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

Optimal. Leaf size=287 \[ -\frac {(b+2 c x) \sqrt {d+e x}}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\sqrt {2} \sqrt {c} \left (4 c d-e \left (2 b-\sqrt {b^2-4 a c}\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {\sqrt {2} \sqrt {c} \left (4 c d-e \left (\sqrt {b^2-4 a c}+2 b\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \]

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Rubi [A]  time = 0.67, antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {736, 826, 1166, 208} \begin {gather*} -\frac {(b+2 c x) \sqrt {d+e x}}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\sqrt {2} \sqrt {c} \left (4 c d-e \left (2 b-\sqrt {b^2-4 a c}\right )\right ) \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 )}{\left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {\sqrt {2} \sqrt {c} \left (4 c d-e \left (\sqrt {b^2-4 a c}+2 b\right )\right ) \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 )}{\left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(((b + 2*c*x)*Sqrt[d + e*x])/((b^2 - 4*a*c)*(a + b*x + c*x^2))) + (Sqrt[2]*Sqrt[c]*(4*c*d - (2*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]])/((b^2 - 4*a*c)^(
3/2)*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) - (Sqrt[2]*Sqrt[c]*(4*c*d - (2*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]])/((b^2 - 4*a*c)^(3/2)*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 736

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^m*(b + 2*
c*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m
- 1)*(b*e*m + 2*c*d*(2*p + 3) + 2*c*e*(m + 2*p + 3)*x)*(a + b*x + c*x^2)^(p + 1), 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] && LtQ[p, -1] && GtQ[m
, 0] && (LtQ[m, 1] || (ILtQ[m + 2*p + 3, 0] && NeQ[m, 2])) && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 826

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(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, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 1166

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

Rubi steps

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

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Mathematica [A]  time = 0.59, size = 283, normalized size = 0.99 \begin {gather*} \frac {-\frac {\sqrt {2} \sqrt {c} \left (e \left (\sqrt {b^2-4 a c}-2 b\right )+4 c d\right ) \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 {b^2-4 a c} \sqrt {e \left (\sqrt {b^2-4 a c}-b\right )+2 c d}}+\frac {\sqrt {2} \sqrt {c} \left (4 c d-e \left (\sqrt {b^2-4 a c}+2 b\right )\right ) \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 {(b+2 c x) \sqrt {d+e x}}{a+x (b+c x)}}{4 a c-b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [C]  time = 1.88, size = 417, normalized size = 1.45 \begin {gather*} \frac {\left (-\sqrt {2} \sqrt {c} e \sqrt {4 a c-b^2}-2 i \sqrt {2} b \sqrt {c} e+4 i \sqrt {2} c^{3/2} d\right ) \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 )}{\left (b^2-4 a c\right ) \sqrt {4 a c-b^2} \sqrt {-i e \sqrt {4 a c-b^2}+b e-2 c d}}+\frac {\left (-\sqrt {2} \sqrt {c} e \sqrt {4 a c-b^2}+2 i \sqrt {2} b \sqrt {c} e-4 i \sqrt {2} c^{3/2} d\right ) \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 )}{\left (b^2-4 a c\right ) \sqrt {4 a c-b^2} \sqrt {i e \sqrt {4 a c-b^2}+b e-2 c d}}-\frac {e \sqrt {d+e x} (b e+2 c (d+e x)-2 c d)}{\left (b^2-4 a c\right ) \left (a e^2+b e (d+e x)-b d e+c d^2-2 c d (d+e x)+c (d+e x)^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((e*Sqrt[d + e*x]*(-2*c*d + b*e + 2*c*(d + e*x)))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2 - 2*c*d*(d + e*x) + b
