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

Optimal. Leaf size=66 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a-d} (b+2 c x)}{\sqrt{b^2-4 c d} \sqrt{a+b x+c x^2}}\right )}{\sqrt{a-d} \sqrt{b^2-4 c d}} \]

[Out]

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

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Rubi [A]  time = 0.0790939, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {982, 208} \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a-d} (b+2 c x)}{\sqrt{b^2-4 c d} \sqrt{a+b x+c x^2}}\right )}{\sqrt{a-d} \sqrt{b^2-4 c d}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 982

Int[1/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[-2*e, Su
bst[Int[1/(e*(b*e - 4*a*f) - (b*d - a*e)*x^2), x], x, (e + 2*f*x)/Sqrt[d + e*x + f*x^2]], x] /; FreeQ[{a, b, c
, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && EqQ[c*e - b*f, 0]

Rule 208

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

Rubi steps

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

Mathematica [B]  time = 0.225642, size = 161, normalized size = 2.44 \[ \frac{\tanh ^{-1}\left (\frac{4 a c-2 c x \sqrt{b^2-4 c d}-b \left (\sqrt{b^2-4 c d}+b\right )}{4 c \sqrt{a-d} \sqrt{a+x (b+c x)}}\right )+\tanh ^{-1}\left (\frac{-2 c \left (2 a+x \sqrt{b^2-4 c d}\right )-b \sqrt{b^2-4 c d}+b^2}{4 c \sqrt{a-d} \sqrt{a+x (b+c x)}}\right )}{\sqrt{a-d} \sqrt{b^2-4 c d}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.327, size = 307, normalized size = 4.7 \begin{align*}{\ln \left ({ \left ( 2\,a-2\,d-\sqrt{{b}^{2}-4\,cd} \left ( x+{\frac{1}{2\,c} \left ( \sqrt{{b}^{2}-4\,cd}+b \right ) } \right ) +2\,\sqrt{a-d}\sqrt{ \left ( x+1/2\,{\frac{\sqrt{{b}^{2}-4\,cd}+b}{c}} \right ) ^{2}c-\sqrt{{b}^{2}-4\,cd} \left ( x+1/2\,{\frac{\sqrt{{b}^{2}-4\,cd}+b}{c}} \right ) +a-d} \right ) \left ( x+{\frac{1}{2\,c} \left ( \sqrt{{b}^{2}-4\,cd}+b \right ) } \right ) ^{-1}} \right ){\frac{1}{\sqrt{{b}^{2}-4\,cd}}}{\frac{1}{\sqrt{a-d}}}}-{\ln \left ({ \left ( 2\,a-2\,d+\sqrt{{b}^{2}-4\,cd} \left ( x-{\frac{1}{2\,c} \left ( -b+\sqrt{{b}^{2}-4\,cd} \right ) } \right ) +2\,\sqrt{a-d}\sqrt{ \left ( x-1/2\,{\frac{-b+\sqrt{{b}^{2}-4\,cd}}{c}} \right ) ^{2}c+\sqrt{{b}^{2}-4\,cd} \left ( x-1/2\,{\frac{-b+\sqrt{{b}^{2}-4\,cd}}{c}} \right ) +a-d} \right ) \left ( x-{\frac{1}{2\,c} \left ( -b+\sqrt{{b}^{2}-4\,cd} \right ) } \right ) ^{-1}} \right ){\frac{1}{\sqrt{{b}^{2}-4\,cd}}}{\frac{1}{\sqrt{a-d}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.45844, size = 1755, normalized size = 26.59 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*log((8*a^2*b^4 + (b^4*c^2 + 24*a*b^2*c^3 + 16*a^2*c^4 + 128*c^4*d^2 - 32*(b^2*c^3 + 4*a*c^4)*d)*x^4 + 2*(
b^5*c + 24*a*b^3*c^2 + 16*a^2*b*c^3 + 128*b*c^3*d^2 - 32*(b^3*c^2 + 4*a*b*c^3)*d)*x^3 + (b^4 + 24*a*b^2*c + 16
*a^2*c^2)*d^2 + (b^6 + 32*a*b^4*c + 48*a^2*b^2*c^2 + 32*(5*b^2*c^2 + 4*a*c^3)*d^2 - 2*(19*b^4*c + 104*a*b^2*c^
2 + 48*a^2*c^3)*d)*x^2 - 4*(2*a*b^3 + 2*(b^2*c^2 + 4*a*c^3 - 8*c^3*d)*x^3 + 3*(b^3*c + 4*a*b*c^2 - 8*b*c^2*d)*
x^2 - (b^3 + 4*a*b*c)*d + (b^4 + 8*a*b^2*c - 2*(5*b^2*c + 4*a*c^2)*d)*x)*sqrt(a*b^2 + 4*c*d^2 - (b^2 + 4*a*c)*
d)*sqrt(c*x^2 + b*x + a) - 8*(a*b^4 + 4*a^2*b^2*c)*d + 2*(4*a*b^5 + 16*a^2*b^3*c + 16*(b^3*c + 4*a*b*c^2)*d^2
- (3*b^5 + 40*a*b^3*c + 48*a^2*b*c^2)*d)*x)/(c^2*x^4 + 2*b*c*x^3 + 2*b*d*x + (b^2 + 2*c*d)*x^2 + d^2))/sqrt(a*
b^2 + 4*c*d^2 - (b^2 + 4*a*c)*d), -sqrt(-a*b^2 - 4*c*d^2 + (b^2 + 4*a*c)*d)*arctan(-1/2*(2*a*b^2 + (b^2*c + 4*
a*c^2 - 8*c^2*d)*x^2 - (b^2 + 4*a*c)*d + (b^3 + 4*a*b*c - 8*b*c*d)*x)*sqrt(-a*b^2 - 4*c*d^2 + (b^2 + 4*a*c)*d)
*sqrt(c*x^2 + b*x + a)/(a^2*b^3 + 4*a*b*c*d^2 + 2*(a*b^2*c^2 + 4*c^3*d^2 - (b^2*c^2 + 4*a*c^3)*d)*x^3 + 3*(a*b
^3*c + 4*b*c^2*d^2 - (b^3*c + 4*a*b*c^2)*d)*x^2 - (a*b^3 + 4*a^2*b*c)*d + (a*b^4 + 2*a^2*b^2*c + 4*(b^2*c + 2*
a*c^2)*d^2 - (b^4 + 6*a*b^2*c + 8*a^2*c^2)*d)*x))/(a*b^2 + 4*c*d^2 - (b^2 + 4*a*c)*d)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 14.6685, size = 620, normalized size = 9.39 \begin{align*} \frac{\log \left ({\left | -3 \, a b^{2} c + 4 \, a^{2} c^{2} + 2 \, b^{2} c d -{\left (b^{2} c + 4 \, a c^{2} - 8 \, c^{2} d + 4 \, \sqrt{a b^{2} - b^{2} d - 4 \, a c d + 4 \, c d^{2}} c^{\frac{3}{2}}\right )}{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{2} - \sqrt{a b^{2} - b^{2} d - 4 \, a c d + 4 \, c d^{2}} b^{2} \sqrt{c} -{\left (b^{3} \sqrt{c} + 4 \, a b c^{\frac{3}{2}} - 8 \, b c^{\frac{3}{2}} d + 4 \, \sqrt{a b^{2} - b^{2} d - 4 \, a c d + 4 \, c d^{2}} b c\right )}{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \right |}\right )}{\sqrt{a b^{2} - b^{2} d - 4 \, a c d + 4 \, c d^{2}}} - \frac{\log \left ({\left | -3 \, a b^{2} c + 4 \, a^{2} c^{2} + 2 \, b^{2} c d -{\left (b^{2} c + 4 \, a c^{2} - 8 \, c^{2} d - 4 \, \sqrt{a b^{2} - b^{2} d - 4 \, a c d + 4 \, c d^{2}} c^{\frac{3}{2}}\right )}{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{2} + \sqrt{a b^{2} - b^{2} d - 4 \, a c d + 4 \, c d^{2}} b^{2} \sqrt{c} -{\left (b^{3} \sqrt{c} + 4 \, a b c^{\frac{3}{2}} - 8 \, b c^{\frac{3}{2}} d - 4 \, \sqrt{a b^{2} - b^{2} d - 4 \, a c d + 4 \, c d^{2}} b c\right )}{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \right |}\right )}{\sqrt{a b^{2} - b^{2} d - 4 \, a c d + 4 \, c d^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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