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

Optimal. Leaf size=112 \[ -\frac{\sqrt{a} \sqrt{b c-a d} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{b^2}+\frac{(b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{2 b^2 \sqrt{d}}+\frac{x \sqrt{c+d x^2}}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.270774, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{\sqrt{a} \sqrt{b c-a d} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{b^2}+\frac{(b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{2 b^2 \sqrt{d}}+\frac{x \sqrt{c+d x^2}}{2 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 40.5569, size = 99, normalized size = 0.88 \[ \frac{\sqrt{a} \sqrt{a d - b c} \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{b^{2}} + \frac{x \sqrt{c + d x^{2}}}{2 b} - \frac{\left (2 a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{2 b^{2} \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.214566, size = 108, normalized size = 0.96 \[ \frac{\frac{(b c-2 a d) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )}{\sqrt{d}}-2 \sqrt{a} \sqrt{b c-a d} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )+b x \sqrt{c+d x^2}}{2 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.018, size = 1010, normalized size = 9. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x*(d*x^2+c)^(1/2)/b+1/2/b*c/d^(1/2)*ln(x*d^(1/2)+(d*x^2+c)^(1/2))-1/2*a/(-a*
b)^(1/2)/b*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*
d-b*c)/b)^(1/2)-1/2*a/b^2*d^(1/2)*ln((d*(-a*b)^(1/2)/b+(x-1/b*(-a*b)^(1/2))*d)/d
^(1/2)+((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*
c)/b)^(1/2))-1/2*a^2/(-a*b)^(1/2)/b^2/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b+2*
d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x-1/b*(-a*b)^(1/2
))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x-1/b*(-a*b)
^(1/2)))*d+1/2*a/(-a*b)^(1/2)/b/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b+2*d*(-a*
b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x-1/b*(-a*b)^(1/2))^2*d
+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x-1/b*(-a*b)^(1/2)
))*c+1/2*a/(-a*b)^(1/2)/b*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-
a*b)^(1/2))-(a*d-b*c)/b)^(1/2)-1/2*a/b^2*d^(1/2)*ln((-d*(-a*b)^(1/2)/b+(x+1/b*(-
a*b)^(1/2))*d)/d^(1/2)+((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b
)^(1/2))-(a*d-b*c)/b)^(1/2))+1/2*a^2/(-a*b)^(1/2)/b^2/(-(a*d-b*c)/b)^(1/2)*ln((-
2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x
+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2
))/(x+1/b*(-a*b)^(1/2)))*d-1/2*a/(-a*b)^(1/2)/b/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d
-b*c)/b-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x+1/b*(
-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x+
1/b*(-a*b)^(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(sqrt(d*x^2 + c)*x^2/(b*x^2 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.331841, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{d x^{2} + c} b \sqrt{d} x -{\left (b c - 2 \, a d\right )} \log \left (2 \, \sqrt{d x^{2} + c} d x -{\left (2 \, d x^{2} + c\right )} \sqrt{d}\right ) + \sqrt{-a b c + a^{2} d} \sqrt{d} \log \left (\frac{{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} - 4 \,{\left ({\left (b c - 2 \, a d\right )} x^{3} - a c x\right )} \sqrt{-a b c + a^{2} d} \sqrt{d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )}{4 \, b^{2} \sqrt{d}}, \frac{2 \, \sqrt{d x^{2} + c} b \sqrt{-d} x + 2 \,{\left (b c - 2 \, a d\right )} \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right ) + \sqrt{-a b c + a^{2} d} \sqrt{-d} \log \left (\frac{{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} - 4 \,{\left ({\left (b c - 2 \, a d\right )} x^{3} - a c x\right )} \sqrt{-a b c + a^{2} d} \sqrt{d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )}{4 \, b^{2} \sqrt{-d}}, \frac{2 \, \sqrt{d x^{2} + c} b \sqrt{d} x + 2 \, \sqrt{a b c - a^{2} d} \sqrt{d} \arctan \left (-\frac{{\left (b c - 2 \, a d\right )} x^{2} - a c}{2 \, \sqrt{a b c - a^{2} d} \sqrt{d x^{2} + c} x}\right ) -{\left (b c - 2 \, a d\right )} \log \left (2 \, \sqrt{d x^{2} + c} d x -{\left (2 \, d x^{2} + c\right )} \sqrt{d}\right )}{4 \, b^{2} \sqrt{d}}, \frac{\sqrt{d x^{2} + c} b \sqrt{-d} x +{\left (b c - 2 \, a d\right )} \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right ) + \sqrt{a b c - a^{2} d} \sqrt{-d} \arctan \left (-\frac{{\left (b c - 2 \, a d\right )} x^{2} - a c}{2 \, \sqrt{a b c - a^{2} d} \sqrt{d x^{2} + c} x}\right )}{2 \, b^{2} \sqrt{-d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.24782, size = 185, normalized size = 1.65 \[ \frac{\sqrt{d x^{2} + c} x}{2 \, b} + \frac{{\left (a b c \sqrt{d} - a^{2} d^{\frac{3}{2}}\right )} \arctan \left (\frac{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt{a b c d - a^{2} d^{2}}}\right )}{\sqrt{a b c d - a^{2} d^{2}} b^{2}} - \frac{{\left (b c - 2 \, a d\right )}{\rm ln}\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2}\right )}{4 \, b^{2} \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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