*e*(d + e*x) + c*(d + e*x)^2))) + (((4*I)*Sqrt[2]*c^(3/2)*d - (2*I)*Sqrt[2]*b*Sqrt[c]*e - Sqrt[2]*Sqrt[c]*Sqrt
[-b^2 + 4*a*c]*e)*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e - I*Sqrt[-b^2 + 4*a*c]*e]])/((b^2 -
4*a*c)*Sqrt[-b^2 + 4*a*c]*Sqrt[-2*c*d + b*e - I*Sqrt[-b^2 + 4*a*c]*e]) + (((-4*I)*Sqrt[2]*c^(3/2)*d + (2*I)*S
qrt[2]*b*Sqrt[c]*e - Sqrt[2]*Sqrt[c]*Sqrt[-b^2 + 4*a*c]*e)*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d
+ b*e + I*Sqrt[-b^2 + 4*a*c]*e]])/((b^2 - 4*a*c)*Sqrt[-b^2 + 4*a*c]*Sqrt[-2*c*d + b*e + I*Sqrt[-b^2 + 4*a*c]*e
])

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fricas [B]  time = 0.58, size = 6393, normalized size = 22.28

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(sqrt(1/2)*(a*b^2 - 4*a^2*c + (b^2*c - 4*a*c^2)*x^2 + (b^3 - 4*a*b*c)*x)*sqrt((32*c^3*d^3 - 48*b*c^2*d^2*e
+ 6*(3*b^2*c + 4*a*c^2)*d*e^2 - (b^3 + 12*a*b*c)*e^3 + sqrt(e^6/((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 6
4*a^3*c^5)*d^4 - 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^3*e + (b^8 - 10*a*b^6*c + 24*a^2*b
^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^2*e^2 - 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d*e^
3 + (a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3)*e^4))*((b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*
a^3*c^4)*d^2 - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*d*e + (a*b^6 - 12*a^2*b^4*c + 48*a^3*b^2*c^2
- 64*a^4*c^3)*e^2))/((b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d^2 - (b^7 - 12*a*b^5*c + 48*a^2*b^
3*c^2 - 64*a^3*b*c^3)*d*e + (a*b^6 - 12*a^2*b^4*c + 48*a^3*b^2*c^2 - 64*a^4*c^3)*e^2))*log(sqrt(1/2)*(2*(b^4*c
- 8*a*b^2*c^2 + 16*a^2*c^3)*d*e^4 - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*e^5 - (8*(b^6*c^3 - 12*a*b^4*c^4 + 48*a^
2*b^2*c^5 - 64*a^3*c^6)*d^4 - 16*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*d^3*e + 3*(3*b^8*c -
32*a*b^6*c^2 + 96*a^2*b^4*c^3 - 256*a^4*c^5)*d^2*e^2 - (b^9 - 96*a^2*b^5*c^2 + 512*a^3*b^3*c^3 - 768*a^4*b*c^
4)*d*e^3 + (a*b^8 - 8*a^2*b^6*c + 128*a^4*b^2*c^3 - 256*a^5*c^4)*e^4)*sqrt(e^6/((b^6*c^2 - 12*a*b^4*c^3 + 48*a
^2*b^2*c^4 - 64*a^3*c^5)*d^4 - 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^3*e + (b^8 - 10*a*b^
6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^2*e^2 - 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a
^4*b*c^3)*d*e^3 + (a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3)*e^4)))*sqrt((32*c^3*d^3 - 48*b*c^2*d^
2*e + 6*(3*b^2*c + 4*a*c^2)*d*e^2 - (b^3 + 12*a*b*c)*e^3 + sqrt(e^6/((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4
- 64*a^3*c^5)*d^4 - 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^3*e + (b^8 - 10*a*b^6*c + 24*a^
2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^2*e^2 - 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d
*e^3 + (a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3)*e^4))*((b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 -
64*a^3*c^4)*d^2 - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*d*e + (a*b^6 - 12*a^2*b^4*c + 48*a^3*b^2*
c^2 - 64*a^4*c^3)*e^2))/((b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d^2 - (b^7 - 12*a*b^5*c + 48*a^2
*b^3*c^2 - 64*a^3*b*c^3)*d*e + (a*b^6 - 12*a^2*b^4*c + 48*a^3*b^2*c^2 - 64*a^4*c^3)*e^2)) + 2*(16*c^3*d^2*e^3
- 16*b*c^2*d*e^4 + (3*b^2*c + 4*a*c^2)*e^5)*sqrt(e*x + d)) - sqrt(1/2)*(a*b^2 - 4*a^2*c + (b^2*c - 4*a*c^2)*x^
2 + (b^3 - 4*a*b*c)*x)*sqrt((32*c^3*d^3 - 48*b*c^2*d^2*e + 6*(3*b^2*c + 4*a*c^2)*d*e^2 - (b^3 + 12*a*b*c)*e^3
+ sqrt(e^6/((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^4 - 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*
