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

Optimal. Leaf size=103 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} c x}{\sqrt{a c^2-d^2}}\right )}{\sqrt{b} \sqrt{a c^2-d^2}}-\frac{\tan ^{-1}\left (\frac{\sqrt{b} d x}{\sqrt{a+b x^2} \sqrt{a c^2-d^2}}\right )}{\sqrt{b} \sqrt{a c^2-d^2}} \]

[Out]

ArcTan[(Sqrt[b]*c*x)/Sqrt[a*c^2 - d^2]]/(Sqrt[b]*Sqrt[a*c^2 - d^2]) - ArcTan[(Sq
rt[b]*d*x)/(Sqrt[a*c^2 - d^2]*Sqrt[a + b*x^2])]/(Sqrt[b]*Sqrt[a*c^2 - d^2])

_______________________________________________________________________________________

Rubi [A]  time = 0.147746, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} c x}{\sqrt{a c^2-d^2}}\right )}{\sqrt{b} \sqrt{a c^2-d^2}}-\frac{\tan ^{-1}\left (\frac{\sqrt{b} d x}{\sqrt{a+b x^2} \sqrt{a c^2-d^2}}\right )}{\sqrt{b} \sqrt{a c^2-d^2}} \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[(Sqrt[b]*c*x)/Sqrt[a*c^2 - d^2]]/(Sqrt[b]*Sqrt[a*c^2 - d^2]) - ArcTan[(Sq
rt[b]*d*x)/(Sqrt[a*c^2 - d^2]*Sqrt[a + b*x^2])]/(Sqrt[b]*Sqrt[a*c^2 - d^2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 23.8457, size = 87, normalized size = 0.84 \[ - \frac{\operatorname{atanh}{\left (\frac{\sqrt{b} c x}{\sqrt{- a c^{2} + d^{2}}} \right )}}{\sqrt{b} \sqrt{- a c^{2} + d^{2}}} + \frac{\operatorname{atanh}{\left (\frac{\sqrt{b} d x}{\sqrt{a + b x^{2}} \sqrt{- a c^{2} + d^{2}}} \right )}}{\sqrt{b} \sqrt{- a c^{2} + d^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0584958, size = 83, normalized size = 0.81 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} c x}{\sqrt{a c^2-d^2}}\right )-\tan ^{-1}\left (\frac{\sqrt{b} d x}{\sqrt{a+b x^2} \sqrt{a c^2-d^2}}\right )}{\sqrt{b} \sqrt{a c^2-d^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.031, size = 2005, normalized size = 19.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*d*b*c^2/(-a*b)^(1/2)/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/((-a*b)^
(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2))*((x-1/b*(-a*b)^(1/2))^2*b+2*(-a*b)^(1/2)*(
x-1/b*(-a*b)^(1/2)))^(1/2)-1/2*d*b^(1/2)*c^2/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^
2))^(1/2))/((-a*b)^(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2))*ln(((x-1/b*(-a*b)^(1/2)
)*b+(-a*b)^(1/2))/b^(1/2)+((x-1/b*(-a*b)^(1/2))^2*b+2*(-a*b)^(1/2)*(x-1/b*(-a*b)
^(1/2)))^(1/2))+1/2*d*b*c^2/(-a*b)^(1/2)/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^
(1/2))/((-a*b)^(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2))*((x+1/b*(-a*b)^(1/2))^2*b-2
*(-a*b)^(1/2)*(x+1/b*(-a*b)^(1/2)))^(1/2)-1/2*d*b^(1/2)*c^2/((-a*b)^(1/2)*c^2+(-
c^2*b*(a*c^2-d^2))^(1/2))/((-a*b)^(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2))*ln(((x+1
/b*(-a*b)^(1/2))*b-(-a*b)^(1/2))/b^(1/2)+((x+1/b*(-a*b)^(1/2))^2*b-2*(-a*b)^(1/2
)*(x+1/b*(-a*b)^(1/2)))^(1/2))+1/2*d*b*c^4/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2)
)^(1/2))/((-a*b)^(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2))/(-c^2*b*(a*c^2-d^2))^(1/2
)*((x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)^2*b+2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x-
(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+d^2/c^2)^(1/2)+1/2*d*b^(1/2)*c^2/((-a*b)^(1/2)
*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/((-a*b)^(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2))*l
n(((-c^2*b*(a*c^2-d^2))^(1/2)/c^2+(x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)*b)/b^(1/2
)+((x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)^2*b+2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x-
(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+d^2/c^2)^(1/2))-1/2*b*c^2/((-a*b)^(1/2)*c^2+(-
c^2*b*(a*c^2-d^2))^(1/2))/((-a*b)^(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2))/(-c^2*b*
(a*c^2-d^2))^(1/2)*d^3/(d^2/c^2)^(1/2)*ln((2*d^2/c^2+2*(-c^2*b*(a*c^2-d^2))^(1/2
)/c^2*(x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+2*(d^2/c^2)^(1/2)*((x-(-c^2*b*(a*c^2-
d^2))^(1/2)/c^2/b)^2*b+2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x-(-c^2*b*(a*c^2-d^2))^
(1/2)/c^2/b)+d^2/c^2)^(1/2))/(x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b))-1/2*d*b*c^4/(
(-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/((-a*b)^(1/2)*c^2-(-c^2*b*(a*c^2-d^
2))^(1/2))/(-c^2*b*(a*c^2-d^2))^(1/2)*((x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)^2*b-
2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+d^2/c^2)^(
1/2)+1/2*d*b^(1/2)*c^2/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/((-a*b)^(1/
2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2))*ln((-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2+(x+(-c^2*
b*(a*c^2-d^2))^(1/2)/c^2/b)*b)/b^(1/2)+((x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)^2*b
-2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+d^2/c^2)^
(1/2))+1/2*b*c^2/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/((-a*b)^(1/2)*c^2
-(-c^2*b*(a*c^2-d^2))^(1/2))/(-c^2*b*(a*c^2-d^2))^(1/2)*d^3/(d^2/c^2)^(1/2)*ln((
2*d^2/c^2-2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+
2*(d^2/c^2)^(1/2)*((x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)^2*b-2*(-c^2*b*(a*c^2-d^2
))^(1/2)/c^2*(x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+d^2/c^2)^(1/2))/(x+(-c^2*b*(a*
c^2-d^2))^(1/2)/c^2/b))+1/(b*(a*c^2-d^2))^(1/2)*arctan(b*c*x/(b*(a*c^2-d^2))^(1/
2))

