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

Optimal. Leaf size=160 \[ \frac{d \sqrt{a+b x^2}}{a x \left (a c^2-d^2\right )}+\frac{\sqrt{b} c^2 \tan ^{-1}\left (\frac{\sqrt{b} d x}{\sqrt{a+b x^2} \sqrt{a c^2-d^2}}\right )}{\left (a c^2-d^2\right )^{3/2}}-\frac{\sqrt{b} c^2 \tan ^{-1}\left (\frac{\sqrt{b} c x}{\sqrt{a c^2-d^2}}\right )}{\left (a c^2-d^2\right )^{3/2}}-\frac{c}{x \left (a c^2-d^2\right )} \]

[Out]

-(c/((a*c^2 - d^2)*x)) + (d*Sqrt[a + b*x^2])/(a*(a*c^2 - d^2)*x) - (Sqrt[b]*c^2*
ArcTan[(Sqrt[b]*c*x)/Sqrt[a*c^2 - d^2]])/(a*c^2 - d^2)^(3/2) + (Sqrt[b]*c^2*ArcT
an[(Sqrt[b]*d*x)/(Sqrt[a*c^2 - d^2]*Sqrt[a + b*x^2])])/(a*c^2 - d^2)^(3/2)

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Rubi [A]  time = 0.505991, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ \frac{d \sqrt{a+b x^2}}{a x \left (a c^2-d^2\right )}+\frac{\sqrt{b} c^2 \tan ^{-1}\left (\frac{\sqrt{b} d x}{\sqrt{a+b x^2} \sqrt{a c^2-d^2}}\right )}{\left (a c^2-d^2\right )^{3/2}}-\frac{\sqrt{b} c^2 \tan ^{-1}\left (\frac{\sqrt{b} c x}{\sqrt{a c^2-d^2}}\right )}{\left (a c^2-d^2\right )^{3/2}}-\frac{c}{x \left (a c^2-d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(c/((a*c^2 - d^2)*x)) + (d*Sqrt[a + b*x^2])/(a*(a*c^2 - d^2)*x) - (Sqrt[b]*c^2*
ArcTan[(Sqrt[b]*c*x)/Sqrt[a*c^2 - d^2]])/(a*c^2 - d^2)^(3/2) + (Sqrt[b]*c^2*ArcT
an[(Sqrt[b]*d*x)/(Sqrt[a*c^2 - d^2]*Sqrt[a + b*x^2])])/(a*c^2 - d^2)^(3/2)

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Rubi in Sympy [A]  time = 50.8058, size = 129, normalized size = 0.81 \[ - \frac{\sqrt{b} c^{2} \operatorname{atanh}{\left (\frac{\sqrt{b} c x}{\sqrt{- a c^{2} + d^{2}}} \right )}}{\left (- a c^{2} + d^{2}\right )^{\frac{3}{2}}} + \frac{\sqrt{b} c^{2} \operatorname{atanh}{\left (\frac{\sqrt{b} d x}{\sqrt{a + b x^{2}} \sqrt{- a c^{2} + d^{2}}} \right )}}{\left (- a c^{2} + d^{2}\right )^{\frac{3}{2}}} + \frac{c}{x \left (- a c^{2} + d^{2}\right )} - \frac{d \sqrt{a + b x^{2}}}{a x \left (- a c^{2} + d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(b)*c**2*atanh(sqrt(b)*c*x/sqrt(-a*c**2 + d**2))/(-a*c**2 + d**2)**(3/2) +
sqrt(b)*c**2*atanh(sqrt(b)*d*x/(sqrt(a + b*x**2)*sqrt(-a*c**2 + d**2)))/(-a*c**2
+ d**2)**(3/2) + c/(x*(-a*c**2 + d**2)) - d*sqrt(a + b*x**2)/(a*x*(-a*c**2 + d*
*2))

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Mathematica [A]  time = 0.17119, size = 139, normalized size = 0.87 \[ \frac{\sqrt{a c^2-d^2} \left (d \sqrt{a+b x^2}-a c\right )+a \sqrt{b} c^2 x \tan ^{-1}\left (\frac{\sqrt{b} d x}{\sqrt{a+b x^2} \sqrt{a c^2-d^2}}\right )-a \sqrt{b} c^2 x \tan ^{-1}\left (\frac{\sqrt{b} c x}{\sqrt{a c^2-d^2}}\right )}{a x \left (a c^2-d^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a*c^2 - d^2]*(-(a*c) + d*Sqrt[a + b*x^2]) - a*Sqrt[b]*c^2*x*ArcTan[(Sqrt[b
]*c*x)/Sqrt[a*c^2 - d^2]] + a*Sqrt[b]*c^2*x*ArcTan[(Sqrt[b]*d*x)/(Sqrt[a*c^2 - d
^2]*Sqrt[a + b*x^2])])/(a*(a*c^2 - d^2)^(3/2)*x)

