3.7 \(\int \frac {1}{\sqrt {d+e x+f x^2} (a e+b e x+b f x^2)^2} \, dx\)

Optimal. Leaf size=162 \[ -\frac {\left (8 a e f-b \left (4 d f+e^2\right )\right ) \tanh ^{-1}\left (\frac {(e+2 f x) \sqrt {b d-a e}}{\sqrt {e} \sqrt {b e-4 a f} \sqrt {d+e x+f x^2}}\right )}{e^{3/2} (b d-a e)^{3/2} (b e-4 a f)^{3/2}}-\frac {b (e+2 f x) \sqrt {d+e x+f x^2}}{e (b d-a e) (b e-4 a f) \left (a e+b e x+b f x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.31, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {974, 12, 982, 208} \[ -\frac {\left (8 a e f-b \left (4 d f+e^2\right )\right ) \tanh ^{-1}\left (\frac {(e+2 f x) \sqrt {b d-a e}}{\sqrt {e} \sqrt {b e-4 a f} \sqrt {d+e x+f x^2}}\right )}{e^{3/2} (b d-a e)^{3/2} (b e-4 a f)^{3/2}}-\frac {b (e+2 f x) \sqrt {d+e x+f x^2}}{e (b d-a e) (b e-4 a f) \left (a e+b e x+b f x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*(e + 2*f*x)*Sqrt[d + e*x + f*x^2])/(e*(b*d - a*e)*(b*e - 4*a*f)*(a*e + b*e*x + b*f*x^2))) - ((8*a*e*f - b
*(e^2 + 4*d*f))*ArcTanh[(Sqrt[b*d - a*e]*(e + 2*f*x))/(Sqrt[e]*Sqrt[b*e - 4*a*f]*Sqrt[d + e*x + f*x^2])])/(e^(
3/2)*(b*d - a*e)^(3/2)*(b*e - 4*a*f)^(3/2))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 208

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

Rule 974

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_.) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp[((2*a
*c^2*e - b^2*c*e + b^3*f + b*c*(c*d - 3*a*f) + c*(2*c^2*d + b^2*f - c*(b*e + 2*a*f))*x)*(a + b*x + c*x^2)^(p +
1)*(d + e*x + f*x^2)^(q + 1))/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)), x] - Dist[1/
((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)), Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*
x^2)^q*Simp[2*c*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1) - (2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(a*f*(
p + 1) - c*d*(p + 2)) - e*(b^2*c*e - 2*a*c^2*e - b^3*f - b*c*(c*d - 3*a*f))*(p + q + 2) + (2*f*(2*a*c^2*e - b^
2*c*e + b^3*f + b*c*(c*d - 3*a*f))*(p + q + 2) - (2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(b*f*(p + 1) - c*e*(2*p +
q + 4)))*x + c*f*(2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e,
f, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 - (b*d - a*e)*(c*e
- b*f), 0] &&  !( !IntegerQ[p] && ILtQ[q, -1]) &&  !IGtQ[q, 0]

Rule 982

Int[1/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[-2*e, Su
bst[Int[1/(e*(b*e - 4*a*f) - (b*d - a*e)*x^2), x], x, (e + 2*f*x)/Sqrt[d + e*x + f*x^2]], x] /; FreeQ[{a, b, c
, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && EqQ[c*e - b*f, 0]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {d+e x+f x^2} \left (a e+b e x+b f x^2\right )^2} \, dx &=-\frac {b (e+2 f x) \sqrt {d+e x+f x^2}}{e (b d-a e) (b e-4 a f) \left (a e+b e x+b f x^2\right )}+\frac {\int \frac {b (b d-a e) f^2 \left (8 a e f-b \left (e^2+4 d f\right )\right )}{2 \sqrt {d+e x+f x^2} \left (a e+b e x+b f x^2\right )} \, dx}{b e (b d-a e)^2 f^2 (b e-4 a f)}\\ &=-\frac {b (e+2 f x) \sqrt {d+e x+f x^2}}{e (b d-a e) (b e-4 a f) \left (a e+b e x+b f x^2\right )}+\frac {\left (8 a e f-b \left (e^2+4 d f\right )\right ) \int \frac {1}{\sqrt {d+e x+f x^2} \left (a e+b e x+b f x^2\right )} \, dx}{2 e (b d-a e) (b e-4 a f)}\\ &=-\frac {b (e+2 f x) \sqrt {d+e x+f x^2}}{e (b d-a e) (b e-4 a f) \left (a e+b e x+b f x^2\right )}-\frac {\left (8 a e f-b \left (e^2+4 d f\right )\right ) \operatorname {Subst}\left (\int \frac {1}{e \left (b e^2-4 a e f\right )-\left (b d e-a e^2\right ) x^2} \, dx,x,\frac {e+2 f x}{\sqrt {d+e x+f x^2}}\right )}{(b d-a e) (b e-4 a f)}\\ &=-\frac {b (e+2 f x) \sqrt {d+e x+f x^2}}{e (b d-a e) (b e-4 a f) \left (a e+b e x+b f x^2\right )}-\frac {\left (8 a e f-b \left (e^2+4 d f\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b d-a e} (e+2 f x)}{\sqrt {e} \sqrt {b e-4 a f} \sqrt {d+e x+f x^2}}\right )}{e^{3/2} (b d-a e)^{3/2} (b e-4 a f)^{3/2}}\\ \end {align*}

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Mathematica [B]  time = 1.96, size = 490, normalized size = 3.02 \[ \frac {2 f \left (-\frac {\left (b \left (4 d f+e^2\right )-8 a e f\right ) \tanh ^{-1}\left (\frac {-\sqrt {e} (e+2 f x) \sqrt {b e-4 a f}-\sqrt {b} \left (e^2-4 d f\right )}{4 f \sqrt {b d-a e} \sqrt {d+x (e+f x)}}\right )}{4 f (b d-a e)^{3/2} (b e-4 a f)^{3/2}}-\frac {e \tanh ^{-1}\left (\frac {\sqrt {b} \left (e^2-4 d f\right )-\sqrt {e} (e+2 f x) \sqrt {b e-4 a f}}{4 f \sqrt {b d-a e} \sqrt {d+x (e+f x)}}\right )}{4 f (b d-a