\(\int \frac {1}{(d+e x)^3 \sqrt {a+b x+c x^2}} \, dx\) [2380]

Optimal result

Integrand size = 22, antiderivative size = 209 \[ \int \frac {1}{(d+e x)^3 \sqrt {a+b x+c x^2}} \, dx=-\frac {e \sqrt {a+b x+c x^2}}{2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac {3 e (2 c d-b e) \sqrt {a+b x+c x^2}}{4 \left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac {\left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{8 \left (c d^2-b d e+a e^2\right )^{5/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 209, 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 = {758, 820, 738, 212} \[ \int \frac {1}{(d+e x)^3 \sqrt {a+b x+c x^2}} \, dx=\frac {\left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{8 \left (a e^2-b d e+c d^2\right )^{5/2}}-\frac {3 e \sqrt {a+b x+c x^2} (2 c d-b e)}{4 (d+e x) \left (a e^2-b d e+c d^2\right )^2}-\frac {e \sqrt {a+b x+c x^2}}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )} \]

[In]

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

[Out]

-1/2*(e*Sqrt[a + b*x + c*x^2])/((c*d^2 - b*d*e + a*e^2)*(d + e*x)^2) - (3*e*(2*c*d - b*e)*Sqrt[a + b*x + c*x^2
])/(4*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)) + ((8*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(2*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])])/(8*(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 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*} \text {integral}& = -\frac {e \sqrt {a+b x+c x^2}}{2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac {\int \frac {\frac {1}{2} (-4 c d+3 b e)+c e x}{(d+e x)^2 \sqrt {a+b x+c x^2}} \, dx}{2 \left (c d^2-b d e+a e^2\right )} \\ & = -\frac {e \sqrt {a+b x+c x^2}}{2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac {3 e (2 c d-b e) \sqrt {a+b x+c x^2}}{4 \left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac {\left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) \int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{8 \left (c d^2-b d e+a e^2\right )^2} \\ & = -\frac {e \sqrt {a+b x+c x^2}}{2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac {3 e (2 c d-b e) \sqrt {a+b x+c x^2}}{4 \left (c d^2-b d e+a e^2\right )^2 (d+e x)}-\frac {\left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a 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 )}{4 \left (c d^2-b d e+a e^2\right )^2} \\ & = -\frac {e \sqrt {a+b x+c x^2}}{2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac {3 e (2 c d-b e) \sqrt {a+b x+c x^2}}{4 \left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac {\left (8 c^2 d^2+3 b^2 e^2-4 c e (2 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 )}{8 \left (c d^2-b d e+a e^2\right )^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.19 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.98 \[ \int \frac {1}{(d+e x)^3 \sqrt {a+b x+c x^2}} \, dx=-\frac {e \sqrt {a+x (b+c x)}}{2 \left (c d^2+e (-b d+a e)\right ) (d+e x)^2}-\frac {3 e (2 c d-b e) \sqrt {a+x (b+c x)}}{4 \left (c d^2+e (-b d+a e)\right )^2 (d+e x)}-\frac {\left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{8 \left (c d^2+e (-b d+a e)\right )^{5/2}} \]

[In]

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

[Out]

-1/2*(e*Sqrt[a + x*(b + c*x)])/((c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^2) - (3*e*(2*c*d - b*e)*Sqrt[a + x*(b + c
*x)])/(4*(c*d^2 + e*(-(b*d) + a*e))^2*(d + e*x)) - ((8*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(2*b*d + a*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)])])/(8*(c*d^2 + e*(-(b*d
) + a*e))^(5/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(560\) vs. \(2(191)=382\).

Time = 0.36 (sec) , antiderivative size = 561, normalized size of antiderivative = 2.68

method	result	size
default	\(\frac {-\frac {e^{2} \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2}}-\frac {3 \left (b e -2 c d \right ) e \left (-\frac {e^{2} \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}+\frac {\left (b e -2 c d \right ) e \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}\right )}{4 \left (a \,e^{2}-b d e +c \,d^{2}\right )}+\frac {c \,e^{2} \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}}{e^{3}}\)	\(561\)

[In]

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

[Out]

1/e^3*(-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)^(1/2
)-3/4*(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)^(1/2)+1/2*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^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)))+1/2*c/(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. 660 vs. \(2 (191) = 382\).

