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

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

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Rubi [A]  time = 0.08, antiderivative size = 66, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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 {gather*}

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 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 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]

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*}

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Mathematica [B]  time = 0.23, size = 161, normalized size = 2.44 \begin {gather*} \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}} \end {gather*}

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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IntegrateAlgebraic [C]  time = 0.44, size = 146, normalized size = 2.21 \begin {gather*} \text {RootSum}\left [\text {$\#$1}^4 c-2 \text {$\#$1}^3 b \sqrt {c}-2 \text {$\#$1}^2 a c+\text {$\#$1}^2 b^2+4 \text {$\#$1}^2 c d+2 \text {$\#$1} a b \sqrt {c}-4 \text {$\#$1} b \sqrt {c} d+a^2 c-a b^2+b^2 d\&,\frac {\log \left (-\text {$\#$1}+\sqrt {a+b x+c x^2}-\sqrt {c} x\right )}{\text {$\#$1}^2 \left (-\sqrt {c}\right )+\text {$\#$1} b+a \sqrt {c}-2 \sqrt {c} d}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

RootSum[-(a*b^2) + a^2*c + b^2*d + 2*a*b*Sqrt[c]*#1 - 4*b*Sqrt[c]*d*#1 + b^2*#1^2 - 2*a*c*#1^2 + 4*c*d*#1^2 -
2*b*Sqrt[c]*#1^3 + c*#1^4 & , Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]/(a*Sqrt[c] - 2*Sqrt[c]*d + b*#1 -
Sqrt[c]*#1^2) & ]

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fricas [B]  time = 0.52, size = 813, normalized size = 12.32 \begin {gather*} \left [\frac {\log \left (\frac {8 \, a^{2} b^{4} + {\left (b^{4} c^{2} + 24 \, a b^{2} c^{3} + 16 \, a^{2} c^{4} + 128 \, c^{4} d^{2} - 32 \, {\left (b^{2} c^{3} + 4 \, a c^{4}\right )} d\right )} x^{4} + 2 \, {\left (b^{5} c + 24 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3} + 128 \, b c^{3} d^{2} - 32 \, {\left (b^{3} c^{2} + 4 \, a b c^{3}\right )} d\right )} x^{3} + {\left (b^{4} + 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{2} + {\left (b^{6} + 32 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} + 32 \, {\left (5 \, b^{2} c^{2} + 4 \, a c^{3}\right )} d^{2} - 2 \, {\left (19 \, b^{4} c + 104 \, a b^{2} c^{2} + 48 \, a^{2} c^{3}\right )} d\right )} x^{2} - 4 \, {\left (2 \, a b^{3} + 2 \, {\left (b^{2} c^{2} + 4 \, a c^{3} - 8 \, c^{3} d\right )} x^{3} + 3 \, {\left (b^{3} c + 4 \, a b c^{2} - 8 \, b c^{2} d\right )} x^{2} - {\left (b^{3} + 4 \, a b c\right )} d + {\left (b^{4} + 8 \, a b^{2} c - 2 \, {\left (5 \, b^{2} c + 4 \, a c^{2}\right )} d\right )} x\right )} \sqrt {a b^{2} + 4 \, c d^{2} - {\left (b^{2} + 4 \, a c\right )} d} \sqrt {c x^{2} + b x + a} - 8 \, {\left (a b^{4} + 4 \, a^{2} b^{2} c\right )} d + 2 \, {\left (4 \, a b^{5} + 16 \, a^{2} b^{3} c + 16 \, {\left (b^{3} c + 4 \, a b c^{2}\right )} d^{2} - {\left (3 \, b^{5} + 40 \, a b^{3} c + 48 \, a^{2} b c^{2}\right )} d\right )} x}{c^{2} x^{4} + 2 \, b c x^{3} + 2 \, b d x + {\left (b^{2} + 2 \, c d\right )} x^{2} + d^{2}}\right )}{2 \, \sqrt {a b^{2} + 4 \, c d^{2} - {\left (b^{2} + 4 \, a c\right )} d}}, -\frac {\sqrt {-a b^{2} - 4 \, c d^{2} + {\left (b^{2} + 4 \, a c\right )} d} \arctan \left (-\frac {{\left (2 \, a b^{2} + {\left (b^{2} c + 4 \, a c^{2} - 8 \, c^{2} d\right )} x^{2} - {\left (b^{2} + 4 \, a c\right )} d + {\left (b^{3} + 4 \, a b c - 8 \, b c d\right )} x\right )} \sqrt {-a b^{2} - 4 \, c d^{2} + {\left (b^{2} + 4 \, a c\right )} d} \sqrt {c x^{2} + b x + a}}{2 \, {\left (a^{2} b^{3} + 4 \, a b c d^{2} + 2 \, {\left (a b^{2} c^{2} + 4 \, c^{3} d^{2} - {\left (b^{2} c^{2} + 4 \, a c^{3}\right )} d\right )} x^{3} + 3 \, {\left (a b^{3} c + 4 \, b c^{2} d^{2} - {\left (b^{3} c + 4 \, a b c^{2}\right )} d\right )} x^{2} - {\left (a b^{3} + 4 \, a^{2} b c\right )} d + {\left (a b^{4} + 2 \, a^{2} b^{2} c + 4 \, {\left (b^{2} c + 2 \, a c^{2}\right )} d^{2} - {\left (b^{4} + 6 \, a b^{2} c + 8 \, a^{2} c^{2}\right )} d\right )} x\right )}}\right )}{a b^{2} + 4 \, c d^{2} - {\left (b^{2} + 4 \, a c\right )} d}\right ] \end {gather*}

