \left (c d^2-b d e+a e^2\right )^4 (d+e x)^2}-\frac {e \left (a+b x+c x^2\right )^{3/2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^5}-\frac {7 e (2 c d-b e) \left (a+b x+c x^2\right )^{3/2}}{40 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}-\frac {e \left (108 c^2 d^2+35 b^2 e^2-4 c e (27 b d+8 a e)\right ) \left (a+b x+c x^2\right )^{3/2}}{240 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^3}-\frac {\left (b^2-4 a c\right ) (2 c d-b e) \left (16 c^2 d^2+7 b^2 e^2-4 c e (4 b d+3 a e)\right ) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{256 \left (c d^2-b d e+a e^2\right )^{9/2}} \]

[Out]

-1/5*e*(c*x^2+b*x+a)^(3/2)/(a*e^2-b*d*e+c*d^2)/(e*x+d)^5-7/40*e*(-b*e+2*c*d)*(c*x^2+b*x+a)^(3/2)/(a*e^2-b*d*e+
c*d^2)^2/(e*x+d)^4-1/240*e*(108*c^2*d^2+35*b^2*e^2-4*c*e*(8*a*e+27*b*d))*(c*x^2+b*x+a)^(3/2)/(a*e^2-b*d*e+c*d^
2)^3/(e*x+d)^3-1/256*(-4*a*c+b^2)*(-b*e+2*c*d)*(16*c^2*d^2+7*b^2*e^2-4*c*e*(3*a*e+4*b*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)^(9/2)+1/128*(-b*e+2*c*d)
*(16*c^2*d^2+7*b^2*e^2-4*c*e*(3*a*e+4*b*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)
^4/(e*x+d)^2

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {758, 848, 820, 734, 738, 212} \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^6} \, dx=-\frac {\left (b^2-4 a c\right ) (2 c d-b e) \left (-4 c e (3 a e+4 b d)+7 b^2 e^2+16 c^2 d^2\right ) \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{256 \left (a e^2-b d e+c d^2\right )^{9/2}}-\frac {e \left (a+b x+c x^2\right )^{3/2} \left (-4 c e (8 a e+27 b d)+35 b^2 e^2+108 c^2 d^2\right )}{240 (d+e x)^3 \left (a e^2-b d e+c d^2\right )^3}+\frac {\sqrt {a+b x+c x^2} (2 c d-b e) \left (-4 c e (3 a e+4 b d)+7 b^2 e^2+16 c^2 d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{128 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^4}-\frac {7 e \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{40 (d+e x)^4 \left (a e^2-b d e+c d^2\right )^2}-\frac {e \left (a+b x+c x^2\right )^{3/2}}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )} \]

[In]

Int[Sqrt[a + b*x + c*x^2]/(d + e*x)^6,x]

[Out]

((2*c*d - b*e)*(16*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(4*b*d + 3*a*e))*(b*d - 2*a*e + (2*c*d - b*e)*x)*Sqrt[a + b*x +
c*x^2])/(128*(c*d^2 - b*d*e + a*e^2)^4*(d + e*x)^2) - (e*(a + b*x + c*x^2)^(3/2))/(5*(c*d^2 - b*d*e + a*e^2)*
(d + e*x)^5) - (7*e*(2*c*d - b*e)*(a + b*x + c*x^2)^(3/2))/(40*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^4) - (e*(10
8*c^2*d^2 + 35*b^2*e^2 - 4*c*e*(27*b*d + 8*a*e))*(a + b*x + c*x^2)^(3/2))/(240*(c*d^2 - b*d*e + a*e^2)^3*(d +
e*x)^3) - ((b^2 - 4*a*c)*(2*c*d - b*e)*(16*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(4*b*d + 3*a*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])])/(256*(c*d^2 - b*d*e + a*e^2)^(9/2))

