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

Optimal. Leaf size=124 \[ -\frac{b^2 \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{a^{3/2} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^2} (b c-2 a d)}{a c^2 x (b c-a d)}-\frac{d}{c x \sqrt{c+d x^2} (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.36288, antiderivative size = 124, 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 \[ -\frac{b^2 \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{a^{3/2} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^2} (b c-2 a d)}{a c^2 x (b c-a d)}-\frac{d}{c x \sqrt{c+d x^2} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 58.2331, size = 100, normalized size = 0.81 \[ \frac{d}{c x \sqrt{c + d x^{2}} \left (a d - b c\right )} - \frac{\sqrt{c + d x^{2}} \left (2 a d - b c\right )}{a c^{2} x \left (a d - b c\right )} + \frac{b^{2} \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{a^{\frac{3}{2}} \left (a d - b c\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.366333, size = 102, normalized size = 0.82 \[ \frac{\frac{d^2 x^2}{b c-a d}-\frac{c+d x^2}{a}}{c^2 x \sqrt{c+d x^2}}-\frac{b^2 \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{a^{3/2} (b c-a d)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.02, size = 695, normalized size = 5.6 \[ -{\frac{1}{acx}{\frac{1}{\sqrt{d{x}^{2}+c}}}}-2\,{\frac{dx}{a{c}^{2}\sqrt{d{x}^{2}+c}}}+{\frac{{b}^{2}}{2\,a \left ( ad-bc \right ) }{\frac{1}{\sqrt{-ab}}}{\frac{1}{\sqrt{ \left ( x-{\frac{1}{b}\sqrt{-ab}} \right ) ^{2}d+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}}}}}-{\frac{bxd}{2\,a \left ( ad-bc \right ) c}{\frac{1}{\sqrt{ \left ( x-{\frac{1}{b}\sqrt{-ab}} \right ) ^{2}d+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}}}}}-{\frac{{b}^{2}}{2\,a \left ( ad-bc \right ) }\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ( x-{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-ab}}}{\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}}-{\frac{{b}^{2}}{2\,a \left ( ad-bc \right ) }{\frac{1}{\sqrt{-ab}}}{\frac{1}{\sqrt{ \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ) ^{2}d-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}}}}}-{\frac{bxd}{2\,a \left ( ad-bc \right ) c}{\frac{1}{\sqrt{ \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ) ^{2}d-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}}}}}+{\frac{{b}^{2}}{2\,a \left ( ad-bc \right ) }\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-ab}}}{\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/a/c/x/(d*x^2+c)^(1/2)-2/a*d/c^2*x/(d*x^2+c)^(1/2)+1/2*b^2/a/(-a*b)^(1/2)/(a*d
-b*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)-1/2*b/a/(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*b^2/a/(-a*b)^(1/2)/(a*d-b*c)/(-(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*b^2/a/(-a*b)^(1/2)/(a*d-b*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
)-1/2*b/a/(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*b^2/a/(-a*b)^(1/2)/(a*d-b*c)/(-(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. \[ \int \frac{1}{{\left (b x^{2} + a\right )}{\left (d x^{2} + c\right )}^{\frac{3}{2}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.358749, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \, \sqrt{-a b c + a^{2} d}{\left (b c^{2} - a c d +{\left (b c d - 2 \, a d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c} +{\left (b^{2} c^{2} d x^{3} + b^{2} c^{3} x\right )} \log \left (\frac{{\left ({\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2}\right )} \sqrt{-a b c + a^{2} d} + 4 \,{\left ({\left (a b^{2} c^{2} - 3 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} x^{3} -{\left (a^{2} b c^{2} - a^{3} c d\right )} x\right )} \sqrt{d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )}{4 \,{\left ({\left (a b c^{3} d - a^{2} c^{2} d^{2}\right )} x^{3} +{\left (a b c^{4} - a^{2} c^{3} d\right )} x\right )} \sqrt{-a b c + a^{2} d}}, -\frac{2 \, \sqrt{a b c - a^{2} d}{\left (b c^{2} - a c d +{\left (b c d - 2 \, a d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c} +{\left (b^{2} c^{2} d x^{3} + b^{2} c^{3} x\right )} \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{2} - a c}{2 \, \sqrt{a b c - a^{2} d} \sqrt{d x^{2} + c} x}\right )}{2 \,{\left ({\left (a b c^{3} d - a^{2} c^{2} d^{2}\right )} x^{3} +{\left (a b c^{4} - a^{2} c^{3} d\right )} x\right )} \sqrt{a b c - a^{2} d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError