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

Optimal. Leaf size=199 \[ \frac {2 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {2 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.30, antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {707, 1093, 208} \begin {gather*} \frac {2 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {2 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 707

Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2*e, Subst[Int[1/(c*d^
2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{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]

Rule 1093

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/(b/
2 - q/2 + c*x^2), x], x] - Dist[c/q, Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*
a*c, 0] && PosQ[b^2 - 4*a*c]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.53, size = 176, normalized size = 0.88 \begin {gather*} \frac {2 \sqrt {2} \sqrt {c} \left (\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {e \sqrt {b^2-4 a c}-b e+2 c d}}\right )}{\sqrt {e \left (\sqrt {b^2-4 a c}-b\right )+2 c d}}\right )}{\sqrt {b^2-4 a c}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [C]  time = 0.72, size = 223, normalized size = 1.12 \begin {gather*} \frac {2 i \sqrt {2} \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {i e \sqrt {4 a c-b^2}+b e-2 c d}}\right )}{\sqrt {4 a c-b^2} \sqrt {i e \sqrt {4 a c-b^2}+b e-2 c d}}-\frac {2 i \sqrt {2} \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-i e \sqrt {4 a c-b^2}+b e-2 c d}}\right )}{\sqrt {4 a c-b^2} \sqrt {-i e \sqrt {4 a c-b^2}+b e-2 c d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*I)*Sqrt[2]*Sqrt[c]*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e - I*Sqrt[-b^2 + 4*a*c]*e]])/(
Sqrt[-b^2 + 4*a*c]*Sqrt[-2*c*d + b*e - I*Sqrt[-b^2 + 4*a*c]*e]) + ((2*I)*Sqrt[2]*Sqrt[c]*ArcTan[(Sqrt[2]*Sqrt[
c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e + I*Sqrt[-b^2 + 4*a*c]*e]])/(Sqrt[-b^2 + 4*a*c]*Sqrt[-2*c*d + b*e + I*Sqrt
[-b^2 + 4*a*c]*e])

________________________________________________________________________________________

fricas [B]  time = 0.49, size = 2641, normalized size = 13.27

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(2)*sqrt((2*c*d - b*e + ((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2)*sqrt(e^
2/((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 -
4*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4)))/((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)
*e^2))*log(4*sqrt(e*x + d)*c*e + sqrt(2)*((b^2 - 4*a*c)*e^2 - (2*(b^2*c^2 - 4*a*c^3)*d^3 - 3*(b^3*c - 4*a*b*c^
2)*d^2*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d*e^2 - (a*b^3 - 4*a^2*b*c)*e^3)*sqrt(e^2/((b^2*c^2 - 4*a*c^3)*d^4 -
2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2 -
4*a^3*c)*e^4)))*sqrt((2*c*d - b*e + ((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2)*sqr
t(e^2/((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^
3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4)))/((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^
2*c)*e^2))) + 1/2*sqrt(2)*sqrt((2*c*d - b*e + ((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)
*e^2)*sqrt(e^2/((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2
- 2*(a*b^3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4)))/((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b
^2 - 4*a^2*c)*e^2))*log(4*sqrt(e*x + d)*c*e - sqrt(2)*((b^2 - 4*a*c)*e^2 - (2*(b^2*c^2 - 4*a*c^3)*d^3 - 3*(b^3
*c - 4*a*b*c^2)*d^2*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d*e^2 - (a*b^3 - 4*a^2*b*c)*e^3)*sqrt(e^2/((b^2*c^2 - 4*
a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e^3
+ (a^2*b^2 - 4*a^3*c)*e^4)))*sqrt((2*c*d - b*e + ((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^
2*c)*e^2)*sqrt(e^2/((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*
e^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4)))/((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e +
(a*b^2 - 4*a^2*c)*e^2))) - 1/2*sqrt(2)*sqrt((2*c*d - b*e - ((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b
^2 - 4*a^2*c)*e^2)*sqrt(e^2/((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*
c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4)))/((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*
c)*d*e + (a*b^2 - 4*a^2*c)*e^2))*log(4*sqrt(e*x + d)*c*e + sqrt(2)*((b^2 - 4*a*c)*e^2 + (2*(b^2*c^2 - 4*a*c^3)
*d^3 - 3*(b^3*c - 4*a*b*c^2)*d^2*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d*e^2 - (a*b^3 - 4*a^2*b*c)*e^3)*sqrt(e^2/(
(b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4*a
^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4)))*sqrt((2*c*d - b*e - ((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e +
(a*b^2 - 4*a^2*c)*e^2)*sqrt(e^2/((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*
a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4)))/((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*
a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2))) + 1/2*sqrt(2)*sqrt((2*c*d - b*e - ((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*
c)*d*e + (a*b^2 - 4*a^2*c)*e^2)*sqrt(e^2/((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b
^2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4)))/((b^2*c - 4*a*c^2)*d^2 -
(b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2))*log(4*sqrt(e*x + d)*c*e - sqrt(2)*((b^2 - 4*a*c)*e^2 + (2*(b^2*c
^2 - 4*a*c^3)*d^3 - 3*(b^3*c - 4*a*b*c^2)*d^2*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d*e^2 - (a*b^3 - 4*a^2*b*c)*e^
3)*sqrt(e^2/((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 - 2
*(a*b^3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4)))*sqrt((2*c*d - b*e - ((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*
a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2)*sqrt(e^2/((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2
*a*b^2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4)))/((b^2*c - 4*a*c^2)*d^
2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2)))

