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

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 = 0.16826, 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 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a-d} (b+2 c x)}{\sqrt{b^2-4 c d} \sqrt{a+b x+c x^2}}\right )}{\sqrt{a-d} \sqrt{b^2-4 c d}} \]

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])

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Rubi in Sympy [A]  time = 20.1515, size = 61, normalized size = 0.92 \[ - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{a - d} \left (b + 2 c x\right )}{\sqrt{b^{2} - 4 c d} \sqrt{a + b x + c x^{2}}} \right )}}{\sqrt{a - d} \sqrt{b^{2} - 4 c d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c*x**2+b*x+d)/(c*x**2+b*x+a)**(1/2),x)

[Out]

-2*atanh(sqrt(a - d)*(b + 2*c*x)/(sqrt(b**2 - 4*c*d)*sqrt(a + b*x + c*x**2)))/(s
qrt(a - d)*sqrt(b**2 - 4*c*d))

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Mathematica [B]  time = 0.472859, size = 249, normalized size = 3.77 \[ \frac{-\log \left (4 c \left (-\sqrt{a-d} \sqrt{b^2-4 c d} \sqrt{a+x (b+c x)}+a \left (-\sqrt{b^2-4 c d}\right )+2 c d x\right )-b^3+b^2 \left (\sqrt{b^2-4 c d}-2 c x\right )+4 b c d\right )+\log \left (-4 c \left (\sqrt{a-d} \sqrt{b^2-4 c d} \sqrt{a+x (b+c x)}+a \sqrt{b^2-4 c d}+2 c d x\right )+b^3+b^2 \left (\sqrt{b^2-4 c d}+2 c x\right )-4 b c d\right )+\log \left (-\sqrt{b^2-4 c d}+b+2 c x\right )-\log \left (\sqrt{b^2-4 c d}+b+2 c x\right )}{\sqrt{a-d} \sqrt{b^2-4 c d}} \]

Antiderivative was successfully verified.

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

[Out]

(Log[b - Sqrt[b^2 - 4*c*d] + 2*c*x] - Log[b + Sqrt[b^2 - 4*c*d] + 2*c*x] - Log[-
b^3 + 4*b*c*d + b^2*(Sqrt[b^2 - 4*c*d] - 2*c*x) + 4*c*(-(a*Sqrt[b^2 - 4*c*d]) +
2*c*d*x - Sqrt[a - d]*Sqrt[b^2 - 4*c*d]*Sqrt[a + x*(b + c*x)])] + Log[b^3 - 4*b*
c*d + b^2*(Sqrt[b^2 - 4*c*d] + 2*c*x) - 4*c*(a*Sqrt[b^2 - 4*c*d] + 2*c*d*x + Sqr
t[a - d]*Sqrt[b^2 - 4*c*d]*Sqrt[a + x*(b + c*x)])])/(Sqrt[a - d]*Sqrt[b^2 - 4*c*
d])

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Maple [B]  time = 0.038, size = 307, normalized size = 4.7 \[ -{1\ln \left ({1 \left ( 2\,a-2\,d+\sqrt{{b}^{2}-4\,cd} \left ( x-{\frac{1}{2\,c} \left ( -b+\sqrt{{b}^{2}-4\,cd} \right ) } \right ) +2\,\sqrt{a-d}\sqrt{ \left ( x-1/2\,{\frac{-b+\sqrt{{b}^{2}-4\,cd}}{c}} \right ) ^{2}c+\sqrt{{b}^{2}-4\,cd} \left ( x-1/2\,{\frac{-b+\sqrt{{b}^{2}-4\,cd}}{c}} \right ) +a-d} \right ) \left ( x-{\frac{1}{2\,c} \left ( -b+\sqrt{{b}^{2}-4\,cd} \right ) } \right ) ^{-1}} \right ){\frac{1}{\sqrt{{b}^{2}-4\,cd}}}{\frac{1}{\sqrt{a-d}}}}+{1\ln \left ({1 \left ( 2\,a-2\,d-\sqrt{{b}^{2}-4\,cd} \left ( x+{\frac{1}{2\,c} \left ( b+\sqrt{{b}^{2}-4\,cd} \right ) } \right ) +2\,\sqrt{a-d}\sqrt{ \left ( x+1/2\,{\frac{b+\sqrt{{b}^{2}-4\,cd}}{c}} \right ) ^{2}c-\sqrt{{b}^{2}-4\,cd} \left ( x+1/2\,{\frac{b+\sqrt{{b}^{2}-4\,cd}}{c}} \right ) +a-d} \right ) \left ( x+{\frac{1}{2\,c} \left ( b+\sqrt{{b}^{2}-4\,cd} \right ) } \right ) ^{-1}} \right ){\frac{1}{\sqrt{{b}^{2}-4\,cd}}}{\frac{1}{\sqrt{a-d}}}} \]

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+(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))+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]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.358473, size = 1, normalized size = 0.02 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^2 + b*x + a)*(c*x^2 + b*x + d)),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 + 1
04*a*b^2*c^2 + 48*a^2*c^3)*d)*x^2 - 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)*sqrt(a*b^2 + 4*c*d^2 - (b^2 + 4*a*c)*d) - 4*(2*a^2*b^5 - 4*(b^3*c + 4*a*b*c^2
)*d^3 + 2*(a*b^4*c^2 + 4*a^2*b^2*c^3 - 32*c^4*d^3 + 12*(b^2*c^3 + 4*a*c^4)*d^2 -
(b^4*c^2 + 16*a*b^2*c^3 + 16*a^2*c^4)*d)*x^3 + (b^5 + 16*a*b^3*c + 16*a^2*b*c^2
)*d^2 + 3*(a*b^5*c + 4*a^2*b^3*c^2 - 32*b*c^3*d^3 + 12*(b^3*c^2 + 4*a*b*c^3)*d^2
- (b^5*c + 16*a*b^3*c^2 + 16*a^2*b*c^3)*d)*x^2 - 3*(a*b^5 + 4*a^2*b^3*c)*d + (a
*b^6 + 8*a^2*b^4*c - 8*(5*b^2*c^2 + 4*a*c^3)*d^3 + 2*(7*b^4*c + 40*a*b^2*c^2 + 1
6*a^2*c^3)*d^2 - (b^6 + 22*a*b^4*c + 40*a^2*b^2*c^2)*d)*x)*sqrt(c*x^2 + b*x + a)
)/(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), 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)/((a*b^3 + 4*b*c*d^2 - (b^3 + 4*a*b*c)*d + 2*(a*b^2*c + 4*c^2*d^2 - (b^2*c
+ 4*a*c^2)*d)*x)*sqrt(c*x^2 + b*x + a)))/sqrt(-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. \[ \int \frac{1}{\sqrt{a + b x + c x^{2}} \left (b x + c x^{2} + d\right )}\, dx \]

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/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: RuntimeError