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 758

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[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)),
Int[(d + e*x)^(m + 1)*Simp[c*d*(m + 1) - b*e*(m + p + 2) - c*e*(m + 2*p + 3)*x, x]*(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] && NeQ[m, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]) || (SumSimplerQ[m, 1] && IntegerQ
[p]) || ILtQ[Simplify[m + 2*p + 3], 0])

Rule 820

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(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[S
implify[m + 2*p + 3], 0]

Rule 848

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

Rubi steps \begin{align*} \text {integral}& = -\frac {e \left (a+b x+c x^2\right )^{3/2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^5}-\frac {\int \frac {\left (\frac {1}{2} (-10 c d+7 b e)+2 c e x\right ) \sqrt {a+b x+c x^2}}{(d+e x)^5} \, dx}{5 \left (c d^2-b d e+a e^2\right )} \\ & = -\frac {e \left (a+b x+c x^2\right )^{3/2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^5}-\frac {7 e (2 c d-b e) \left (a+b x+c x^2\right )^{3/2}}{40 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}+\frac {\int \frac {\left (\frac {1}{4} \left (80 c^2 d^2+35 b^2 e^2-2 c e (47 b d+16 a e)\right )-\frac {7}{2} c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{(d+e x)^4} \, dx}{20 \left (c d^2-b d e+a e^2\right )^2} \\ & = -\frac {e \left (a+b x+c x^2\right )^{3/2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^5}-\frac {7 e (2 c d-b e) \left (a+b x+c x^2\right )^{3/2}}{40 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}-\frac {e \left (108 c^2 d^2+35 b^2 e^2-4 c e (27 b d+8 a e)\right ) \left (a+b x+c x^2\right )^{3/2}}{240 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^3}+\frac {\left ((2 c d-b e) \left (16 c^2 d^2+7 b^2 e^2-4 c e (4 b d+3 a e)\right )\right ) \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^3} \, dx}{32 \left (c d^2-b d e+a e^2\right )^3} \\ & = \frac {(2 c d-b e) \left (16 c^2 d^2+7 b^2 e^2-4 c e (4 b d+3 a e)\right ) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{128 \left (c d^2-b d e+a e^2\right )^4 (d+e x)^2}-\frac {e \left (a+b x+c x^2\right )^{3/2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^5}-\frac {7 e (2 c d-b e) \left (a+b x+c x^2\right )^{3/2}}{40 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}-\frac {e \left (108 c^2 d^2+35 b^2 e^2-4 c e (27 b d+8 a e)\right ) \left (a+b x+c x^2\right )^{3/2}}{240 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^3}-\frac {\left (\left (b^2-4 a c\right ) (2 c d-b e) \left (16 c^2 d^2+7 b^2 e^2-4 c e (4 b d+3 a e)\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{256 \left (c d^2-b d e+a e^2\right )^4} \\ & = \frac {(2 c d-b e) \left (16 c^2 d^2+7 b^2 e^2-4 c e (4 b d+3 a e)\right ) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{128 \left (c d^2-b d e+a e^2\right )^4 (d+e x)^2}-\frac {e \left (a+b x+c x^2\right )^{3/2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^5}-\frac {7 e (2 c d-b e) \left (a+b x+c x^2\right )^{3/2}}{40 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}-\frac {e \left (108 c^2 d^2+35 b^2 e^2-4 c e (27 b d+8 a e)\right ) \left (a+b x+c x^2\right )^{3/2}}{240 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^3}+\frac {\left (\left (b^2-4 a c\right ) (2 c d-b e) \left (16 c^2 d^2+7 b^2 e^2-4 c e (4 b d+3 a e)\right )\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 )}{128 \left (c d^2-b d e+a e^2\right )^4} \\ & = \frac {(2 c d-b e) \left (16 c^2 d^2+7 b^2 e^2-4 c e (4 b d+3 a e)\right ) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{128 \left (c d^2-b d e+a e^2\right )^4 (d+e x)^2}-\frac {e \left (a+b x+c x^2\right )^{3/2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^5}-\frac {7 e (2 c d-b e) \left (a+b x+c x^2\right )^{3/2}}{40 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}-\frac {e \left (108 c^2 d^2+35 b^2 e^2-4 c e (27 b d+8 a e)\right ) \left (a+b x+c x^2\right )^{3/2}}{240 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^3}-\frac {\left (b^2-4 a c\right ) (2 c d-b e) \left (16 c^2 d^2+7 b^2 e^2-4 c e (4 b d+3 a e)\right ) \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 )}{256 \left (c d^2-b d e+a e^2\right )^{9/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 11.65 (sec) , antiderivative size = 367, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^6} \, dx=-\frac {\frac {e (a+x (b+c x))^{3/2}}{(d+e x)^5}+\frac {7 e (2 c d-b e) (a+x (b+c x))^{3/2}}{8 \left (c d^2+e (-b d+a e)\right ) (d+e x)^4}-\frac {-\frac {e \left (108 c^2 d^2+35 b^2 e^2-4 c e (27 b d+8 a e)\right ) (a+x (b+c x))^{3/2}}{2 (d+e x)^3}+\frac {15}{4} (2 c d-b e) \left (16 c^2 d^2+7 b^2 e^2-4 c e (4 b d+3 a e)\right ) \left (\frac {\sqrt {a+x (b+c x)} (-2 a e+2 c d x+b (d-e x))}{4 \left (c d^2+e (-b d+a e)\right ) (d+e x)^2}+\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{8 \left (c d^2+e (-b d+a e)\right )^{3/2}}\right )}{24 \left (c d^2+e (-b d+a e)\right )^2}}{5 \left (c d^2+e (-b d+a e)\right )} \]