________________________________________________________________________________________

giac [A]  time = 0.24, size = 288, normalized size = 1.45 \begin {gather*} -\frac {2 \, \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} c \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x e + d}}{\sqrt {-\frac {2 \, c d - b e + \sqrt {{\left (2 \, c d - b e\right )}^{2} - 4 \, {\left (c d^{2} - b d e + a e^{2}\right )} c}}{c}}}\right )}{{\left (2 \, \sqrt {b^{2} - 4 \, a c} c d + {\left (b^{2} - 4 \, a c - \sqrt {b^{2} - 4 \, a c} b\right )} e\right )} {\left | c \right |}} + \frac {2 \, \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} c \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x e + d}}{\sqrt {-\frac {2 \, c d - b e - \sqrt {{\left (2 \, c d - b e\right )}^{2} - 4 \, {\left (c d^{2} - b d e + a e^{2}\right )} c}}{c}}}\right )}{{\left (2 \, \sqrt {b^{2} - 4 \, a c} c d - {\left (b^{2} - 4 \, a c + \sqrt {b^{2} - 4 \, a c} b\right )} e\right )} {\left | c \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*c*arctan(2*sqrt(1/2)*sqrt(x*e + d)/sqrt(-(2*c*d - b*e + sq
rt((2*c*d - b*e)^2 - 4*(c*d^2 - b*d*e + a*e^2)*c))/c))/((2*sqrt(b^2 - 4*a*c)*c*d + (b^2 - 4*a*c - sqrt(b^2 - 4
*a*c)*b)*e)*abs(c)) + 2*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*c*arctan(2*sqrt(1/2)*sqrt(x*e + d)/sq
rt(-(2*c*d - b*e - sqrt((2*c*d - b*e)^2 - 4*(c*d^2 - b*d*e + a*e^2)*c))/c))/((2*sqrt(b^2 - 4*a*c)*c*d - (b^2 -
4*a*c + sqrt(b^2 - 4*a*c)*b)*e)*abs(c))

________________________________________________________________________________________

maple [A]  time = 0.12, size = 194, normalized size = 0.97 \begin {gather*} -\frac {2 \sqrt {2}\, c e \arctanh \left (\frac {\sqrt {e x +d}\, \sqrt {2}\, c}{\sqrt {\left (-b e +2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}\right )}{\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\, \sqrt {\left (-b e +2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}-\frac {2 \sqrt {2}\, c e \arctan \left (\frac {\sqrt {e x +d}\, \sqrt {2}\, c}{\sqrt {\left (b e -2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}\right )}{\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\, \sqrt {\left (b e -2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*c*e/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*
2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)-2*c*e/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((b*e-2*c*d+
(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)
*c)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 1.93, size = 4449, normalized size = 22.36

