3.8.95 \(\int \frac {1}{\sqrt {1-d x} \sqrt {1+d x} (a+b x+c x^2)} \, dx\) [795]

Optimal. Leaf size=282 \[ -\frac {\sqrt {2} c \tanh ^{-1}\left (\frac {2 c+\left (b-\sqrt {b^2-4 a c}\right ) d^2 x}{\sqrt {2} \sqrt {2 c^2+2 a c d^2-b \left (b-\sqrt {b^2-4 a c}\right ) d^2} \sqrt {1-d^2 x^2}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c^2+2 a c d^2-b \left (b-\sqrt {b^2-4 a c}\right ) d^2}}+\frac {\sqrt {2} c \tanh ^{-1}\left (\frac {2 c+\left (b+\sqrt {b^2-4 a c}\right ) d^2 x}{\sqrt {2} \sqrt {2 c^2+2 a c d^2-b \left (b+\sqrt {b^2-4 a c}\right ) d^2} \sqrt {1-d^2 x^2}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c^2+2 a c d^2-b \left (b+\sqrt {b^2-4 a c}\right ) d^2}} \]

[Out]

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

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Rubi [A]
time = 0.34, antiderivative size = 282, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {913, 999, 739, 212} \begin {gather*} \frac {\sqrt {2} c \tanh ^{-1}\left (\frac {d^2 x \left (\sqrt {b^2-4 a c}+b\right )+2 c}{\sqrt {2} \sqrt {1-d^2 x^2} \sqrt {-b d^2 \left (\sqrt {b^2-4 a c}+b\right )+2 a c d^2+2 c^2}}\right )}{\sqrt {b^2-4 a c} \sqrt {-b d^2 \left (\sqrt {b^2-4 a c}+b\right )+2 a c d^2+2 c^2}}-\frac {\sqrt {2} c \tanh ^{-1}\left (\frac {d^2 x \left (b-\sqrt {b^2-4 a c}\right )+2 c}{\sqrt {2} \sqrt {1-d^2 x^2} \sqrt {-b d^2 \left (b-\sqrt {b^2-4 a c}\right )+2 a c d^2+2 c^2}}\right )}{\sqrt {b^2-4 a c} \sqrt {-b d^2 \left (b-\sqrt {b^2-4 a c}\right )+2 a c d^2+2 c^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[2]*c*ArcTanh[(2*c + (b - Sqrt[b^2 - 4*a*c])*d^2*x)/(Sqrt[2]*Sqrt[2*c^2 + 2*a*c*d^2 - b*(b - Sqrt[b^2 -
4*a*c])*d^2]*Sqrt[1 - d^2*x^2])])/(Sqrt[b^2 - 4*a*c]*Sqrt[2*c^2 + 2*a*c*d^2 - b*(b - Sqrt[b^2 - 4*a*c])*d^2])
) + (Sqrt[2]*c*ArcTanh[(2*c + (b + Sqrt[b^2 - 4*a*c])*d^2*x)/(Sqrt[2]*Sqrt[2*c^2 + 2*a*c*d^2 - b*(b + Sqrt[b^2
- 4*a*c])*d^2]*Sqrt[1 - d^2*x^2])])/(Sqrt[b^2 - 4*a*c]*Sqrt[2*c^2 + 2*a*c*d^2 - b*(b + Sqrt[b^2 - 4*a*c])*d^2
])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 739

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

Rule 913

Int[((d_) + (e_.)*(x_))^(m_)*((f_) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :>
Int[(d*f + e*g*x^2)^m*(a + b*x + c*x^2)^p, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[m - n, 0] &&
EqQ[e*f + d*g, 0] && (IntegerQ[m] || (GtQ[d, 0] && GtQ[f, 0]))