e)^{3/2} \sqrt {b e-4 a f}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {e} (e+2 f x) \sqrt {b e-4 a f}-\sqrt {b} \left (e^2-4 d f\right )}{4 f \sqrt {b d-a e} \sqrt {d+x (e+f x)}}\right )}{\sqrt {b d-a e} (b e-4 a f)^{3/2}}-\frac {\sqrt {b} \sqrt {e} \sqrt {d+x (e+f x)}}{(b d-a e) (b e-4 a f) \left (\sqrt {b} (e+2 f x)-\sqrt {e} \sqrt {b e-4 a f}\right )}-\frac {\sqrt {b} \sqrt {e} \sqrt {d+x (e+f x)}}{(b d-a e) (b e-4 a f) \left (\sqrt {e} \sqrt {b e-4 a f}+\sqrt {b} (e+2 f x)\right )}\right )}{e^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*f*(-((Sqrt[b]*Sqrt[e]*Sqrt[d + x*(e + f*x)])/((b*d - a*e)*(b*e - 4*a*f)*(-(Sqrt[e]*Sqrt[b*e - 4*a*f]) + Sqr
t[b]*(e + 2*f*x)))) - (Sqrt[b]*Sqrt[e]*Sqrt[d + x*(e + f*x)])/((b*d - a*e)*(b*e - 4*a*f)*(Sqrt[e]*Sqrt[b*e - 4
*a*f] + Sqrt[b]*(e + 2*f*x))) - ((-8*a*e*f + b*(e^2 + 4*d*f))*ArcTanh[(-(Sqrt[b]*(e^2 - 4*d*f)) - Sqrt[e]*Sqrt
[b*e - 4*a*f]*(e + 2*f*x))/(4*Sqrt[b*d - a*e]*f*Sqrt[d + x*(e + f*x)])])/(4*(b*d - a*e)^(3/2)*f*(b*e - 4*a*f)^
(3/2)) - (e*ArcTanh[(Sqrt[b]*(e^2 - 4*d*f) - Sqrt[e]*Sqrt[b*e - 4*a*f]*(e + 2*f*x))/(4*Sqrt[b*d - a*e]*f*Sqrt[
d + x*(e + f*x)])])/(4*(b*d - a*e)^(3/2)*f*Sqrt[b*e - 4*a*f]) + ArcTanh[(-(Sqrt[b]*(e^2 - 4*d*f)) + Sqrt[e]*Sq
rt[b*e - 4*a*f]*(e + 2*f*x))/(4*Sqrt[b*d - a*e]*f*Sqrt[d + x*(e + f*x)])]/(Sqrt[b*d - a*e]*(b*e - 4*a*f)^(3/2)
)))/e^(3/2)

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fricas [B]  time = 3.07, size = 2005, normalized size = 12.38 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*(sqrt(b^2*d*e^2 - a*b*e^3 - 4*(a*b*d*e - a^2*e^2)*f)*(a*b*e^3 + (b^2*e^2*f + 4*(b^2*d - 2*a*b*e)*f^2)*x^
2 + 4*(a*b*d*e - 2*a^2*e^2)*f + (b^2*e^3 + 4*(b^2*d*e - 2*a*b*e^2)*f)*x)*log((8*b^2*d^2*e^4 - 8*a*b*d*e^5 + a^
2*e^6 + 16*a^2*d^2*e^2*f^2 + (b^2*e^4*f^2 + 16*(b^2*d^2 - 8*a*b*d*e + 8*a^2*e^2)*f^4 + 8*(3*b^2*d*e^2 - 4*a*b*
e^3)*f^3)*x^4 + 2*(b^2*e^5*f + 16*(b^2*d^2*e - 8*a*b*d*e^2 + 8*a^2*e^3)*f^3 + 8*(3*b^2*d*e^3 - 4*a*b*e^4)*f^2)
*x^3 + (b^2*e^6 - 32*(3*a*b*d^2*e - 4*a^2*d*e^2)*f^3 + 16*(3*b^2*d^2*e^2 - 13*a*b*d*e^3 + 10*a^2*e^4)*f^2 + 2*
(16*b^2*d*e^4 - 19*a*b*e^5)*f)*x^2 - 4*sqrt(b^2*d*e^2 - a*b*e^3 - 4*(a*b*d*e - a^2*e^2)*f)*(2*b*d*e^3 - a*e^4
