3.24.41 \(\int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^5} \, dx\) [2341]

Optimal. Leaf size=308 \[ \frac {\left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{64 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}-\frac {e \left (a+b x+c x^2\right )^{3/2}}{4 \left (c d^2-b d e+a e^2\right ) (d+e x)^4}-\frac {5 e (2 c d-b e) \left (a+b x+c x^2\right )^{3/2}}{24 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^3}-\frac {\left (b^2-4 a c\right ) \left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right ) \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{128 \left (c d^2-b d e+a e^2\right )^{7/2}} \]

[Out]

-1/4*e*(c*x^2+b*x+a)^(3/2)/(a*e^2-b*d*e+c*d^2)/(e*x+d)^4-5/24*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)^3-1/128*(-4*a*c+b^2)*(16*c^2*d^2+5*b^2*e^2-4*c*e*(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)^(7/2)+1/64*(16*c^2*d^2+5*b^2*e^2-4*
c*e*(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)^3/(e*x+d)^2

________________________________________________________________________________________

Rubi [A]
time = 0.27, antiderivative size = 308, normalized size of antiderivative = 1.00, 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 = {758, 820, 734, 738, 212} \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{64 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^3}-\frac {\left (b^2-4 a c\right ) \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right ) \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 )}{128 \left (a e^2-b d e+c d^2\right )^{7/2}}-\frac {5 e \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{24 (d+e x)^3 \left (a e^2-b d e+c d^2\right )^2}-\frac {e \left (a+b x+c x^2\right )^{3/2}}{4 (d+e x)^4 \left (a e^2-b d e+c d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((16*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(4*b*d + a*e))*(b*d - 2*a*e + (2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(64*(c*
d^2 - b*d*e + a*e^2)^3*(d + e*x)^2) - (e*(a + b*x + c*x^2)^(3/2))/(4*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^4) - (5
*e*(2*c*d - b*e)*(a + b*x + c*x^2)^(3/2))/(24*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^3) - ((b^2 - 4*a*c)*(16*c^2*
d^2 + 5*b^2*e^2 - 4*c*e*(4*b*d + 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])])/(128*(c*d^2 - b*d*e + a*e^2)^(7/2))

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]

Rubi steps

\begin {align*} \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^5} \, dx &=-\frac {e \left (a+b x+c x^2\right )^{3/2}}{4 \left (c d^2-b d e+a e^2\right ) (d+e x)^4}-\frac {\int \frac {\left (\frac {1}{2} (-8 c d+5 b e)+c e x\right ) \sqrt {a+b x+c x^2}}{(d+e x)^4} \, dx}{4 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {e \left (a+b x+c x^2\right )^{3/2}}{4 \left (c d^2-b d e+a e^2\right ) (d+e x)^4}-\frac {5 e (2 c d-b e) \left (a+b x+c x^2\right )^{3/2}}{24 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^3}+\frac {\left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right ) \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^3} \, dx}{16 \left (c d^2-b d e+a e^2\right )^2}\\ &=\frac {\left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{64 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}-\frac {e \left (a+b x+c x^2\right )^{3/2}}{4 \left (c d^2-b d e+a e^2\right ) (d+e x)^4}-\frac {5 e (2 c d-b e) \left (a+b x+c x^2\right )^{3/2}}{24 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^3}-\frac {\left (\left (b^2-4 a c\right ) \left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{128 \left (c d^2-b d e+a e^2\right )^3}\\ &=\frac {\left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{64 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}-\frac {e \left (a+b x+c x^2\right )^{3/2}}{4 \left (c d^2-b d e+a e^2\right ) (d+e x)^4}-\frac {5 e (2 c d-b e) \left (a+b x+c x^2\right )^{3/2}}{24 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^3}+\frac {\left (\left (b^2-4 a c\right ) \left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+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 )}{64 \left (c d^2-b d e+a e^2\right )^3}\\ &=\frac {\left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{64 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}-\frac {e \left (a+b x+c x^2\right )^{3/2}}{4 \left (c d^2-b d e+a e^2\right ) (d+e x)^4}-\frac {5 e (2 c d-b e) \left (a+b x+c x^2\right )^{3/2}}{24 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^3}-\frac {\left (b^2-4 a c\right ) \left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+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 )}{128 \left (c d^2-b d e+a e^2\right )^{7/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 10.48, size = 276, normalized size = 0.90 \begin {gather*} \frac {-\frac {6 e \left (c d^2+e (-b d+a e)\right ) (a+x (b+c x))^{3/2}}{(d+e x)^4}-\frac {5 e (2 c d-b e) (a+x (b+c x))^{3/2}}{(d+e x)^3}+3 \left (8 c^2 d^2+\frac {5 b^2 e^2}{2}-2 c e (4 b d+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 ) \tanh ^{-1}\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-6*e*(c*d^2 + e*(-(b*d) + a*e))*(a + x*(b + c*x))^(3/2))/(d + e*x)^4 - (5*e*(2*c*d - b*e)*(a + x*(b + c*x))^
(3/2))/(d + e*x)^3 + 3*(8*c^2*d^2 + (5*b^2*e^2)/2 - 2*c*e*(4*b*d + 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)))
)/(24*(c*d^2 + e*(-(b*d) + a*e))^2)