Rule 999

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

Rubi steps

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

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Mathematica [A]
time = 10.42, size = 455, normalized size = 1.61 \begin {gather*} \frac {\sqrt {2} c \left (\sqrt {2 c^2+2 a c d^2-b \left (b+\sqrt {b^2-4 a c}\right ) d^2} \log \left (-b+\sqrt {b^2-4 a c}-2 c x\right )-\sqrt {2 c^2+2 a c d^2+b \left (-b+\sqrt {b^2-4 a c}\right ) d^2} \log \left (b+\sqrt {b^2-4 a c}+2 c x\right )-\sqrt {2 c^2+2 a c d^2-b \left (b+\sqrt {b^2-4 a c}\right ) d^2} \log \left (-2 c-b d^2 x+\sqrt {b^2-4 a c} d^2 x-\sqrt {4 c^2+4 a c d^2+2 b \left (-b+\sqrt {b^2-4 a c}\right ) d^2} \sqrt {1-d^2 x^2}\right )+\sqrt {2 c^2+2 a c d^2+b \left (-b+\sqrt {b^2-4 a c}\right ) d^2} \log \left (2 c+b d^2 x+\sqrt {b^2-4 a c} d^2 x+\sqrt {4 c^2+4 a c d^2-2 b \left (b+\sqrt {b^2-4 a c}\right ) d^2} \sqrt {1-d^2 x^2}\right )\right )}{\sqrt {b^2-4 a c} \sqrt {2 c^2+2 a c d^2+b \left (-b+\sqrt {b^2-4 a c}\right ) d^2} \sqrt {2 c^2+2 a c d^2-b \left (b+\sqrt {b^2-4 a c}\right ) d^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2]*c*(Sqrt[2*c^2 + 2*a*c*d^2 - b*(b + Sqrt[b^2 - 4*a*c])*d^2]*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x] - Sqrt
[2*c^2 + 2*a*c*d^2 + b*(-b + Sqrt[b^2 - 4*a*c])*d^2]*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x] - Sqrt[2*c^2 + 2*a*c*d
^2 - b*(b + Sqrt[b^2 - 4*a*c])*d^2]*Log[-2*c - b*d^2*x + Sqrt[b^2 - 4*a*c]*d^2*x - Sqrt[4*c^2 + 4*a*c*d^2 + 2*
b*(-b + Sqrt[b^2 - 4*a*c])*d^2]*Sqrt[1 - d^2*x^2]] + Sqrt[2*c^2 + 2*a*c*d^2 + b*(-b + Sqrt[b^2 - 4*a*c])*d^2]*
Log[2*c + b*d^2*x + Sqrt[b^2 - 4*a*c]*d^2*x + Sqrt[4*c^2 + 4*a*c*d^2 - 2*b*(b + Sqrt[b^2 - 4*a*c])*d^2]*Sqrt[1
- d^2*x^2]]))/(Sqrt[b^2 - 4*a*c]*Sqrt[2*c^2 + 2*a*c*d^2 + b*(-b + Sqrt[b^2 - 4*a*c])*d^2]*Sqrt[2*c^2 + 2*a*c*
d^2 - b*(b + Sqrt[b^2 - 4*a*c])*d^2])

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.22, size = 1759, normalized size = 6.24

method	result	size
default	\(\text {Expression too large to display}\)	\(1759\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

