\(\int \frac {1}{(d+e x)^4 \sqrt {a+b x+c x^2}} \, dx\) [2381]

Optimal result

Integrand size = 22, antiderivative size = 293 \[ \int \frac {1}{(d+e x)^4 \sqrt {a+b x+c x^2}} \, dx=-\frac {e \sqrt {a+b x+c x^2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}-\frac {5 e (2 c d-b e) \sqrt {a+b x+c x^2}}{12 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}-\frac {e \left (44 c^2 d^2+15 b^2 e^2-4 c e (11 b d+4 a e)\right ) \sqrt {a+b x+c x^2}}{24 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}+\frac {(2 c d-b e) \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{16 \left (c d^2-b d e+a e^2\right )^{7/2}} \]

[Out]

1/16*(-b*e+2*c*d)*(8*c^2*d^2+5*b^2*e^2-4*c*e*(3*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)^(7/2)-1/3*e*(c*x^2+b*x+a)^(1/2)/(a*e^2-b*d*e+c*d^2)/(e
*x+d)^3-5/12*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)^2-1/24*e*(44*c^2*d^2+15*b^2*e^2-
4*c*e*(4*a*e+11*b*d))*(c*x^2+b*x+a)^(1/2)/(a*e^2-b*d*e+c*d^2)^3/(e*x+d)

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {758, 848, 820, 738, 212} \[ \int \frac {1}{(d+e x)^4 \sqrt {a+b x+c x^2}} \, dx=\frac {(2 c d-b e) \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right ) \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{16 \left (a e^2-b d e+c d^2\right )^{7/2}}-\frac {e \sqrt {a+b x+c x^2} \left (-4 c e (4 a e+11 b d)+15 b^2 e^2+44 c^2 d^2\right )}{24 (d+e x) \left (a e^2-b d e+c d^2\right )^3}-\frac {5 e \sqrt {a+b x+c x^2} (2 c d-b e)}{12 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2}-\frac {e \sqrt {a+b x+c x^2}}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )} \]

[In]

Int[1/((d + e*x)^4*Sqrt[a + b*x + c*x^2]),x]

[Out]

-1/3*(e*Sqrt[a + b*x + c*x^2])/((c*d^2 - b*d*e + a*e^2)*(d + e*x)^3) - (5*e*(2*c*d - b*e)*Sqrt[a + b*x + c*x^2
])/(12*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^2) - (e*(44*c^2*d^2 + 15*b^2*e^2 - 4*c*e*(11*b*d + 4*a*e))*Sqrt[a +
b*x + c*x^2])/(24*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)) + ((2*c*d - b*e)*(8*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(2*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])])/(16
*(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 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 \sqrt {a+b x+c x^2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}-\frac {\int \frac {\frac {1}{2} (-6 c d+5 b e)+2 c e x}{(d+e x)^3 \sqrt {a+b x+c x^2}} \, dx}{3 \left (c d^2-b d e+a e^2\right )} \\ & = -\frac {e \sqrt {a+b x+c x^2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}-\frac {5 e (2 c d-b e) \sqrt {a+b x+c x^2}}{12 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac {\int \frac {\frac {1}{4} \left (24 c^2 d^2+15 b^2 e^2-2 c e (17 b d+8 a e)\right )-\frac {5}{2} c e (2 c d-b e) x}{(d+e x)^2 \sqrt {a+b x+c x^2}} \, dx}{6 \left (c d^2-b d e+a e^2\right )^2} \\ & = -\frac {e \sqrt {a+b x+c x^2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}-\frac {5 e (2 c d-b e) \sqrt {a+b x+c x^2}}{12 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}-\frac {e \left (44 c^2 d^2+15 b^2 e^2-4 c e (11 b d+4 a e)\right ) \sqrt {a+b x+c x^2}}{24 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}+\frac {\left ((2 c d-b e) \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right )\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 )^3} \\ & = -\frac {e \sqrt {a+b x+c x^2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}-\frac {5 e (2 c d-b e) \sqrt {a+b x+c x^2}}{12 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}-\frac {e \left (44 c^2 d^2+15 b^2 e^2-4 c e (11 b d+4 a e)\right ) \sqrt {a+b x+c x^2}}{24 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac {\left ((2 c d-b e) \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 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 )}{8 \left (c d^2-b d e+a e^2\right )^3} \\ & = -\frac {e \sqrt {a+b x+c x^2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}-\frac {5 e (2 c d-b e) \sqrt {a+b x+c x^2}}{12 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}-\frac {e \left (44 c^2 d^2+15 b^2 e^2-4 c e (11 b d+4 a e)\right ) \sqrt {a+b x+c x^2}}{24 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}+\frac {(2 c d-b e) \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 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 )}{16 \left (c d^2-b d e+a e^2\right )^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.39 (sec) , antiderivative size = 288, normalized size of antiderivative = 0.98 \[ \int \frac {1}{(d+e x)^4 \sqrt {a+b x+c x^2}} \, dx=-\frac {\frac {2 e \left (c d^2+e (-b d+a e)\right ) \sqrt {a+x (b+c x)}}{(d+e x)^3}+\frac {5 e (2 c d-b e) \sqrt {a+x (b+c x)}}{2 (d+e x)^2}+\frac {e \left (44 c^2 d^2+15 b^2 e^2-4 c e (11 b d+4 a e)\right ) \sqrt {a+x (b+c x)}}{4 \left (c d^2+e (-b d+a e)\right ) (d+e x)}+\frac {3 (2 c d-b e) \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{8 \left (c d^2+e (-b d+a e)\right )^{3/2}}}{6 \left (c d^2+e (-b d+a e)\right )^2} \]