c^3 - 64*a^3*b*c^4)*d^3*e + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^2*e^2 - 2*(a*
b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d*e^3 + (a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*
c^3)*e^4))*((b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d^2 - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64
*a^3*b*c^3)*d*e + (a*b^6 - 12*a^2*b^4*c + 48*a^3*b^2*c^2 - 64*a^4*c^3)*e^2))/((b^6*c - 12*a*b^4*c^2 + 48*a^2*b
^2*c^3 - 64*a^3*c^4)*d^2 - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*d*e + (a*b^6 - 12*a^2*b^4*c + 48
*a^3*b^2*c^2 - 64*a^4*c^3)*e^2))*log(-sqrt(1/2)*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d*e^4 - (b^5 - 8*a*b^3*c
+ 16*a^2*b*c^2)*e^5 - (8*(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*d^4 - 16*(b^7*c^2 - 12*a*b^5*
c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*d^3*e + 3*(3*b^8*c - 32*a*b^6*c^2 + 96*a^2*b^4*c^3 - 256*a^4*c^5)*d^2*e^2
- (b^9 - 96*a^2*b^5*c^2 + 512*a^3*b^3*c^3 - 768*a^4*b*c^4)*d*e^3 + (a*b^8 - 8*a^2*b^6*c + 128*a^4*b^2*c^3 - 2
56*a^5*c^4)*e^4)*sqrt(e^6/((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^4 - 2*(b^7*c - 12*a*b^5*c^
2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^3*e + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*
d^2*e^2 - 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d*e^3 + (a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^
2*c^2 - 64*a^5*c^3)*e^4)))*sqrt((32*c^3*d^3 - 48*b*c^2*d^2*e + 6*(3*b^2*c + 4*a*c^2)*d*e^2 - (b^3 + 12*a*b*c)*
e^3 + sqrt(e^6/((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^4 - 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*
b^3*c^3 - 64*a^3*b*c^4)*d^3*e + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^2*e^2 - 2
*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d*e^3 + (a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*
a^5*c^3)*e^4))*((b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d^2 - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2
- 64*a^3*b*c^3)*d*e + (a*b^6 - 12*a^2*b^4*c + 48*a^3*b^2*c^2 - 64*a^4*c^3)*e^2))/((b^6*c - 12*a*b^4*c^2 + 48*a
^2*b^2*c^3 - 64*a^3*c^4)*d^2 - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*d*e + (a*b^6 - 12*a^2*b^4*c
+ 48*a^3*b^2*c^2 - 64*a^4*c^3)*e^2)) + 2*(16*c^3*d^2*e^3 - 16*b*c^2*d*e^4 + (3*b^2*c + 4*a*c^2)*e^5)*sqrt(e*x
+ d)) + sqrt(1/2)*(a*b^2 - 4*a^2*c + (b^2*c - 4*a*c^2)*x^2 + (b^3 - 4*a*b*c)*x)*sqrt((32*c^3*d^3 - 48*b*c^2*d^
2*e + 6*(3*b^2*c + 4*a*c^2)*d*e^2 - (b^3 + 12*a*b*c)*e^3 - sqrt(e^6/((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4
- 64*a^3*c^5)*d^4 - 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^3*e + (b^8 - 10*a*b^6*c + 24*a^
2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^2*e^2 - 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d
*e^3 + (a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3)*e^4))*((b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 -
64*a^3*c^4)*d^2 - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*d*e + (a*b^6 - 12*a^2*b^4*c + 48*a^3*b^2*
c^2 - 64*a^4*c^3)*e^2))/((b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d^2 - (b^7 - 12*a*b^5*c + 48*a^2
*b^3*c^2 - 64*a^3*b*c^3)*d*e + (a*b^6 - 12*a^2*b^4*c + 48*a^3*b^2*c^2 - 64*a^4*c^3)*e^2))*log(sqrt(1/2)*(2*(b^
4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d*e^4 - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*e^5 + (8*(b^6*c^3 - 12*a*b^4*c^4 + 48
*a^2*b^2*c^5 - 64*a^3*c^6)*d^4 - 16*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*d^3*e + 3*(3*b^8*
c - 32*a*b^6*c^2 + 96*a^2*b^4*c^3 - 256*a^4*c^5)*d^2*e^2 - (b^9 - 96*a^2*b^5*c^2 + 512*a^3*b^3*c^3 - 768*a^4*b