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2524\) vs. \(2(286)=572\).
time = 0.90, size = 2525, normalized size = 8.20

method	result	size
default	\(\text {Expression too large to display}\)	\(2525\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^(1/2)/(e*x+d)^5,x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(1/2)/(e*x+d)^5,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(c*d^2-%e*b*d+%e^2*a>0)', see `
assume?` for

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1850 vs. \(2 (298) = 596\).
time = 36.53, size = 3743, normalized size = 12.15 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(1/2)/(e*x+d)^5,x, algorithm="fricas")

[Out]

[1/768*(3*(16*(b^2*c^2 - 4*a*c^3)*d^6 + (5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*x^4*e^6 - 4*(4*(b^3*c - 4*a*b*c^2)*d
*x^4 - (5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*d*x^3)*e^5 + 2*(8*(b^2*c^2 - 4*a*c^3)*d^2*x^4 - 32*(b^3*c - 4*a*b*c^2
)*d^2*x^3 + 3*(5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*d^2*x^2)*e^4 + 4*(16*(b^2*c^2 - 4*a*c^3)*d^3*x^3 - 24*(b^3*c -
4*a*b*c^2)*d^3*x^2 + (5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*d^3*x)*e^3 + (96*(b^2*c^2 - 4*a*c^3)*d^4*x^2 - 64*(b^3
*c - 4*a*b*c^2)*d^4*x + (5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*d^4)*e^2 + 16*(4*(b^2*c^2 - 4*a*c^3)*d^5*x - (b^3*c
- 4*a*b*c^2)*d^5)*e)*sqrt(c*d^2 - b*d*e + a*e^2)*log(-(8*c^2*d^2*x^2 + 8*b*c*d^2*x + (b^2 + 4*a*c)*d^2 - 4*sqr
t(c*d^2 - b*d*e + a*e^2)*(2*c*d*x + b*d - (b*x + 2*a)*e)*sqrt(c*x^2 + b*x + a) + (8*a*b*x + (b^2 + 4*a*c)*x^2
+ 8*a^2)*e^2 - 2*(4*b*c*d*x^2 + 4*a*b*d + (3*b^2 + 4*a*c)*d*x)*e)/(x^2*e^2 + 2*d*x*e + d^2)) + 4*(96*c^4*d^7*x
+ 48*b*c^3*d^7 - (8*a^3*b*x + 48*a^4 + (15*a*b^3 - 52*a^2*b*c)*x^3 - 2*(5*a^2*b^2 - 12*a^3*c)*x^2)*e^7 + (184
*a^3*b*d + (15*b^4 - 14*a*b^2*c - 104*a^2*c^2)*d*x^3 - (65*a*b^3 - 188*a^2*b*c)*d*x^2 + 4*(11*a^2*b^2 - 8*a^3*
c)*d*x)*e^6 - ((53*b^3*c - 132*a*b*c^2)*d^2*x^3 - (55*b^4 - 14*a*b^2*c - 344*a^2*c^2)*d^2*x^2 + (109*a*b^3 - 1
48*a^2*b*c)*d^2*x + 2*(127*a^2*b^2 + 100*a^3*c)*d^2)*e^5 + (2*(31*b^2*c^2 - 44*a*c^3)*d^3*x^3 - 5*(39*b^3*c -
76*a*b*c^2)*d^3*x^2 + (73*b^4 + 110*a*b^2*c - 328*a^2*c^2)*d^3*x + 7*(19*a*b^3 + 84*a^2*b*c)*d^3)*e^4 - (40*b*