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atan(((((d + e*x)^(1/2)*(-(b^3*e + e*(-(4*a*c - b^2)^3)^(1/2) + 8*a*c^2*d - 2*b^2*c*d - 4*a*b*c*e)/(2*(a*b^4*e
^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e - 8*a*b^2*c^2*d^2 - 8*a^2*b^2*c*e^2 - 16*a^2*b*c^2*
d*e + 8*a*b^3*c*d*e)))^(1/2)*(8*b^3*c^2*e^3 - 16*b^2*c^3*d*e^2 - 32*a*b*c^3*e^3 + 64*a*c^4*d*e^2) - 32*a*c^3*e
^3 + 8*b^2*c^2*e^3)*(-(b^3*e + e*(-(4*a*c - b^2)^3)^(1/2) + 8*a*c^2*d - 2*b^2*c*d - 4*a*b*c*e)/(2*(a*b^4*e^2 +
b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e - 8*a*b^2*c^2*d^2 - 8*a^2*b^2*c*e^2 - 16*a^2*b*c^2*d*e
+ 8*a*b^3*c*d*e)))^(1/2) + 16*c^3*e^2*(d + e*x)^(1/2))*(-(b^3*e + e*(-(4*a*c - b^2)^3)^(1/2) + 8*a*c^2*d - 2*b
^2*c*d - 4*a*b*c*e)/(2*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e - 8*a*b^2*c^2*d^2 -
8*a^2*b^2*c*e^2 - 16*a^2*b*c^2*d*e + 8*a*b^3*c*d*e)))^(1/2)*1i + ((32*a*c^3*e^3 + (d + e*x)^(1/2)*(-(b^3*e + e
*(-(4*a*c - b^2)^3)^(1/2) + 8*a*c^2*d - 2*b^2*c*d - 4*a*b*c*e)/(2*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16
*a^3*c^2*e^2 - b^5*d*e - 8*a*b^2*c^2*d^2 - 8*a^2*b^2*c*e^2 - 16*a^2*b*c^2*d*e + 8*a*b^3*c*d*e)))^(1/2)*(8*b^3*
c^2*e^3 - 16*b^2*c^3*d*e^2 - 32*a*b*c^3*e^3 + 64*a*c^4*d*e^2) - 8*b^2*c^2*e^3)*(-(b^3*e + e*(-(4*a*c - b^2)^3)
^(1/2) + 8*a*c^2*d - 2*b^2*c*d - 4*a*b*c*e)/(2*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*
d*e - 8*a*b^2*c^2*d^2 - 8*a^2*b^2*c*e^2 - 16*a^2*b*c^2*d*e + 8*a*b^3*c*d*e)))^(1/2) + 16*c^3*e^2*(d + e*x)^(1/
2))*(-(b^3*e + e*(-(4*a*c - b^2)^3)^(1/2) + 8*a*c^2*d - 2*b^2*c*d - 4*a*b*c*e)/(2*(a*b^4*e^2 + b^4*c*d^2 + 16*
a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e - 8*a*b^2*c^2*d^2 - 8*a^2*b^2*c*e^2 - 16*a^2*b*c^2*d*e + 8*a*b^3*c*d*e)
))^(1/2)*1i)/((((d + e*x)^(1/2)*(-(b^3*e + e*(-(4*a*c - b^2)^3)^(1/2) + 8*a*c^2*d - 2*b^2*c*d - 4*a*b*c*e)/(2*
(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e - 8*a*b^2*c^2*d^2 - 8*a^2*b^2*c*e^2 - 16*a^
2*b*c^2*d*e + 8*a*b^3*c*d*e)))^(1/2)*(8*b^3*c^2*e^3 - 16*b^2*c^3*d*e^2 - 32*a*b*c^3*e^3 + 64*a*c^4*d*e^2) - 32
*a*c^3*e^3 + 8*b^2*c^2*e^3)*(-(b^3*e + e*(-(4*a*c - b^2)^3)^(1/2) + 8*a*c^2*d - 2*b^2*c*d - 4*a*b*c*e)/(2*(a*b
^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e - 8*a*b^2*c^2*d^2 - 8*a^2*b^2*c*e^2 - 16*a^2*b*
c^2*d*e + 8*a*b^3*c*d*e)))^(1/2) + 16*c^3*e^2*(d + e*x)^(1/2))*(-(b^3*e + e*(-(4*a*c - b^2)^3)^(1/2) + 8*a*c^2
*d - 2*b^2*c*d - 4*a*b*c*e)/(2*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e - 8*a*b^2*c^
