3.17.15 \(\int \frac {b+2 c x}{\sqrt {d+e x} (a+b x+c x^2)} \, dx\) [1615]

Optimal. Leaf size=175 \[ -\frac {2 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {2 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.13, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {840, 1180, 214} \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 (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\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 (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

Rule 214

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

Rule 840

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 1180

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

________________________________________________________________________________________

Mathematica [A]
time = 0.53, size = 161, normalized size = 0.92 \begin {gather*} 2 \sqrt {2} \sqrt {c} \left (\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-\sqrt {b^2-4 a c} e}}\right )}{\sqrt {-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\tan ^{-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 {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*Sqrt[2]*Sqrt[c]*(ArcTan[(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] + ArcTan[(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])

________________________________________________________________________________________

Maple [A]
time = 1.10, size = 159, normalized size = 0.91

method	result	size
derivativedivides	\(8 c \left (-\frac {\sqrt {2}\, \arctanh \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )\)	\(159\)
default	\(8 c \left (-\frac {\sqrt {2}\, \arctanh \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )\)	\(159\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1289 vs. \(2 (145) = 290\).
time = 1.09, size = 1289, normalized size = 7.37 \begin {gather*} \frac {1}{2} \, \sqrt {2} \sqrt {\frac {{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt {\frac {b^{2} - 4 \, a c}{c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + a^{2} e^{4}}} e + 2 \, c d - b e}{c d^{2} - b d e + a e^{2}}} \log \left (\sqrt {2} {\left ({\left (c d^{2} - b d e + a e^{2}\right )} \sqrt {\frac {b^{2} - 4 \, a c}{c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + a^{2} e^{4}}} e - 2 \, c d + b e\right )} \sqrt {\frac {{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt {\frac {b^{2} - 4 \, a c}{c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + a^{2} e^{4}}} e + 2 \, c d - b e}{c d^{2} - b d e + a e^{2}}} + 4 \, \sqrt {x e + d} c\right ) - \frac {1}{2} \, \sqrt {2} \sqrt {\frac {{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt {\frac {b^{2} - 4 \, a c}{c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + a^{2} e^{4}}} e + 2 \, c d - b e}{c d^{2} - b d e + a e^{2}}} \log \left (-\sqrt {2} {\left ({\left (c d^{2} - b d e + a e^{2}\right )} \sqrt {\frac {b^{2} - 4 \, a c}{c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + a^{2} e^{4}}} e - 2 \, c d + b e\right )} \sqrt {\frac {{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt {\frac {b^{2} - 4 \, a c}{c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + a^{2} e^{4}}} e + 2 \, c d - b e}{c d^{2} - b d e + a e^{2}}} + 4 \, \sqrt {x e + d} c\right ) - \frac {1}{2} \, \sqrt {2} \sqrt {-\frac {{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt {\frac {b^{2} - 4 \, a c}{c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + a^{2} e^{4}}} e - 2 \, c d + b e}{c d^{2} - b d e + a e^{2}}} \log \left (\sqrt {2} {\left ({\left (c d^{2} - b d e + a e^{2}\right )} \sqrt {\frac {b^{2} - 4 \, a c}{c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + a^{2} e^{4}}} e + 2 \, c d - b e\right )} \sqrt {-\frac {{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt {\frac {b^{2} - 4 \, a c}{c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + a^{2} e^{4}}} e - 2 \, c d + b e}{c d^{2} - b d e + a e^{2}}} + 4 \, \sqrt {x e + d} c\right ) + \frac {1}{2} \, \sqrt {2} \sqrt {-\frac {{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt {\frac {b^{2} - 4 \, a c}{c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + a^{2} e^{4}}} e - 2 \, c d + b e}{c d^{2} - b d e + a e^{2}}} \log \left (-\sqrt {2} {\left ({\left (c d^{2} - b d e + a e^{2}\right )} \sqrt {\frac {b^{2} - 4 \, a c}{c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + a^{2} e^{4}}} e + 2 \, c d - b e\right )} \sqrt {-\frac {{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt {\frac {b^{2} - 4 \, a c}{c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + a^{2} e^{4}}} e - 2 \, c d + b e}{c d^{2} - b d e + a e^{2}}} + 4 \, \sqrt {x e + d} c\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {b + 2 c x}{\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((2*c*x+b)/(c*x**2+b*x+a)/(e*x+d)**(1/2),x)

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 2.36, size = 254, 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 \, c d - {\left (b - \sqrt {b^{2} - 4 \, a c}\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 \, c d - {\left (b + \sqrt {b^{2} - 4 \, a c}\right )} e\right )} {\left | c \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

Mupad [B]
time = 2.37, size = 205, normalized size = 1.17 \begin {gather*} \mathrm {atan}\left (\sqrt {\frac {2\,c\,d-b\,e+e\,\sqrt {b^2-4\,a\,c}}{2\,c\,d^2-2\,b\,d\,e+2\,a\,e^2}}\,\sqrt {d+e\,x}\,1{}\mathrm {i}\right )\,\sqrt {\frac {2\,c\,d-b\,e+e\,\sqrt {b^2-4\,a\,c}}{2\,c\,d^2-2\,b\,d\,e+2\,a\,e^2}}\,2{}\mathrm {i}+\mathrm {atan}\left (\sqrt {-\frac {b\,e-2\,c\,d+e\,\sqrt {b^2-4\,a\,c}}{2\,c\,d^2-2\,b\,d\,e+2\,a\,e^2}}\,\sqrt {d+e\,x}\,1{}\mathrm {i}\right )\,\sqrt {-\frac {b\,e-2\,c\,d+e\,\sqrt {b^2-4\,a\,c}}{2\,c\,d^2-2\,b\,d\,e+2\,a\,e^2}}\,2{}\mathrm {i} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________