32*(-d*x+1)^(1/2)*(d*x+1)^(1/2)*csgn(d)^2*c^2*(ln(2*((-4*a*c+b^2)^(1/2)*d^2*x+b*d^2*x+(-d^2*x^2+1)^(1/2)*(-(b*
(-4*a*c+b^2)^(1/2)-2*a*c+b^2)*(2*a^2*d^2+b*(-4*a*c+b^2)^(1/2)+2*a*c-b^2)/a^2/c^2)^(1/2)*c+2*c)/(b+2*c*x+(-4*a*
c+b^2)^(1/2)))*a^2*d^4*(-(b*(-4*a*c+b^2)^(1/2)+2*a*c-b^2)*(-2*a^2*d^2+b*(-4*a*c+b^2)^(1/2)-2*a*c+b^2)/a^2/c^2)
^(1/2)-ln(2*(b*d^2*x-(-4*a*c+b^2)^(1/2)*d^2*x+(-d^2*x^2+1)^(1/2)*(-(b*(-4*a*c+b^2)^(1/2)+2*a*c-b^2)*(-2*a^2*d^
2+b*(-4*a*c+b^2)^(1/2)-2*a*c+b^2)/a^2/c^2)^(1/2)*c+2*c)/(b+2*c*x-(-4*a*c+b^2)^(1/2)))*a^2*d^4*(-(b*(-4*a*c+b^2
)^(1/2)-2*a*c+b^2)*(2*a^2*d^2+b*(-4*a*c+b^2)^(1/2)+2*a*c-b^2)/a^2/c^2)^(1/2)+2*ln(2*((-4*a*c+b^2)^(1/2)*d^2*x+
b*d^2*x+(-d^2*x^2+1)^(1/2)*(-(b*(-4*a*c+b^2)^(1/2)-2*a*c+b^2)*(2*a^2*d^2+b*(-4*a*c+b^2)^(1/2)+2*a*c-b^2)/a^2/c
^2)^(1/2)*c+2*c)/(b+2*c*x+(-4*a*c+b^2)^(1/2)))*a*c*d^2*(-(b*(-4*a*c+b^2)^(1/2)+2*a*c-b^2)*(-2*a^2*d^2+b*(-4*a*
c+b^2)^(1/2)-2*a*c+b^2)/a^2/c^2)^(1/2)-ln(2*((-4*a*c+b^2)^(1/2)*d^2*x+b*d^2*x+(-d^2*x^2+1)^(1/2)*(-(b*(-4*a*c+
b^2)^(1/2)-2*a*c+b^2)*(2*a^2*d^2+b*(-4*a*c+b^2)^(1/2)+2*a*c-b^2)/a^2/c^2)^(1/2)*c+2*c)/(b+2*c*x+(-4*a*c+b^2)^(
1/2)))*b^2*d^2*(-(b*(-4*a*c+b^2)^(1/2)+2*a*c-b^2)*(-2*a^2*d^2+b*(-4*a*c+b^2)^(1/2)-2*a*c+b^2)/a^2/c^2)^(1/2)-2
*ln(2*(b*d^2*x-(-4*a*c+b^2)^(1/2)*d^2*x+(-d^2*x^2+1)^(1/2)*(-(b*(-4*a*c+b^2)^(1/2)+2*a*c-b^2)*(-2*a^2*d^2+b*(-
4*a*c+b^2)^(1/2)-2*a*c+b^2)/a^2/c^2)^(1/2)*c+2*c)/(b+2*c*x-(-4*a*c+b^2)^(1/2)))*a*c*d^2*(-(b*(-4*a*c+b^2)^(1/2
)-2*a*c+b^2)*(2*a^2*d^2+b*(-4*a*c+b^2)^(1/2)+2*a*c-b^2)/a^2/c^2)^(1/2)+ln(2*(b*d^2*x-(-4*a*c+b^2)^(1/2)*d^2*x+
(-d^2*x^2+1)^(1/2)*(-(b*(-4*a*c+b^2)^(1/2)+2*a*c-b^2)*(-2*a^2*d^2+b*(-4*a*c+b^2)^(1/2)-2*a*c+b^2)/a^2/c^2)^(1/
2)*c+2*c)/(b+2*c*x-(-4*a*c+b^2)^(1/2)))*b^2*d^2*(-(b*(-4*a*c+b^2)^(1/2)-2*a*c+b^2)*(2*a^2*d^2+b*(-4*a*c+b^2)^(
1/2)+2*a*c-b^2)/a^2/c^2)^(1/2)+ln(2*((-4*a*c+b^2)^(1/2)*d^2*x+b*d^2*x+(-d^2*x^2+1)^(1/2)*(-(b*(-4*a*c+b^2)^(1/
2)-2*a*c+b^2)*(2*a^2*d^2+b*(-4*a*c+b^2)^(1/2)+2*a*c-b^2)/a^2/c^2)^(1/2)*c+2*c)/(b+2*c*x+(-4*a*c+b^2)^(1/2)))*c
^2*(-(b*(-4*a*c+b^2)^(1/2)+2*a*c-b^2)*(-2*a^2*d^2+b*(-4*a*c+b^2)^(1/2)-2*a*c+b^2)/a^2/c^2)^(1/2)-ln(2*(b*d^2*x
-(-4*a*c+b^2)^(1/2)*d^2*x+(-d^2*x^2+1)^(1/2)*(-(b*(-4*a*c+b^2)^(1/2)+2*a*c-b^2)*(-2*a^2*d^2+b*(-4*a*c+b^2)^(1/
2)-2*a*c+b^2)/a^2/c^2)^(1/2)*c+2*c)/(b+2*c*x-(-4*a*c+b^2)^(1/2)))*c^2*(-(b*(-4*a*c+b^2)^(1/2)-2*a*c+b^2)*(2*a^
2*d^2+b*(-4*a*c+b^2)^(1/2)+2*a*c-b^2)/a^2/c^2)^(1/2))/(-d^2*x^2+1)^(1/2)/(b*d-d*(-4*a*c+b^2)^(1/2)-2*c)/(b*d-d
*(-4*a*c+b^2)^(1/2)+2*c)/(-4*a*c+b^2)^(1/2)/(-(b*(-4*a*c+b^2)^(1/2)+2*a*c-b^2)*(-2*a^2*d^2+b*(-4*a*c+b^2)^(1/2
)-2*a*c+b^2)/a^2/c^2)^(1/2)/(d*(-4*a*c+b^2)^(1/2)+b*d-2*c)/(d*(-4*a*c+b^2)^(1/2)+b*d+2*c)/(-(b*(-4*a*c+b^2)^(1
/2)-2*a*c+b^2)*(2*a^2*d^2+b*(-4*a*c+b^2)^(1/2)+2*a*c-b^2)/a^2/c^2)^(1/2)

