3.101 \(\int \frac{1}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=79 \[ \frac{b \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{\sqrt{a} (b c-a d)^{3/2}}-\frac{d x}{c \sqrt{c+d x^2} (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.121559, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{b \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{\sqrt{a} (b c-a d)^{3/2}}-\frac{d x}{c \sqrt{c+d x^2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x^2)*(c + d*x^2)^(3/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 20.6043, size = 66, normalized size = 0.84 \[ \frac{d x}{c \sqrt{c + d x^{2}} \left (a d - b c\right )} - \frac{b \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{\sqrt{a} \left (a d - b c\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**2+a)/(d*x**2+c)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.151802, size = 78, normalized size = 0.99 \[ \frac{d x}{c \sqrt{c+d x^2} (a d-b c)}+\frac{b \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{\sqrt{a} (b c-a d)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x^2)*(c + d*x^2)^(3/2)),x]

[Out]

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

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Maple [B]  time = 0.054, size = 628, normalized size = 8. \[ -{\frac{b}{2\,ad-2\,bc}{\frac{1}{\sqrt{-ab}}}{\frac{1}{\sqrt{ \left ( x-{\frac{1}{b}\sqrt{-ab}} \right ) ^{2}d+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}}}}}+{\frac{dx}{ \left ( 2\,ad-2\,bc \right ) c}{\frac{1}{\sqrt{ \left ( x-{\frac{1}{b}\sqrt{-ab}} \right ) ^{2}d+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}}}}}+{\frac{b}{2\,ad-2\,bc}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ( x-{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-ab}}}{\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}}+{\frac{b}{2\,ad-2\,bc}{\frac{1}{\sqrt{-ab}}}{\frac{1}{\sqrt{ \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ) ^{2}d-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}}}}}+{\frac{dx}{ \left ( 2\,ad-2\,bc \right ) c}{\frac{1}{\sqrt{ \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ) ^{2}d-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}}}}}-{\frac{b}{2\,ad-2\,bc}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-ab}}}{\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^2+a)/(d*x^2+c)^(3/2),x)

[Out]

-1/2/(-a*b)^(1/2)/(a*d-b*c)*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*d-b*c)/c/((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*d+1/2/(-a*b)^(1/2)/(a*
d-b*c)*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*b)^(1/2)/(a*
d-b*c)*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*d-b*c)/c/((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*d-1/2/(-a*b)^(1/2)/(a*d-b*c)*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(1/((b*x^2 + a)*(d*x^2 + c)^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.309897, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \, \sqrt{-a b c + a^{2} d} \sqrt{d x^{2} + c} d x +{\left (b c d x^{2} + b c^{2}\right )} \log \left (\frac{{\left ({\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2}\right )} \sqrt{-a b c + a^{2} d} - 4 \,{\left ({\left (a b^{2} c^{2} - 3 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} x^{3} -{\left (a^{2} b c^{2} - a^{3} c d\right )} x\right )} \sqrt{d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )}{4 \,{\left (b c^{3} - a c^{2} d +{\left (b c^{2} d - a c d^{2}\right )} x^{2}\right )} \sqrt{-a b c + a^{2} d}}, -\frac{2 \, \sqrt{a b c - a^{2} d} \sqrt{d x^{2} + c} d x -{\left (b c d x^{2} + b c^{2}\right )} \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{2} - a c}{2 \, \sqrt{a b c - a^{2} d} \sqrt{d x^{2} + c} x}\right )}{2 \,{\left (b c^{3} - a c^{2} d +{\left (b c^{2} d - a c d^{2}\right )} x^{2}\right )} \sqrt{a b c - a^{2} d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)*(d*x^2 + c)^(3/2)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x**2+a)/(d*x**2+c)**(3/2),x)

[Out]

Integral(1/((a + b*x**2)*(c + d*x**2)**(3/2)), x)

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GIAC/XCAS [A]  time = 0.230092, size = 146, normalized size = 1.85 \[ -\frac{b \sqrt{d} \arctan \left (\frac{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt{a b c d - a^{2} d^{2}}}\right )}{\sqrt{a b c d - a^{2} d^{2}}{\left (b c - a d\right )}} - \frac{d x}{{\left (b c^{2} - a c d\right )} \sqrt{d x^{2} + c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)*(d*x^2 + c)^(3/2)),x, algorithm="giac")

[Out]

-b*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*c - a*d)) - d*x/((b*c^2 - a*c*d)*sq
rt(d*x^2 + c))