Time = 1.17 (sec) , antiderivative size = 1362, normalized size of antiderivative = 6.52 \[ \int \frac {1}{(d+e x)^3 \sqrt {a+b x+c x^2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[-1/16*((8*c^2*d^4 - 8*b*c*d^3*e + (3*b^2 - 4*a*c)*d^2*e^2 + (8*c^2*d^2*e^2 - 8*b*c*d*e^3 + (3*b^2 - 4*a*c)*e^
4)*x^2 + 2*(8*c^2*d^3*e - 8*b*c*d^2*e^2 + (3*b^2 - 4*a*c)*d*e^3)*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*(8*c^2*d^4*e - 13*b*c*d^3*e^2 - 7*a*b*d*e^4 + 2*a^2*e^5 + 5*(b^2 + 2*a*c)*d^2
*e^3 + 3*(2*c^2*d^3*e^2 - 3*b*c*d^2*e^3 - a*b*e^5 + (b^2 + 2*a*c)*d*e^4)*x)*sqrt(c*x^2 + b*x + a))/(c^3*d^8 -
3*b*c^2*d^7*e - 3*a^2*b*d^3*e^5 + a^3*d^2*e^6 + 3*(b^2*c + a*c^2)*d^6*e^2 - (b^3 + 6*a*b*c)*d^5*e^3 + 3*(a*b^2
+ a^2*c)*d^4*e^4 + (c^3*d^6*e^2 - 3*b*c^2*d^5*e^3 - 3*a^2*b*d*e^7 + a^3*e^8 + 3*(b^2*c + a*c^2)*d^4*e^4 - (b^
3 + 6*a*b*c)*d^3*e^5 + 3*(a*b^2 + a^2*c)*d^2*e^6)*x^2 + 2*(c^3*d^7*e - 3*b*c^2*d^6*e^2 - 3*a^2*b*d^2*e^6 + a^3
*d*e^7 + 3*(b^2*c + a*c^2)*d^5*e^3 - (b^3 + 6*a*b*c)*d^4*e^4 + 3*(a*b^2 + a^2*c)*d^3*e^5)*x), 1/8*((8*c^2*d^4
- 8*b*c*d^3*e + (3*b^2 - 4*a*c)*d^2*e^2 + (8*c^2*d^2*e^2 - 8*b*c*d*e^3 + (3*b^2 - 4*a*c)*e^4)*x^2 + 2*(8*c^2*d
^3*e - 8*b*c*d^2*e^2 + (3*b^2 - 4*a*c)*d*e^3)*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*(8*c^2*d^4*e - 13*b*c*d^3*e^2 - 7*a*b*d*e^4 + 2*a^2*
e^5 + 5*(b^2 + 2*a*c)*d^2*e^3 + 3*(2*c^2*d^3*e^2 - 3*b*c*d^2*e^3 - a*b*e^5 + (b^2 + 2*a*c)*d*e^4)*x)*sqrt(c*x^
2 + b*x + a))/(c^3*d^8 - 3*b*c^2*d^7*e - 3*a^2*b*d^3*e^5 + a^3*d^2*e^6 + 3*(b^2*c + a*c^2)*d^6*e^2 - (b^3 + 6*
a*b*c)*d^5*e^3 + 3*(a*b^2 + a^2*c)*d^4*e^4 + (c^3*d^6*e^2 - 3*b*c^2*d^5*e^3 - 3*a^2*b*d*e^7 + a^3*e^8 + 3*(b^2
*c + a*c^2)*d^4*e^4 - (b^3 + 6*a*b*c)*d^3*e^5 + 3*(a*b^2 + a^2*c)*d^2*e^6)*x^2 + 2*(c^3*d^7*e - 3*b*c^2*d^6*e^
2 - 3*a^2*b*d^2*e^6 + a^3*d*e^7 + 3*(b^2*c + a*c^2)*d^5*e^3 - (b^3 + 6*a*b*c)*d^4*e^4 + 3*(a*b^2 + a^2*c)*d^3*
e^5)*x)]