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 4313 vs. \(2 (248) = 496\).
time = 2.60, size = 4313, normalized size = 15.29 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(2)*sqrt(-((b^2 - 2*a*c)*d^2 - 2*c^2 - ((a^2*b^2 - 4*a^3*c)*d^4 + b^2*c^2 - 4*a*c^3 - (b^4 - 6*a*b^2*c
+ 8*a^2*c^2)*d^2)*sqrt(b^2*d^4/((a^4*b^2 - 4*a^5*c)*d^8 - 2*(a^2*b^4 - 6*a^3*b^2*c + 8*a^4*c^2)*d^6 + b^2*c^4
- 4*a*c^5 + (b^6 - 8*a*b^4*c + 22*a^2*b^2*c^2 - 24*a^3*c^3)*d^4 - 2*(b^4*c^2 - 6*a*b^2*c^3 + 8*a^2*c^4)*d^2))
)/((a^2*b^2 - 4*a^3*c)*d^4 + b^2*c^2 - 4*a*c^3 - (b^4 - 6*a*b^2*c + 8*a^2*c^2)*d^2))*log((4*sqrt(d*x + 1)*sqrt
(-d*x + 1)*a*b*c*d^2 - 2*b^2*c*d^2*x - 4*a*b*c*d^2 + 2*(b^2*c^3 - 4*a*c^4 + (a^2*b^2*c - 4*a^3*c^2)*d^4 - (b^4
*c - 6*a*b^2*c^2 + 8*a^2*c^3)*d^2)*sqrt(b^2*d^4/((a^4*b^2 - 4*a^5*c)*d^8 - 2*(a^2*b^4 - 6*a^3*b^2*c + 8*a^4*c^
2)*d^6 + b^2*c^4 - 4*a*c^5 + (b^6 - 8*a*b^4*c + 22*a^2*b^2*c^2 - 24*a^3*c^3)*d^4 - 2*(b^4*c^2 - 6*a*b^2*c^3 +
8*a^2*c^4)*d^2))*x + sqrt(2)*(((a^3*b^3 - 4*a^4*b*c)*d^6 - b^3*c^3 + 4*a*b*c^4 - (a*b^5 - 5*a^2*b^3*c + 4*a^3*
b*c^2)*d^4 + (b^5*c - 5*a*b^3*c^2 + 4*a^2*b*c^3)*d^2)*sqrt(b^2*d^4/((a^4*b^2 - 4*a^5*c)*d^8 - 2*(a^2*b^4 - 6*a
^3*b^2*c + 8*a^4*c^2)*d^6 + b^2*c^4 - 4*a*c^5 + (b^6 - 8*a*b^4*c + 22*a^2*b^2*c^2 - 24*a^3*c^3)*d^4 - 2*(b^4*c
^2 - 6*a*b^2*c^3 + 8*a^2*c^4)*d^2))*x + ((a*b^3 - 4*a^2*b*c)*d^4 + (b^3*c - 4*a*b*c^2)*d^2)*x)*sqrt(-((b^2 - 2
*a*c)*d^2 - 2*c^2 - ((a^2*b^2 - 4*a^3*c)*d^4 + b^2*c^2 - 4*a*c^3 - (b^4 - 6*a*b^2*c + 8*a^2*c^2)*d^2)*sqrt(b^2
*d^4/((a^4*b^2 - 4*a^5*c)*d^8 - 2*(a^2*b^4 - 6*a^3*b^2*c + 8*a^4*c^2)*d^6 + b^2*c^4 - 4*a*c^5 + (b^6 - 8*a*b^4
*c + 22*a^2*b^2*c^2 - 24*a^3*c^3)*d^4 - 2*(b^4*c^2 - 6*a*b^2*c^3 + 8*a^2*c^4)*d^2)))/((a^2*b^2 - 4*a^3*c)*d^4
+ b^2*c^2 - 4*a*c^3 - (b^4 - 6*a*b^2*c + 8*a^2*c^2)*d^2)))/x) - 1/2*sqrt(2)*sqrt(-((b^2 - 2*a*c)*d^2 - 2*c^2 -
((a^2*b^2 - 4*a^3*c)*d^4 + b^2*c^2 - 4*a*c^3 - (b^4 - 6*a*b^2*c + 8*a^2*c^2)*d^2)*sqrt(b^2*d^4/((a^4*b^2 - 4*
a^5*c)*d^8 - 2*(a^2*b^4 - 6*a^3*b^2*c + 8*a^4*c^2)*d^6 + b^2*c^4 - 4*a*c^5 + (b^6 - 8*a*b^4*c + 22*a^2*b^2*c^2
- 24*a^3*c^3)*d^4 - 2*(b^4*c^2 - 6*a*b^2*c^3 + 8*a^2*c^4)*d^2)))/((a^2*b^2 - 4*a^3*c)*d^4 + b^2*c^2 - 4*a*c^3
- (b^4 - 6*a*b^2*c + 8*a^2*c^2)*d^2))*log((4*sqrt(d*x + 1)*sqrt(-d*x + 1)*a*b*c*d^2 - 2*b^2*c*d^2*x - 4*a*b*c
*d^2 + 2*(b^2*c^3 - 4*a*c^4 + (a^2*b^2*c - 4*a^3*c^2)*d^4 - (b^4*c - 6*a*b^2*c^2 + 8*a^2*c^3)*d^2)*sqrt(b^2*d^
4/((a^4*b^2 - 4*a^5*c)*d^8 - 2*(a^2*b^4 - 6*a^3*b^2*c + 8*a^4*c^2)*d^6 + b^2*c^4 - 4*a*c^5 + (b^6 - 8*a*b^4*c
+ 22*a^2*b^2*c^2 - 24*a^3*c^3)*d^4 - 2*(b^4*c^2 - 6*a*b^2*c^3 + 8*a^2*c^4)*d^2))*x - sqrt(2)*(((a^3*b^3 - 4*a^
4*b*c)*d^6 - b^3*c^3 + 4*a*b*c^4 - (a*b^5 - 5*a^2*b^3*c + 4*a^3*b*c^2)*d^4 + (b^5*c - 5*a*b^3*c^2 + 4*a^2*b*c^
3)*d^2)*sqrt(b^2*d^4/((a^4*b^2 - 4*a^5*c)*d^8 - 2*(a^2*b^4 - 6*a^3*b^2*c + 8*a^4*c^2)*d^6 + b^2*c^4 - 4*a*c^5