[In]

Integrate[1/((d + e*x)^4*Sqrt[a + b*x + c*x^2]),x]

[Out]

-1/6*((2*e*(c*d^2 + e*(-(b*d) + a*e))*Sqrt[a + x*(b + c*x)])/(d + e*x)^3 + (5*e*(2*c*d - b*e)*Sqrt[a + x*(b +
c*x)])/(2*(d + e*x)^2) + (e*(44*c^2*d^2 + 15*b^2*e^2 - 4*c*e*(11*b*d + 4*a*e))*Sqrt[a + x*(b + c*x)])/(4*(c*d^
2 + e*(-(b*d) + a*e))*(d + e*x)) + (3*(2*c*d - b*e)*(8*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(2*b*d + 3*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))^(3/2)))/(c*d^2 + e*(-(b*d) + a*e))^2

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(965\) vs. \(2(271)=542\).

Time = 0.40 (sec) , antiderivative size = 966, normalized size of antiderivative = 3.30

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}}}}{3 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{3}}-\frac {5 \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}}}}{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}}}}\right )}{6 \left (a \,e^{2}-b d e +c \,d^{2}\right )}-\frac {2 c \,e^{2} \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 )}{3 \left (a \,e^{2}-b d e +c \,d^{2}\right )}}{e^{4}}\)	\(966\)

[In]

int(1/(e*x+d)^4/(c*x^2+b*x+a)^(1/2),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)^(1/2
)-5/6*(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)^(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)))-2/3*c/(a*e^2-b*d*e+c*d^2)*
e^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))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1323 vs. \(2 (271) = 542\).