- 4*a*d*e^2*f + 2*(b*e^2*f^2 + 4*(b*d - 2*a*e)*f^3)*x^3 + 3*(b*e^3*f + 4*(b*d*e - 2*a*e^2)*f^2)*x^2 + (b*e^4 -
8*a*d*e*f^2 + 2*(4*b*d*e^2 - 5*a*e^3)*f)*x)*sqrt(f*x^2 + e*x + d) - 8*(4*a*b*d^2*e^3 - 3*a^2*d*e^4)*f + 2*(4*
b^2*d*e^5 - 3*a*b*e^6 - 16*(3*a*b*d^2*e^2 - 4*a^2*d*e^3)*f^2 + 8*(2*b^2*d^2*e^3 - 5*a*b*d*e^4 + 2*a^2*e^5)*f)*
x)/(b^2*f^2*x^4 + 2*b^2*e*f*x^3 + 2*a*b*e^2*x + a^2*e^2 + (b^2*e^2 + 2*a*b*e*f)*x^2)) + 4*(b^3*d*e^3 - a*b^2*e
^4 - 4*(a*b^2*d*e^2 - a^2*b*e^3)*f - 2*(4*(a*b^2*d*e - a^2*b*e^2)*f^2 - (b^3*d*e^2 - a*b^2*e^3)*f)*x)*sqrt(f*x
^2 + e*x + d))/(a*b^4*d^2*e^5 - 2*a^2*b^3*d*e^6 + a^3*b^2*e^7 + 16*(a^3*b^2*d^2*e^3 - 2*a^4*b*d*e^4 + a^5*e^5)
*f^2 + (16*(a^2*b^3*d^2*e^2 - 2*a^3*b^2*d*e^3 + a^4*b*e^4)*f^3 - 8*(a*b^4*d^2*e^3 - 2*a^2*b^3*d*e^4 + a^3*b^2*
e^5)*f^2 + (b^5*d^2*e^4 - 2*a*b^4*d*e^5 + a^2*b^3*e^6)*f)*x^2 - 8*(a^2*b^3*d^2*e^4 - 2*a^3*b^2*d*e^5 + a^4*b*e
^6)*f + (b^5*d^2*e^5 - 2*a*b^4*d*e^6 + a^2*b^3*e^7 + 16*(a^2*b^3*d^2*e^3 - 2*a^3*b^2*d*e^4 + a^4*b*e^5)*f^2 -
8*(a*b^4*d^2*e^4 - 2*a^2*b^3*d*e^5 + a^3*b^2*e^6)*f)*x), 1/2*(sqrt(-b^2*d*e^2 + a*b*e^3 + 4*(a*b*d*e - a^2*e^2
)*f)*(a*b*e^3 + (b^2*e^2*f + 4*(b^2*d - 2*a*b*e)*f^2)*x^2 + 4*(a*b*d*e - 2*a^2*e^2)*f + (b^2*e^3 + 4*(b^2*d*e
- 2*a*b*e^2)*f)*x)*arctan(-1/2*sqrt(-b^2*d*e^2 + a*b*e^3 + 4*(a*b*d*e - a^2*e^2)*f)*(2*b*d*e^2 - a*e^3 - 4*a*d
*e*f + (b*e^2*f + 4*(b*d - 2*a*e)*f^2)*x^2 + (b*e^3 + 4*(b*d*e - 2*a*e^2)*f)*x)*sqrt(f*x^2 + e*x + d)/(b^2*d^2
*e^3 - a*b*d*e^4 - 2*(4*(a*b*d*e - a^2*e^2)*f^3 - (b^2*d*e^2 - a*b*e^3)*f^2)*x^3 - 3*(4*(a*b*d*e^2 - a^2*e^3)*
f^2 - (b^2*d*e^3 - a*b*e^4)*f)*x^2 - 4*(a*b*d^2*e^2 - a^2*d*e^3)*f + (b^2*d*e^4 - a*b*e^5 - 8*(a*b*d^2*e - a^2
*d*e^2)*f^2 + 2*(b^2*d^2*e^2 - 3*a*b*d*e^3 + 2*a^2*e^4)*f)*x)) - 2*(b^3*d*e^3 - a*b^2*e^4 - 4*(a*b^2*d*e^2 - a
^2*b*e^3)*f - 2*(4*(a*b^2*d*e - a^2*b*e^2)*f^2 - (b^3*d*e^2 - a*b^2*e^3)*f)*x)*sqrt(f*x^2 + e*x + d))/(a*b^4*d
^2*e^5 - 2*a^2*b^3*d*e^6 + a^3*b^2*e^7 + 16*(a^3*b^2*d^2*e^3 - 2*a^4*b*d*e^4 + a^5*e^5)*f^2 + (16*(a^2*b^3*d^2
*e^2 - 2*a^3*b^2*d*e^3 + a^4*b*e^4)*f^3 - 8*(a*b^4*d^2*e^3 - 2*a^2*b^3*d*e^4 + a^3*b^2*e^5)*f^2 + (b^5*d^2*e^4
- 2*a*b^4*d*e^5 + a^2*b^3*e^6)*f)*x^2 - 8*(a^2*b^3*d^2*e^4 - 2*a^3*b^2*d*e^5 + a^4*b*e^6)*f + (b^5*d^2*e^5 -
2*a*b^4*d*e^6 + a^2*b^3*e^7 + 16*(a^2*b^3*d^2*e^3 - 2*a^3*b^2*d*e^4 + a^4*b*e^5)*f^2 - 8*(a*b^4*d^2*e^4 - 2*a^