+ (b^6 - 8*a*b^4*c + 22*a^2*b^2*c^2 - 24*a^3*c^3)*d^4 - 2*(b^4*c^2 - 6*a*b^2*c^3 + 8*a^2*c^4)*d^2))*x + ((a*b^
3 - 4*a^2*b*c)*d^4 + (b^3*c - 4*a*b*c^2)*d^2)*x)*sqrt(-((b^2 - 2*a*c)*d^2 - 2*c^2 - ((a^2*b^2 - 4*a^3*c)*d^4 +
b^2*c^2 - 4*a*c^3 - (b^4 - 6*a*b^2*c + 8*a^2*c^2)*d^2)*sqrt(b^2*d^4/((a^4*b^2 - 4*a^5*c)*d^8 - 2*(a^2*b^4 - 6
*a^3*b^2*c + 8*a^4*c^2)*d^6 + b^2*c^4 - 4*a*c^5 + (b^6 - 8*a*b^4*c + 22*a^2*b^2*c^2 - 24*a^3*c^3)*d^4 - 2*(b^4
*c^2 - 6*a*b^2*c^3 + 8*a^2*c^4)*d^2)))/((a^2*b^2 - 4*a^3*c)*d^4 + b^2*c^2 - 4*a*c^3 - (b^4 - 6*a*b^2*c + 8*a^2
*c^2)*d^2)))/x) - 1/2*sqrt(2)*sqrt(-((b^2 - 2*a*c)*d^2 - 2*c^2 + ((a^2*b^2 - 4*a^3*c)*d^4 + b^2*c^2 - 4*a*c^3
- (b^4 - 6*a*b^2*c + 8*a^2*c^2)*d^2)*sqrt(b^2*d^4/((a^4*b^2 - 4*a^5*c)*d^8 - 2*(a^2*b^4 - 6*a^3*b^2*c + 8*a^4*
c^2)*d^6 + b^2*c^4 - 4*a*c^5 + (b^6 - 8*a*b^4*c + 22*a^2*b^2*c^2 - 24*a^3*c^3)*d^4 - 2*(b^4*c^2 - 6*a*b^2*c^3
+ 8*a^2*c^4)*d^2)))/((a^2*b^2 - 4*a^3*c)*d^4 + b^2*c^2 - 4*a*c^3 - (b^4 - 6*a*b^2*c + 8*a^2*c^2)*d^2))*log((4*
sqrt(d*x + 1)*sqrt(-d*x + 1)*a*b*c*d^2 - 2*b^2*c*d^2*x - 4*a*b*c*d^2 - 2*(b^2*c^3 - 4*a*c^4 + (a^2*b^2*c - 4*a
^3*c^2)*d^4 - (b^4*c - 6*a*b^2*c^2 + 8*a^2*c^3)*d^2)*sqrt(b^2*d^4/((a^4*b^2 - 4*a^5*c)*d^8 - 2*(a^2*b^4 - 6*a^
3*b^2*c + 8*a^4*c^2)*d^6 + b^2*c^4 - 4*a*c^5 + (b^6 - 8*a*b^4*c + 22*a^2*b^2*c^2 - 24*a^3*c^3)*d^4 - 2*(b^4*c^
2 - 6*a*b^2*c^3 + 8*a^2*c^4)*d^2))*x + sqrt(2)*(((a^3*b^3 - 4*a^4*b*c)*d^6 - b^3*c^3 + 4*a*b*c^4 - (a*b^5 - 5*
a^2*b^3*c + 4*a^3*b*c^2)*d^4 + (b^5*c - 5*a*b^3*c^2 + 4*a^2*b*c^3)*d^2)*sqrt(b^2*d^4/((a^4*b^2 - 4*a^5*c)*d^8
- 2*(a^2*b^4 - 6*a^3*b^2*c + 8*a^4*c^2)*d^6 + b^2*c^4 - 4*a*c^5 + (b^6 - 8*a*b^4*c + 22*a^2*b^2*c^2 - 24*a^3*c
^3)*d^4 - 2*(b^4*c^2 - 6*a*b^2*c^3 + 8*a^2*c^4)*d^2))*x - ((a*b^3 - 4*a^2*b*c)*d^4 + (b^3*c - 4*a*b*c^2)*d^2)*
x)*sqrt(-((b^2 - 2*a*c)*d^2 - 2*c^2 + ((a^2*b^2 - 4*a^3*c)*d^4 + b^2*c^2 - 4*a*c^3 - (b^4 - 6*a*b^2*c + 8*a^2*
c^2)*d^2)*sqrt(b^2*d^4/((a^4*b^2 - 4*a^5*c)*d^8 - 2*(a^2*b^4 - 6*a^3*b^2*c + 8*a^4*c^2)*d^6 + b^2*c^4 - 4*a*c^
5 + (b^6 - 8*a*b^4*c + 22*a^2*b^2*c^2 - 24*a^3*c^3)*d^4 - 2*(b^4*c^2 - 6*a*b^2*c^3 + 8*a^2*c^4)*d^2)))/((a^2*b
^2 - 4*a^3*c)*d^4 + b^2*c^2 - 4*a*c^3 - (b^4 - 6*a*b^2*c + 8*a^2*c^2)*d^2)))/x) + 1/2*sqrt(2)*sqrt(-((b^2 - 2*
a*c)*d^2 - 2*c^2 + ((a^2*b^2 - 4*a^3*c)*d^4 + b^2*c^2 - 4*a*c^3 - (b^4 - 6*a*b^2*c + 8*a^2*c^2)*d^2)*sqrt(b^2*
d^4/((a^4*b^2 - 4*a^5*c)*d^8 - 2*(a^2*b^4 - 6*a...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 681 vs. \(2 (248) = 496\).