Time = 5.31 (sec) , antiderivative size = 2688, normalized size of antiderivative = 9.17 \[ \int \frac {1}{(d+e x)^4 \sqrt {a+b x+c x^2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/96*(3*(16*c^3*d^6 - 24*b*c^2*d^5*e + 6*(3*b^2*c - 4*a*c^2)*d^4*e^2 - (5*b^3 - 12*a*b*c)*d^3*e^3 + (16*c^3*d
^3*e^3 - 24*b*c^2*d^2*e^4 + 6*(3*b^2*c - 4*a*c^2)*d*e^5 - (5*b^3 - 12*a*b*c)*e^6)*x^3 + 3*(16*c^3*d^4*e^2 - 24
*b*c^2*d^3*e^3 + 6*(3*b^2*c - 4*a*c^2)*d^2*e^4 - (5*b^3 - 12*a*b*c)*d*e^5)*x^2 + 3*(16*c^3*d^5*e - 24*b*c^2*d^
4*e^2 + 6*(3*b^2*c - 4*a*c^2)*d^3*e^3 - (5*b^3 - 12*a*b*c)*d^2*e^4)*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*(72*c^3*d^6*e - 162*b*c^2*d^5*e^2 - 34*a^2*b*d*e^6 + 8*a^3*e^7 + (123*b^2*
c + 92*a*c^2)*d^4*e^3 - (33*b^3 + 136*a*b*c)*d^3*e^4 + (59*a*b^2 + 28*a^2*c)*d^2*e^5 + (44*c^3*d^4*e^3 - 88*b*
c^2*d^3*e^4 + (59*b^2*c + 28*a*c^2)*d^2*e^5 - (15*b^3 + 28*a*b*c)*d*e^6 + (15*a*b^2 - 16*a^2*c)*e^7)*x^2 + 2*(
54*c^3*d^5*e^2 - 113*b*c^2*d^4*e^3 - 5*a^2*b*e^7 + (79*b^2*c + 48*a*c^2)*d^3*e^4 - 2*(10*b^3 + 29*a*b*c)*d^2*e
^5 + (25*a*b^2 - 6*a^2*c)*d*e^6)*x)*sqrt(c*x^2 + b*x + a))/(c^4*d^11 - 4*b*c^3*d^10*e - 4*a^3*b*d^4*e^7 + a^4*
d^3*e^8 + 2*(3*b^2*c^2 + 2*a*c^3)*d^9*e^2 - 4*(b^3*c + 3*a*b*c^2)*d^8*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^7
*e^4 - 4*(a*b^3 + 3*a^2*b*c)*d^6*e^5 + 2*(3*a^2*b^2 + 2*a^3*c)*d^5*e^6 + (c^4*d^8*e^3 - 4*b*c^3*d^7*e^4 - 4*a^
3*b*d*e^10 + a^4*e^11 + 2*(3*b^2*c^2 + 2*a*c^3)*d^6*e^5 - 4*(b^3*c + 3*a*b*c^2)*d^5*e^6 + (b^4 + 12*a*b^2*c +
6*a^2*c^2)*d^4*e^7 - 4*(a*b^3 + 3*a^2*b*c)*d^3*e^8 + 2*(3*a^2*b^2 + 2*a^3*c)*d^2*e^9)*x^3 + 3*(c^4*d^9*e^2 - 4
*b*c^3*d^8*e^3 - 4*a^3*b*d^2*e^9 + a^4*d*e^10 + 2*(3*b^2*c^2 + 2*a*c^3)*d^7*e^4 - 4*(b^3*c + 3*a*b*c^2)*d^6*e^
5 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^5*e^6 - 4*(a*b^3 + 3*a^2*b*c)*d^4*e^7 + 2*(3*a^2*b^2 + 2*a^3*c)*d^3*e^8)*
x^2 + 3*(c^4*d^10*e - 4*b*c^3*d^9*e^2 - 4*a^3*b*d^3*e^8 + a^4*d^2*e^9 + 2*(3*b^2*c^2 + 2*a*c^3)*d^8*e^3 - 4*(b