2*d^2 - 8*a^2*b^2*c*e^2 - 16*a^2*b*c^2*d*e + 8*a*b^3*c*d*e)))^(1/2) - ((32*a*c^3*e^3 + (d + e*x)^(1/2)*(-(b^3*
e + e*(-(4*a*c - b^2)^3)^(1/2) + 8*a*c^2*d - 2*b^2*c*d - 4*a*b*c*e)/(2*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2
+ 16*a^3*c^2*e^2 - b^5*d*e - 8*a*b^2*c^2*d^2 - 8*a^2*b^2*c*e^2 - 16*a^2*b*c^2*d*e + 8*a*b^3*c*d*e)))^(1/2)*(8
*b^3*c^2*e^3 - 16*b^2*c^3*d*e^2 - 32*a*b*c^3*e^3 + 64*a*c^4*d*e^2) - 8*b^2*c^2*e^3)*(-(b^3*e + e*(-(4*a*c - b^
2)^3)^(1/2) + 8*a*c^2*d - 2*b^2*c*d - 4*a*b*c*e)/(2*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 -
b^5*d*e - 8*a*b^2*c^2*d^2 - 8*a^2*b^2*c*e^2 - 16*a^2*b*c^2*d*e + 8*a*b^3*c*d*e)))^(1/2) + 16*c^3*e^2*(d + e*x
)^(1/2))*(-(b^3*e + e*(-(4*a*c - b^2)^3)^(1/2) + 8*a*c^2*d - 2*b^2*c*d - 4*a*b*c*e)/(2*(a*b^4*e^2 + b^4*c*d^2
+ 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e - 8*a*b^2*c^2*d^2 - 8*a^2*b^2*c*e^2 - 16*a^2*b*c^2*d*e + 8*a*b^3*c
*d*e)))^(1/2)))*(-(b^3*e + e*(-(4*a*c - b^2)^3)^(1/2) + 8*a*c^2*d - 2*b^2*c*d - 4*a*b*c*e)/(2*(a*b^4*e^2 + b^4
*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e - 8*a*b^2*c^2*d^2 - 8*a^2*b^2*c*e^2 - 16*a^2*b*c^2*d*e + 8*
a*b^3*c*d*e)))^(1/2)*2i + atan(((((d + e*x)^(1/2)*((e*(-(4*a*c - b^2)^3)^(1/2) - b^3*e - 8*a*c^2*d + 2*b^2*c*d
+ 4*a*b*c*e)/(2*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e - 8*a*b^2*c^2*d^2 - 8*a^2*
b^2*c*e^2 - 16*a^2*b*c^2*d*e + 8*a*b^3*c*d*e)))^(1/2)*(8*b^3*c^2*e^3 - 16*b^2*c^3*d*e^2 - 32*a*b*c^3*e^3 + 64*
a*c^4*d*e^2) - 32*a*c^3*e^3 + 8*b^2*c^2*e^3)*((e*(-(4*a*c - b^2)^3)^(1/2) - b^3*e - 8*a*c^2*d + 2*b^2*c*d + 4*
a*b*c*e)/(2*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e - 8*a*b^2*c^2*d^2 - 8*a^2*b^2*c
*e^2 - 16*a^2*b*c^2*d*e + 8*a*b^3*c*d*e)))^(1/2) + 16*c^3*e^2*(d + e*x)^(1/2))*((e*(-(4*a*c - b^2)^3)^(1/2) -
b^3*e - 8*a*c^2*d + 2*b^2*c*d + 4*a*b*c*e)/(2*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d
*e - 8*a*b^2*c^2*d^2 - 8*a^2*b^2*c*e^2 - 16*a^2*b*c^2*d*e + 8*a*b^3*c*d*e)))^(1/2)*1i + ((32*a*c^3*e^3 + (d +
e*x)^(1/2)*((e*(-(4*a*c - b^2)^3)^(1/2) - b^3*e - 8*a*c^2*d + 2*b^2*c*d + 4*a*b*c*e)/(2*(a*b^4*e^2 + b^4*c*d^2
+ 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e - 8*a*b^2*c^2*d^2 - 8*a^2*b^2*c*e^2 - 16*a^2*b*c^2*d*e + 8*a*b^3*
c*d*e)))^(1/2)*(8*b^3*c^2*e^3 - 16*b^2*c^3*d*e^2 - 32*a*b*c^3*e^3 + 64*a*c^4*d*e^2) - 8*b^2*c^2*e^3)*((e*(-(4*
a*c - b^2)^3)^(1/2) - b^3*e - 8*a*c^2*d + 2*b^2*c*d + 4*a*b*c*e)/(2*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 +