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Maple [B]  time = 0.044, size = 2289, normalized size = 14.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*c^2/d^2/(b*(a*c^2-d^2))^(1/2)*arctan(b*c*x/(b*(a*c^2-d^2))^(1/2))-a*c^4/(a*c^2
-d^2)*b/d^2/(b*(a*c^2-d^2))^(1/2)*arctan(b*c*x/(b*(a*c^2-d^2))^(1/2))-c/(a*c^2-d
^2)/x+d/a^2/(a*c^2-d^2)/x*(b*x^2+a)^(3/2)-d/a^2/(a*c^2-d^2)*b*x*(b*x^2+a)^(1/2)-
d/a/(a*c^2-d^2)*b^(1/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))+1/2*d*b^2*c^2/a/(-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^(3/2)*c^2/a/((-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^2*c^2/a/(-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^(3/2)*c^2/a/((-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^2*c^6/(a*c^2-d^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)
*((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^(3/2)*c^4/(a*c^2-d^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^2*c^4/(a*c^2-
d^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^2*c^6/(a*c^2-d^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)*((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^(3/2)*c^4/(a*c^2-d^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^2*c^4/(a*c^2-d^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))

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

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^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.39092, size = 1, normalized size = 0.01 \[ \left [-\frac{a c^{2} x \sqrt{-\frac{b}{a c^{2} - d^{2}}} \log \left (\frac{a^{4} c^{4} - 2 \, a^{3} c^{2} d^{2} + a^{2} d^{4} +{\left (a^{2} b^{2} c^{4} - 8 \, a b^{2} c^{2} d^{2} + 8 \, b^{2} d^{4}\right )} x^{4} + 2 \,{\left (a^{3} b c^{4} - 5 \, a^{2} b c^{2} d^{2} + 4 \, a b d^{4}\right )} x^{2} + 4 \,{\left ({\left (a^{2} b c^{4} d - 3 \, a b c^{2} d^{3} + 2 \, b d^{5}\right )} x^{3} +{\left (a^{3} c^{4} d - 2 \, a^{2} c^{2} d^{3} + a d^{5}\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{-\frac{b}{a c^{2} - d^{2}}}}{b^{2} c^{4} x^{4} + a^{2} c^{4} - 2 \, a c^{2} d^{2} + d^{4} + 2 \,{\left (a b c^{4} - b c^{2} d^{2}\right )} x^{2}}\right ) + 2 \, a c^{2} x \sqrt{-\frac{b}{a c^{2} - d^{2}}} \log \left (\frac{b c^{2} x^{2} - a c^{2} + 2 \,{\left (a c^{3} - c d^{2}\right )} x \sqrt{-\frac{b}{a c^{2} - d^{2}}} + d^{2}}{b c^{2} x^{2} + a c^{2} - d^{2}}\right ) + 4 \, a c - 4 \, \sqrt{b x^{2} + a} d}{4 \,{\left (a^{2} c^{2} - a d^{2}\right )} x}, \frac{2 \, a c^{2} x \sqrt{\frac{b}{a c^{2} - d^{2}}} \arctan \left (-\frac{b c x}{{\left (a c^{2} - d^{2}\right )} \sqrt{\frac{b}{a c^{2} - d^{2}}}}\right ) - a c^{2} x \sqrt{\frac{b}{a c^{2} - d^{2}}} \arctan \left (\frac{a^{2} c^{2} - a d^{2} +{\left (a b c^{2} - 2 \, b d^{2}\right )} x^{2}}{2 \,{\left (a c^{2} d - d^{3}\right )} \sqrt{b x^{2} + a} x \sqrt{\frac{b}{a c^{2} - d^{2}}}}\right ) - 2 \, a c + 2 \, \sqrt{b x^{2} + a} d}{2 \,{\left (a^{2} c^{2} - a d^{2}\right )} x}\right ] \]

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^2),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.284054, size = 285, normalized size = 1.78 \[ -b^{\frac{3}{2}} d{\left (\frac{c^{2} \arctan \left (\frac{{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} c^{2} + a c^{2} - 2 \, d^{2}}{2 \, \sqrt{a c^{2} - d^{2}} d}\right )}{{\left (a b c^{2} - b d^{2}\right )} \sqrt{a c^{2} - d^{2}} d} + \frac{2}{{\left (a b c^{2} - b d^{2}\right )}{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}}\right )} - \frac{b c^{2} \arctan \left (\frac{b c x}{\sqrt{a b c^{2} - b d^{2}}}\right )}{\sqrt{a b c^{2} - b d^{2}}{\left (a c^{2} - d^{2}\right )}} - \frac{c}{{\left (a c^{2} - d^{2}\right )} x} \]

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^2),x, algorithm="giac")

[Out]

-b^(3/2)*d*(c^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))/((a*b*c^2 - b*d^2)*sqrt(a*c^2 - d^2)*d) + 2/((a*b*c^2 -
b*d^2)*((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a))) - b*c^2*arctan(b*c*x/sqrt(a*b*c^2
- b*d^2))/(sqrt(a*b*c^2 - b*d^2)*(a*c^2 - d^2)) - c/((a*c^2 - d^2)*x)