^3*c + 3*a*b*c^2)*d^7*e^4 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^6*e^5 - 4*(a*b^3 + 3*a^2*b*c)*d^5*e^6 + 2*(3*a^2*
b^2 + 2*a^3*c)*d^4*e^7)*x), 1/48*(3*(16*c^3*d^6 - 24*b*c^2*d^5*e + 6*(3*b^2*c - 4*a*c^2)*d^4*e^2 - (5*b^3 - 12
*a*b*c)*d^3*e^3 + (16*c^3*d^3*e^3 - 24*b*c^2*d^2*e^4 + 6*(3*b^2*c - 4*a*c^2)*d*e^5 - (5*b^3 - 12*a*b*c)*e^6)*x
^3 + 3*(16*c^3*d^4*e^2 - 24*b*c^2*d^3*e^3 + 6*(3*b^2*c - 4*a*c^2)*d^2*e^4 - (5*b^3 - 12*a*b*c)*d*e^5)*x^2 + 3*
(16*c^3*d^5*e - 24*b*c^2*d^4*e^2 + 6*(3*b^2*c - 4*a*c^2)*d^3*e^3 - (5*b^3 - 12*a*b*c)*d^2*e^4)*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*(72
*c^3*d^6*e - 162*b*c^2*d^5*e^2 - 34*a^2*b*d*e^6 + 8*a^3*e^7 + (123*b^2*c + 92*a*c^2)*d^4*e^3 - (33*b^3 + 136*a
*b*c)*d^3*e^4 + (59*a*b^2 + 28*a^2*c)*d^2*e^5 + (44*c^3*d^4*e^3 - 88*b*c^2*d^3*e^4 + (59*b^2*c + 28*a*c^2)*d^2
*e^5 - (15*b^3 + 28*a*b*c)*d*e^6 + (15*a*b^2 - 16*a^2*c)*e^7)*x^2 + 2*(54*c^3*d^5*e^2 - 113*b*c^2*d^4*e^3 - 5*
a^2*b*e^7 + (79*b^2*c + 48*a*c^2)*d^3*e^4 - 2*(10*b^3 + 29*a*b*c)*d^2*e^5 + (25*a*b^2 - 6*a^2*c)*d*e^6)*x)*sqr
t(c*x^2 + b*x + a))/(c^4*d^11 - 4*b*c^3*d^10*e - 4*a^3*b*d^4*e^7 + a^4*d^3*e^8 + 2*(3*b^2*c^2 + 2*a*c^3)*d^9*e
^2 - 4*(b^3*c + 3*a*b*c^2)*d^8*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^7*e^4 - 4*(a*b^3 + 3*a^2*b*c)*d^6*e^5 +
2*(3*a^2*b^2 + 2*a^3*c)*d^5*e^6 + (c^4*d^8*e^3 - 4*b*c^3*d^7*e^4 - 4*a^3*b*d*e^10 + a^4*e^11 + 2*(3*b^2*c^2 +
2*a*c^3)*d^6*e^5 - 4*(b^3*c + 3*a*b*c^2)*d^5*e^6 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^7 - 4*(a*b^3 + 3*a^2*b
*c)*d^3*e^8 + 2*(3*a^2*b^2 + 2*a^3*c)*d^2*e^9)*x^3 + 3*(c^4*d^9*e^2 - 4*b*c^3*d^8*e^3 - 4*a^3*b*d^2*e^9 + a^4*
d*e^10 + 2*(3*b^2*c^2 + 2*a*c^3)*d^7*e^4 - 4*(b^3*c + 3*a*b*c^2)*d^6*e^5 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^5*
e^6 - 4*(a*b^3 + 3*a^2*b*c)*d^4*e^7 + 2*(3*a^2*b^2 + 2*a^3*c)*d^3*e^8)*x^2 + 3*(c^4*d^10*e - 4*b*c^3*d^9*e^2 -
4*a^3*b*d^3*e^8 + a^4*d^2*e^9 + 2*(3*b^2*c^2 + 2*a*c^3)*d^8*e^3 - 4*(b^3*c + 3*a*b*c^2)*d^7*e^4 + (b^4 + 12*a
*b^2*c + 6*a^2*c^2)*d^6*e^5 - 4*(a*b^3 + 3*a^2*b*c)*d^5*e^6 + 2*(3*a^2*b^2 + 2*a^3*c)*d^4*e^7)*x)]