*c^4)*d*e^3 + (a*b^8 - 8*a^2*b^6*c + 128*a^4*b^2*c^3 - 256*a^5*c^4)*e^4)*sqrt(e^6/((b^6*c^2 - 12*a*b^4*c^3 + 4
8*a^2*b^2*c^4 - 64*a^3*c^5)*d^4 - 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^3*e + (b^8 - 10*a
*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^2*e^2 - 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 6
4*a^4*b*c^3)*d*e^3 + (a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3)*e^4)))*sqrt((32*c^3*d^3 - 48*b*c^2
*d^2*e + 6*(3*b^2*c + 4*a*c^2)*d*e^2 - (b^3 + 12*a*b*c)*e^3 - sqrt(e^6/((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c
^4 - 64*a^3*c^5)*d^4 - 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^3*e + (b^8 - 10*a*b^6*c + 24
*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^2*e^2 - 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3
)*d*e^3 + (a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3)*e^4))*((b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3
- 64*a^3*c^4)*d^2 - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*d*e + (a*b^6 - 12*a^2*b^4*c + 48*a^3*b
^2*c^2 - 64*a^4*c^3)*e^2))/((b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d^2 - (b^7 - 12*a*b^5*c + 48*
a^2*b^3*c^2 - 64*a^3*b*c^3)*d*e + (a*b^6 - 12*a^2*b^4*c + 48*a^3*b^2*c^2 - 64*a^4*c^3)*e^2)) + 2*(16*c^3*d^2*e
^3 - 16*b*c^2*d*e^4 + (3*b^2*c + 4*a*c^2)*e^5)*sqrt(e*x + d)) - sqrt(1/2)*(a*b^2 - 4*a^2*c + (b^2*c - 4*a*c^2)
*x^2 + (b^3 - 4*a*b*c)*x)*sqrt((32*c^3*d^3 - 48*b*c^2*d^2*e + 6*(3*b^2*c + 4*a*c^2)*d*e^2 - (b^3 + 12*a*b*c)*e
^3 - sqrt(e^6/((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^4 - 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b
^3*c^3 - 64*a^3*b*c^4)*d^3*e + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^2*e^2 - 2*
(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d*e^3 + (a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a
^5*c^3)*e^4))*((b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d^2 - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 -
64*a^3*b*c^3)*d*e + (a*b^6 - 12*a^2*b^4*c + 48*a^3*b^2*c^2 - 64*a^4*c^3)*e^2))/((b^6*c - 12*a*b^4*c^2 + 48*a^
2*b^2*c^3 - 64*a^3*c^4)*d^2 - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*d*e + (a*b^6 - 12*a^2*b^4*c +
48*a^3*b^2*c^2 - 64*a^4*c^3)*e^2))*log(-sqrt(1/2)*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d*e^4 - (b^5 - 8*a*b^
3*c + 16*a^2*b*c^2)*e^5 + (8*(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*d^4 - 16*(b^7*c^2 - 12*a*b
^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*d^3*e + 3*(3*b^8*c - 32*a*b^6*c^2 + 96*a^2*b^4*c^3 - 256*a^4*c^5)*d^2*
e^2 - (b^9 - 96*a^2*b^5*c^2 + 512*a^3*b^3*c^3 - 768*a^4*b*c^4)*d*e^3 + (a*b^8 - 8*a^2*b^6*c + 128*a^4*b^2*c^3
- 256*a^5*c^4)*e^4)*sqrt(e^6/((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^4 - 2*(b^7*c - 12*a*b^5
*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^3*e + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^
4)*d^2*e^2 - 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d*e^3 + (a^2*b^6 - 12*a^3*b^4*c + 48*a^4
*b^2*c^2 - 64*a^5*c^3)*e^4)))*sqrt((32*c^3*d^3 - 48*b*c^2*d^2*e + 6*(3*b^2*c + 4*a*c^2)*d*e^2 - (b^3 + 12*a*b*
c)*e^3 - sqrt(e^6/((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^4 - 2*(b^7*c - 12*a*b^5*c^2 + 48*a
^2*b^3*c^3 - 64*a^3*b*c^4)*d^3*e + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^2*e^2
- 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d*e^3 + (a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 -
64*a^5*c^3)*e^4))*((b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d^2 - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c