[In]

Integrate[Sqrt[a + b*x + c*x^2]/(d + e*x)^6,x]

[Out]

-1/5*((e*(a + x*(b + c*x))^(3/2))/(d + e*x)^5 + (7*e*(2*c*d - b*e)*(a + x*(b + c*x))^(3/2))/(8*(c*d^2 + e*(-(b
*d) + a*e))*(d + e*x)^4) - (-1/2*(e*(108*c^2*d^2 + 35*b^2*e^2 - 4*c*e*(27*b*d + 8*a*e))*(a + x*(b + c*x))^(3/2
))/(d + e*x)^3 + (15*(2*c*d - b*e)*(16*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(4*b*d + 3*a*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))))/4)/(24*(c*d^2 + e*(-(b*d) + a*e))^2))/(c*d^2 + e*(-(b*d) + a*e))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3911\) vs. \(2(376)=752\).

Time = 0.69 (sec) , antiderivative size = 3912, normalized size of antiderivative = 9.73

method	result	size
default	\(\text {Expression too large to display}\)	\(3912\)

[In]

int((c*x^2+b*x+a)^(1/2)/(e*x+d)^6,x,method=_RETURNVERBOSE)

[Out]

1/e^6*(-1/5/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)^5*((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
)-7/10*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(-1/4/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)^4*((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)-5/8*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(-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)))))-1/4*c/(a*e^2-b*d*e+c*d^2)*e^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)))))-2/5*c/(a*e^2-b*d*e+c*d^2)*e^2*(
-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. 3166 vs. \(2 (376) = 752\).

Time = 58.98 (sec) , antiderivative size = 6374, normalized size of antiderivative = 15.86 \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^6} \, dx=\text {Too large to display} \]

[In]

integrate((c*x^2+b*x+a)^(1/2)/(e*x+d)^6,x, algorithm="fricas")

[Out]

Too large to include

Sympy [F]

\[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^6} \, dx=\int \frac {\sqrt {a + b x + c x^{2}}}{\left (d + e x\right )^{6}}\, dx \]

[In]

integrate((c*x**2+b*x+a)**(1/2)/(e*x+d)**6,x)

[Out]

Integral(sqrt(a + b*x + c*x**2)/(d + e*x)**6, x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^6} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((c*x^2+b*x+a)^(1/2)/(e*x+d)^6,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. 8697 vs. \(2 (376) = 752\).