time = 5.46, size = 681, normalized size = 2.41 \begin {gather*} -{\left (\frac {{\left (a d^{2} - b d + c\right )} {\left (\frac {a d^{2} - c + \sqrt {-{\left (a d^{2} + b d + c\right )} {\left (a d^{2} - b d + c\right )} + {\left (a d^{2} - c\right )}^{2}}}{a d^{2} - b d + c} - 1\right )} \sqrt {\frac {a d^{2} - c + \sqrt {-{\left (a d^{2} + b d + c\right )} {\left (a d^{2} - b d + c\right )} + {\left (a d^{2} - c\right )}^{2}}}{a d^{2} - b d + c}} \arctan \left (-\frac {\frac {\sqrt {2} - \sqrt {-d x + 1}}{\sqrt {d x + 1}} - \frac {\sqrt {d x + 1}}{\sqrt {2} - \sqrt {-d x + 1}}}{2 \, \sqrt {\frac {a d^{2} - c + \sqrt {-{\left (a d^{2} + b d + c\right )} {\left (a d^{2} - b d + c\right )} + {\left (a d^{2} - c\right )}^{2}}}{a d^{2} - b d + c}}}\right )}{{\left (a d^{2} - c + \sqrt {-{\left (a d^{2} + b d + c\right )} {\left (a d^{2} - b d + c\right )} + {\left (a d^{2} - c\right )}^{2}}\right )} \sqrt {-{\left (a d^{2} + b d + c\right )} {\left (a d^{2} - b d + c\right )} + {\left (a d^{2} - c\right )}^{2}}} - \frac {{\left (a d^{2} - b d + c\right )} {\left (\frac {a d^{2} - c - \sqrt {-{\left (a d^{2} + b d + c\right )} {\left (a d^{2} - b d + c\right )} + {\left (a d^{2} - c\right )}^{2}}}{a d^{2} - b d + c} - 1\right )} \sqrt {\frac {a d^{2} - c - \sqrt {-{\left (a d^{2} + b d + c\right )} {\left (a d^{2} - b d + c\right )} + {\left (a d^{2} - c\right )}^{2}}}{a d^{2} - b d + c}} \arctan \left (-\frac {\frac {\sqrt {2} - \sqrt {-d x + 1}}{\sqrt {d x + 1}} - \frac {\sqrt {d x + 1}}{\sqrt {2} - \sqrt {-d x + 1}}}{2 \, \sqrt {\frac {a d^{2} - c - \sqrt {-{\left (a d^{2} + b d + c\right )} {\left (a d^{2} - b d + c\right )} + {\left (a d^{2} - c\right )}^{2}}}{a d^{2} - b d + c}}}\right )}{{\left (a d^{2} - c - \sqrt {-{\left (a d^{2} + b d + c\right )} {\left (a d^{2} - b d + c\right )} + {\left (a d^{2} - c\right )}^{2}}\right )} \sqrt {-{\left (a d^{2} + b d + c\right )} {\left (a d^{2} - b d + c\right )} + {\left (a d^{2} - c\right )}^{2}}}\right )} d \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((a*d^2 - b*d + c)*((a*d^2 - c + sqrt(-(a*d^2 + b*d + c)*(a*d^2 - b*d + c) + (a*d^2 - c)^2))/(a*d^2 - b*d + c
) - 1)*sqrt((a*d^2 - c + sqrt(-(a*d^2 + b*d + c)*(a*d^2 - b*d + c) + (a*d^2 - c)^2))/(a*d^2 - b*d + c))*arctan
(-1/2*((sqrt(2) - sqrt(-d*x + 1))/sqrt(d*x + 1) - sqrt(d*x + 1)/(sqrt(2) - sqrt(-d*x + 1)))/sqrt((a*d^2 - c +
sqrt(-(a*d^2 + b*d + c)*(a*d^2 - b*d + c) + (a*d^2 - c)^2))/(a*d^2 - b*d + c)))/((a*d^2 - c + sqrt(-(a*d^2 + b
*d + c)*(a*d^2 - b*d + c) + (a*d^2 - c)^2))*sqrt(-(a*d^2 + b*d + c)*(a*d^2 - b*d + c) + (a*d^2 - c)^2)) - (a*d
^2 - b*d + c)*((a*d^2 - c - sqrt(-(a*d^2 + b*d + c)*(a*d^2 - b*d + c) + (a*d^2 - c)^2))/(a*d^2 - b*d + c) - 1)
*sqrt((a*d^2 - c - sqrt(-(a*d^2 + b*d + c)*(a*d^2 - b*d + c) + (a*d^2 - c)^2))/(a*d^2 - b*d + c))*arctan(-1/2*
((sqrt(2) - sqrt(-d*x + 1))/sqrt(d*x + 1) - sqrt(d*x + 1)/(sqrt(2) - sqrt(-d*x + 1)))/sqrt((a*d^2 - c - sqrt(-
(a*d^2 + b*d + c)*(a*d^2 - b*d + c) + (a*d^2 - c)^2))/(a*d^2 - b*d + c)))/((a*d^2 - c - sqrt(-(a*d^2 + b*d + c
)*(a*d^2 - b*d + c) + (a*d^2 - c)^2))*sqrt(-(a*d^2 + b*d + c)*(a*d^2 - b*d + c) + (a*d^2 - c)^2)))*d