^2 - 64*a^3*b*c^3)*d*e + (a*b^6 - 12*a^2*b^4*c + 48*a^3*b^2*c^2 - 64*a^4*c^3)*e^2))/((b^6*c - 12*a*b^4*c^2 + 4
8*a^2*b^2*c^3 - 64*a^3*c^4)*d^2 - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*d*e + (a*b^6 - 12*a^2*b^4
*c + 48*a^3*b^2*c^2 - 64*a^4*c^3)*e^2)) + 2*(16*c^3*d^2*e^3 - 16*b*c^2*d*e^4 + (3*b^2*c + 4*a*c^2)*e^5)*sqrt(e
*x + d)) - 2*(2*c*x + b)*sqrt(e*x + d))/(a*b^2 - 4*a^2*c + (b^2*c - 4*a*c^2)*x^2 + (b^3 - 4*a*b*c)*x)

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giac [B]  time = 1.28, size = 1041, normalized size = 3.63 \begin {gather*} -\frac {2 \, {\left (x e + d\right )}^{\frac {3}{2}} c e - 2 \, \sqrt {x e + d} c d e + \sqrt {x e + d} b e^{2}}{{\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} + {\left (x e + d\right )} b e - b d e + a e^{2}\right )} {\left (b^{2} - 4 \, a c\right )}} - \frac {{\left (\sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left (b^{2} e - 4 \, a c e\right )}^{2} e + {\left (2 \, \sqrt {b^{2} - 4 \, a c} c d e - \sqrt {b^{2} - 4 \, a c} b e^{2}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left | b^{2} e - 4 \, a c e \right |} - 2 \, {\left (4 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{2} e - 4 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} d e^{2} + {\left (b^{4} - 4 \, a b^{2} c\right )} e^{3}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x e + d}}{\sqrt {-\frac {2 \, b^{2} c d - 8 \, a c^{2} d - b^{3} e + 4 \, a b c e + \sqrt {{\left (2 \, b^{2} c d - 8 \, a c^{2} d - b^{3} e + 4 \, a b c e\right )}^{2} - 4 \, {\left (b^{2} c d^{2} - 4 \, a c^{2} d^{2} - b^{3} d e + 4 \, a b c d e + a b^{2} e^{2} - 4 \, a^{2} c e^{2}\right )} {\left (b^{2} c - 4 \, a c^{2}\right )}}}{b^{2} c - 4 \, a c^{2}}}}\right )}{4 \, {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {b^{2} - 4 \, a c} d^{2} - {\left (b^{3} - 4 \, a b c\right )} \sqrt {b^{2} - 4 \, a c} d e + {\left (a b^{2} - 4 \, a^{2} c\right )} \sqrt {b^{2} - 4 \, a c} e^{2}\right )} {\left | b^{2} e - 4 \, a c e \right |} {\left | c \right |}} + \frac {{\left (\sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left (b^{2} e - 4 \, a c e\right )}^{2} e - {\left (2 \, \sqrt {b^{2} - 4 \, a c} c d e - \sqrt {b^{2} - 4 \, a c} b e^{2}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left | b^{2} e - 4 \, a c e \right |} - 2 \, {\left (4 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{2} e - 4 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} d e^{2} + {\left (b^{4} - 4 \, a b^{2} c\right )} e^{3}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x e + d}}{\sqrt {-\frac {2 \, b^{2} c d - 8 \, a c^{2} d - b^{3} e + 4 \, a b c e - \sqrt {{\left (2 \, b^{2} c d - 8 \, a c^{2} d - b^{3} e + 4 \, a b c e\right )}^{2} - 4 \, {\left (b^{2} c d^{2} - 4 \, a c^{2} d^{2} - b^{3} d e + 4 \, a b c d e + a b^{2} e^{2} - 4 \, a^{2} c e^{2}\right )} {\left (b^{2} c - 4 \, a c^{2}\right )}}}{b^{2} c - 4 \, a c^{2}}}}\right )}{4 \, {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {b^{2} - 4 \, a c} d^{2} - {\left (b^{3} - 4 \, a b c\right )} \sqrt {b^{2} - 4 \, a c} d e + {\left (a b^{2} - 4 \, a^{2} c\right )} \sqrt {b^{2} - 4 \, a c} e^{2}\right )} {\left | b^{2} e - 4 \, a c e \right |} {\left | c \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.34, size = 869, normalized size = 3.03 \begin {gather*} \frac {2 \sqrt {2}\, b c \,e^{2} \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 )}{\left (4 a c -b^{2}\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}\, b c \,e^{2} \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 )}{\left (4 a c -b^{2}\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 {4 \sqrt {2}\, c^{2} d 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 )}{\left (4 a c -b^{2}\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 {4 \sqrt {2}\, c^{2} d 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 )}{\left (4 a c -b^{2}\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 {\sqrt {2}\, \sqrt {-4 a c \,e^{2}+b^{2} e^{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 )}{\left (4 a c -b^{2}\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 {\sqrt {2}\, \sqrt {-4 a c \,e^{2}+b^{2} e^{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 )}{\left (4 a c -b^{2}\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 {\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, \sqrt {e x +d}\, e}{\left (4 a c -b^{2}\right ) \sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\, \left (e x +\frac {b e}{2 c}-\frac {\sqrt {\left (-4 a c +b^{2}\right ) e^{2}}}{2 c}\right )}+\frac {\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, \sqrt {e x +d}\, e}{\left (4 a c -b^{2}\right ) \sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\, \left (e x +\frac {b e}{2 c}+\frac {\sqrt {\left (-4 a c +b^{2}\right ) e^{2}}}{2 c}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 7.61, size = 5740, normalized size = 20.00

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((2*c*e*(d + e*x)^(3/2))/(4*a*c - b^2) + (e*(b*e - 2*c*d)*(d + e*x)^(1/2))/(4*a*c - b^2))/((b*e - 2*c*d)*(d +
e*x) + c*(d + e*x)^2 + a*e^2 + c*d^2 - b*d*e) - log(((4*c^2*e^3*(b*e - 2*c*d) - 8*c^2*e^2*(4*a*c - b^2)*(b*e -
2*c*d)*(d + e*x)^(1/2)*(((e^3*(-(4*a*c - b^2)^9)^(1/2))/8 - (b^9*e^3)/8 - 256*a^3*c^6*d^3 + 4*b^6*c^3*d^3 - 4
8*a*b^4*c^4*d^3 + 96*a^4*b*c^4*e^3 - 192*a^4*c^5*d*e^2 - 6*b^7*c^2*d^2*e + 192*a^2*b^2*c^5*d^3 + 12*a^2*b^5*c^
2*e^3 - 64*a^3*b^3*c^3*e^3 + (9*b^8*c*d*e^2)/4 + 72*a*b^5*c^3*d^2*e - 24*a*b^6*c^2*d*e^2 + 384*a^3*b*c^5*d^2*e
- 288*a^2*b^3*c^4*d^2*e + 72*a^2*b^4*c^3*d*e^2)/((4*a*c - b^2)^6*(a*e^2 + c*d^2 - b*d*e)))^(1/2))*(((e^3*(-(4
*a*c - b^2)^9)^(1/2))/8 - (b^9*e^3)/8 - 256*a^3*c^6*d^3 + 4*b^6*c^3*d^3 - 48*a*b^4*c^4*d^3 + 96*a^4*b*c^4*e^3
- 192*a^4*c^5*d*e^2 - 6*b^7*c^2*d^2*e + 192*a^2*b^2*c^5*d^3 + 12*a^2*b^5*c^2*e^3 - 64*a^3*b^3*c^3*e^3 + (9*b^8
*c*d*e^2)/4 + 72*a*b^5*c^3*d^2*e - 24*a*b^6*c^2*d*e^2 + 384*a^3*b*c^5*d^2*e - 288*a^2*b^3*c^4*d^2*e + 72*a^2*b
^4*c^3*d*e^2)/((4*a*c - b^2)^6*(a*e^2 + c*d^2 - b*d*e)))^(1/2) + (4*c^3*e^2*(d + e*x)^(1/2)*(5*b^2*e^2 + 16*c^
2*d^2 - 4*a*c*e^2 - 16*b*c*d*e))/(4*a*c - b^2)^2)*(((e^3*(-(4*a*c - b^2)^9)^(1/2))/8 - (b^9*e^3)/8 - 256*a^3*c
^6*d^3 + 4*b^6*c^3*d^3 - 48*a*b^4*c^4*d^3 + 96*a^4*b*c^4*e^3 - 192*a^4*c^5*d*e^2 - 6*b^7*c^2*d^2*e + 192*a^2*b
^2*c^5*d^3 + 12*a^2*b^5*c^2*e^3 - 64*a^3*b^3*c^3*e^3 + (9*b^8*c*d*e^2)/4 + 72*a*b^5*c^3*d^2*e - 24*a*b^6*c^2*d
*e^2 + 384*a^3*b*c^5*d^2*e - 288*a^2*b^3*c^4*d^2*e + 72*a^2*b^4*c^3*d*e^2)/((4*a*c - b^2)^6*(a*e^2 + c*d^2 - b
*d*e)))^(1/2) - (2*c^3*e^3*(3*b^2*e^2 + 16*c^2*d^2 + 4*a*c*e^2 - 16*b*c*d*e))/(4*a*c - b^2)^3)*(((e^3*(-(4*a*c
- b^2)^9)^(1/2))/8 - (b^9*e^3)/8 - 256*a^3*c^6*d^3 + 4*b^6*c^3*d^3 - 48*a*b^4*c^4*d^3 + 96*a^4*b*c^4*e^3 - 19
2*a^4*c^5*d*e^2 - 6*b^7*c^2*d^2*e + 192*a^2*b^2*c^5*d^3 + 12*a^2*b^5*c^2*e^3 - 64*a^3*b^3*c^3*e^3 + (9*b^8*c*d
*e^2)/4 + 72*a*b^5*c^3*d^2*e - 24*a*b^6*c^2*d*e^2 + 384*a^3*b*c^5*d^2*e - 288*a^2*b^3*c^4*d^2*e + 72*a^2*b^4*c
^3*d*e^2)/(a*b^12*e^2 + b^12*c*d^2 + 4096*a^6*c^7*d^2 + 4096*a^7*c^6*e^2 - b^13*d*e - 24*a*b^10*c^2*d^2 - 24*a
^2*b^10*c*e^2 + 240*a^2*b^8*c^3*d^2 - 1280*a^3*b^6*c^4*d^2 + 3840*a^4*b^4*c^5*d^2 - 6144*a^5*b^2*c^6*d^2 + 240
*a^3*b^8*c^2*e^2 - 1280*a^4*b^6*c^3*e^2 + 3840*a^5*b^4*c^4*e^2 - 6144*a^6*b^2*c^5*e^2 - 4096*a^6*b*c^6*d*e - 2