Time = 0.69 (sec) , antiderivative size = 8697, normalized size of antiderivative = 21.63 \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^6} \, dx=\text {Too large to display} \]

[In]

integrate((c*x^2+b*x+a)^(1/2)/(e*x+d)^6,x, algorithm="giac")

[Out]

-1/128*(32*b^2*c^3*d^3 - 128*a*c^4*d^3 - 48*b^3*c^2*d^2*e + 192*a*b*c^3*d^2*e + 30*b^4*c*d*e^2 - 144*a*b^2*c^2
*d*e^2 + 96*a^2*c^3*d*e^2 - 7*b^5*e^3 + 40*a*b^3*c*e^3 - 48*a^2*b*c^2*e^3)*arctan(-((sqrt(c)*x - sqrt(c*x^2 +
b*x + a))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e - a*e^2))/((c^4*d^8 - 4*b*c^3*d^7*e + 6*b^2*c^2*d^6*e^2 + 4*a*c^3
*d^6*e^2 - 4*b^3*c*d^5*e^3 - 12*a*b*c^2*d^5*e^3 + b^4*d^4*e^4 + 12*a*b^2*c*d^4*e^4 + 6*a^2*c^2*d^4*e^4 - 4*a*b
^3*d^3*e^5 - 12*a^2*b*c*d^3*e^5 + 6*a^2*b^2*d^2*e^6 + 4*a^3*c*d^2*e^6 - 4*a^3*b*d*e^7 + a^4*e^8)*sqrt(-c*d^2 +
b*d*e - a*e^2)) + 1/1920*(480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*b^2*c^3*d^3*e^6 - 1920*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))^9*a*c^4*d^3*e^6 - 720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*b^3*c^2*d^2*e^7 + 2880*(sqrt(c)*
x - sqrt(c*x^2 + b*x + a))^9*a*b*c^3*d^2*e^7 + 450*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*b^4*c*d*e^8 - 2160*(s
qrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a*b^2*c^2*d*e^8 + 1440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a^2*c^3*d*e^8
- 105*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*b^5*e^9 + 600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a*b^3*c*e^9 -
720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a^2*b*c^2*e^9 + 4320*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*b^2*c^(7
/2)*d^4*e^5 - 17280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a*c^(9/2)*d^4*e^5 - 6480*(sqrt(c)*x - sqrt(c*x^2 + b
*x + a))^8*b^3*c^(5/2)*d^3*e^6 + 25920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a*b*c^(7/2)*d^3*e^6 + 4050*(sqrt(
c)*x - sqrt(c*x^2 + b*x + a))^8*b^4*c^(3/2)*d^2*e^7 - 19440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a*b^2*c^(5/2
)*d^2*e^7 + 12960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a^2*c^(7/2)*d^2*e^7 - 945*(sqrt(c)*x - sqrt(c*x^2 + b*
x + a))^8*b^5*sqrt(c)*d*e^8 + 5400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a*b^3*c^(3/2)*d*e^8 - 6480*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^8*a^2*b*c^(5/2)*d*e^8 + 15040*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*b^2*c^4*d^5*e^4
- 60160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*c^5*d^5*e^4 - 20320*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*b^3*
c^3*d^4*e^5 + 81280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*b*c^4*d^4*e^5 + 10740*(sqrt(c)*x - sqrt(c*x^2 + b*
x + a))^7*b^4*c^2*d^3*e^6 - 56480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*b^2*c^3*d^3*e^6 + 54080*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^7*a^2*c^4*d^3*e^6 - 1190*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*b^5*c*d^2*e^7 + 12080*(
sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*b^3*c^2*d^2*e^7 - 29280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^2*b*c^3
*d^2*e^7 - 490*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*b^6*d*e^8 + 700*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*b
^4*c*d*e^8 + 6720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^2*b^2*c^2*d*e^8 - 6720*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^7*a^3*c^3*d*e^8 + 490*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*b^5*e^9 - 2800*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^7*a^2*b^3*c*e^9 + 3360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^3*b*c^2*e^9 + 7680*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))^6*c^(13/2)*d^8*e - 30720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b*c^(11/2)*d^7*e^2 + 70720*(s
qrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b^2*c^(9/2)*d^6*e^3 - 67840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a*c^(11/
2)*d^6*e^3 - 52000*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b^3*c^(7/2)*d^5*e^4 - 7040*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^6*a*b*c^(9/2)*d^5*e^4 + 7260*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b^4*c^(5/2)*d^4*e^5 + 59680*(sqrt
(c)*x - sqrt(c*x^2 + b*x + a))^6*a*b^2*c^(7/2)*d^4*e^5 + 182720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^2*c^(9
/2)*d^4*e^5 + 9310*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b^5*c^(3/2)*d^3*e^6 - 46960*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^6*a*b^3*c^(5/2)*d^3*e^6 - 176160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^2*b*c^(7/2)*d^3*e^6 - 3430
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b^6*sqrt(c)*d^2*e^7 + 4900*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a*b^4*
c^(3/2)*d^2*e^7 + 93120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^2*b^2*c^(5/2)*d^2*e^7 - 16320*(sqrt(c)*x - sqr
t(c*x^2 + b*x + a))^6*a^3*c^(7/2)*d^2*e^7 + 3430*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a*b^5*sqrt(c)*d*e^8 - 1
9600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^2*b^3*c^(3/2)*d*e^8 - 7200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*
a^3*b*c^(5/2)*d*e^8 + 7680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^4*c^(5/2)*e^9 + 3072*(sqrt(c)*x - sqrt(c*x^
2 + b*x + a))^5*c^7*d^9 + 9216*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b*c^6*d^8*e - 50048*(sqrt(c)*x - sqrt(c*x
^2 + b*x + a))^5*b^2*c^5*d^7*e^2 - 57856*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*c^6*d^7*e^2 + 129280*(sqrt(c)
*x - sqrt(c*x^2 + b*x + a))^5*b^3*c^4*d^6*e^3 - 1024*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b*c^5*d^6*e^3 - 1
20680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^4*c^3*d^5*e^4 - 122240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b
^2*c^4*d^5*e^4 + 226432*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^2*c^5*d^5*e^4 + 47944*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^5*b^5*c^2*d^4*e^5 + 167520*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^3*c^3*d^4*e^5 - 17920*(sqrt(c
)*x - sqrt(c*x^2 + b*x + a))^5*a^2*b*c^4*d^4*e^5 - 4658*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^6*c*d^3*e^6 -
80020*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^4*c^2*d^3*e^6 - 155840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a
^2*b^2*c^3*d^3*e^6 - 120640*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^3*c^4*d^3*e^6 - 896*(sqrt(c)*x - sqrt(c*x^
2 + b*x + a))^5*b^7*d^2*e^7 + 5938*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^5*c*d^2*e^7 + 105040*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^5*a^2*b^3*c^2*d^2*e^7 + 132000*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^3*b*c^3*d^2*e^7
+ 1792*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^6*d*e^8 - 6400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^2*b^4
*c*d*e^8 - 92160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^3*b^2*c^2*d*e^8 + 15360*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^5*a^4*c^3*d*e^8 - 896*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^2*b^5*e^9 + 5120*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^5*a^3*b^3*c*e^9 + 15360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^4*b*c^2*e^9 + 7680*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))^4*b*c^(13/2)*d^9 - 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^2*c^(11/2)*d^8*e - 7680*(
sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*c^(13/2)*d^8*e - 17600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^3*c^(9/2