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Mupad [B]
time = 82.37, size = 2500, normalized size = 8.87 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- atan(((-(8*a*c^3 - 2*b^2*c^2 + b^4*d^2 + b*d^2*(-(4*a*c - b^2)^3)^(1/2) + 8*a^2*c^2*d^2 - 6*a*b^2*c*d^2)/(2*
(16*a^2*c^4 + b^4*c^2 - b^6*d^2 - 8*a*b^2*c^3 + a^2*b^4*d^4 + 32*a^3*c^3*d^2 + 16*a^4*c^2*d^4 - 8*a^3*b^2*c*d^
4 - 32*a^2*b^2*c^2*d^2 + 10*a*b^4*c*d^2)))^(1/2)*((-(8*a*c^3 - 2*b^2*c^2 + b^4*d^2 + b*d^2*(-(4*a*c - b^2)^3)^
(1/2) + 8*a^2*c^2*d^2 - 6*a*b^2*c*d^2)/(2*(16*a^2*c^4 + b^4*c^2 - b^6*d^2 - 8*a*b^2*c^3 + a^2*b^4*d^4 + 32*a^3
*c^3*d^2 + 16*a^4*c^2*d^4 - 8*a^3*b^2*c*d^4 - 32*a^2*b^2*c^2*d^2 + 10*a*b^4*c*d^2)))^(1/2)*((-(8*a*c^3 - 2*b^2
*c^2 + b^4*d^2 + b*d^2*(-(4*a*c - b^2)^3)^(1/2) + 8*a^2*c^2*d^2 - 6*a*b^2*c*d^2)/(2*(16*a^2*c^4 + b^4*c^2 - b^
6*d^2 - 8*a*b^2*c^3 + a^2*b^4*d^4 + 32*a^3*c^3*d^2 + 16*a^4*c^2*d^4 - 8*a^3*b^2*c*d^4 - 32*a^2*b^2*c^2*d^2 + 1
0*a*b^4*c*d^2)))^(1/2)*((-(8*a*c^3 - 2*b^2*c^2 + b^4*d^2 + b*d^2*(-(4*a*c - b^2)^3)^(1/2) + 8*a^2*c^2*d^2 - 6*
a*b^2*c*d^2)/(2*(16*a^2*c^4 + b^4*c^2 - b^6*d^2 - 8*a*b^2*c^3 + a^2*b^4*d^4 + 32*a^3*c^3*d^2 + 16*a^4*c^2*d^4
- 8*a^3*b^2*c*d^4 - 32*a^2*b^2*c^2*d^2 + 10*a*b^4*c*d^2)))^(1/2)*((-(8*a*c^3 - 2*b^2*c^2 + b^4*d^2 + b*d^2*(-(
4*a*c - b^2)^3)^(1/2) + 8*a^2*c^2*d^2 - 6*a*b^2*c*d^2)/(2*(16*a^2*c^4 + b^4*c^2 - b^6*d^2 - 8*a*b^2*c^3 + a^2*
b^4*d^4 + 32*a^3*c^3*d^2 + 16*a^4*c^2*d^4 - 8*a^3*b^2*c*d^4 - 32*a^2*b^2*c^2*d^2 + 10*a*b^4*c*d^2)))^(1/2)*((-
(8*a*c^3 - 2*b^2*c^2 + b^4*d^2 + b*d^2*(-(4*a*c - b^2)^3)^(1/2) + 8*a^2*c^2*d^2 - 6*a*b^2*c*d^2)/(2*(16*a^2*c^
4 + b^4*c^2 - b^6*d^2 - 8*a*b^2*c^3 + a^2*b^4*d^4 + 32*a^3*c^3*d^2 + 16*a^4*c^2*d^4 - 8*a^3*b^2*c*d^4 - 32*a^2
*b^2*c^2*d^2 + 10*a*b^4*c*d^2)))^(1/2)*((((1 - d*x)^(1/2) - 1)^2*(1073741824*a*b^10*d^12 - 2147483648*a^3*b^8*
d^14 + 1073741824*a^5*b^6*d^16 - 36283883716608*a^3*c^8*d^6 + 36283883716608*a^4*c^7*d^8 + 210900074102784*a^5
*c^6*d^10 + 167812962189312*a^6*c^5*d^12 + 29480655519744*a^7*c^4*d^14 - 2267742732288*a*b^4*c^6*d^6 + 7602092
11392*a*b^6*c^4*d^8 + 1504312295424*a*b^8*c^2*d^10 + 75161927680*a^2*b^8*c*d^12 - 66571993088*a^4*b^6*c*d^14 -
8589934592*a^6*b^4*c*d^16 + 18141941858304*a^2*b^2*c^7*d^6 - 3813930958848*a^2*b^4*c^5*d^8 - 5978594476032*a^
3*b^2*c^6*d^8 - 21930103013376*a^2*b^6*c^3*d^10 + 116415088558080*a^3*b^4*c^4*d^10 - 263779711451136*a^4*b^2*c
^5*d^10 - 4173634469888*a^3*b^6*c^2*d^12 + 39994735460352*a^4*b^4*c^3*d^12 - 140239272148992*a^5*b^2*c^4*d^12
+ 2478196129792*a^5*b^4*c^2*d^14 - 16080357556224*a^6*b^2*c^3*d^14 + 17179869184*a^7*b^2*c^2*d^16))/((d*x + 1)