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{b c x^{2} + a c + \sqrt{b x^{2} + a} d}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.305486, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \log \left (\frac{2 \,{\left (a b c^{3} - b c d^{2}\right )} x +{\left (b c^{2} x^{2} - a c^{2} + d^{2}\right )} \sqrt{-a b c^{2} + b d^{2}}}{b c^{2} x^{2} + a c^{2} - d^{2}}\right ) + \log \left (\frac{{\left (a^{4} c^{4} - 2 \, a^{3} c^{2} d^{2} + a^{2} d^{4} +{\left (a^{2} b^{2} c^{4} - 8 \, a b^{2} c^{2} d^{2} + 8 \, b^{2} d^{4}\right )} x^{4} + 2 \,{\left (a^{3} b c^{4} - 5 \, a^{2} b c^{2} d^{2} + 4 \, a b d^{4}\right )} x^{2}\right )} \sqrt{-a b c^{2} + b d^{2}} + 4 \,{\left ({\left (a^{2} b^{2} c^{4} d - 3 \, a b^{2} c^{2} d^{3} + 2 \, b^{2} d^{5}\right )} x^{3} +{\left (a^{3} b c^{4} d - 2 \, a^{2} b c^{2} d^{3} + a b d^{5}\right )} x\right )} \sqrt{b x^{2} + a}}{b^{2} c^{4} x^{4} + a^{2} c^{4} - 2 \, a c^{2} d^{2} + d^{4} + 2 \,{\left (a b c^{4} - b c^{2} d^{2}\right )} x^{2}}\right )}{4 \, \sqrt{-a b c^{2} + b d^{2}}}, -\frac{2 \, \arctan \left (-\frac{\sqrt{a b c^{2} - b d^{2}} c x}{a c^{2} - d^{2}}\right ) - \arctan \left (\frac{{\left (a^{2} c^{2} - a d^{2} +{\left (a b c^{2} - 2 \, b d^{2}\right )} x^{2}\right )} \sqrt{a b c^{2} - b d^{2}}}{2 \,{\left (a b c^{2} d - b d^{3}\right )} \sqrt{b x^{2} + a} x}\right )}{2 \, \sqrt{a b c^{2} - b d^{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*(2*log((2*(a*b*c^3 - b*c*d^2)*x + (b*c^2*x^2 - a*c^2 + d^2)*sqrt(-a*b*c^2 +
b*d^2))/(b*c^2*x^2 + a*c^2 - d^2)) + log(((a^4*c^4 - 2*a^3*c^2*d^2 + a^2*d^4 +
(a^2*b^2*c^4 - 8*a*b^2*c^2*d^2 + 8*b^2*d^4)*x^4 + 2*(a^3*b*c^4 - 5*a^2*b*c^2*d^2
+ 4*a*b*d^4)*x^2)*sqrt(-a*b*c^2 + b*d^2) + 4*((a^2*b^2*c^4*d - 3*a*b^2*c^2*d^3
+ 2*b^2*d^5)*x^3 + (a^3*b*c^4*d - 2*a^2*b*c^2*d^3 + a*b*d^5)*x)*sqrt(b*x^2 + a))
/(b^2*c^4*x^4 + a^2*c^4 - 2*a*c^2*d^2 + d^4 + 2*(a*b*c^4 - b*c^2*d^2)*x^2)))/sqr
t(-a*b*c^2 + b*d^2), -1/2*(2*arctan(-sqrt(a*b*c^2 - b*d^2)*c*x/(a*c^2 - d^2)) -
arctan(1/2*(a^2*c^2 - a*d^2 + (a*b*c^2 - 2*b*d^2)*x^2)*sqrt(a*b*c^2 - b*d^2)/((a
*b*c^2*d - b*d^3)*sqrt(b*x^2 + a)*x)))/sqrt(a*b*c^2 - b*d^2)]

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{a c + b c x^{2} + d \sqrt{a + b x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.279895, size = 144, normalized size = 1.4 \[ \frac{\arctan \left (\frac{b c x}{\sqrt{a b c^{2} - b d^{2}}}\right )}{\sqrt{a b c^{2} - b d^{2}}} + \frac{\arctan \left (\frac{{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} c^{2} + a c^{2} - 2 \, d^{2}}{2 \, \sqrt{a c^{2} - d^{2}} d}\right )}{\sqrt{a c^{2} - d^{2}} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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