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

Optimal. Leaf size=147 \[ \frac{\sqrt{a c^2-d^2} \tan ^{-1}\left (\frac{\sqrt{b} d x}{\sqrt{a+b x^2} \sqrt{a c^2-d^2}}\right )}{b^{3/2} c^2}-\frac{\sqrt{a c^2-d^2} \tan ^{-1}\left (\frac{\sqrt{b} c x}{\sqrt{a c^2-d^2}}\right )}{b^{3/2} c^2}-\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{3/2} c^2}+\frac{x}{b c} \]

[Out]

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

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Rubi [A]  time = 0.478813, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241 \[ \frac{\sqrt{a c^2-d^2} \tan ^{-1}\left (\frac{\sqrt{b} d x}{\sqrt{a+b x^2} \sqrt{a c^2-d^2}}\right )}{b^{3/2} c^2}-\frac{\sqrt{a c^2-d^2} \tan ^{-1}\left (\frac{\sqrt{b} c x}{\sqrt{a c^2-d^2}}\right )}{b^{3/2} c^2}-\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{3/2} c^2}+\frac{x}{b c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 45.5677, size = 128, normalized size = 0.87 \[ \frac{x}{b c} - \frac{d \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{b^{\frac{3}{2}} c^{2}} - \frac{\sqrt{- a c^{2} + d^{2}} \operatorname{atanh}{\left (\frac{\sqrt{b} c x}{\sqrt{- a c^{2} + d^{2}}} \right )}}{b^{\frac{3}{2}} c^{2}} + \frac{\sqrt{- a c^{2} + d^{2}} \operatorname{atanh}{\left (\frac{\sqrt{b} d x}{\sqrt{a + b x^{2}} \sqrt{- a c^{2} + d^{2}}} \right )}}{b^{\frac{3}{2}} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.190331, size = 157, normalized size = 1.07 \[ \frac{\sqrt{a c^2-d^2} \left (\sqrt{b} c x-d \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )\right )+\left (a c^2-d^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} d x}{\sqrt{a+b x^2} \sqrt{a c^2-d^2}}\right )+\left (d^2-a c^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} c x}{\sqrt{a c^2-d^2}}\right )}{b^{3/2} c^2 \sqrt{a c^2-d^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.043, size = 3501, normalized size = 23.8 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*d*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*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))/b^(1/2)-1/2*d*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*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))/b^(1/2)-1/2*d*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)*a+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))/(-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)*d^3-1/2*d*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))/b^(1/2)*a+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))*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))/b^(1/2)*d^3+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))/(-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))*a-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))/(-c^2*b*(a*c^2-d^2))^(1/2)*d^5/(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*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)*a-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))/(-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)*d^3-1/2*d*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))/b^(1/2)*a+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))*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))/b^(1/2)*d^3-1/2*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))/
(-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))*a+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))/(-c^2*b*(a*c^2-d^2))^(1/2)*d^5/(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))-a/b/(b*(a*c^2-d^2))^(1/2)*arctan(b*c*x/(b*(a*c^2-d^2))^(1/2))+x/b/c+1/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))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.422737, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.286272, size = 244, normalized size = 1.66 \[ \frac{x}{b c} - \frac{{\left (a c^{2} - d^{2}\right )} \arctan \left (\frac{b c x}{\sqrt{a b c^{2} - b d^{2}}}\right )}{\sqrt{a b c^{2} - b d^{2}} b c^{2}} + \frac{d{\rm ln}\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2}\right )}{2 \, b^{\frac{3}{2}} c^{2}} - \frac{{\left (a c^{2} d - d^{3}\right )} \arctan \left (\frac{{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} c^{2} + a c^{2} - 2 \, d^{2}}{2 \, \sqrt{a c^{2} - d^{2}} d}\right )}{\sqrt{a c^{2} - d^{2}} b^{\frac{3}{2}} c^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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