^(1/2) - 1)^2 + 1073741824*a*b^10*d^12 + (((1 - d*x)^(1/2) - 1)*(1176821039104*a*b^7*c^3*d^9 - 21440476741632*
a^3*b*c^7*d^7 - 1340029796352*a*b^5*c^5*d^7 - 11544872091648*a^4*b*c^6*d^9 + 42193758715904*a^5*b*c^5*d^11 - 2
10453397504*a^3*b^7*c*d^13 + 32985348833280*a^6*b*c^4*d^13 + 42949672960*a^5*b^5*c*d^15 + 687194767360*a^7*b*c
^3*d^15 + 10720238370816*a^2*b^3*c^6*d^7 - 10136122818560*a^2*b^5*c^4*d^9 + 24601572671488*a^3*b^3*c^5*d^9 - 3
646427234304*a^2*b^7*c^2*d^11 + 23768349016064*a^3*b^5*c^3*d^11 - 57999238365184*a^4*b^3*c^4*d^11 + 3745211482
112*a^4*b^5*c^2*d^13 - 19859928776704*a^5*b^3*c^3*d^13 - 343597383680*a^6*b^3*c^2*d^15 + 167503724544*a*b^9*c*
d^11))/((d*x + 1)^(1/2) - 1) - 2147483648*a^3*b^8*d^14 + 1073741824*a^5*b^6*d^16 + 1099511627776*a^3*c^8*d^6 -
4947802324992*a^4*c^7*d^8 - 1580547964928*a^5*c^6*d^10 + 16080357556224*a^6*c^5*d^12 + 11613591568384*a^7*c^4
*d^14 + 68719476736*a*b^4*c^6*d^6 - 115964116992*a*b^6*c^4*d^8 + 48318382080*a*b^8*c^2*d^10 + 23622320128*a^2*
b^8*c*d^12 - 15032385536*a^4*b^6*c*d^14 - 8589934592*a^6*b^4*c*d^16 - 549755813888*a^2*b^2*c^7*d^6 + 618475290
624*a^2*b^4*c^5*d^8 + 618475290624*a^3*b^2*c^6*d^8 - 77309411328*a^2*b^6*c^3*d^10 - 1799591297024*a^3*b^4*c^4*
d^10 + 5738076307456*a^4*b^2*c^5*d^10 - 1081258016768*a^3*b^6*c^2*d^12 + 8246337208320*a^4*b^4*c^3*d^12 - 2149
2016349184*a^5*b^2*c^4*d^12 + 949187772416*a^5*b^4*c^2*d^14 - 6322191859712*a^6*b^2*c^3*d^14 + 17179869184*a^7
*b^2*c^2*d^16) + (((1 - d*x)^(1/2) - 1)^2*(1778116460544*a*b^5*c^4*d^8 + 28449863368704*a^3*b*c^6*d^8 - 176737
9042304*a*b^7*c^2*d^10 + 57312043597824*a^4*b*c^5*d^10 - 47244640256*a^2*b^7*c*d^12 + 29618094473216*a^5*b*c^4
*d^12 + 47244640256*a^4*b^5*c*d^14 + 755914244096*a^6*b*c^3*d^14 - 14224931684352*a^2*b^3*c^5*d^8 + 1772103506
3296*a^2*b^5*c^3*d^10 - 56934086475776*a^3*b^3*c^4*d^10 + 2229088026624*a^3*b^5*c^2*d^12 - 15564961480704*a^4*
b^3*c^3*d^12 - 377957122048*a^5*b^3*c^2*d^14))/((d*x + 1)^(1/2) - 1)^2 + (((1 - d*x)^(1/2) - 1)*(3023656976384
0*a^3*c^7*d^7 + 57449482551296*a^4*c^6*d^9 + 24189255811072*a^5*c^5*d^11 - 3023656976384*a^6*c^4*d^13 + 188978
5610240*a*b^4*c^5*d^7 - 1778116460544*a*b^6*c^3*d^9 + 128849018880*a^3*b^6*c*d^13 - 15118284881920*a^2*b^2*c^6
*d^7 + 17815524343808*a^2*b^4*c^4*d^9 - 57174604644352*a^3*b^2*c^5*d^9 + 1494648619008*a^2*b^6*c^2*d^11 - 4260
607557632*a^3*b^4*c^3*d^11 - 4672924418048*a^4*b^2*c^4*d^11 - 1219770712064*a^4*b^4*c^2*d^13 + 3573412790272*a
^5*b^2*c^3*d^13 - 128849018880*a*b^8*c*d^11))/((d*x + 1)^(1/2) - 1) + 77309411328*a*b^5*c^4*d^8 + 123695058124
8*a^3*b*c^6*d^8 - 88046829568*a*b^7*c^2*d^10 + ...

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