Sympy [F]

\[ \int \frac {1}{(d+e x)^3 \sqrt {a+b x+c x^2}} \, dx=\int \frac {1}{\left (d + e x\right )^{3} \sqrt {a + b x + c x^{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(d+e x)^3 \sqrt {a+b x+c x^2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(1/(e*x+d)^3/(c*x^2+b*x+a)^(1/2),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. 795 vs. \(2 (191) = 382\).

Time = 0.30 (sec) , antiderivative size = 795, normalized size of antiderivative = 3.80 \[ \int \frac {1}{(d+e x)^3 \sqrt {a+b x+c x^2}} \, dx=\frac {{\left (8 \, c^{2} d^{2} - 8 \, b c d e + 3 \, b^{2} e^{2} - 4 \, a c e^{2}\right )} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} + b d e - a e^{2}}}\right )}{4 \, {\left (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}\right )} \sqrt {-c d^{2} + b d e - a e^{2}}} - \frac {8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} c^{2} d^{2} e - 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} b c d e^{2} + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} b^{2} e^{3} - 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} a c e^{3} + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} c^{\frac {5}{2}} d^{3} - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} b c^{\frac {3}{2}} d^{2} e + 9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} b^{2} \sqrt {c} d e^{2} - 12 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a c^{\frac {3}{2}} d e^{2} + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b c^{2} d^{3} - 20 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b^{2} c d^{2} e - 40 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a c^{2} d^{2} e + 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b^{3} d e^{2} + 28 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a b c d e^{2} - 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a b^{2} e^{3} - 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a^{2} c e^{3} + 6 \, b^{2} c^{\frac {3}{2}} d^{3} - 3 \, b^{3} \sqrt {c} d^{2} e - 20 \, a b c^{\frac {3}{2}} d^{2} e + 11 \, a b^{2} \sqrt {c} d e^{2} + 12 \, a^{2} c^{\frac {3}{2}} d e^{2} - 8 \, a^{2} b \sqrt {c} e^{3}}{4 \, {\left (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}\right )} {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} e + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} d + b d - a e\right )}^{2}} \]

[In]

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

[Out]

1/4*(8*c^2*d^2 - 8*b*c*d*e + 3*b^2*e^2 - 4*a*c*e^2)*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^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)
*sqrt(-c*d^2 + b*d*e - a*e^2)) - 1/4*(8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*c^2*d^2*e - 8*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))^3*b*c*d*e^2 + 3*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^2*e^3 - 4*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^3*a*c*e^3 + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*c^(5/2)*d^3 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^2*b*c^(3/2)*d^2*e + 9*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^2*sqrt(c)*d*e^2 - 12*(sqrt(c)*x - sqrt(c*x
^2 + b*x + a))^2*a*c^(3/2)*d*e^2 + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b*c^2*d^3 - 20*(sqrt(c)*x - sqrt(c*x
^2 + b*x + a))*b^2*c*d^2*e - 40*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*c^2*d^2*e + 5*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))*b^3*d*e^2 + 28*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b*c*d*e^2 - 5*(sqrt(c)*x - sqrt(c*x^2 + b*x + a
))*a*b^2*e^3 - 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*c*e^3 + 6*b^2*c^(3/2)*d^3 - 3*b^3*sqrt(c)*d^2*e - 20*
a*b*c^(3/2)*d^2*e + 11*a*b^2*sqrt(c)*d*e^2 + 12*a^2*c^(3/2)*d*e^2 - 8*a^2*b*sqrt(c)*e^3)/((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)*((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)^2)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^3 \sqrt {a+b x+c x^2}} \, dx=\int \frac {1}{{\left (d+e\,x\right )}^3\,\sqrt {c\,x^2+b\,x+a}} \,d x \]

[In]

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

[Out]

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