)*d^7*e^2 - 113920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b*c^(11/2)*d^7*e^2 + 81920*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^4*b^4*c^(7/2)*d^6*e^3 + 114240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^2*c^(9/2)*d^6*e^3 + 16768
0*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*c^(11/2)*d^6*e^3 - 85780*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^5
*c^(5/2)*d^5*e^4 - 237120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^3*c^(7/2)*d^5*e^4 + 63040*(sqrt(c)*x - sqr
t(c*x^2 + b*x + a))^4*a^2*b*c^(9/2)*d^5*e^4 + 35330*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^6*c^(3/2)*d^4*e^5
+ 244660*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^4*c^(5/2)*d^4*e^5 + 32960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a
))^4*a^2*b^2*c^(7/2)*d^4*e^5 - 178880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^3*c^(9/2)*d^4*e^5 - 4480*(sqrt(c
)*x - sqrt(c*x^2 + b*x + a))^4*b^7*sqrt(c)*d^3*e^6 - 101890*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^5*c^(3/2
)*d^3*e^6 - 179920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*b^3*c^(5/2)*d^3*e^6 + 56160*(sqrt(c)*x - sqrt(c*x
^2 + b*x + a))^4*a^3*b*c^(7/2)*d^3*e^6 + 8960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^6*sqrt(c)*d^2*e^7 + 11
3920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*b^4*c^(3/2)*d^2*e^7 + 51200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
^4*a^3*b^2*c^(5/2)*d^2*e^7 + 71680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^4*c^(7/2)*d^2*e^7 - 4480*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^4*a^2*b^5*sqrt(c)*d*e^8 - 71680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^3*b^3*c^(3/2
)*d*e^8 - 33280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^4*b*c^(5/2)*d*e^8 + 24320*(sqrt(c)*x - sqrt(c*x^2 + b*
x + a))^4*a^4*b^2*c^(3/2)*e^9 + 2560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^5*c^(5/2)*e^9 + 7680*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^3*b^2*c^6*d^9 - 11520*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^3*c^5*d^8*e - 15360*(sqr
t(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b*c^6*d^8*e + 14080*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^4*c^4*d^7*e^2
- 90880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^2*c^5*d^7*e^2 + 15360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*
a^2*c^6*d^7*e^2 + 14080*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^5*c^3*d^6*e^3 + 88320*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^3*a*b^3*c^4*d^6*e^3 + 281600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b*c^5*d^6*e^3 - 25220*(sqrt
(c)*x - sqrt(c*x^2 + b*x + a))^3*b^6*c^2*d^5*e^4 - 124960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^4*c^3*d^5*
e^4 - 239360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b^2*c^4*d^5*e^4 - 160000*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^3*a^3*c^5*d^5*e^4 + 10510*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^7*c*d^4*e^5 + 120280*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))^3*a*b^5*c^2*d^4*e^5 + 200320*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b^3*c^3*d^4*e^5 + 42
240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^3*b*c^4*d^4*e^5 - 790*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^8*d^
3*e^6 - 47320*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^6*c*d^3*e^6 - 160420*(sqrt(c)*x - sqrt(c*x^2 + b*x + a
))^3*a^2*b^4*c^2*d^3*e^6 - 79200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^3*b^2*c^3*d^3*e^6 + 93120*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^3*a^4*c^4*d^3*e^6 + 2370*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^7*d^2*e^7 + 72310*