c^3*d^4*x^3 - 4*(61*b^2*c^2 - 64*a*c^3)*d^4*x^2 + (271*b^3*c - 244*a*b*c^2)*d^4*x + (15*b^4 + 466*a*b^2*c + 37
6*a^2*c^2)*d^4)*e^3 + (16*c^4*d^5*x^3 - 168*b*c^3*d^5*x^2 + 2*(187*b^2*c^2 - 100*a*c^3)*d^5*x + (63*b^3*c + 57
2*a*b*c^2)*d^5)*e^2 + 16*(4*c^4*d^6*x^2 - 17*b*c^3*d^6*x - 2*(3*b^2*c^2 + 7*a*c^3)*d^6)*e)*sqrt(c*x^2 + b*x +
a))/(c^4*d^12 + a^4*x^4*e^12 - 4*(a^3*b*d*x^4 - a^4*d*x^3)*e^11 - 2*(8*a^3*b*d^2*x^3 - 3*a^4*d^2*x^2 - (3*a^2*
b^2 + 2*a^3*c)*d^2*x^4)*e^10 - 4*(6*a^3*b*d^3*x^2 - a^4*d^3*x + (a*b^3 + 3*a^2*b*c)*d^3*x^4 - 2*(3*a^2*b^2 + 2
*a^3*c)*d^3*x^3)*e^9 - (16*a^3*b*d^4*x - (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*x^4 - a^4*d^4 + 16*(a*b^3 + 3*a^2*
b*c)*d^4*x^3 - 12*(3*a^2*b^2 + 2*a^3*c)*d^4*x^2)*e^8 - 4*((b^3*c + 3*a*b*c^2)*d^5*x^4 + a^3*b*d^5 - (b^4 + 12*
a*b^2*c + 6*a^2*c^2)*d^5*x^3 + 6*(a*b^3 + 3*a^2*b*c)*d^5*x^2 - 2*(3*a^2*b^2 + 2*a^3*c)*d^5*x)*e^7 + 2*((3*b^2*
c^2 + 2*a*c^3)*d^6*x^4 - 8*(b^3*c + 3*a*b*c^2)*d^6*x^3 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^6*x^2 - 8*(a*b^3 +
3*a^2*b*c)*d^6*x + (3*a^2*b^2 + 2*a^3*c)*d^6)*e^6 - 4*(b*c^3*d^7*x^4 - 2*(3*b^2*c^2 + 2*a*c^3)*d^7*x^3 + 6*(b
^3*c + 3*a*b*c^2)*d^7*x^2 - (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^7*x + (a*b^3 + 3*a^2*b*c)*d^7)*e^5 + (c^4*d^8*x^4
- 16*b*c^3*d^8*x^3 + 12*(3*b^2*c^2 + 2*a*c^3)*d^8*x^2 - 16*(b^3*c + 3*a*b*c^2)*d^8*x + (b^4 + 12*a*b^2*c + 6*
a^2*c^2)*d^8)*e^4 + 4*(c^4*d^9*x^3 - 6*b*c^3*d^9*x^2 + 2*(3*b^2*c^2 + 2*a*c^3)*d^9*x - (b^3*c + 3*a*b*c^2)*d^9
)*e^3 + 2*(3*c^4*d^10*x^2 - 8*b*c^3*d^10*x + (3*b^2*c^2 + 2*a*c^3)*d^10)*e^2 + 4*(c^4*d^11*x - b*c^3*d^11)*e),
-1/384*(3*(16*(b^2*c^2 - 4*a*c^3)*d^6 + (5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*x^4*e^6 - 4*(4*(b^3*c - 4*a*b*c^2)*
d*x^4 - (5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*d*x^3)*e^5 + 2*(8*(b^2*c^2 - 4*a*c^3)*d^2*x^4 - 32*(b^3*c - 4*a*b*c^
2)*d^2*x^3 + 3*(5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*d^2*x^2)*e^4 + 4*(16*(b^2*c^2 - 4*a*c^3)*d^3*x^3 - 24*(b^3*c
- 4*a*b*c^2)*d^3*x^2 + (5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*d^3*x)*e^3 + (96*(b^2*c^2 - 4*a*c^3)*d^4*x^2 - 64*(b^
3*c - 4*a*b*c^2)*d^4*x + (5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*d^4)*e^2 + 16*(4*(b^2*c^2 - 4*a*c^3)*d^5*x - (b^3*c