2*b^3*d*e^5 + a^3*b^2*e^6)*f)*x)]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab
le to divide, perhaps due to rounding error%%%{%%%{1,[2]%%%},[8,2,0,0,0]%%%}+%%%{%%{[%%%{-4,[1]%%%},0]:[1,0,%%
%{-1,[1]%%%}]%%},[7,2,1,0,0]%%%}+%%%{%%%{6,[1]%%%},[6,2,2,0,0]%%%}+%%%{%%%{-4,[2]%%%},[6,2,0,0,1]%%%}+%%%{%%%{
8,[2]%%%},[6,1,1,1,0]%%%}+%%%{%%{[-4,0]:[1,0,%%%{-1,[1]%%%}]%%},[5,2,3,0,0]%%%}+%%%{%%{[%%%{12,[1]%%%},0]:[1,0
,%%%{-1,[1]%%%}]%%},[5,2,1,0,1]%%%}+%%%{%%{[%%%{-24,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,1,2,1,0]%%%}+%%%{1,[
4,2,4,0,0]%%%}+%%%{%%%{-14,[1]%%%},[4,2,2,0,1]%%%}+%%%{%%%{6,[2]%%%},[4,2,0,0,2]%%%}+%%%{%%%{26,[1]%%%},[4,1,3
,1,0]%%%}+%%%{%%%{-16,[2]%%%},[4,1,1,1,1]%%%}+%%%{%%%{16,[2]%%%},[4,0,2,2,0]%%%}+%%%{%%{[8,0]:[1,0,%%%{-1,[1]%
%%}]%%},[3,2,3,0,1]%%%}+%%%{%%{[%%%{-12,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,2,1,0,2]%%%}+%%%{%%{[-12,0]:[1,0
,%%%{-1,[1]%%%}]%%},[3,1,4,1,0]%%%}+%%%{%%{[%%%{32,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,1,2,1,1]%%%}+%%%{%%{[
%%%{-32,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,0,3,2,0]%%%}+%%%{-2,[2,2,4,0,1]%%%}+%%%{%%%{10,[1]%%%},[2,2,2,0,
2]%%%}+%%%{%%%{-4,[2]%%%},[2,2,0,0,3]%%%}+%%%{2,[2,1,5,1,0]%%%}+%%%{%%%{-28,[1]%%%},[2,1,3,1,1]%%%}+%%%{%%%{8,
[2]%%%},[2,1,1,1,2]%%%}+%%%{%%%{24,[1]%%%},[2,0,4,2,0]%%%}+%%%{%%{[-4,0]:[1,0,%%%{-1,[1]%%%}]%%},[1,2,3,0,2]%%
%}+%%%{%%{[%%%{4,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,2,1,0,3]%%%}+%%%{%%{[12,0]:[1,0,%%%{-1,[1]%%%}]%%},[1,1
,4,1,1]%%%}+%%%{%%{[%%%{-8,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,1,2,1,2]%%%}+%%%{%%{[-8,0]:[1,0,%%%{-1,[1]%%%
}]%%},[1,0,5,2,0]%%%}+%%%{1,[0,2,4,0,2]%%%}+%%%{%%%{-2,[1]%%%},[0,2,2,0,3]%%%}+%%%{%%%{1,[2]%%%},[0,2,0,0,4]%%
%}+%%%{-2,[0,1,5,1,1]%%%}+%%%{%%%{2,[1]%%%},[0,1,3,1,2]%%%}+%%%{1,[0,0,6,2,0]%%%} / %%%{%%%{1,[3]%%%},[8,2,0,0
,0]%%%}+%%%{%%{poly1[%%%{-4,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,2,1,0,0]%%%}+%%%{%%%{6,[2]%%%},[6,2,2,0,0]%%
%}+%%%{%%%{-4,[3]%%%},[6,2,0,0,1]%%%}+%%%{%%%{8,[3]%%%},[6,1,1,1,0]%%%}+%%%{%%{poly1[%%%{-4,[1]%%%},0]:[1,0,%%
%{-1,[1]%%%}]%%},[5,2,3,0,0]%%%}+%%%{%%{poly1[%%%{12,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,2,1,0,1]%%%}+%%%{%%