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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giac [B]  time = 1.90, size = 703, normalized size = 10.65 \begin {gather*} -\frac {\log \left ({\left | -{\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} b^{2} c - 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a c^{2} + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} c^{2} d - {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b^{3} \sqrt {c} - 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a b c^{\frac {3}{2}} + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b c^{\frac {3}{2}} d - 3 \, a b^{2} c + 4 \, \sqrt {a b^{2} - b^{2} d - 4 \, a c d + 4 \, c d^{2}} {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} c^{\frac {3}{2}} + 4 \, a^{2} c^{2} + 2 \, b^{2} c d + 4 \, \sqrt {a b^{2} - b^{2} d - 4 \, a c d + 4 \, c d^{2}} {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b c + \sqrt {a b^{2} - b^{2} d - 4 \, a c d + 4 \, c d^{2}} b^{2} \sqrt {c} \right |}\right )}{\sqrt {a b^{2} - b^{2} d - 4 \, a c d + 4 \, c d^{2}}} + \frac {\log \left ({\left | -{\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} b^{2} c - 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a c^{2} + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} c^{2} d - {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b^{3} \sqrt {c} - 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a b c^{\frac {3}{2}} + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b c^{\frac {3}{2}} d - 3 \, a b^{2} c - 4 \, \sqrt {a b^{2} - b^{2} d - 4 \, a c d + 4 \, c d^{2}} {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} c^{\frac {3}{2}} + 4 \, a^{2} c^{2} + 2 \, b^{2} c d - 4 \, \sqrt {a b^{2} - b^{2} d - 4 \, a c d + 4 \, c d^{2}} {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b c - \sqrt {a b^{2} - b^{2} d - 4 \, a c d + 4 \, c d^{2}} b^{2} \sqrt {c} \right |}\right )}{\sqrt {a b^{2} - b^{2} d - 4 \, a c d + 4 \, c d^{2}}} \end {gather*}

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

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maple [B]  time = 0.03, size = 307, normalized size = 4.65 \begin {gather*} -\frac {\ln \left (\frac {2 a -2 d +\sqrt {b^{2}-4 c d}\, \left (x -\frac {-b +\sqrt {b^{2}-4 c d}}{2 c}\right )+2 \sqrt {a -d}\, \sqrt {a +\left (x -\frac {-b +\sqrt {b^{2}-4 c d}}{2 c}\right )^{2} c -d +\sqrt {b^{2}-4 c d}\, \left (x -\frac {-b +\sqrt {b^{2}-4 c d}}{2 c}\right )}}{x -\frac {-b +\sqrt {b^{2}-4 c d}}{2 c}}\right )}{\sqrt {b^{2}-4 c d}\, \sqrt {a -d}}+\frac {\ln \left (\frac {2 a -2 d -\sqrt {b^{2}-4 c d}\, \left (x +\frac {b +\sqrt {b^{2}-4 c d}}{2 c}\right )+2 \sqrt {a -d}\, \sqrt {a +\left (x +\frac {b +\sqrt {b^{2}-4 c d}}{2 c}\right )^{2} c -d -\sqrt {b^{2}-4 c d}\, \left (x +\frac {b +\sqrt {b^{2}-4 c d}}{2 c}\right )}}{x +\frac {b +\sqrt {b^{2}-4 c d}}{2 c}}\right )}{\sqrt {b^{2}-4 c d}\, \sqrt {a -d}} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*c*d-b^2>0)', see `assume?` f
or more details)Is 4*c*d-b^2 positive, negative or zero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{\sqrt {c\,x^2+b\,x+a}\,\left (c\,x^2+b\,x+d\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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