16*a^3*c^2*e^2 - b^5*d*e - 8*a*b^2*c^2*d^2 - 8*a^2*b^2*c*e^2 - 16*a^2*b*c^2*d*e + 8*a*b^3*c*d*e)))^(1/2) + 16*
c^3*e^2*(d + e*x)^(1/2))*((e*(-(4*a*c - b^2)^3)^(1/2) - b^3*e - 8*a*c^2*d + 2*b^2*c*d + 4*a*b*c*e)/(2*(a*b^4*e
^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e - 8*a*b^2*c^2*d^2 - 8*a^2*b^2*c*e^2 - 16*a^2*b*c^2*
d*e + 8*a*b^3*c*d*e)))^(1/2)*1i)/((((d + e*x)^(1/2)*((e*(-(4*a*c - b^2)^3)^(1/2) - b^3*e - 8*a*c^2*d + 2*b^2*c
*d + 4*a*b*c*e)/(2*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e - 8*a*b^2*c^2*d^2 - 8*a^
2*b^2*c*e^2 - 16*a^2*b*c^2*d*e + 8*a*b^3*c*d*e)))^(1/2)*(8*b^3*c^2*e^3 - 16*b^2*c^3*d*e^2 - 32*a*b*c^3*e^3 + 6
4*a*c^4*d*e^2) - 32*a*c^3*e^3 + 8*b^2*c^2*e^3)*((e*(-(4*a*c - b^2)^3)^(1/2) - b^3*e - 8*a*c^2*d + 2*b^2*c*d +
4*a*b*c*e)/(2*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e - 8*a*b^2*c^2*d^2 - 8*a^2*b^2
*c*e^2 - 16*a^2*b*c^2*d*e + 8*a*b^3*c*d*e)))^(1/2) + 16*c^3*e^2*(d + e*x)^(1/2))*((e*(-(4*a*c - b^2)^3)^(1/2)
- b^3*e - 8*a*c^2*d + 2*b^2*c*d + 4*a*b*c*e)/(2*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5
*d*e - 8*a*b^2*c^2*d^2 - 8*a^2*b^2*c*e^2 - 16*a^2*b*c^2*d*e + 8*a*b^3*c*d*e)))^(1/2) - ((32*a*c^3*e^3 + (d + e
*x)^(1/2)*((e*(-(4*a*c - b^2)^3)^(1/2) - b^3*e - 8*a*c^2*d + 2*b^2*c*d + 4*a*b*c*e)/(2*(a*b^4*e^2 + b^4*c*d^2
+ 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e - 8*a*b^2*c^2*d^2 - 8*a^2*b^2*c*e^2 - 16*a^2*b*c^2*d*e + 8*a*b^3*c
*d*e)))^(1/2)*(8*b^3*c^2*e^3 - 16*b^2*c^3*d*e^2 - 32*a*b*c^3*e^3 + 64*a*c^4*d*e^2) - 8*b^2*c^2*e^3)*((e*(-(4*a
*c - b^2)^3)^(1/2) - b^3*e - 8*a*c^2*d + 2*b^2*c*d + 4*a*b*c*e)/(2*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 1
6*a^3*c^2*e^2 - b^5*d*e - 8*a*b^2*c^2*d^2 - 8*a^2*b^2*c*e^2 - 16*a^2*b*c^2*d*e + 8*a*b^3*c*d*e)))^(1/2) + 16*c
^3*e^2*(d + e*x)^(1/2))*((e*(-(4*a*c - b^2)^3)^(1/2) - b^3*e - 8*a*c^2*d + 2*b^2*c*d + 4*a*b*c*e)/(2*(a*b^4*e^
2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e - 8*a*b^2*c^2*d^2 - 8*a^2*b^2*c*e^2 - 16*a^2*b*c^2*d
*e + 8*a*b^3*c*d*e)))^(1/2)))*((e*(-(4*a*c - b^2)^3)^(1/2) - b^3*e - 8*a*c^2*d + 2*b^2*c*d + 4*a*b*c*e)/(2*(a*
b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e - 8*a*b^2*c^2*d^2 - 8*a^2*b^2*c*e^2 - 16*a^2*b
*c^2*d*e + 8*a*b^3*c*d*e)))^(1/2)*2i

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {d + e x} \left (a + b x + c x^{2}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________