{poly1[%%%{-24,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,1,2,1,0]%%%}+%%%{%%%{1,[1]%%%},[4,2,4,0,0]%%%}+%%%{%%%{-1
4,[2]%%%},[4,2,2,0,1]%%%}+%%%{%%%{6,[3]%%%},[4,2,0,0,2]%%%}+%%%{%%%{26,[2]%%%},[4,1,3,1,0]%%%}+%%%{%%%{-16,[3]
%%%},[4,1,1,1,1]%%%}+%%%{%%%{16,[3]%%%},[4,0,2,2,0]%%%}+%%%{%%{poly1[%%%{8,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},
[3,2,3,0,1]%%%}+%%%{%%{poly1[%%%{-12,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,2,1,0,2]%%%}+%%%{%%{poly1[%%%{-12,[
1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,1,4,1,0]%%%}+%%%{%%{poly1[%%%{32,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,1,
2,1,1]%%%}+%%%{%%{poly1[%%%{-32,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,0,3,2,0]%%%}+%%%{%%%{-2,[1]%%%},[2,2,4,0
,1]%%%}+%%%{%%%{10,[2]%%%},[2,2,2,0,2]%%%}+%%%{%%%{-4,[3]%%%},[2,2,0,0,3]%%%}+%%%{%%%{2,[1]%%%},[2,1,5,1,0]%%%
}+%%%{%%%{-28,[2]%%%},[2,1,3,1,1]%%%}+%%%{%%%{8,[3]%%%},[2,1,1,1,2]%%%}+%%%{%%%{24,[2]%%%},[2,0,4,2,0]%%%}+%%%
{%%{poly1[%%%{-4,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,2,3,0,2]%%%}+%%%{%%{poly1[%%%{4,[2]%%%},0]:[1,0,%%%{-1,
[1]%%%}]%%},[1,2,1,0,3]%%%}+%%%{%%{poly1[%%%{12,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,1,4,1,1]%%%}+%%%{%%{poly
1[%%%{-8,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,1,2,1,2]%%%}+%%%{%%{poly1[%%%{-8,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}
]%%},[1,0,5,2,0]%%%}+%%%{%%%{1,[1]%%%},[0,2,4,0,2]%%%}+%%%{%%%{-2,[2]%%%},[0,2,2,0,3]%%%}+%%%{%%%{1,[3]%%%},[0
,2,0,0,4]%%%}+%%%{%%%{-2,[1]%%%},[0,1,5,1,1]%%%}+%%%{%%%{2,[2]%%%},[0,1,3,1,2]%%%}+%%%{%%%{1,[1]%%%},[0,0,6,2,
0]%%%} Error: Bad Argument Value

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maple [B]  time = 0.03, size = 1377, normalized size = 8.50 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b f x^{2} + b e x + a e\right )}^{2} \sqrt {f x^{2} + e x + d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((b*f*x^2 + b*e*x + a*e)^2*sqrt(f*x^2 + e*x + d)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (b\,f\,x^2+b\,e\,x+a\,e\right )}^2\,\sqrt {f\,x^2+e\,x+d}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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