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]

-((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))

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Rubi [A]  time = 0.305681, antiderivative size = 162, normalized size of antiderivative = 1., 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 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 12

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

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]

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]

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*}

Mathematica [B]  time = 2.12731, 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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Maple [B]  time = 0.352, size = 1377, normalized size = 8.5 \begin{align*} \text{result too large to display} \end{align*}

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b f x^{2} + b e x + a e\right )}^{2} \sqrt{f x^{2} + e x + d}}\,{d x} \end{align*}

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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Fricas [B]  time = 8.12704, size = 4059, normalized size = 25.06 \begin{align*} \text{result too large to display} \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \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 \end{align*}

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]

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

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

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