40*a^2*b^9*c^2*d*e + 1280*a^3*b^7*c^3*d*e - 3840*a^4*b^5*c^4*d*e + 6144*a^5*b^3*c^5*d*e + 24*a*b^11*c*d*e))^(1
/2) - log(((4*c^2*e^3*(b*e - 2*c*d) - 8*c^2*e^2*(4*a*c - b^2)*(b*e - 2*c*d)*(d + e*x)^(1/2)*(-((b^9*e^3)/8 + (
e^3*(-(4*a*c - b^2)^9)^(1/2))/8 + 256*a^3*c^6*d^3 - 4*b^6*c^3*d^3 + 48*a*b^4*c^4*d^3 - 96*a^4*b*c^4*e^3 + 192*
a^4*c^5*d*e^2 + 6*b^7*c^2*d^2*e - 192*a^2*b^2*c^5*d^3 - 12*a^2*b^5*c^2*e^3 + 64*a^3*b^3*c^3*e^3 - (9*b^8*c*d*e
^2)/4 - 72*a*b^5*c^3*d^2*e + 24*a*b^6*c^2*d*e^2 - 384*a^3*b*c^5*d^2*e + 288*a^2*b^3*c^4*d^2*e - 72*a^2*b^4*c^3
*d*e^2)/((4*a*c - b^2)^6*(a*e^2 + c*d^2 - b*d*e)))^(1/2))*(-((b^9*e^3)/8 + (e^3*(-(4*a*c - b^2)^9)^(1/2))/8 +
256*a^3*c^6*d^3 - 4*b^6*c^3*d^3 + 48*a*b^4*c^4*d^3 - 96*a^4*b*c^4*e^3 + 192*a^4*c^5*d*e^2 + 6*b^7*c^2*d^2*e -
192*a^2*b^2*c^5*d^3 - 12*a^2*b^5*c^2*e^3 + 64*a^3*b^3*c^3*e^3 - (9*b^8*c*d*e^2)/4 - 72*a*b^5*c^3*d^2*e + 24*a*
b^6*c^2*d*e^2 - 384*a^3*b*c^5*d^2*e + 288*a^2*b^3*c^4*d^2*e - 72*a^2*b^4*c^3*d*e^2)/((4*a*c - b^2)^6*(a*e^2 +
c*d^2 - b*d*e)))^(1/2) + (4*c^3*e^2*(d + e*x)^(1/2)*(5*b^2*e^2 + 16*c^2*d^2 - 4*a*c*e^2 - 16*b*c*d*e))/(4*a*c
- b^2)^2)*(-((b^9*e^3)/8 + (e^3*(-(4*a*c - b^2)^9)^(1/2))/8 + 256*a^3*c^6*d^3 - 4*b^6*c^3*d^3 + 48*a*b^4*c^4*d
^3 - 96*a^4*b*c^4*e^3 + 192*a^4*c^5*d*e^2 + 6*b^7*c^2*d^2*e - 192*a^2*b^2*c^5*d^3 - 12*a^2*b^5*c^2*e^3 + 64*a^
3*b^3*c^3*e^3 - (9*b^8*c*d*e^2)/4 - 72*a*b^5*c^3*d^2*e + 24*a*b^6*c^2*d*e^2 - 384*a^3*b*c^5*d^2*e + 288*a^2*b^
3*c^4*d^2*e - 72*a^2*b^4*c^3*d*e^2)/((4*a*c - b^2)^6*(a*e^2 + c*d^2 - b*d*e)))^(1/2) - (2*c^3*e^3*(3*b^2*e^2 +
16*c^2*d^2 + 4*a*c*e^2 - 16*b*c*d*e))/(4*a*c - b^2)^3)*(-((b^9*e^3)/8 + (e^3*(-(4*a*c - b^2)^9)^(1/2))/8 + 25
6*a^3*c^6*d^3 - 4*b^6*c^3*d^3 + 48*a*b^4*c^4*d^3 - 96*a^4*b*c^4*e^3 + 192*a^4*c^5*d*e^2 + 6*b^7*c^2*d^2*e - 19
2*a^2*b^2*c^5*d^3 - 12*a^2*b^5*c^2*e^3 + 64*a^3*b^3*c^3*e^3 - (9*b^8*c*d*e^2)/4 - 72*a*b^5*c^3*d^2*e + 24*a*b^
6*c^2*d*e^2 - 384*a^3*b*c^5*d^2*e + 288*a^2*b^3*c^4*d^2*e - 72*a^2*b^4*c^3*d*e^2)/(a*b^12*e^2 + b^12*c*d^2 + 4
096*a^6*c^7*d^2 + 4096*a^7*c^6*e^2 - b^13*d*e - 24*a*b^10*c^2*d^2 - 24*a^2*b^10*c*e^2 + 240*a^2*b^8*c^3*d^2 -
1280*a^3*b^6*c^4*d^2 + 3840*a^4*b^4*c^5*d^2 - 6144*a^5*b^2*c^6*d^2 + 240*a^3*b^8*c^2*e^2 - 1280*a^4*b^6*c^3*e^
2 + 3840*a^5*b^4*c^4*e^2 - 6144*a^6*b^2*c^5*e^2 - 4096*a^6*b*c^6*d*e - 240*a^2*b^9*c^2*d*e + 1280*a^3*b^7*c^3*
d*e - 3840*a^4*b^5*c^4*d*e + 6144*a^5*b^3*c^5*d*e + 24*a*b^11*c*d*e))^(1/2) + log((2^(1/2)*((2^(1/2)*(4*c^2*e^
3*(b*e - 2*c*d) + 2*2^(1/2)*c^2*e^2*(4*a*c - b^2)*(b*e - 2*c*d)*(d + e*x)^(1/2)*(-(b^9*e^3 + e^3*(-(4*a*c - b^
2)^9)^(1/2) + 2048*a^3*c^6*d^3 - 32*b^6*c^3*d^3 + 384*a*b^4*c^4*d^3 - 768*a^4*b*c^4*e^3 + 1536*a^4*c^5*d*e^2 +
48*b^7*c^2*d^2*e - 1536*a^2*b^2*c^5*d^3 - 96*a^2*b^5*c^2*e^3 + 512*a^3*b^3*c^3*e^3 - 18*b^8*c*d*e^2 - 576*a*b
^5*c^3*d^2*e + 192*a*b^6*c^2*d*e^2 - 3072*a^3*b*c^5*d^2*e + 2304*a^2*b^3*c^4*d^2*e - 576*a^2*b^4*c^3*d*e^2)/((
4*a*c - b^2)^6*(a*e^2 + c*d^2 - b*d*e)))^(1/2))*(-(b^9*e^3 + e^3*(-(4*a*c - b^2)^9)^(1/2) + 2048*a^3*c^6*d^3 -
32*b^6*c^3*d^3 + 384*a*b^4*c^4*d^3 - 768*a^4*b*c^4*e^3 + 1536*a^4*c^5*d*e^2 + 48*b^7*c^2*d^2*e - 1536*a^2*b^2
*c^5*d^3 - 96*a^2*b^5*c^2*e^3 + 512*a^3*b^3*c^3*e^3 - 18*b^8*c*d*e^2 - 576*a*b^5*c^3*d^2*e + 192*a*b^6*c^2*d*e
^2 - 3072*a^3*b*c^5*d^2*e + 2304*a^2*b^3*c^4*d^2*e - 576*a^2*b^4*c^3*d*e^2)/((4*a*c - b^2)^6*(a*e^2 + c*d^2 -
b*d*e)))^(1/2))/4 - (4*c^3*e^2*(d + e*x)^(1/2)*(5*b^2*e^2 + 16*c^2*d^2 - 4*a*c*e^2 - 16*b*c*d*e))/(4*a*c - b^2
)^2)*(-(b^9*e^3 + e^3*(-(4*a*c - b^2)^9)^(1/2) + 2048*a^3*c^6*d^3 - 32*b^6*c^3*d^3 + 384*a*b^4*c^4*d^3 - 768*a