Sympy [F]

\[ \int \frac {1}{(d+e x)^4 \sqrt {a+b x+c x^2}} \, dx=\int \frac {1}{\left (d + e x\right )^{4} \sqrt {a + b x + c x^{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(d+e x)^4 \sqrt {a+b x+c x^2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(1/(e*x+d)^4/(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. 2133 vs. \(2 (271) = 542\).

Time = 0.33 (sec) , antiderivative size = 2133, normalized size of antiderivative = 7.28 \[ \int \frac {1}{(d+e x)^4 \sqrt {a+b x+c x^2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/8*(16*c^3*d^3 - 24*b*c^2*d^2*e + 18*b^2*c*d*e^2 - 24*a*c^2*d*e^2 - 5*b^3*e^3 + 12*a*b*c*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^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c
*d^4*e^2 + 3*a*c^2*d^4*e^2 - b^3*d^3*e^3 - 6*a*b*c*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a^2*c*d^2*e^4 - 3*a^2*b*d*e^5
+ a^3*e^6)*sqrt(-c*d^2 + b*d*e - a*e^2)) - 1/24*(48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*c^3*d^3*e^2 - 72*(s
qrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b*c^2*d^2*e^3 + 54*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^2*c*d*e^4 - 72*
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*c^2*d*e^4 - 15*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^3*e^5 + 36*(sqr
t(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b*c*e^5 + 240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*c^(7/2)*d^4*e - 360*(s
qrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b*c^(5/2)*d^3*e^2 + 270*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^2*c^(3/2)*
d^2*e^3 - 360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*c^(5/2)*d^2*e^3 - 75*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
^4*b^3*sqrt(c)*d*e^4 + 180*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b*c^(3/2)*d*e^4 + 352*(sqrt(c)*x - sqrt(c*x
^2 + b*x + a))^3*c^4*d^5 - 400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b*c^3*d^4*e + 204*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^3*b^2*c^2*d^3*e^2 - 656*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*c^3*d^3*e^2 + 34*(sqrt(c)*x - sqr
t(c*x^2 + b*x + a))^3*b^3*c*d^2*e^3 + 264*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b*c^2*d^2*e^3 - 40*(sqrt(c)*
x - sqrt(c*x^2 + b*x + a))^3*b^4*d*e^4 - 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^2*c*d*e^4 + 192*(sqrt(c)
*x - sqrt(c*x^2 + b*x + a))^3*a^2*c^2*d*e^4 + 40*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^3*e^5 - 96*(sqrt(c)
*x - sqrt(c*x^2 + b*x + a))^3*a^2*b*c*e^5 + 528*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b*c^(7/2)*d^5 - 756*(sqr
t(c)*x - sqrt(c*x^2 + b*x + a))^2*b^2*c^(5/2)*d^4*e - 816*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*c^(7/2)*d^4*
e + 498*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^3*c^(3/2)*d^3*e^2 + 648*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*
a*b*c^(5/2)*d^3*e^2 - 120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^4*sqrt(c)*d^2*e^3 - 432*(sqrt(c)*x - sqrt(c*
x^2 + b*x + a))^2*a*b^2*c^(3/2)*d^2*e^3 + 288*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*c^(5/2)*d^2*e^3 + 120*
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^3*sqrt(c)*d*e^4 - 96*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*c^(3/
2)*e^5 + 264*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^2*c^3*d^5 - 336*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^3*c^2
*d^4*e - 816*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b*c^3*d^4*e + 180*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^4*c
*d^3*e^2 + 900*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^2*c^2*d^3*e^2 + 480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
*a^2*c^3*d^3*e^2 - 33*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^5*d^2*e^3 - 450*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)*a*b^3*c*d^2*e^3 - 432*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b*c^2*d^2*e^3 + 66*(sqrt(c)*x - sqrt(c*x^2 + b
*x + a))*a*b^4*d*e^4 + 306*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^2*c*d*e^4 - 120*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))*a^3*c^2*d*e^4 - 33*(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 + 44*b^3*c^(5/2)*d^5 - 44*b^4*c^(3/2)*d^4*e - 204*a*b^2*c^(5/2)*d^4*e + 15*b^5*sqrt(c)*d^
3*e^2 + 206*a*b^3*c^(3/2)*d^3*e^2 + 240*a^2*b*c^(5/2)*d^3*e^2 - 78*a*b^4*sqrt(c)*d^2*e^3 - 222*a^2*b^2*c^(3/2)
*d^2*e^3 - 88*a^3*c^(5/2)*d^2*e^3 + 111*a^2*b^3*sqrt(c)*d*e^4 + 28*a^3*b*c^(3/2)*d*e^4 - 48*a^3*b^2*sqrt(c)*e^
5 + 32*a^4*c^(3/2)*e^5)/((c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 + 3*a*c^2*d^4*e^2 - b^3*d^3*e^3 - 6*a*b*c*
d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a^2*c*d^2*e^4 - 3*a^2*b*d*e^5 + a^3*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. \[ \int \frac {1}{(d+e x)^4 \sqrt {a+b x+c x^2}} \, dx=\int \frac {1}{{\left (d+e\,x\right )}^4\,\sqrt {c\,x^2+b\,x+a}} \,d x \]

[In]

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

[Out]

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