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b^5*c*d^2*e^7 + 71280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^3*b^3*
c^2*d^2*e^7 + 3680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^4*b*c^3*d^2*e^7 - 2370*(sqrt(c)*x - sqrt(c*x^2 + b*
x + a))^3*a^2*b^6*d*e^8 - 44700*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^3*b^4*c*d*e^8 - 17920*(sqrt(c)*x - sqr
t(c*x^2 + b*x + a))^3*a^4*b^2*c^2*d*e^8 - 13760*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^5*c^3*d*e^8 + 790*(sqr
t(c)*x - sqrt(c*x^2 + b*x + a))^3*a^3*b^5*e^9 + 9200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^4*b^3*c*e^9 + 120
00*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^5*b*c^2*e^9 + 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^3*c^(11/
2)*d^9 - 7200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^4*c^(9/2)*d^8*e - 11520*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^2*a*b^2*c^(11/2)*d^8*e + 13760*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^5*c^(7/2)*d^7*e^2 - 37760*(sqrt(c)*
x - sqrt(c*x^2 + b*x + a))^2*a*b^3*c^(9/2)*d^7*e^2 + 23040*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b*c^(11/2
)*d^7*e^2 - 6340*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^6*c^(5/2)*d^6*e^3 + 22000*(sqrt(c)*x - sqrt(c*x^2 + b
*x + a))^2*a*b^4*c^(7/2)*d^6*e^3 + 176640*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b^2*c^(9/2)*d^6*e^3 - 7680
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*c^(11/2)*d^6*e^3 - 1750*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^7*c
^(3/2)*d^5*e^4 - 13120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^5*c^(5/2)*d^5*e^4 - 187840*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))^2*a^2*b^3*c^(7/2)*d^5*e^4 - 216960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*b*c^(9/2)*d^5*e
^4 + 1470*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^8*sqrt(c)*d^4*e^5 + 16760*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)^2*a*b^6*c^(3/2)*d^4*e^5 + 112660*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b^4*c^(5/2)*d^4*e^5 + 212960*(sqr
t(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*b^2*c^(7/2)*d^4*e^5 + 89920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^4*c^
(9/2)*d^4*e^5 - 8250*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^7*sqrt(c)*d^3*e^6 - 46750*(sqrt(c)*x - sqrt(c*x
^2 + b*x + a))^2*a^2*b^5*c^(3/2)*d^3*e^6 - 154800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*b^3*c^(5/2)*d^3*e^
6 - 40160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^4*b*c^(7/2)*d^3*e^6 + 15930*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^2*a^2*b^6*sqrt(c)*d^2*e^7 + 55340*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*b^4*c^(3/2)*d^2*e^7 + 58880*(s
qrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^4*b^2*c^(5/2)*d^2*e^7 - 20800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^5*
c^(7/2)*d^2*e^7 - 12990*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*b^5*sqrt(c)*d*e^8 - 28720*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))^2*a^4*b^3*c^(3/2)*d*e^8 + 160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^5*b*c^(5/2)*d*e^8 + 38
40*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^4*b^4*sqrt(c)*e^9 + 5120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^5*
b^2*c^(3/2)*e^9 + 2560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^6*c^(5/2)*e^9 + 960*(sqrt(c)*x - sqrt(c*x^2 + b
*x + a))*b^4*c^5*d^9 - 1920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^5*c^4*d^8*e - 3840*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))*a*b^3*c^5*d^8*e + 4280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^6*c^3*d^7*e^2 - 8480*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))*a*b^4*c^4*d^7*e^2 + 11520*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^2*c^5*d^7*e^2 - 3080*(s
