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

Optimal. Leaf size=82 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{b \sqrt{d}}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{b \sqrt{b c-a d}} \]

[Out]

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

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Rubi [A]  time = 0.155655, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{b \sqrt{d}}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{b \sqrt{b c-a d}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 23.7468, size = 70, normalized size = 0.85 \[ - \frac{\sqrt{a} \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{b \sqrt{a d - b c}} + \frac{\operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{b \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0846598, size = 85, normalized size = 1.04 \[ \frac{\log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )}{b \sqrt{d}}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{b \sqrt{b c-a d}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.015, size = 337, normalized size = 4.1 \[{\frac{1}{b}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){\frac{1}{\sqrt{d}}}}+{\frac{a}{2\,b}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ( x-{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-ab}}}{\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}}-{\frac{a}{2\,b}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-ab}}}{\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/b*ln(x*d^(1/2)+(d*x^2+c)^(1/2))/d^(1/2)+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)))-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)))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.355237, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{d} \sqrt{-\frac{a}{b c - a d}} \log \left (\frac{{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} - 4 \,{\left ({\left (b^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{3} -{\left (a b c^{2} - a^{2} c d\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{-\frac{a}{b c - a d}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 2 \, \log \left (-2 \, \sqrt{d x^{2} + c} d x -{\left (2 \, d x^{2} + c\right )} \sqrt{d}\right )}{4 \, b \sqrt{d}}, \frac{\sqrt{-d} \sqrt{-\frac{a}{b c - a d}} \log \left (\frac{{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} - 4 \,{\left ({\left (b^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{3} -{\left (a b c^{2} - a^{2} c d\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{-\frac{a}{b c - a d}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \, \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right )}{4 \, b \sqrt{-d}}, -\frac{\sqrt{d} \sqrt{\frac{a}{b c - a d}} \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{2} - a c}{2 \, \sqrt{d x^{2} + c}{\left (b c - a d\right )} x \sqrt{\frac{a}{b c - a d}}}\right ) - \log \left (-2 \, \sqrt{d x^{2} + c} d x -{\left (2 \, d x^{2} + c\right )} \sqrt{d}\right )}{2 \, b \sqrt{d}}, -\frac{\sqrt{-d} \sqrt{\frac{a}{b c - a d}} \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{2} - a c}{2 \, \sqrt{d x^{2} + c}{\left (b c - a d\right )} x \sqrt{\frac{a}{b c - a d}}}\right ) - 2 \, \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right )}{2 \, b \sqrt{-d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.263037, size = 138, normalized size = 1.68 \[ \frac{a \sqrt{d} \arctan \left (\frac{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt{a b c d - a^{2} d^{2}}}\right )}{\sqrt{a b c d - a^{2} d^{2}} b} - \frac{{\rm ln}\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2}\right )}{2 \, b \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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