^4*b*c^4*e^3 + 1536*a^4*c^5*d*e^2 + 48*b^7*c^2*d^2*e - 1536*a^2*b^2*c^5*d^3 - 96*a^2*b^5*c^2*e^3 + 512*a^3*b^3
*c^3*e^3 - 18*b^8*c*d*e^2 - 576*a*b^5*c^3*d^2*e + 192*a*b^6*c^2*d*e^2 - 3072*a^3*b*c^5*d^2*e + 2304*a^2*b^3*c^
4*d^2*e - 576*a^2*b^4*c^3*d*e^2)/((4*a*c - b^2)^6*(a*e^2 + c*d^2 - b*d*e)))^(1/2))/4 - (2*c^3*e^3*(3*b^2*e^2 +
16*c^2*d^2 + 4*a*c*e^2 - 16*b*c*d*e))/(4*a*c - b^2)^3)*(-(b^9*e^3 + e^3*(-(4*a*c - b^2)^9)^(1/2) + 2048*a^3*c
^6*d^3 - 32*b^6*c^3*d^3 + 384*a*b^4*c^4*d^3 - 768*a^4*b*c^4*e^3 + 1536*a^4*c^5*d*e^2 + 48*b^7*c^2*d^2*e - 1536
*a^2*b^2*c^5*d^3 - 96*a^2*b^5*c^2*e^3 + 512*a^3*b^3*c^3*e^3 - 18*b^8*c*d*e^2 - 576*a*b^5*c^3*d^2*e + 192*a*b^6
*c^2*d*e^2 - 3072*a^3*b*c^5*d^2*e + 2304*a^2*b^3*c^4*d^2*e - 576*a^2*b^4*c^3*d*e^2)/(8*(a*b^12*e^2 + b^12*c*d^
2 + 4096*a^6*c^7*d^2 + 4096*a^7*c^6*e^2 - b^13*d*e - 24*a*b^10*c^2*d^2 - 24*a^2*b^10*c*e^2 + 240*a^2*b^8*c^3*d
^2 - 1280*a^3*b^6*c^4*d^2 + 3840*a^4*b^4*c^5*d^2 - 6144*a^5*b^2*c^6*d^2 + 240*a^3*b^8*c^2*e^2 - 1280*a^4*b^6*c
^3*e^2 + 3840*a^5*b^4*c^4*e^2 - 6144*a^6*b^2*c^5*e^2 - 4096*a^6*b*c^6*d*e - 240*a^2*b^9*c^2*d*e + 1280*a^3*b^7
*c^3*d*e - 3840*a^4*b^5*c^4*d*e + 6144*a^5*b^3*c^5*d*e + 24*a*b^11*c*d*e)))^(1/2) + log((2^(1/2)*((2^(1/2)*(4*
c^2*e^3*(b*e - 2*c*d) + 2*2^(1/2)*c^2*e^2*(4*a*c - b^2)*(b*e - 2*c*d)*(d + e*x)^(1/2)*((e^3*(-(4*a*c - b^2)^9)
^(1/2) - b^9*e^3 - 2048*a^3*c^6*d^3 + 32*b^6*c^3*d^3 - 384*a*b^4*c^4*d^3 + 768*a^4*b*c^4*e^3 - 1536*a^4*c^5*d*
e^2 - 48*b^7*c^2*d^2*e + 1536*a^2*b^2*c^5*d^3 + 96*a^2*b^5*c^2*e^3 - 512*a^3*b^3*c^3*e^3 + 18*b^8*c*d*e^2 + 57
6*a*b^5*c^3*d^2*e - 192*a*b^6*c^2*d*e^2 + 3072*a^3*b*c^5*d^2*e - 2304*a^2*b^3*c^4*d^2*e + 576*a^2*b^4*c^3*d*e^
2)/((4*a*c - b^2)^6*(a*e^2 + c*d^2 - b*d*e)))^(1/2))*((e^3*(-(4*a*c - b^2)^9)^(1/2) - b^9*e^3 - 2048*a^3*c^6*d
^3 + 32*b^6*c^3*d^3 - 384*a*b^4*c^4*d^3 + 768*a^4*b*c^4*e^3 - 1536*a^4*c^5*d*e^2 - 48*b^7*c^2*d^2*e + 1536*a^2
*b^2*c^5*d^3 + 96*a^2*b^5*c^2*e^3 - 512*a^3*b^3*c^3*e^3 + 18*b^8*c*d*e^2 + 576*a*b^5*c^3*d^2*e - 192*a*b^6*c^2
*d*e^2 + 3072*a^3*b*c^5*d^2*e - 2304*a^2*b^3*c^4*d^2*e + 576*a^2*b^4*c^3*d*e^2)/((4*a*c - b^2)^6*(a*e^2 + c*d^
2 - b*d*e)))^(1/2))/4 - (4*c^3*e^2*(d + e*x)^(1/2)*(5*b^2*e^2 + 16*c^2*d^2 - 4*a*c*e^2 - 16*b*c*d*e))/(4*a*c -
b^2)^2)*((e^3*(-(4*a*c - b^2)^9)^(1/2) - b^9*e^3 - 2048*a^3*c^6*d^3 + 32*b^6*c^3*d^3 - 384*a*b^4*c^4*d^3 + 76
8*a^4*b*c^4*e^3 - 1536*a^4*c^5*d*e^2 - 48*b^7*c^2*d^2*e + 1536*a^2*b^2*c^5*d^3 + 96*a^2*b^5*c^2*e^3 - 512*a^3*
b^3*c^3*e^3 + 18*b^8*c*d*e^2 + 576*a*b^5*c^3*d^2*e - 192*a*b^6*c^2*d*e^2 + 3072*a^3*b*c^5*d^2*e - 2304*a^2*b^3
*c^4*d^2*e + 576*a^2*b^4*c^3*d*e^2)/((4*a*c - b^2)^6*(a*e^2 + c*d^2 - b*d*e)))^(1/2))/4 - (2*c^3*e^3*(3*b^2*e^
2 + 16*c^2*d^2 + 4*a*c*e^2 - 16*b*c*d*e))/(4*a*c - b^2)^3)*((e^3*(-(4*a*c - b^2)^9)^(1/2) - b^9*e^3 - 2048*a^3
*c^6*d^3 + 32*b^6*c^3*d^3 - 384*a*b^4*c^4*d^3 + 768*a^4*b*c^4*e^3 - 1536*a^4*c^5*d*e^2 - 48*b^7*c^2*d^2*e + 15
36*a^2*b^2*c^5*d^3 + 96*a^2*b^5*c^2*e^3 - 512*a^3*b^3*c^3*e^3 + 18*b^8*c*d*e^2 + 576*a*b^5*c^3*d^2*e - 192*a*b
^6*c^2*d*e^2 + 3072*a^3*b*c^5*d^2*e - 2304*a^2*b^3*c^4*d^2*e + 576*a^2*b^4*c^3*d*e^2)/(8*(a*b^12*e^2 + b^12*c*
d^2 + 4096*a^6*c^7*d^2 + 4096*a^7*c^6*e^2 - b^13*d*e - 24*a*b^10*c^2*d^2 - 24*a^2*b^10*c*e^2 + 240*a^2*b^8*c^3
*d^2 - 1280*a^3*b^6*c^4*d^2 + 3840*a^4*b^4*c^5*d^2 - 6144*a^5*b^2*c^6*d^2 + 240*a^3*b^8*c^2*e^2 - 1280*a^4*b^6
*c^3*e^2 + 3840*a^5*b^4*c^4*e^2 - 6144*a^6*b^2*c^5*e^2 - 4096*a^6*b*c^6*d*e - 240*a^2*b^9*c^2*d*e + 1280*a^3*b
^7*c^3*d*e - 3840*a^4*b^5*c^4*d*e + 6144*a^5*b^3*c^5*d*e + 24*a*b^11*c*d*e)))^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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