qrt(c)*x - sqrt(c*x^2 + b*x + a))*b^7*c^2*d^6*e^3 + 480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^5*c^3*d^6*e^3
+ 49280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^3*c^4*d^6*e^3 - 7680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3
*b*c^5*d^6*e^3 + 600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^8*c*d^5*e^4 + 5380*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))*a*b^6*c^2*d^5*e^4 - 46840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^4*c^3*d^5*e^4 - 98880*(sqrt(c)*x - sqr
t(c*x^2 + b*x + a))*a^3*b^2*c^4*d^5*e^4 + 1920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^4*c^5*d^5*e^4 + 105*(sqrt
(c)*x - sqrt(c*x^2 + b*x + a))*b^9*d^4*e^5 - 1950*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^7*c*d^4*e^5 + 16960*
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^5*c^2*d^4*e^5 + 96640*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*b^3*c^
3*d^4*e^5 + 85120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^4*b*c^4*d^4*e^5 - 420*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))*a*b^8*d^3*e^6 - 990*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^6*c*d^3*e^6 - 49820*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))*a^3*b^4*c^2*d^3*e^6 - 74400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^4*b^2*c^3*d^3*e^6 - 24640*(sqrt(
c)*x - sqrt(c*x^2 + b*x + a))*a^5*c^4*d^3*e^6 + 630*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^7*d^2*e^7 + 8670
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*b^5*c*d^2*e^7 + 43360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^4*b^3*c^2
*d^2*e^7 + 16160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^5*b*c^3*d^2*e^7 - 420*(sqrt(c)*x - sqrt(c*x^2 + b*x + a
))*a^3*b^6*d*e^8 - 9570*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^4*b^4*c*d*e^8 - 13520*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))*a^5*b^2*c^2*d*e^8 + 3680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^6*c^3*d*e^8 + 105*(sqrt(c)*x - sqrt(c
*x^2 + b*x + a))*a^4*b^5*e^9 + 3240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^5*b^3*c*e^9 + 720*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))*a^6*b*c^2*e^9 + 96*b^5*c^(9/2)*d^9 - 192*b^6*c^(7/2)*d^8*e - 480*a*b^4*c^(9/2)*d^8*e + 476*b
^7*c^(5/2)*d^7*e^2 - 848*a*b^5*c^(7/2)*d^7*e^2 + 1920*a^2*b^3*c^(9/2)*d^7*e^2 - 380*b^8*c^(3/2)*d^6*e^3 - 332*
a*b^6*c^(5/2)*d^6*e^3 + 5200*a^2*b^4*c^(7/2)*d^6*e^3 - 1920*a^3*b^2*c^(9/2)*d^6*e^3 + 105*b^9*sqrt(c)*d^5*e^4
+ 990*a*b^7*c^(3/2)*d^5*e^4 - 3244*a^2*b^5*c^(5/2)*d^5*e^4 - 15200*a^3*b^3*c^(7/2)*d^5*e^4 + 960*a^4*b*c^(9/2)
*d^5*e^4 - 420*a*b^8*sqrt(c)*d^4*e^5 - 90*a^2*b^6*c^(3/2)*d^4*e^5 + 11420*a^3*b^4*c^(5/2)*d^4*e^5 + 20320*a^4*
b^2*c^(7/2)*d^4*e^5 - 192*a^5*c^(9/2)*d^4*e^5 + 630*a^2*b^7*sqrt(c)*d^3*e^6 - 3150*a^3*b^5*c^(3/2)*d^3*e^6 - 1
6000*a^4*b^3*c^(5/2)*d^3*e^6 - 11936*a^5*b*c^(7/2)*d^3*e^6 - 420*a^3*b^6*sqrt(c)*d^2*e^7 + 5790*a^4*b^4*c^(3/2
)*d^2*e^7 + 9008*a^5*b^2*c^(5/2)*d^2*e^7 + 2656*a^6*c^(7/2)*d^2*e^7 + 105*a^4*b^5*sqrt(c)*d*e^8 - 4440*a^5*b^3
*c^(3/2)*d*e^8 - 816*a^6*b*c^(5/2)*d*e^8 + 1280*a^6*b^2*c^(3/2)*e^9 - 512*a^7*c^(5/2)*e^9)/((c^4*d^8*e^2 - 4*b
*c^3*d^7*e^3 + 6*b^2*c^2*d^6*e^4 + 4*a*c^3*d^6*e^4 - 4*b^3*c*d^5*e^5 - 12*a*b*c^2*d^5*e^5 + b^4*d^4*e^6 + 12*a
*b^2*c*d^4*e^6 + 6*a^2*c^2*d^4*e^6 - 4*a*b^3*d^3*e^7 - 12*a^2*b*c*d^3*e^7 + 6*a^2*b^2*d^2*e^8 + 4*a^3*c*d^2*e^
8 - 4*a^3*b*d*e^9 + a^4*e^10)*((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)^5)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^6} \, dx=\int \frac {\sqrt {c\,x^2+b\,x+a}}{{\left (d+e\,x\right )}^6} \,d x \]

[In]

int((a + b*x + c*x^2)^(1/2)/(d + e*x)^6,x)

[Out]

int((a + b*x + c*x^2)^(1/2)/(d + e*x)^6, x)