- 4*a*b*c^2)*d^5)*e)*sqrt(-c*d^2 + b*d*e - a*e^2)*arctan(-1/2*sqrt(-c*d^2 + b*d*e - a*e^2)*(2*c*d*x + b*d - (
b*x + 2*a)*e)*sqrt(c*x^2 + b*x + a)/(c^2*d^2*x^2 + b*c*d^2*x + a*c*d^2 + (a*c*x^2 + a*b*x + a^2)*e^2 - (b*c*d*
x^2 + b^2*d*x + a*b*d)*e)) - 2*(96*c^4*d^7*x + 48*b*c^3*d^7 - (8*a^3*b*x + 48*a^4 + (15*a*b^3 - 52*a^2*b*c)*x^
3 - 2*(5*a^2*b^2 - 12*a^3*c)*x^2)*e^7 + (184*a^3*b*d + (15*b^4 - 14*a*b^2*c - 104*a^2*c^2)*d*x^3 - (65*a*b^3 -
188*a^2*b*c)*d*x^2 + 4*(11*a^2*b^2 - 8*a^3*c)*d*x)*e^6 - ((53*b^3*c - 132*a*b*c^2)*d^2*x^3 - (55*b^4 - 14*a*b
^2*c - 344*a^2*c^2)*d^2*x^2 + (109*a*b^3 - 148*a^2*b*c)*d^2*x + 2*(127*a^2*b^2 + 100*a^3*c)*d^2)*e^5 + (2*(31*
b^2*c^2 - 44*a*c^3)*d^3*x^3 - 5*(39*b^3*c - 76*a*b*c^2)*d^3*x^2 + (73*b^4 + 110*a*b^2*c - 328*a^2*c^2)*d^3*x +
7*(19*a*b^3 + 84*a^2*b*c)*d^3)*e^4 - (40*b*c^3*d^4*x^3 - 4*(61*b^2*c^2 - 64*a*c^3)*d^4*x^2 + (271*b^3*c - 244
*a*b*c^2)*d^4*x + (15*b^4 + 466*a*b^2*c + 376*a^2*c^2)*d^4)*e^3 + (16*c^4*d^5*x^3 - 168*b*c^3*d^5*x^2 + 2*(187
*b^2*c^2 - 100*a*c^3)*d^5*x + (63*b^3*c + 572*a*b*c^2)*d^5)*e^2 + 16*(4*c^4*d^6*x^2 - 17*b*c^3*d^6*x - 2*(3*b^
2*c^2 + 7*a*c^3)*d^6)*e)*sqrt(c*x^2 + b*x + a))/(c^4*d^12 + a^4*x^4*e^12 - 4*(a^3*b*d*x^4 - a^4*d*x^3)*e^11 -
2*(8*a^3*b*d^2*x^3 - 3*a^4*d^2*x^2 - (3*a^2*b^2 + 2*a^3*c)*d^2*x^4)*e^10 - 4*(6*a^3*b*d^3*x^2 - a^4*d^3*x + (a
*b^3 + 3*a^2*b*c)*d^3*x^4 - 2*(3*a^2*b^2 + 2*a^3*c)*d^3*x^3)*e^9 - (16*a^3*b*d^4*x - (b^4 + 12*a*b^2*c + 6*a^2
*c^2)*d^4*x^4 - a^4*d^4 + 16*(a*b^3 + 3*a^2*b*c...

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a + b x + c x^{2}}}{\left (d + e x\right )^{5}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**(1/2)/(e*x+d)**5,x)

[Out]

Integral(sqrt(a + b*x + c*x**2)/(d + e*x)**5, x)

________________________________________________________________________________________

Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(1/2)/(e*x+d)^5,x, algorithm="giac")

[Out]

Timed out

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {c\,x^2+b\,x+a}}{{\left (d+e\,x\right )}^5} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x + c*x^2)^(1/2)/(d + e*x)^5,x)

[Out]

int((a + b*x + c*x^2)^(1/2)/(d + e*x)^5, x)

________________________________________________________________________________________